Subgroups — C₇×#24207 #13201423

Order 1792.

1076 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#464	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2582	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#2042	C₂	1	1	yes	maximal, characteristic
7	C₇×#2016	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2015	C₂	1	1	yes	maximal, characteristic
9	C₇×#1985	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1959	C₂	1	1	yes	maximal, characteristic
11	C₇×#1878	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1877	C₂	1	1	yes	maximal, characteristic
13	C₇×#1865	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1864	C₂	1	1	yes	maximal, characteristic
15	C₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₇×SD₁₆	C₂²	1	1	yes	characteristic
19	C₇×#556	—	2	4	no	—
20	C₇×#556	—	2	4	no	—
21	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₇×#126	—	2	1	no	—
23	C₂×C₇×#126	—	2	1	no	—
24	C₂×C₇×#124	C₂²	1	1	yes	characteristic
25	C₂×C₇×#124	C₂²	1	1	yes	characteristic
26	C₂×C₇×C₂²⋊Q₈	—	2	1	no	—
27	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
32	C₂×C₇×#117	—	2	1	no	—
33	C₇×#464	—	2	4	no	—
34	C₇×#441	—	2	4	no	—
35	C₂×C₇×#106	C₂²	1	1	yes	characteristic
36	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₇×#103	C₂²	1	1	yes	characteristic
38	C₂×C₇×#103	C₂²	1	1	yes	characteristic
39	C₂×C₇×#103	C₂²	1	1	yes	characteristic
40	C₂×C₇×#103	C₂²	1	1	yes	characteristic
41	C₂×C₇×#103	C₂²	1	1	yes	characteristic
42	C₂×C₇×#103	C₂²	1	1	yes	characteristic
43	C₂×C₇×#103	C₂²	1	1	yes	characteristic
44	C₂×C₇×#102	C₂²	1	1	yes	characteristic
45	C₂×C₇×#102	C₂²	1	1	yes	characteristic
46	C₂×C₇×#102	C₂²	1	1	yes	characteristic
47	C₂×C₇×#97	C₂²	1	1	yes	characteristic
48	C₇×#398	—	2	1	no	—
49	C₇×#398	—	2	1	no	—
50	C₇×#391	—	2	1	no	—
51	C₇×#391	C₂²	1	1	yes	characteristic
52	C₇×#391	C₂²	1	1	yes	characteristic
53	C₇×#389	—	2	1	no	—
54	C₇×#389	C₂²	1	1	yes	characteristic
55	C₇×#389	C₂²	1	1	yes	characteristic
56	C₇×#384	C₂²	1	1	yes	characteristic
57	C₇×#340	C₂²	1	1	yes	characteristic
58	C₇×#340	C₂²	1	1	yes	characteristic
59	C₇×#336	C₂²	1	1	yes	characteristic
60	C₇×#336	C₂²	1	1	yes	characteristic
61	C₇×#336	C₂²	1	1	yes	characteristic
62	C₇×#336	C₂²	1	1	yes	characteristic
63	#24207	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
65	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
66	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
67	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
68	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₇	—	2	1	no	—
70	C₂×C₇×Q₁₆	—	2	2	no	—
71	C₂×C₇×Q₁₆	—	2	2	no	—
72	C₂×C₇×Q₁₆	—	2	2	no	—
73	C₂×C₇×Q₁₆	—	2	2	no	—
74	C₂×C₇×Q₁₆	—	2	2	no	—
75	C₂×C₇×Q₁₆	—	4	1	no	—
76	C₂×C₇×SD₁₆	—	2	2	no	—
77	C₂×C₇×SD₁₆	—	2	2	no	—
78	C₂×C₇×SD₁₆	—	4	1	no	—
79	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
80	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
81	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
82	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
83	C₇×C₄⋊Q₈	—	2	2	no	—
84	C₇×C₄⋊Q₈	—	2	2	no	—
85	C₇×#126	—	2	8	no	—
86	C₇×#126	—	2	8	no	—
87	C₇×#124	—	2	2	no	—
88	C₇×#124	—	2	2	no	—
89	C₇×#124	—	2	2	no	—
90	C₇×#124	D₄	1	4	yes	—
91	C₇×C₂²⋊Q₈	—	2	8	no	—
92	C₄×C₇×Q₈	—	2	2	no	—
93	C₄×C₇×Q₈	—	2	2	no	—
94	C₄×C₇×Q₈	—	2	2	no	—
95	C₄×C₇×Q₈	—	2	2	no	—
96	C₄×C₇×Q₈	—	2	2	no	—
97	C₄×C₇×Q₈	—	2	2	no	—
98	C₄×C₇×Q₈	—	2	2	no	—
99	C₄×C₇×Q₈	D₄	1	4	yes	—
100	C₄×C₇×D₄	—	2	2	no	—
101	C₄×C₇×D₄	—	2	2	no	—
102	C₇×#117	—	2	8	no	—
103	C₂×C₇×C₄⋊C₄	—	2	1	no	—
104	C₂×C₇×C₄⋊C₄	—	2	1	no	—
105	C₂×C₇×C₄⋊C₄	—	2	1	no	—
106	C₂×C₇×C₄⋊C₄	—	2	1	no	—
107	C₂×C₇×C₄⋊C₄	—	2	1	no	—
108	C₂×C₇×C₄⋊C₄	—	2	1	no	—
109	C₂×C₇×C₄⋊C₄	—	2	1	no	—
110	C₂×C₇×C₄⋊C₄	—	2	1	no	—
111	C₂×C₇×C₄⋊C₄	—	2	1	no	—
112	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
113	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
114	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
115	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
116	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
117	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
118	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
119	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
120	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
121	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
122	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
123	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₄²×C₇	—	2	1	no	—
125	C₂×C₄²×C₇	—	2	1	no	—
126	C₂×C₄²×C₇	—	2	1	no	—
127	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
130	C₇×#106	—	2	2	no	—
131	C₇×C₄⋊C₈	—	2	2	no	—
132	C₇×#103	—	2	2	no	—
133	C₇×#103	—	2	2	no	—
134	C₇×#103	—	2	2	no	—
135	C₇×#103	—	2	2	no	—
136	C₇×#103	—	4	1	no	—
137	C₇×#103	—	4	1	no	—
138	C₇×#103	—	4	1	no	—
139	C₇×#102	—	2	2	no	—
140	C₇×#102	—	2	2	no	—
141	C₇×#102	—	4	1	no	—
142	C₇×#97	—	2	2	no	—
143	C₇×#95	—	2	1	no	—
144	C₇×#95	—	2	1	no	—
145	C₇×#95	—	2	1	no	—
146	C₇×#95	—	2	1	no	—
147	C₇×#95	—	2	1	no	—
148	C₇×#95	—	2	1	no	—
149	C₂×#556	C₂×C₇	1	1	yes	characteristic
150	C₂×#464	C₂×C₇	1	1	yes	characteristic
151	C₂×#441	C₂×C₇	1	1	yes	characteristic, complemented
152	#2582	C₂×C₇	1	1	yes	characteristic, complemented
153	#2043	C₂×C₇	1	1	yes	characteristic, complemented
154	#2042	C₂×C₇	1	1	yes	characteristic
155	#2016	C₂×C₇	1	1	yes	characteristic, complemented
156	#2015	C₂×C₇	1	1	yes	characteristic
157	#1985	C₂×C₇	1	1	yes	characteristic, complemented
158	#1959	C₂×C₇	1	1	yes	characteristic
159	#1878	C₂×C₇	1	1	yes	characteristic, complemented
160	#1877	C₂×C₇	1	1	yes	characteristic
161	#1865	C₂×C₇	1	1	yes	characteristic, complemented
162	#1864	C₂×C₇	1	1	yes	characteristic
163	#1803	C₂×C₇	1	1	yes	characteristic, complemented
164	C₂⁴×C₇	—	2	1	no	—
165	C₂×C₇×Q₈	—	2	2	no	—
166	C₂×C₇×Q₈	—	2	2	no	—
167	C₂×C₇×Q₈	—	2	1	no	—
168	C₂×C₇×Q₈	—	2	1	no	—
169	C₂×C₇×Q₈	—	2	1	no	—
170	C₂×C₇×Q₈	—	2	1	no	—
171	C₂×C₇×Q₈	—	2	1	no	—
172	C₂×C₇×Q₈	—	2	1	no	—
173	C₂×C₇×Q₈	—	2	1	no	—
174	C₂×C₇×Q₈	—	2	1	no	—
175	C₂×C₇×Q₈	—	2	1	no	—
176	C₂×C₇×Q₈	—	2	1	no	—
177	C₂×C₇×Q₈	—	2	1	no	—
178	C₂×C₇×Q₈	—	2	1	no	—
179	C₂×C₇×Q₈	—	2	1	no	—
180	C₂×C₇×Q₈	—	2	1	no	—
181	C₂×C₇×Q₈	—	2	1	no	—
182	C₂×C₇×Q₈	—	4	1	no	—
183	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
184	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
185	C₂×C₇×Q₈	#41	1	2	yes	—
186	C₂×C₇×D₄	—	2	1	no	—
187	C₂×C₇×D₄	—	2	1	no	—
188	C₂×C₇×D₄	—	2	1	no	—
189	C₂×C₇×D₄	—	2	1	no	—
190	C₂×C₇×D₄	—	2	1	no	—
191	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
192	C₂²×C₄×C₇	—	2	4	no	—
193	C₂²×C₄×C₇	—	2	1	no	—
194	C₂²×C₄×C₇	—	2	1	no	—
195	C₂²×C₄×C₇	—	2	1	no	—
196	C₂²×C₄×C₇	—	2	1	no	—
197	C₂²×C₄×C₇	—	2	1	no	—
198	C₂²×C₄×C₇	—	2	1	no	—
199	C₂²×C₄×C₇	—	2	1	no	—
200	C₂²×C₄×C₇	—	4	2	no	—
201	C₂²×C₄×C₇	—	4	2	no	—
202	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
209	C₇×Q₁₆	—	2	8	no	—
210	C₇×Q₁₆	—	4	4	no	—
211	C₇×SD₁₆	—	4	4	no	—
212	C₂×C₇×C₈	—	2	1	no	—
213	C₂×C₇×C₈	—	2	1	no	—
214	C₂×C₇×C₈	—	2	1	no	—
215	C₂×C₇×C₈	—	2	1	no	—
216	C₂×C₇×C₈	—	2	1	no	—
217	C₂×C₇×C₈	—	2	1	no	—
218	C₂×C₇×C₈	—	2	1	no	—
219	C₂×C₇×C₈	—	2	1	no	—
220	C₂×C₇×C₈	—	2	1	no	—
221	C₂×C₇×C₈	—	2	1	no	—
222	C₂×C₇×C₈	—	2	1	no	—
223	C₂×C₇×C₈	#41	1	2	yes	—
224	C₇×C₄⋊C₄	—	2	4	no	—
225	C₇×C₄⋊C₄	—	2	4	no	—
226	C₇×C₄⋊C₄	—	2	4	no	—
227	C₇×C₄⋊C₄	—	2	4	no	—
228	C₇×C₄⋊C₄	—	2	2	no	—
229	C₇×C₄⋊C₄	—	2	2	no	—
230	C₇×C₄⋊C₄	—	2	2	no	—
231	C₇×C₄⋊C₄	—	2	1	no	—
232	C₇×C₄⋊C₄	—	2	1	no	—
233	C₇×C₄⋊C₄	—	2	1	no	—
234	C₇×C₄⋊C₄	—	2	1	no	—
235	C₇×C₄⋊C₄	—	2	1	no	—
236	C₇×C₄⋊C₄	—	2	1	no	—
237	C₇×C₄⋊C₄	—	2	1	no	—
238	C₇×C₄⋊C₄	—	4	2	no	—
239	C₇×C₄⋊C₄	—	4	2	no	—
240	C₇×C₄⋊C₄	—	8	1	no	—
241	C₇×C₄⋊C₄	—	8	1	no	—
242	C₇×C₄⋊C₄	—	8	1	no	—
243	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
244	C₇×C₂²⋊C₄	—	2	4	no	—
245	C₇×C₂²⋊C₄	—	2	4	no	—
246	C₇×C₂²⋊C₄	—	2	4	no	—
247	C₇×C₂²⋊C₄	—	2	4	no	—
248	C₇×C₂²⋊C₄	—	2	4	no	—
249	C₄²×C₇	—	2	4	no	—
250	C₄²×C₇	—	2	4	no	—
251	C₄²×C₇	—	2	1	no	—
252	C₄²×C₇	—	2	1	no	—
253	C₄²×C₇	—	2	1	no	—
254	C₄²×C₇	—	2	1	no	—
255	C₄²×C₇	—	4	2	no	—
256	C₄²×C₇	C₂×D₄	1	2	yes	—
257	C₄²×C₇	C₂×D₄	1	2	yes	—
258	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
259	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
260	C₂²×SD₁₆	C₂²×C₇	1	1	yes	characteristic
261	#556	—	2	4	no	—
262	#556	—	2	4	no	—
263	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
264	C₂×#126	—	2	1	no	—
265	C₂×#126	—	2	1	no	—
266	C₂×#124	C₂²×C₇	1	1	yes	characteristic
267	C₂×#124	C₂²×C₇	1	1	yes	characteristic
268	C₂×C₂²⋊Q₈	—	2	1	no	—
269	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
270	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
271	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
272	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
274	C₂×#117	—	2	1	no	—
275	#464	—	2	4	no	—
276	#441	—	2	4	no	—
277	C₂×#106	C₂²×C₇	1	1	yes	characteristic
278	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
279	C₂×#103	C₂²×C₇	1	1	yes	characteristic
280	C₂×#103	C₂²×C₇	1	1	yes	characteristic
281	C₂×#103	C₂²×C₇	1	1	yes	characteristic
282	C₂×#103	C₂²×C₇	1	1	yes	characteristic
283	C₂×#103	C₂²×C₇	1	1	yes	characteristic
284	C₂×#103	C₂²×C₇	1	1	yes	characteristic
285	C₂×#103	C₂²×C₇	1	1	yes	characteristic
286	C₂×#102	C₂²×C₇	1	1	yes	characteristic
287	C₂×#102	C₂²×C₇	1	1	yes	characteristic
288	C₂×#102	C₂²×C₇	1	1	yes	characteristic
289	C₂×#97	C₂²×C₇	1	1	yes	characteristic
290	#398	—	2	1	no	—
291	#398	—	2	1	no	—
292	#391	—	2	1	no	—
293	#391	C₂²×C₇	1	1	yes	characteristic
294	#391	C₂²×C₇	1	1	yes	characteristic
295	#389	—	2	1	no	—
296	#389	C₂²×C₇	1	1	yes	characteristic
297	#389	C₂²×C₇	1	1	yes	characteristic
298	#384	C₂²×C₇	1	1	yes	characteristic
299	#340	C₂²×C₇	1	1	yes	characteristic
300	#340	C₂²×C₇	1	1	yes	characteristic
301	#336	C₂²×C₇	1	1	yes	characteristic
302	#336	C₂²×C₇	1	1	yes	characteristic
303	#336	C₂²×C₇	1	1	yes	characteristic
304	#336	C₂²×C₇	1	1	yes	characteristic
305	C₂³×C₇	—	2	2	no	—
306	C₂³×C₇	—	4	1	no	—
307	C₂³×C₇	—	4	1	no	—
308	C₂³×C₇	—	4	1	no	—
309	C₂³×C₇	—	4	1	no	—
310	C₂³×C₇	—	4	1	no	—
311	C₂³×C₇	—	4	1	no	—
312	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
313	C₇×Q₈	—	2	2	no	—
314	C₇×Q₈	—	2	2	no	—
315	C₇×Q₈	—	2	2	no	—
316	C₇×Q₈	—	2	2	no	—
317	C₇×Q₈	—	2	2	no	—
318	C₇×Q₈	—	2	2	no	—
319	C₇×Q₈	—	4	4	no	—
320	C₇×Q₈	—	4	1	no	—
321	C₇×Q₈	—	4	1	no	—
322	C₇×Q₈	—	4	1	no	—
323	C₇×Q₈	—	4	1	no	—
324	C₇×Q₈	—	4	1	no	—
325	C₇×Q₈	—	4	1	no	—
326	C₇×D₄	—	2	2	no	—
327	C₇×D₄	—	2	2	no	—
328	C₇×D₄	—	4	1	no	—
329	C₇×D₄	—	4	1	no	—
330	C₂×C₄×C₇	—	2	1	no	—
331	C₂×C₄×C₇	—	2	1	no	—
332	C₂×C₄×C₇	—	2	1	no	—
333	C₂×C₄×C₇	—	2	1	no	—
334	C₂×C₄×C₇	—	2	1	no	—
335	C₂×C₄×C₇	—	2	1	no	—
336	C₂×C₄×C₇	—	2	1	no	—
337	C₂×C₄×C₇	—	2	2	no	—
338	C₂×C₄×C₇	—	2	1	no	—
339	C₂×C₄×C₇	—	2	2	no	—
340	C₂×C₄×C₇	—	2	2	no	—
341	C₂×C₄×C₇	—	2	1	no	—
342	C₂×C₄×C₇	—	2	2	no	—
343	C₂×C₄×C₇	—	2	1	no	—
344	C₂×C₄×C₇	—	2	1	no	—
345	C₂×C₄×C₇	—	2	1	no	—
346	C₂×C₄×C₇	—	2	1	no	—
347	C₂×C₄×C₇	—	2	1	no	—
348	C₂×C₄×C₇	—	2	1	no	—
349	C₂×C₄×C₇	—	4	8	no	—
350	C₂×C₄×C₇	—	4	1	no	—
351	C₂×C₄×C₇	—	4	1	no	—
352	C₂×C₄×C₇	—	4	1	no	—
353	C₂×C₄×C₇	—	4	1	no	—
354	C₂×C₄×C₇	—	4	1	no	—
355	C₂×C₄×C₇	—	4	1	no	—
356	C₂×C₄×C₇	—	4	1	no	—
357	C₂×C₄×C₇	—	4	1	no	—
358	C₂×C₄×C₇	—	4	1	no	—
359	C₂×C₄×C₇	—	4	1	no	—
360	C₂×C₄×C₇	—	4	1	no	—
361	C₂×C₄×C₇	—	4	1	no	—
362	C₂×C₄×C₇	—	4	1	no	—
363	C₂×C₄×C₇	—	4	1	no	—
364	C₂×C₄×C₇	—	4	1	no	—
365	C₂×C₄×C₇	—	4	1	no	—
366	C₂×C₄×C₇	—	4	1	no	—
367	C₂×C₄×C₇	#120	1	2	yes	—
368	C₂×C₄×C₇	#120	1	2	yes	—
369	C₂×C₄×C₇	#137	1	2	yes	—
370	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
371	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
372	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
373	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
374	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
375	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
376	C₇×C₈	—	2	2	no	—
377	C₇×C₈	—	4	1	no	—
378	C₇×C₈	—	4	1	no	—
379	C₇×C₈	—	4	1	no	—
380	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
381	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
382	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
383	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
384	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
385	C₂³×C₄	—	2	1	no	—
386	C₂×Q₁₆	—	2	2	no	—
387	C₂×Q₁₆	—	2	2	no	—
388	C₂×Q₁₆	—	2	2	no	—
389	C₂×Q₁₆	—	2	2	no	—
390	C₂×Q₁₆	—	2	2	no	—
391	C₂×Q₁₆	—	4	1	no	—
392	C₂×SD₁₆	—	2	2	no	—
393	C₂×SD₁₆	—	2	2	no	—
394	C₂×SD₁₆	—	4	1	no	—
395	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
396	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
397	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
399	C₄⋊Q₈	—	2	2	no	—
400	C₄⋊Q₈	—	2	2	no	—
401	#126	—	2	8	no	—
402	#126	—	2	8	no	—
403	#124	—	2	2	no	—
404	#124	—	2	2	no	—
405	#124	—	2	2	no	—
406	#124	C₇×D₄	1	4	yes	—
407	C₂²⋊Q₈	—	2	8	no	—
408	C₄×Q₈	—	2	2	no	—
409	C₄×Q₈	—	2	2	no	—
410	C₄×Q₈	—	2	2	no	—
411	C₄×Q₈	—	2	2	no	—
412	C₄×Q₈	—	2	2	no	—
413	C₄×Q₈	—	2	2	no	—
414	C₄×Q₈	—	2	2	no	—
415	C₄×Q₈	C₇×D₄	1	4	yes	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	—	2	2	no	—
418	#117	—	2	8	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	—	2	1	no	—
443	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
446	#106	—	2	2	no	—
447	C₄⋊C₈	—	2	2	no	—
448	#103	—	2	2	no	—
449	#103	—	2	2	no	—
450	#103	—	2	2	no	—
451	#103	—	2	2	no	—
452	#103	—	4	1	no	—
453	#103	—	4	1	no	—
454	#103	—	4	1	no	—
455	#102	—	2	2	no	—
456	#102	—	2	2	no	—
457	#102	—	4	1	no	—
458	#97	—	2	2	no	—
459	#95	—	2	1	no	—
460	#95	—	2	1	no	—
461	#95	—	2	1	no	—
462	#95	—	2	1	no	—
463	#95	—	2	1	no	—
464	#95	—	2	1	no	—
465	C₂²×C₇	—	4	2	no	—
466	C₂²×C₇	—	4	2	no	—
467	C₂²×C₇	—	4	2	no	—
468	C₂²×C₇	—	8	1	no	—
469	C₂²×C₇	—	8	1	no	—
470	C₂²×C₇	—	8	1	no	—
471	C₂²×C₇	—	8	1	no	—
472	C₂²×C₇	C₂×#120	1	1	yes	characteristic
473	C₂²×C₇	#548	1	1	yes	characteristic
474	C₂²×C₇	#550	1	1	yes	characteristic
475	C₂²×C₇	C₂×#137	1	1	yes	characteristic
476	C₂²×C₇	#575	1	1	yes	characteristic
477	C₂²×C₇	#577	1	1	yes	characteristic
478	C₂²×C₇	#578	1	1	yes	characteristic
479	C₄×C₇	—	2	2	no	—
480	C₄×C₇	—	2	2	no	—
481	C₄×C₇	—	2	2	no	—
482	C₄×C₇	—	4	2	no	—
483	C₄×C₇	—	4	2	no	—
484	C₄×C₇	—	4	2	no	—
485	C₄×C₇	—	4	1	no	—
486	C₄×C₇	—	4	2	no	—
487	C₄×C₇	—	4	1	no	—
488	C₄×C₇	—	4	1	no	—
489	C₄×C₇	—	8	1	no	—
490	C₄×C₇	—	8	1	no	—
491	C₄×C₇	—	8	1	no	—
492	C₄×C₇	#525	1	1	yes	characteristic
493	C₄×C₇	#537	1	1	yes	characteristic
494	C₄×C₇	#546	1	1	yes	characteristic
495	C₄×C₇	#547	1	1	yes	characteristic
496	C₂⁴	—	2	1	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	2	1	no	—
502	C₂×Q₈	—	2	1	no	—
503	C₂×Q₈	—	2	1	no	—
504	C₂×Q₈	—	2	1	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	—	2	1	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	—	2	1	no	—
511	C₂×Q₈	—	2	1	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	4	1	no	—
515	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
516	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
517	C₂×Q₈	C₇×#41	1	2	yes	—
518	C₂×D₄	—	2	1	no	—
519	C₂×D₄	—	2	1	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	1	no	—
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	4	2	no	—
533	C₂²×C₄	—	4	2	no	—
534	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
535	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
536	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
537	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
538	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
539	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
540	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
541	Q₁₆	—	2	8	no	—
542	Q₁₆	—	4	4	no	—
543	SD₁₆	—	4	4	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	C₇×#41	1	2	yes	—
556	C₄⋊C₄	—	2	4	no	—
557	C₄⋊C₄	—	2	4	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	4	2	no	—
571	C₄⋊C₄	—	4	2	no	—
572	C₄⋊C₄	—	8	1	no	—
573	C₄⋊C₄	—	8	1	no	—
574	C₄⋊C₄	—	8	1	no	—
575	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
576	C₂²⋊C₄	—	2	4	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	4	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₄²	—	2	4	no	—
582	C₄²	—	2	4	no	—
583	C₄²	—	2	1	no	—
584	C₄²	—	2	1	no	—
585	C₄²	—	2	1	no	—
586	C₄²	—	2	1	no	—
587	C₄²	—	4	2	no	—
588	C₄²	C₂×C₇×D₄	1	2	yes	—
589	C₄²	C₂×C₇×D₄	1	2	yes	—
590	C₂×C₇	—	8	2	no	—
591	C₂×C₇	#2407	1	1	yes	characteristic
592	C₂×C₇	#3007	1	1	yes	characteristic
593	C₂×C₇	#3013	1	1	yes	characteristic
594	C₂×C₇	#3369	1	1	yes	characteristic
595	C₂×C₇	#3377	1	1	yes	characteristic
596	C₂×C₇	#3398	1	1	yes	characteristic
597	C₂×C₇	#3401	1	1	yes	characteristic
598	C₂³	—	2	2	no	—
599	C₂³	—	4	1	no	—
600	C₂³	—	4	1	no	—
601	C₂³	—	4	1	no	—
602	C₂³	—	4	1	no	—
603	C₂³	—	4	1	no	—
604	C₂³	—	4	1	no	—
605	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
606	Q₈	—	2	2	no	—
607	Q₈	—	2	2	no	—
608	Q₈	—	2	2	no	—
609	Q₈	—	2	2	no	—
610	Q₈	—	2	2	no	—
611	Q₈	—	2	2	no	—
612	Q₈	—	4	4	no	—
613	Q₈	—	4	1	no	—
614	Q₈	—	4	1	no	—
615	Q₈	—	4	1	no	—
616	Q₈	—	4	1	no	—
617	Q₈	—	4	1	no	—
618	Q₈	—	4	1	no	—
619	D₄	—	2	2	no	—
620	D₄	—	2	2	no	—
621	D₄	—	4	1	no	—
622	D₄	—	4	1	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	4	8	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	C₇×#120	1	2	yes	—
661	C₂×C₄	C₇×#120	1	2	yes	—
662	C₂×C₄	C₇×#137	1	2	yes	—
663	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
664	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
665	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
666	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
667	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
668	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
669	C₈	—	2	2	no	—
670	C₈	—	4	1	no	—
671	C₈	—	4	1	no	—
672	C₈	—	4	1	no	—
673	C₇	#24207	1	1	yes	characteristic, direct product factor, Sylow
674	C₂²	—	4	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	4	2	no	—
677	C₂²	—	8	1	no	—
678	C₂²	—	8	1	no	—
679	C₂²	—	8	1	no	—
680	C₂²	—	8	1	no	—
681	C₂²	C₂×C₇×#120	1	1	yes	characteristic
682	C₂²	C₇×#548	1	1	yes	characteristic
683	C₂²	C₇×#550	1	1	yes	characteristic
684	C₂²	C₂×C₇×#137	1	1	yes	characteristic
685	C₂²	C₇×#575	1	1	yes	characteristic
686	C₂²	C₇×#577	1	1	yes	characteristic
687	C₂²	C₇×#578	1	1	yes	characteristic
688	C₄	—	2	2	no	—
689	C₄	—	2	2	no	—
690	C₄	—	2	2	no	—
691	C₄	—	4	2	no	—
692	C₄	—	4	2	no	—
693	C₄	—	4	2	no	—
694	C₄	—	4	1	no	—
695	C₄	—	4	2	no	—
696	C₄	—	4	1	no	—
697	C₄	—	4	1	no	—
698	C₄	—	8	1	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	C₇×#525	1	1	yes	characteristic
702	C₄	C₇×#537	1	1	yes	characteristic
703	C₄	C₇×#546	1	1	yes	characteristic
704	C₄	C₇×#547	1	1	yes	characteristic
705	C₂	—	8	2	no	—
706	C₂	C₇×#2407	1	1	yes	characteristic
707	C₂	C₇×#3007	1	1	yes	characteristic
708	C₂	C₇×#3013	1	1	yes	characteristic
709	C₂	C₇×#3369	1	1	yes	characteristic
710	C₂	C₇×#3377	1	1	yes	characteristic
711	C₂	C₇×#3398	1	1	yes	characteristic
712	C₂	C₇×#3401	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₇×#441	C₂	semidirect
4. C₇×#2582	C₂	semidirect
5. C₇×#2043	C₂	semidirect
7. C₇×#2016	C₂	semidirect
9. C₇×#1985	C₂	semidirect
11. C₇×#1878	C₂	semidirect
13. C₇×#1865	C₂	semidirect
15. C₇×#1803	C₂	semidirect
63. #24207	C₇	direct (N × H)
151. C₂×#441	C₂×C₇	semidirect
152. #2582	C₂×C₇	semidirect
153. #2043	C₂×C₇	semidirect
155. #2016	C₂×C₇	semidirect
157. #1985	C₂×C₇	semidirect
159. #1878	C₂×C₇	semidirect
161. #1865	C₂×C₇	semidirect
163. #1803	C₂×C₇	semidirect
673. C₇	#24207	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#556	2
C₂×C₇×#464	2
C₂×C₇×#441	2
C₇×#2582	2
C₇×#2043	2
C₇×#2042	2
C₇×#2016	2
C₇×#2015	2
C₇×#1985	2
C₇×#1959	2
C₇×#1878	2
C₇×#1877	2
C₇×#1865	2
C₇×#1864	2
C₇×#1803	2
#24207	7

Subgroups — C7×#24207 #13201423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24207 #13201423