Subgroups — C₇×#21504 #13198720

Order 1792.

1092 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#461	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#440	C₂	1	2	yes	maximal
3	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
5	C₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1928	C₂	1	2	yes	maximal, complemented
8	C₇×#1877	C₂	1	2	yes	maximal
9	C₇×#1802	C₂	1	1	yes	maximal, characteristic
10	C₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1763	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
15	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₇	—	2	1	no	—
19	C₇×#461	C₄	1	8	yes	complemented
20	C₇×#440	—	2	8	no	—
21	C₂×C₇×#107	C₂²	1	2	yes	—
22	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₇×#103	C₂²	1	2	yes	—
25	C₂×C₇×#103	C₂²	1	2	yes	—
26	C₂×C₇×#103	C₂²	1	2	yes	—
27	C₂×C₇×#102	C₂²	1	2	yes	—
28	C₂×C₇×#97	C₂²	1	2	yes	—
29	C₇×#391	C₂²	1	2	yes	—
30	C₇×#389	—	2	1	no	—
31	C₇×#389	—	2	1	no	—
32	C₇×#384	C₂²	1	1	yes	characteristic
33	C₄×C₇×C₄⋊C₄	—	2	1	no	—
34	C₄×C₇×C₄⋊C₄	—	2	1	no	—
35	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
38	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
39	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
40	C₄×C₇×C₂²⋊C₄	—	2	1	no	—
41	C₄³×C₇	C₂²	1	1	yes	characteristic
42	C₇×#340	C₂²	1	2	yes	—
43	C₇×#340	C₂²	1	2	yes	—
44	C₇×#336	C₂²	1	2	yes	—
45	#21504	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂²×C₇×Q₈	C₂³	1	2	yes	—
47	C₂²×C₇×D₄	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₇	—	2	1	no	—
49	C₂³×C₄×C₇	—	2	1	no	—
50	C₂³×C₄×C₇	—	2	1	no	—
51	C₂×C₇×SD₁₆	—	2	4	no	—
52	C₂×C₇×SD₁₆	—	2	4	no	—
53	C₂×C₇×SD₁₆	C₂×C₄	1	8	yes	—
54	C₂²×C₇×C₈	C₂³	1	2	yes	—
55	C₂²×C₇×C₈	C₂³	1	2	yes	—
56	C₇×C₄⋊Q₈	—	2	2	no	—
57	C₇×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
58	C₄×C₇×Q₈	—	2	4	no	—
59	C₄×C₇×Q₈	—	2	4	no	—
60	C₄×C₇×D₄	—	2	2	no	—
61	C₄×C₇×D₄	—	2	2	no	—
62	C₄×C₇×D₄	—	2	2	no	—
63	C₄×C₇×D₄	—	2	2	no	—
64	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
65	C₄×C₇×D₄	D₄	1	4	yes	—
66	C₂×C₇×C₄⋊C₄	—	2	1	no	—
67	C₂×C₇×C₄⋊C₄	—	2	1	no	—
68	C₂×C₇×C₄⋊C₄	—	2	1	no	—
69	C₂×C₇×C₄⋊C₄	—	2	1	no	—
70	C₂×C₇×C₄⋊C₄	—	2	2	no	—
71	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
72	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
73	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
74	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
76	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
77	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
78	C₂×C₄²×C₇	—	2	1	no	—
79	C₂×C₄²×C₇	—	2	1	no	—
80	C₂×C₄²×C₇	—	2	1	no	—
81	C₂×C₄²×C₇	—	2	2	no	—
82	C₂×C₄²×C₇	—	4	4	no	—
83	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
85	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
90	C₇×#107	—	2	4	no	—
91	C₇×C₄⋊C₈	—	2	2	no	—
92	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
93	C₇×#103	—	2	4	no	—
94	C₇×#103	—	2	4	no	—
95	C₇×#103	C₂×C₄	1	8	yes	complemented
96	C₇×#102	—	2	4	no	—
97	C₇×#97	—	2	4	no	—
98	C₇×#95	—	2	1	no	—
99	C₇×#95	—	2	1	no	—
100	C₇×#95	—	2	1	no	—
101	C₇×#95	—	2	2	no	—
102	C₇×#95	—	2	2	no	—
103	C₂×#461	C₂×C₇	1	1	yes	characteristic
104	C₂×#440	C₂×C₇	1	2	yes	—
105	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
106	C₄²×D₄	C₂×C₇	1	1	yes	characteristic
107	#1981	C₂×C₇	1	1	yes	characteristic, complemented
108	#1947	C₂×C₇	1	1	yes	characteristic, complemented
109	#1928	C₂×C₇	1	2	yes	complemented
110	#1877	C₂×C₇	1	2	yes	—
111	#1802	C₂×C₇	1	1	yes	characteristic
112	#1767	C₂×C₇	1	1	yes	characteristic, complemented
113	#1763	C₂×C₇	1	2	yes	complemented
114	C₂⁴×C₇	—	2	1	no	—
115	C₂×C₇×Q₈	—	2	2	no	—
116	C₂×C₇×Q₈	—	2	2	no	—
117	C₂×C₇×Q₈	—	2	2	no	—
118	C₂×C₇×Q₈	—	2	2	no	—
119	C₂×C₇×Q₈	—	2	2	no	—
120	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
121	C₂×C₇×D₄	—	2	1	no	—
122	C₂×C₇×D₄	—	2	1	no	—
123	C₂×C₇×D₄	—	2	1	no	—
124	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
125	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
126	C₂×C₇×D₄	#41	1	2	yes	—
127	C₂²×C₄×C₇	—	2	4	no	—
128	C₂²×C₄×C₇	—	2	1	no	—
129	C₂²×C₄×C₇	—	2	1	no	—
130	C₂²×C₄×C₇	—	2	1	no	—
131	C₂²×C₄×C₇	—	2	2	no	—
132	C₂²×C₄×C₇	—	2	4	no	—
133	C₂²×C₄×C₇	—	2	4	no	—
134	C₂²×C₄×C₇	—	2	2	no	—
135	C₂²×C₄×C₇	—	4	2	no	—
136	C₂²×C₄×C₇	—	4	2	no	—
137	C₂²×C₄×C₇	—	4	2	no	—
138	C₂²×C₄×C₇	—	4	2	no	—
139	C₂²×C₄×C₇	—	4	2	no	—
140	C₂²×C₄×C₇	—	4	2	no	—
141	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
148	C₇×SD₁₆	—	2	16	no	—
149	C₂×C₇×C₈	—	2	2	no	—
150	C₂×C₇×C₈	—	2	2	no	—
151	C₂×C₇×C₈	—	2	2	no	—
152	C₂×C₇×C₈	—	2	2	no	—
153	C₂×C₇×C₈	—	2	2	no	—
154	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
155	C₇×C₄⋊C₄	—	2	4	no	—
156	C₇×C₄⋊C₄	—	2	4	no	—
157	C₇×C₄⋊C₄	—	2	1	no	—
158	C₇×C₄⋊C₄	—	2	1	no	—
159	C₇×C₄⋊C₄	—	2	1	no	—
160	C₇×C₄⋊C₄	—	2	2	no	—
161	C₇×C₄⋊C₄	—	2	2	no	—
162	C₇×C₄⋊C₄	—	4	2	no	—
163	C₇×C₄⋊C₄	—	4	4	no	—
164	C₇×C₄⋊C₄	—	4	2	no	—
165	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
166	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
167	C₇×C₄⋊C₄	#41	1	2	yes	—
168	C₇×C₂²⋊C₄	—	2	4	no	—
169	C₇×C₂²⋊C₄	—	2	4	no	—
170	C₇×C₂²⋊C₄	—	2	4	no	—
171	C₄²×C₇	—	2	2	no	—
172	C₄²×C₇	—	2	2	no	—
173	C₄²×C₇	—	2	2	no	—
174	C₄²×C₇	—	2	1	no	—
175	C₄²×C₇	—	2	1	no	—
176	C₄²×C₇	—	4	1	no	—
177	C₄²×C₇	—	4	2	no	—
178	C₄²×C₇	—	4	2	no	—
179	C₄²×C₇	—	4	4	no	—
180	C₄²×C₇	—	4	2	no	—
181	C₄²×C₇	C₂²×C₄	1	2	yes	—
182	C₄²×C₇	C₂×D₄	1	2	yes	—
183	C₄²×C₇	#41	1	2	yes	—
184	C₄²×C₇	#41	1	2	yes	—
185	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
186	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
187	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
188	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
189	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
190	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
191	C₂²×C₄²	—	2	1	no	—
192	#461	C₄×C₇	1	8	yes	complemented
193	#440	—	2	8	no	—
194	C₂×#107	C₂²×C₇	1	2	yes	—
195	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
196	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
197	C₂×#103	C₂²×C₇	1	2	yes	—
198	C₂×#103	C₂²×C₇	1	2	yes	—
199	C₂×#103	C₂²×C₇	1	2	yes	—
200	C₂×#102	C₂²×C₇	1	2	yes	—
201	C₂×#97	C₂²×C₇	1	2	yes	—
202	#391	C₂²×C₇	1	2	yes	—
203	#389	—	2	1	no	—
204	#389	—	2	1	no	—
205	#384	C₂²×C₇	1	1	yes	characteristic
206	C₄×C₄⋊C₄	—	2	1	no	—
207	C₄×C₄⋊C₄	—	2	1	no	—
208	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
209	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
210	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
211	C₄×C₂²⋊C₄	—	2	1	no	—
212	C₄×C₂²⋊C₄	—	2	1	no	—
213	C₄×C₂²⋊C₄	—	2	1	no	—
214	C₄³	C₂²×C₇	1	1	yes	characteristic
215	#340	C₂²×C₇	1	2	yes	—
216	#340	C₂²×C₇	1	2	yes	—
217	#336	C₂²×C₇	1	2	yes	—
218	C₂³×C₇	—	2	2	no	—
219	C₂³×C₇	—	2	2	no	—
220	C₂³×C₇	—	2	2	no	—
221	C₂³×C₇	—	4	1	no	—
222	C₂³×C₇	—	4	1	no	—
223	C₂³×C₇	—	4	1	no	—
224	C₂³×C₇	—	4	1	no	—
225	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
226	C₇×Q₈	—	2	4	no	—
227	C₇×Q₈	—	2	4	no	—
228	C₇×Q₈	—	4	2	no	—
229	C₇×Q₈	—	4	2	no	—
230	C₇×D₄	—	2	2	no	—
231	C₇×D₄	—	2	2	no	—
232	C₇×D₄	—	2	2	no	—
233	C₇×D₄	C₄×D₄	1	4	yes	—
234	C₂×C₄×C₇	—	2	1	no	—
235	C₂×C₄×C₇	—	2	1	no	—
236	C₂×C₄×C₇	—	2	1	no	—
237	C₂×C₄×C₇	—	2	1	no	—
238	C₂×C₄×C₇	—	2	1	no	—
239	C₂×C₄×C₇	—	2	1	no	—
240	C₂×C₄×C₇	—	2	1	no	—
241	C₂×C₄×C₇	—	2	1	no	—
242	C₂×C₄×C₇	—	2	2	no	—
243	C₂×C₄×C₇	—	2	4	no	—
244	C₂×C₄×C₇	—	2	2	no	—
245	C₂×C₄×C₇	—	2	1	no	—
246	C₂×C₄×C₇	—	2	2	no	—
247	C₂×C₄×C₇	—	2	1	no	—
248	C₂×C₄×C₇	—	2	1	no	—
249	C₂×C₄×C₇	—	4	8	no	—
250	C₂×C₄×C₇	—	4	8	no	—
251	C₂×C₄×C₇	—	4	2	no	—
252	C₂×C₄×C₇	—	4	2	no	—
253	C₂×C₄×C₇	—	4	1	no	—
254	C₂×C₄×C₇	—	4	1	no	—
255	C₂×C₄×C₇	—	4	1	no	—
256	C₂×C₄×C₇	—	4	1	no	—
257	C₂×C₄×C₇	—	4	1	no	—
258	C₂×C₄×C₇	—	4	1	no	—
259	C₂×C₄×C₇	—	4	8	no	—
260	C₂×C₄×C₇	—	4	2	no	—
261	C₂×C₄×C₇	—	4	2	no	—
262	C₂×C₄×C₇	—	4	2	no	—
263	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
264	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
265	C₂×C₄×C₇	#121	1	2	yes	—
266	C₂×C₄×C₇	#121	1	2	yes	—
267	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
268	C₂×C₄×C₇	#137	1	2	yes	—
269	C₂×C₄×C₇	#137	1	2	yes	—
270	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
271	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
275	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
276	C₇×C₈	—	2	4	no	—
277	C₇×C₈	—	4	2	no	—
278	C₂²×Q₈	C₂³×C₇	1	2	yes	—
279	C₂²×D₄	C₂³×C₇	1	1	yes	characteristic
280	C₂³×C₄	—	2	1	no	—
281	C₂³×C₄	—	2	1	no	—
282	C₂³×C₄	—	2	1	no	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×SD₁₆	—	2	4	no	—
285	C₂×SD₁₆	C₂×C₄×C₇	1	8	yes	—
286	C₂²×C₈	C₂³×C₇	1	2	yes	—
287	C₂²×C₈	C₂³×C₇	1	2	yes	—
288	C₄⋊Q₈	—	2	2	no	—
289	C₄⋊Q₈	C₂×C₄×C₇	1	4	yes	complemented
290	C₄×Q₈	—	2	4	no	—
291	C₄×Q₈	—	2	4	no	—
292	C₄×D₄	—	2	2	no	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	—	2	2	no	—
295	C₄×D₄	—	2	2	no	—
296	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
297	C₄×D₄	C₇×D₄	1	4	yes	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	2	1	no	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	—	2	1	no	—
313	C₂×C₄²	—	2	1	no	—
314	C₂×C₄²	—	4	4	no	—
315	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
316	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
317	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
318	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
322	#107	—	2	4	no	—
323	C₄⋊C₈	—	2	2	no	—
324	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
325	#103	—	2	4	no	—
326	#103	—	2	4	no	—
327	#103	C₂×C₄×C₇	1	8	yes	complemented
328	#102	—	2	4	no	—
329	#97	—	2	4	no	—
330	#95	—	2	1	no	—
331	#95	—	2	1	no	—
332	#95	—	2	1	no	—
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	C₂²×C₇	—	2	4	no	—
336	C₂²×C₇	—	4	2	no	—
337	C₂²×C₇	—	4	2	no	—
338	C₂²×C₇	—	4	2	no	—
339	C₂²×C₇	—	4	2	no	—
340	C₂²×C₇	—	4	2	no	—
341	C₂²×C₇	—	4	2	no	—
342	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
343	C₂²×C₇	C₂×#121	1	1	yes	characteristic
344	C₂²×C₇	#525	1	1	yes	characteristic
345	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
346	C₂²×C₇	C₂×#137	1	1	yes	characteristic
347	C₂²×C₇	C₂×#137	1	1	yes	characteristic
348	C₂²×C₇	#575	1	1	yes	characteristic
349	C₄×C₇	—	2	2	no	—
350	C₄×C₇	—	2	2	no	—
351	C₄×C₇	—	2	2	no	—
352	C₄×C₇	—	2	2	no	—
353	C₄×C₇	—	4	2	no	—
354	C₄×C₇	—	4	1	no	—
355	C₄×C₇	—	4	2	no	—
356	C₄×C₇	—	4	2	no	—
357	C₄×C₇	—	4	4	no	—
358	C₄×C₇	—	8	2	no	—
359	C₄×C₇	#440	1	4	yes	complemented
360	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
361	C₄×C₇	C₄×#41	1	1	yes	characteristic
362	C₄×C₇	#529	1	1	yes	characteristic
363	C₄×C₇	#532	1	1	yes	characteristic
364	C₂⁴	—	2	1	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
375	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
376	C₂×D₄	C₇×#41	1	2	yes	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	4	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	—	4	2	no	—
388	C₂²×C₄	—	4	2	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
392	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
393	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
394	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
395	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
396	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
398	SD₁₆	—	2	16	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	4	2	no	—
413	C₄⋊C₄	—	4	4	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
416	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
417	C₄⋊C₄	C₇×#41	1	2	yes	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	2	4	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	4	1	no	—
427	C₄²	—	4	2	no	—
428	C₄²	—	4	2	no	—
429	C₄²	—	4	4	no	—
430	C₄²	—	4	2	no	—
431	C₄²	C₂²×C₄×C₇	1	2	yes	—
432	C₄²	C₂×C₇×D₄	1	2	yes	—
433	C₄²	C₇×#41	1	2	yes	—
434	C₄²	C₇×#41	1	2	yes	—
435	C₂×C₇	—	4	4	no	—
436	C₂×C₇	C₄×#121	1	1	yes	characteristic
437	C₂×C₇	C₂×#440	1	1	yes	characteristic
438	C₂×C₇	C₄×#137	1	1	yes	characteristic
439	C₂×C₇	#3036	1	1	yes	characteristic
440	C₂×C₇	#3039	1	1	yes	characteristic
441	C₂×C₇	#3110	1	1	yes	characteristic
442	C₂×C₇	#3116	1	1	yes	characteristic
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	4	1	no	—
447	C₂³	—	4	1	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	4	1	no	—
450	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
451	Q₈	—	2	4	no	—
452	Q₈	—	2	4	no	—
453	Q₈	—	4	2	no	—
454	Q₈	—	4	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	2	2	no	—
458	D₄	C₄×C₇×D₄	1	4	yes	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	4	8	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	8	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
489	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
490	C₂×C₄	C₇×#121	1	2	yes	—
491	C₂×C₄	C₇×#121	1	2	yes	—
492	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
493	C₂×C₄	C₇×#137	1	2	yes	—
494	C₂×C₄	C₇×#137	1	2	yes	—
495	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
496	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
497	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
498	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
499	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
500	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
501	C₈	—	2	4	no	—
502	C₈	—	4	2	no	—
503	C₇	#21504	1	1	yes	characteristic, direct product factor, Sylow
504	C₂²	—	2	4	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
512	C₂²	C₂×C₇×#121	1	1	yes	characteristic
513	C₂²	C₇×#525	1	1	yes	characteristic
514	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
515	C₂²	C₂×C₇×#137	1	1	yes	characteristic
516	C₂²	C₂×C₇×#137	1	1	yes	characteristic
517	C₂²	C₇×#575	1	1	yes	characteristic
518	C₄	—	2	2	no	—
519	C₄	—	2	2	no	—
520	C₄	—	2	2	no	—
521	C₄	—	2	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	4	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	1	no	—
527	C₄	—	8	2	no	—
528	C₄	C₇×#440	1	4	yes	complemented
529	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
530	C₄	C₄×C₇×#41	1	1	yes	characteristic
531	C₄	C₇×#529	1	1	yes	characteristic
532	C₄	C₇×#532	1	1	yes	characteristic
533	C₂	—	4	4	no	—
534	C₂	C₄×C₇×#121	1	1	yes	characteristic
535	C₂	C₂×C₇×#440	1	1	yes	characteristic
536	C₂	C₄×C₇×#137	1	1	yes	characteristic
537	C₂	C₇×#3036	1	1	yes	characteristic
538	C₂	C₇×#3039	1	1	yes	characteristic
539	C₂	C₇×#3110	1	1	yes	characteristic
540	C₂	C₇×#3116	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₇×C₄⋊Q₈	C₂	semidirect
5. C₇×#1981	C₂	semidirect
6. C₇×#1947	C₂	semidirect
7. C₇×#1928	C₂	semidirect
10. C₇×#1767	C₂	semidirect
11. C₇×#1763	C₂	semidirect
19. C₇×#461	C₄	semidirect
45. #21504	C₇	direct (N × H)
57. C₇×C₄⋊Q₈	C₂×C₄	semidirect
92. C₇×C₄⋊C₈	C₂×C₄	semidirect
95. C₇×#103	C₂×C₄	semidirect
105. C₄×C₄⋊Q₈	C₂×C₇	semidirect
107. #1981	C₂×C₇	semidirect
108. #1947	C₂×C₇	semidirect
109. #1928	C₂×C₇	semidirect
112. #1767	C₂×C₇	semidirect
113. #1763	C₂×C₇	semidirect
192. #461	C₄×C₇	semidirect
289. C₄⋊Q₈	C₂×C₄×C₇	semidirect
324. C₄⋊C₈	C₂×C₄×C₇	semidirect
327. #103	C₂×C₄×C₇	semidirect
359. C₄×C₇	#440	semidirect
503. C₇	#21504	direct (N × H)
528. C₄	C₇×#440	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#461	2
C₂×C₇×#440	2
C₄×C₇×C₄⋊Q₈	2
C₄²×C₇×D₄	2
C₇×#1981	2
C₇×#1947	2
C₇×#1928	2
C₇×#1877	2
C₇×#1802	2
C₇×#1767	2
C₇×#1763	2
#21504	7

Subgroups — C7×#21504 #13198720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21504 #13198720