Subgroups — C₇×#24781 #13201997

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2417	C₂	1	2	yes	maximal
6	C₇×#2068	C₂	1	2	yes	maximal, complemented
7	C₇×#2036	C₂	1	2	yes	maximal, complemented
8	C₇×#1969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1942	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1872	C₂	1	2	yes	maximal, complemented
11	C₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
18	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₇×#467	—	2	4	no	—
20	C₇×#467	—	2	4	no	—
21	C₇×C₈×D₄	—	2	4	no	—
22	C₂×C₇×#107	—	2	4	no	—
23	C₂×C₇×#107	C₂²	1	1	yes	characteristic
24	C₂×C₇×#107	C₂²	1	2	yes	—
25	C₂×C₇×#107	C₂²	1	1	yes	characteristic
26	C₂×C₇×#107	C₂²	1	1	yes	characteristic
27	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₇×#103	C₂²	1	2	yes	—
29	C₂×C₇×#103	C₂²	1	2	yes	—
30	C₂×C₇×#103	C₂²	1	2	yes	—
31	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
34	C₇×#398	—	2	2	no	—
35	C₇×#396	—	2	2	no	—
36	C₇×#391	C₂²	1	2	yes	—
37	C₇×#389	C₂²	1	2	yes	—
38	C₇×#386	—	2	2	no	—
39	C₇×#380	—	2	2	no	—
40	C₇×#380	C₂²	1	2	yes	—
41	C₇×#340	C₂²	1	2	yes	—
42	C₇×#340	C₂²	1	2	yes	—
43	#24781	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂²×C₇×Q₈	C₂³	1	2	yes	—
45	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₇	—	2	2	no	—
47	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂×C₇×Q₁₆	—	2	2	no	—
50	C₂×C₇×Q₁₆	—	2	2	no	—
51	C₂×C₇×Q₁₆	—	2	2	no	—
52	C₂²×C₇×C₈	—	2	2	no	—
53	C₂²×C₇×C₈	—	2	2	no	—
54	C₂²×C₇×C₈	—	2	2	no	—
55	C₂²×C₇×C₈	—	2	2	no	—
56	C₂²×C₇×C₈	—	2	2	no	—
57	C₂²×C₇×C₈	D₄	1	4	yes	—
58	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
59	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
60	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₇×C₂²⋊Q₈	—	2	4	no	—
63	C₇×C₂²⋊Q₈	—	2	4	no	—
64	C₇×C₂²⋊Q₈	—	2	4	no	—
65	C₇×C₂²⋊Q₈	—	4	2	no	—
66	C₄×C₇×D₄	—	2	2	no	—
67	C₄×C₇×D₄	—	2	2	no	—
68	C₂×C₇×C₄⋊C₄	—	2	4	no	—
69	C₂×C₇×C₄⋊C₄	—	2	2	no	—
70	C₂×C₇×C₄⋊C₄	—	2	2	no	—
71	C₂×C₇×C₄⋊C₄	—	2	2	no	—
72	C₂×C₇×C₄⋊C₄	—	2	2	no	—
73	C₂×C₇×C₄⋊C₄	—	4	2	no	—
74	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
75	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
76	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
77	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
78	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
79	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
80	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
81	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
82	C₂×C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
84	C₇×#107	—	2	2	no	—
85	C₇×#107	—	2	2	no	—
86	C₇×#107	—	2	4	no	—
87	C₇×#107	D₄	1	4	yes	complemented
88	C₇×C₄⋊C₈	—	2	2	no	—
89	C₇×#103	—	2	4	no	—
90	C₇×#103	—	2	4	no	—
91	C₇×#103	—	4	2	no	—
92	C₇×C₂²⋊C₈	—	2	2	no	—
93	C₇×C₂²⋊C₈	—	2	2	no	—
94	C₄×C₇×C₈	—	2	2	no	—
95	C₇×#95	—	2	2	no	—
96	C₇×#95	—	2	2	no	—
97	C₇×#95	—	2	2	no	—
98	C₇×#95	—	2	2	no	—
99	C₂×#467	C₂×C₇	1	1	yes	characteristic, complemented
100	C₂×#467	C₂×C₇	1	1	yes	characteristic, complemented
101	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
102	C₂²×#107	C₂×C₇	1	1	yes	characteristic, complemented
103	#2417	C₂×C₇	1	2	yes	—
104	#2068	C₂×C₇	1	2	yes	complemented
105	#2036	C₂×C₇	1	2	yes	complemented
106	#1969	C₂×C₇	1	1	yes	characteristic, complemented
107	#1942	C₂×C₇	1	1	yes	characteristic, complemented
108	#1872	C₂×C₇	1	2	yes	complemented
109	#1808	C₂×C₇	1	1	yes	characteristic, complemented
110	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
111	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇×Q₈	—	2	2	no	—
113	C₂×C₇×Q₈	—	2	2	no	—
114	C₂×C₇×Q₈	—	2	2	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×D₄	—	2	2	no	—
119	C₂×C₇×D₄	—	2	2	no	—
120	C₂×C₇×D₄	Q₁₆	1	4	yes	complemented
121	C₂²×C₄×C₇	—	2	1	no	—
122	C₂²×C₄×C₇	—	2	1	no	—
123	C₂²×C₄×C₇	—	2	1	no	—
124	C₂²×C₄×C₇	—	2	1	no	—
125	C₂²×C₄×C₇	—	2	2	no	—
126	C₂²×C₄×C₇	—	2	1	no	—
127	C₂²×C₄×C₇	—	2	1	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	1	no	—
132	C₂²×C₄×C₇	—	2	2	no	—
133	C₂²×C₄×C₇	—	2	2	no	—
134	C₂²×C₄×C₇	—	2	2	no	—
135	C₂²×C₄×C₇	—	4	4	no	—
136	C₂²×C₄×C₇	—	4	4	no	—
137	C₂²×C₄×C₇	—	8	2	no	—
138	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₇	#41	1	2	yes	—
144	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
145	C₇×Q₁₆	—	4	4	no	—
146	C₂×C₇×C₈	—	2	1	no	—
147	C₂×C₇×C₈	—	2	1	no	—
148	C₂×C₇×C₈	—	2	8	no	—
149	C₂×C₇×C₈	—	2	1	no	—
150	C₂×C₇×C₈	—	2	1	no	—
151	C₂×C₇×C₈	—	2	1	no	—
152	C₂×C₇×C₈	—	2	1	no	—
153	C₂×C₇×C₈	—	2	1	no	—
154	C₂×C₇×C₈	—	2	1	no	—
155	C₂×C₇×C₈	—	4	4	no	—
156	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
157	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
158	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
159	C₂×C₇×C₈	#41	1	2	yes	—
160	C₇×C₄⋊C₄	—	2	2	no	—
161	C₇×C₄⋊C₄	—	2	2	no	—
162	C₇×C₄⋊C₄	—	2	2	no	—
163	C₇×C₄⋊C₄	—	2	2	no	—
164	C₇×C₄⋊C₄	—	2	2	no	—
165	C₇×C₄⋊C₄	—	2	2	no	—
166	C₇×C₄⋊C₄	—	4	4	no	—
167	C₇×C₄⋊C₄	—	4	4	no	—
168	C₇×C₄⋊C₄	—	4	4	no	—
169	C₇×C₄⋊C₄	—	8	2	no	—
170	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
171	C₇×C₂²⋊C₄	—	2	2	no	—
172	C₇×C₂²⋊C₄	—	2	2	no	—
173	C₇×C₂²⋊C₄	—	4	4	no	—
174	C₇×C₂²⋊C₄	—	4	4	no	—
175	C₇×C₂²⋊C₄	—	8	2	no	—
176	C₄²×C₇	—	2	1	no	—
177	C₄²×C₇	—	2	1	no	—
178	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
179	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
180	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
181	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
182	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
183	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
184	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
185	#467	—	2	4	no	—
186	#467	—	2	4	no	—
187	C₈×D₄	—	2	4	no	—
188	C₂×#107	—	2	4	no	—
189	C₂×#107	C₂²×C₇	1	1	yes	characteristic
190	C₂×#107	C₂²×C₇	1	1	yes	characteristic
191	C₂×#107	C₂²×C₇	1	2	yes	—
192	C₂×#107	C₂²×C₇	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
194	C₂×#103	C₂²×C₇	1	2	yes	—
195	C₂×#103	C₂²×C₇	1	2	yes	—
196	C₂×#103	C₂²×C₇	1	2	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
198	C₂×C₂²⋊C₈	C₂²×C₇	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
200	#398	—	2	2	no	—
201	#396	—	2	2	no	—
202	#391	C₂²×C₇	1	2	yes	—
203	#389	C₂²×C₇	1	2	yes	—
204	#386	—	2	2	no	—
205	#380	—	2	2	no	—
206	#380	C₂²×C₇	1	2	yes	—
207	#340	C₂²×C₇	1	2	yes	—
208	#340	C₂²×C₇	1	2	yes	—
209	C₂³×C₇	—	2	1	no	—
210	C₂³×C₇	—	2	1	no	—
211	C₂³×C₇	—	2	1	no	—
212	C₂³×C₇	—	2	1	no	—
213	C₂³×C₇	—	2	1	no	—
214	C₂³×C₇	—	2	1	no	—
215	C₂³×C₇	—	2	1	no	—
216	C₂³×C₇	—	2	1	no	—
217	C₂³×C₇	—	2	1	no	—
218	C₂³×C₇	—	2	1	no	—
219	C₂³×C₇	#121	1	2	yes	—
220	C₂³×C₇	C₂×Q₁₆	1	2	yes	—
221	C₂³×C₇	C₂×Q₁₆	1	2	yes	—
222	C₂³×C₇	#135	1	2	yes	—
223	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
224	C₇×Q₈	—	4	2	no	—
225	C₇×Q₈	—	4	2	no	—
226	C₇×Q₈	—	4	2	no	—
227	C₇×Q₈	—	4	2	no	—
228	C₇×D₄	—	2	8	no	—
229	C₂×C₄×C₇	—	2	1	no	—
230	C₂×C₄×C₇	—	2	1	no	—
231	C₂×C₄×C₇	—	2	1	no	—
232	C₂×C₄×C₇	—	2	1	no	—
233	C₂×C₄×C₇	—	2	2	no	—
234	C₂×C₄×C₇	—	2	2	no	—
235	C₂×C₄×C₇	—	2	2	no	—
236	C₂×C₄×C₇	—	2	2	no	—
237	C₂×C₄×C₇	—	2	1	no	—
238	C₂×C₄×C₇	—	2	1	no	—
239	C₂×C₄×C₇	—	2	4	no	—
240	C₂×C₄×C₇	—	2	1	no	—
241	C₂×C₄×C₇	—	2	1	no	—
242	C₂×C₄×C₇	—	2	1	no	—
243	C₂×C₄×C₇	—	2	1	no	—
244	C₂×C₄×C₇	—	4	2	no	—
245	C₂×C₄×C₇	—	4	2	no	—
246	C₂×C₄×C₇	—	4	1	no	—
247	C₂×C₄×C₇	—	4	1	no	—
248	C₂×C₄×C₇	—	4	1	no	—
249	C₂×C₄×C₇	—	4	1	no	—
250	C₂×C₄×C₇	—	4	2	no	—
251	C₂×C₄×C₇	—	4	2	no	—
252	C₂×C₄×C₇	—	4	2	no	—
253	C₂×C₄×C₇	—	4	2	no	—
254	C₂×C₄×C₇	—	4	2	no	—
255	C₂×C₄×C₇	—	4	2	no	—
256	C₂×C₄×C₇	—	4	2	no	—
257	C₂×C₄×C₇	—	4	2	no	—
258	C₂×C₄×C₇	—	4	2	no	—
259	C₂×C₄×C₇	—	8	8	no	—
260	C₂×C₄×C₇	#121	1	2	yes	—
261	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
262	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
266	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
267	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
268	C₇×C₈	—	2	2	no	—
269	C₇×C₈	—	4	1	no	—
270	C₇×C₈	—	4	1	no	—
271	C₇×C₈	#121	1	4	yes	—
272	C₂²×Q₈	C₂³×C₇	1	2	yes	—
273	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
274	C₂³×C₄	—	2	2	no	—
275	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
276	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
277	C₂×Q₁₆	—	2	2	no	—
278	C₂×Q₁₆	—	2	2	no	—
279	C₂×Q₁₆	—	2	2	no	—
280	C₂²×C₈	—	2	2	no	—
281	C₂²×C₈	—	2	2	no	—
282	C₂²×C₈	—	2	2	no	—
283	C₂²×C₈	—	2	2	no	—
284	C₂²×C₈	—	2	2	no	—
285	C₂²×C₈	C₇×D₄	1	4	yes	—
286	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
290	C₂²⋊Q₈	—	2	4	no	—
291	C₂²⋊Q₈	—	2	4	no	—
292	C₂²⋊Q₈	—	2	4	no	—
293	C₂²⋊Q₈	—	4	2	no	—
294	C₄×D₄	—	2	2	no	—
295	C₄×D₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	4	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	4	2	no	—
302	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
305	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
310	C₂×C₂²⋊C₄	C₇×D₄	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
312	#107	—	2	2	no	—
313	#107	—	2	2	no	—
314	#107	—	2	4	no	—
315	#107	C₇×D₄	1	4	yes	complemented
316	C₄⋊C₈	—	2	2	no	—
317	#103	—	2	4	no	—
318	#103	—	2	4	no	—
319	#103	—	4	2	no	—
320	C₂²⋊C₈	—	2	2	no	—
321	C₂²⋊C₈	—	2	2	no	—
322	C₄×C₈	—	2	2	no	—
323	#95	—	2	2	no	—
324	#95	—	2	2	no	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	C₂²×C₇	—	2	2	no	—
328	C₂²×C₇	—	2	2	no	—
329	C₂²×C₇	—	2	2	no	—
330	C₂²×C₇	—	2	2	no	—
331	C₂²×C₇	—	2	2	no	—
332	C₂²×C₇	—	2	2	no	—
333	C₂²×C₇	—	2	2	no	—
334	C₂²×C₇	—	2	2	no	—
335	C₂²×C₇	—	2	2	no	—
336	C₂²×C₇	—	4	1	no	—
337	C₂²×C₇	—	4	1	no	—
338	C₂²×C₇	—	4	1	no	—
339	C₂²×C₇	—	4	1	no	—
340	C₂²×C₇	#467	1	4	yes	complemented
341	C₂²×C₇	C₂×#121	1	1	yes	characteristic
342	C₂²×C₇	D₄²	1	1	yes	characteristic
343	C₂²×C₇	#547	1	1	yes	characteristic
344	C₂²×C₇	C₂²×Q₁₆	1	1	yes	characteristic
345	C₂²×C₇	C₂×#135	1	1	yes	characteristic
346	C₂²×C₇	#576	1	1	yes	characteristic
347	C₂²×C₇	#578	1	1	yes	characteristic
348	C₄×C₇	—	2	2	no	—
349	C₄×C₇	—	2	2	no	—
350	C₄×C₇	—	4	1	no	—
351	C₄×C₇	—	4	4	no	—
352	C₄×C₇	—	4	1	no	—
353	C₄×C₇	—	8	2	no	—
354	C₄×C₇	—	8	2	no	—
355	C₄×C₇	—	8	2	no	—
356	C₄×C₇	C₂×#121	1	1	yes	characteristic
357	C₄×C₇	#537	1	1	yes	characteristic
358	C₄×C₇	#548	1	1	yes	characteristic
359	C₄×C₇	#553	1	1	yes	characteristic
360	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
361	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	C₇×Q₁₆	1	4	yes	complemented
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	4	4	no	—
387	C₂²×C₄	—	8	2	no	—
388	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
389	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
390	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
391	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
392	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
393	C₂²×C₄	C₇×#41	1	2	yes	—
394	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
395	Q₁₆	—	4	4	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	8	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	4	4	no	—
406	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
407	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
408	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
409	C₂×C₈	C₇×#41	1	2	yes	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	4	4	no	—
417	C₄⋊C₄	—	4	4	no	—
418	C₄⋊C₄	—	4	4	no	—
419	C₄⋊C₄	—	8	2	no	—
420	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
421	C₂²⋊C₄	—	2	2	no	—
422	C₂²⋊C₄	—	2	2	no	—
423	C₂²⋊C₄	—	4	4	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	—	8	2	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₂×C₇	—	2	4	no	—
429	C₂×C₇	—	4	2	no	—
430	C₂×C₇	#2416	1	1	yes	characteristic
431	C₂×C₇	C₂×#467	1	1	yes	characteristic
432	C₂×C₇	#3061	1	1	yes	characteristic
433	C₂×C₇	#3280	1	1	yes	characteristic
434	C₂×C₇	D₄×Q₁₆	1	1	yes	characteristic
435	C₂×C₇	#3336	1	1	yes	characteristic
436	C₂×C₇	#3338	1	1	yes	characteristic
437	C₂³	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	2	1	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	1	no	—
447	C₂³	C₇×#121	1	2	yes	—
448	C₂³	C₂×C₇×Q₁₆	1	2	yes	—
449	C₂³	C₂×C₇×Q₁₆	1	2	yes	—
450	C₂³	C₇×#135	1	2	yes	—
451	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	Q₈	—	4	2	no	—
455	Q₈	—	4	2	no	—
456	D₄	—	2	8	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	8	8	no	—
488	C₂×C₄	C₇×#121	1	2	yes	—
489	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
490	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
491	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
492	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
493	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
494	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
495	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
496	C₈	—	2	2	no	—
497	C₈	—	4	1	no	—
498	C₈	—	4	1	no	—
499	C₈	C₇×#121	1	4	yes	—
500	C₇	#24781	1	1	yes	characteristic, direct product factor, Sylow
501	C₂²	—	2	2	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	2	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	2	2	no	—
508	C₂²	—	2	2	no	—
509	C₂²	—	2	2	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	1	no	—
512	C₂²	—	4	1	no	—
513	C₂²	—	4	1	no	—
514	C₂²	C₇×#467	1	4	yes	complemented
515	C₂²	C₂×C₇×#121	1	1	yes	characteristic
516	C₂²	C₇×D₄²	1	1	yes	characteristic
517	C₂²	C₇×#547	1	1	yes	characteristic
518	C₂²	C₂²×C₇×Q₁₆	1	1	yes	characteristic
519	C₂²	C₂×C₇×#135	1	1	yes	characteristic
520	C₂²	C₇×#576	1	1	yes	characteristic
521	C₂²	C₇×#578	1	1	yes	characteristic
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	4	1	no	—
525	C₄	—	4	4	no	—
526	C₄	—	4	1	no	—
527	C₄	—	8	2	no	—
528	C₄	—	8	2	no	—
529	C₄	—	8	2	no	—
530	C₄	C₂×C₇×#121	1	1	yes	characteristic
531	C₄	C₇×#537	1	1	yes	characteristic
532	C₄	C₇×#548	1	1	yes	characteristic
533	C₄	C₇×#553	1	1	yes	characteristic
534	C₂	—	2	4	no	—
535	C₂	—	4	2	no	—
536	C₂	C₇×#2416	1	1	yes	characteristic
537	C₂	C₂×C₇×#467	1	1	yes	characteristic
538	C₂	C₇×#3061	1	1	yes	characteristic
539	C₂	C₇×#3280	1	1	yes	characteristic
540	C₂	C₇×D₄×Q₁₆	1	1	yes	characteristic
541	C₂	C₇×#3336	1	1	yes	characteristic
542	C₂	C₇×#3338	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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#467	C₂	semidirect
2. C₂×C₇×#467	C₂	semidirect
4. C₂²×C₇×#107	C₂	semidirect
6. C₇×#2068	C₂	semidirect
7. C₇×#2036	C₂	semidirect
8. C₇×#1969	C₂	semidirect
9. C₇×#1942	C₂	semidirect
10. C₇×#1872	C₂	semidirect
11. C₇×#1808	C₂	semidirect
43. #24781	C₇	direct (N × H)
87. C₇×#107	D₄	semidirect
99. C₂×#467	C₂×C₇	semidirect
100. C₂×#467	C₂×C₇	semidirect
102. C₂²×#107	C₂×C₇	semidirect
104. #2068	C₂×C₇	semidirect
105. #2036	C₂×C₇	semidirect
106. #1969	C₂×C₇	semidirect
107. #1942	C₂×C₇	semidirect
108. #1872	C₂×C₇	semidirect
109. #1808	C₂×C₇	semidirect
120. C₂×C₇×D₄	Q₁₆	semidirect
315. #107	C₇×D₄	semidirect
340. C₂²×C₇	#467	semidirect
370. C₂×D₄	C₇×Q₁₆	semidirect
500. C₇	#24781	direct (N × H)
514. C₂²	C₇×#467	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#467	2
C₂×C₇×C₈×D₄	2
C₂²×C₇×#107	2
C₇×#2417	2
C₇×#2068	2
C₇×#2036	2
C₇×#1969	2
C₇×#1942	2
C₇×#1872	2
C₇×#1808	2
#24781	7

Subgroups — C7×#24781 #13201997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#24781 #13201997