Subgroups — C₇×#19814 #13197030

Order 1792.

590 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3384	C₂	1	1	yes	maximal, characteristic
2	C₇×#3380	C₂	1	1	yes	maximal, characteristic
3	C₇×#3338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3299	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1723	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1696	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1675	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1672	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1644	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1637	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1632	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1632	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1584	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1580	C₂	1	1	yes	maximal, characteristic
16	C₇×#554	C₂²	1	1	yes	characteristic
17	C₇×#554	C₂²	1	1	yes	characteristic
18	C₇×#554	C₂²	1	1	yes	characteristic
19	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
20	C₇×#501	—	2	1	no	—
21	C₇×#500	C₂²	1	1	yes	characteristic
22	C₇×#500	C₂²	1	1	yes	characteristic
23	C₇×#499	—	2	1	no	—
24	C₇×#498	C₂²	1	1	yes	characteristic
25	C₇×#494	C₂²	1	1	yes	characteristic
26	C₇×#487	C₂²	1	1	yes	characteristic
27	C₇×#487	C₂²	1	1	yes	characteristic
28	C₇×#484	—	2	1	no	—
29	C₇×#484	—	2	1	no	—
30	C₇×#484	—	2	1	no	—
31	C₇×#478	—	2	1	no	—
32	C₇×#478	—	2	1	no	—
33	C₇×#478	—	2	1	no	—
34	C₇×#477	C₂²	1	1	yes	characteristic
35	C₇×#477	C₂²	1	1	yes	characteristic
36	C₇×#477	C₂²	1	1	yes	characteristic
37	C₇×#476	—	2	1	no	—
38	C₇×#476	C₂²	1	1	yes	characteristic
39	C₇×#476	C₂²	1	1	yes	characteristic
40	C₇×#476	C₂²	1	1	yes	characteristic
41	C₇×#474	C₂²	1	1	yes	characteristic
42	C₇×#467	—	2	1	no	—
43	C₇×#467	—	2	1	no	—
44	C₇×#462	C₂²	1	1	yes	characteristic
45	C₇×#451	—	2	1	no	—
46	C₇×#446	C₂²	1	1	yes	characteristic
47	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
48	C₇×#442	—	2	1	no	—
49	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
50	C₄×C₇×D₈	—	2	1	no	—
51	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
52	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
53	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
54	C₂×C₇×#107	—	2	1	no	—
55	C₂×C₇×#103	—	2	1	no	—
56	C₇×#335	C₂²	1	1	yes	characteristic
57	C₇×#334	C₂²	1	1	yes	characteristic
58	C₇×#333	C₂²	1	1	yes	characteristic
59	C₇×#333	C₂²	1	1	yes	characteristic
60	C₇×#333	C₂²	1	1	yes	characteristic
61	C₇×#332	C₂²	1	1	yes	characteristic
62	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₇×#322	C₂²	1	1	yes	characteristic
67	#19814	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₇×Q₈	—	2	1	no	—
69	C₂×C₇×Q₁₆	—	2	1	no	—
70	C₂×C₇×Q₁₆	—	4	1	no	—
71	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
72	C₂×C₇×D₈	D₄	1	1	yes	characteristic
73	C₂²×C₇×C₈	—	2	1	no	—
74	C₂²×C₇×C₈	—	2	1	no	—
75	C₇×C₄⋊Q₈	—	2	1	no	—
76	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₇×#125	—	2	1	no	—
81	C₇×#125	—	2	1	no	—
82	C₇×#125	—	2	1	no	—
83	C₇×C₂²⋊Q₈	—	2	1	no	—
84	C₇×C₂²⋊Q₈	—	2	1	no	—
85	C₇×C₂²⋊Q₈	—	2	1	no	—
86	C₇×C₂²⋊Q₈	—	2	1	no	—
87	C₇×C₂²⋊Q₈	—	4	1	no	—
88	C₇×C₂²⋊Q₈	—	4	1	no	—
89	C₇×C₂²⋊Q₈	—	4	1	no	—
90	C₇×C₂²⋊Q₈	—	4	1	no	—
91	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
92	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
93	C₄×C₇×D₄	—	2	1	no	—
94	C₄×C₇×D₄	—	2	1	no	—
95	C₄×C₇×D₄	—	2	1	no	—
96	C₄×C₇×D₄	—	2	1	no	—
97	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
98	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
99	C₂×C₇×C₄⋊C₄	—	2	1	no	—
100	C₂×C₇×C₄⋊C₄	—	2	1	no	—
101	C₂×C₇×C₄⋊C₄	—	2	1	no	—
102	C₇×#107	—	2	1	no	—
103	C₇×#107	—	2	1	no	—
104	C₇×#107	—	2	1	no	—
105	C₇×#107	—	2	1	no	—
106	C₇×#107	—	2	1	no	—
107	C₇×#107	—	2	1	no	—
108	C₇×#107	—	2	1	no	—
109	C₇×#107	—	2	1	no	—
110	C₇×#107	D₄	1	1	yes	characteristic
111	C₇×#107	D₄	1	1	yes	characteristic
112	C₇×#106	—	2	1	no	—
113	C₇×#106	—	2	1	no	—
114	C₇×#106	—	2	1	no	—
115	C₇×#106	—	2	1	no	—
116	C₇×#106	—	2	1	no	—
117	C₇×C₄⋊C₈	—	2	1	no	—
118	C₇×C₄⋊C₈	—	2	1	no	—
119	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₇×#103	—	2	1	no	—
124	C₇×#103	—	2	1	no	—
125	C₇×#103	—	2	1	no	—
126	C₇×#103	—	2	1	no	—
127	C₇×#103	—	2	1	no	—
128	C₇×#103	—	2	1	no	—
129	C₇×#103	—	4	1	no	—
130	C₇×#103	D₄	1	1	yes	characteristic
131	C₇×#103	D₄	1	1	yes	characteristic
132	C₇×#102	—	2	1	no	—
133	C₇×#102	—	2	1	no	—
134	C₇×#102	—	2	1	no	—
135	C₇×#102	—	2	1	no	—
136	C₇×#102	D₄	1	1	yes	characteristic
137	C₇×#102	D₄	1	1	yes	characteristic
138	C₇×C₂²⋊C₈	—	2	1	no	—
139	C₇×C₂²⋊C₈	—	2	1	no	—
140	C₇×#97	—	2	1	no	—
141	C₄×C₇×C₈	—	2	1	no	—
142	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
145	#3384	C₂×C₇	1	1	yes	characteristic
146	#3380	C₂×C₇	1	1	yes	characteristic
147	#3338	C₂×C₇	1	1	yes	characteristic, complemented
148	#3299	C₂×C₇	1	1	yes	characteristic, complemented
149	#1723	C₂×C₇	1	1	yes	characteristic, complemented
150	#1696	C₂×C₇	1	1	yes	characteristic, complemented
151	#1683	C₂×C₇	1	1	yes	characteristic, complemented
152	#1675	C₂×C₇	1	1	yes	characteristic, complemented
153	#1672	C₂×C₇	1	1	yes	characteristic, complemented
154	#1644	C₂×C₇	1	1	yes	characteristic, complemented
155	#1637	C₂×C₇	1	1	yes	characteristic, complemented
156	#1632	C₂×C₇	1	1	yes	characteristic, complemented
157	#1632	C₂×C₇	1	1	yes	characteristic, complemented
158	#1584	C₂×C₇	1	1	yes	characteristic, complemented
159	#1580	C₂×C₇	1	1	yes	characteristic
160	C₂×C₇×Q₈	—	2	1	no	—
161	C₂×C₇×Q₈	—	2	1	no	—
162	C₂×C₇×Q₈	—	2	1	no	—
163	C₂×C₇×Q₈	—	2	1	no	—
164	C₂×C₇×Q₈	—	4	1	no	—
165	C₂×C₇×Q₈	—	4	2	no	—
166	C₂×C₇×Q₈	—	8	1	no	—
167	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
168	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
169	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
170	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₇	—	2	1	no	—
172	C₂²×C₄×C₇	—	2	1	no	—
173	C₂²×C₄×C₇	—	4	1	no	—
174	C₂²×C₄×C₇	—	4	1	no	—
175	C₂²×C₄×C₇	—	4	1	no	—
176	C₂²×C₄×C₇	—	4	1	no	—
177	C₇×Q₁₆	—	4	1	no	—
178	C₇×Q₁₆	—	8	1	no	—
179	C₇×Q₁₆	—	8	2	no	—
180	C₇×D₈	Q₁₆	1	4	yes	complemented
181	C₂×C₇×C₈	—	2	1	no	—
182	C₂×C₇×C₈	—	2	1	no	—
183	C₂×C₇×C₈	—	2	1	no	—
184	C₂×C₇×C₈	—	2	1	no	—
185	C₂×C₇×C₈	—	2	1	no	—
186	C₂×C₇×C₈	—	4	2	no	—
187	C₂×C₇×C₈	—	8	1	no	—
188	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
189	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
190	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
191	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
192	C₇×C₄⋊C₄	—	2	1	no	—
193	C₇×C₄⋊C₄	—	2	1	no	—
194	C₇×C₄⋊C₄	—	2	1	no	—
195	C₇×C₄⋊C₄	—	2	1	no	—
196	C₇×C₄⋊C₄	—	2	1	no	—
197	C₇×C₄⋊C₄	—	2	1	no	—
198	C₇×C₄⋊C₄	—	2	1	no	—
199	C₇×C₄⋊C₄	—	2	1	no	—
200	C₇×C₄⋊C₄	—	2	1	no	—
201	C₇×C₄⋊C₄	—	2	1	no	—
202	C₇×C₄⋊C₄	—	4	1	no	—
203	C₇×C₄⋊C₄	—	4	1	no	—
204	C₇×C₄⋊C₄	—	4	1	no	—
205	C₇×C₄⋊C₄	—	4	1	no	—
206	C₇×C₄⋊C₄	—	4	1	no	—
207	C₇×C₄⋊C₄	—	4	1	no	—
208	C₇×C₄⋊C₄	—	4	1	no	—
209	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
210	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
211	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
212	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
213	C₇×C₂²⋊C₄	—	2	1	no	—
214	C₇×C₂²⋊C₄	—	2	1	no	—
215	C₇×C₂²⋊C₄	—	4	1	no	—
216	C₇×C₂²⋊C₄	—	4	1	no	—
217	C₇×C₂²⋊C₄	—	4	1	no	—
218	C₇×C₂²⋊C₄	—	4	1	no	—
219	C₄²×C₇	—	2	1	no	—
220	C₄²×C₇	—	2	1	no	—
221	C₄²×C₇	C₂⁴	1	1	yes	characteristic
222	#554	C₂²×C₇	1	1	yes	characteristic
223	#554	C₂²×C₇	1	1	yes	characteristic
224	#554	C₂²×C₇	1	1	yes	characteristic
225	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
226	#501	—	2	1	no	—
227	#500	C₂²×C₇	1	1	yes	characteristic
228	#500	C₂²×C₇	1	1	yes	characteristic
229	#499	—	2	1	no	—
230	#498	C₂²×C₇	1	1	yes	characteristic
231	#494	C₂²×C₇	1	1	yes	characteristic
232	#487	C₂²×C₇	1	1	yes	characteristic
233	#487	C₂²×C₇	1	1	yes	characteristic
234	#484	—	2	1	no	—
235	#484	—	2	1	no	—
236	#484	—	2	1	no	—
237	#478	—	2	1	no	—
238	#478	—	2	1	no	—
239	#478	—	2	1	no	—
240	#477	C₂²×C₇	1	1	yes	characteristic
241	#477	C₂²×C₇	1	1	yes	characteristic
242	#477	C₂²×C₇	1	1	yes	characteristic
243	#476	—	2	1	no	—
244	#476	C₂²×C₇	1	1	yes	characteristic
245	#476	C₂²×C₇	1	1	yes	characteristic
246	#476	C₂²×C₇	1	1	yes	characteristic
247	#474	C₂²×C₇	1	1	yes	characteristic
248	#467	—	2	1	no	—
249	#467	—	2	1	no	—
250	#462	C₂²×C₇	1	1	yes	characteristic
251	#451	—	2	1	no	—
252	#446	C₂²×C₇	1	1	yes	characteristic
253	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
254	#442	—	2	1	no	—
255	C₄×Q₁₆	C₂²×C₇	1	1	yes	characteristic
256	C₄×D₈	—	2	1	no	—
257	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
258	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
259	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
260	C₂×#107	—	2	1	no	—
261	C₂×#103	—	2	1	no	—
262	#335	C₂²×C₇	1	1	yes	characteristic
263	#334	C₂²×C₇	1	1	yes	characteristic
264	#333	C₂²×C₇	1	1	yes	characteristic
265	#333	C₂²×C₇	1	1	yes	characteristic
266	#333	C₂²×C₇	1	1	yes	characteristic
267	#332	C₂²×C₇	1	1	yes	characteristic
268	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
269	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
270	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
271	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
272	#322	C₂²×C₇	1	1	yes	characteristic
273	C₂³×C₇	—	2	1	no	—
274	C₂³×C₇	—	2	1	no	—
275	C₇×Q₈	—	2	1	no	—
276	C₇×Q₈	—	2	1	no	—
277	C₇×Q₈	—	2	1	no	—
278	C₇×Q₈	—	2	1	no	—
279	C₇×Q₈	—	4	1	no	—
280	C₇×Q₈	—	4	1	no	—
281	C₇×Q₈	—	8	1	no	—
282	C₇×Q₈	—	8	1	no	—
283	C₇×Q₈	—	8	1	no	—
284	C₇×Q₈	—	8	2	no	—
285	C₇×D₄	—	2	1	no	—
286	C₇×D₄	—	2	1	no	—
287	C₇×D₄	C₂×Q₁₆	1	2	yes	—
288	C₇×D₄	C₂×Q₁₆	1	2	yes	—
289	C₂×C₄×C₇	—	2	1	no	—
290	C₂×C₄×C₇	—	2	1	no	—
291	C₂×C₄×C₇	—	2	1	no	—
292	C₂×C₄×C₇	—	2	1	no	—
293	C₂×C₄×C₇	—	2	1	no	—
294	C₂×C₄×C₇	—	2	1	no	—
295	C₂×C₄×C₇	—	4	1	no	—
296	C₂×C₄×C₇	—	4	1	no	—
297	C₂×C₄×C₇	—	4	1	no	—
298	C₂×C₄×C₇	—	4	1	no	—
299	C₂×C₄×C₇	—	4	1	no	—
300	C₂×C₄×C₇	—	4	1	no	—
301	C₂×C₄×C₇	—	4	1	no	—
302	C₂×C₄×C₇	—	4	1	no	—
303	C₂×C₄×C₇	—	8	2	no	—
304	C₂×C₄×C₇	—	16	1	no	—
305	C₂×C₄×C₇	—	16	1	no	—
306	C₂×C₄×C₇	—	16	1	no	—
307	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
308	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
309	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
310	C₇×C₈	—	2	1	no	—
311	C₇×C₈	—	2	1	no	—
312	C₇×C₈	—	2	1	no	—
313	C₇×C₈	—	4	1	no	—
314	C₇×C₈	—	4	1	no	—
315	C₇×C₈	—	4	1	no	—
316	C₇×C₈	—	4	1	no	—
317	C₇×C₈	—	4	1	no	—
318	C₇×C₈	C₂×Q₁₆	1	2	yes	—
319	C₂²×Q₈	—	2	1	no	—
320	C₂×Q₁₆	—	2	1	no	—
321	C₂×Q₁₆	—	4	1	no	—
322	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
323	C₂×D₈	C₇×D₄	1	1	yes	characteristic
324	C₂²×C₈	—	2	1	no	—
325	C₂²×C₈	—	2	1	no	—
326	C₄⋊Q₈	—	2	1	no	—
327	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
328	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
329	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
330	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
331	#125	—	2	1	no	—
332	#125	—	2	1	no	—
333	#125	—	2	1	no	—
334	C₂²⋊Q₈	—	2	1	no	—
335	C₂²⋊Q₈	—	2	1	no	—
336	C₂²⋊Q₈	—	2	1	no	—
337	C₂²⋊Q₈	—	2	1	no	—
338	C₂²⋊Q₈	—	4	1	no	—
339	C₂²⋊Q₈	—	4	1	no	—
340	C₂²⋊Q₈	—	4	1	no	—
341	C₂²⋊Q₈	—	4	1	no	—
342	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
343	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
344	C₄×D₄	—	2	1	no	—
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	1	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
349	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	#107	—	2	1	no	—
354	#107	—	2	1	no	—
355	#107	—	2	1	no	—
356	#107	—	2	1	no	—
357	#107	—	2	1	no	—
358	#107	—	2	1	no	—
359	#107	—	2	1	no	—
360	#107	—	2	1	no	—
361	#107	C₇×D₄	1	1	yes	characteristic
362	#107	C₇×D₄	1	1	yes	characteristic
363	#106	—	2	1	no	—
364	#106	—	2	1	no	—
365	#106	—	2	1	no	—
366	#106	—	2	1	no	—
367	#106	—	2	1	no	—
368	C₄⋊C₈	—	2	1	no	—
369	C₄⋊C₈	—	2	1	no	—
370	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
371	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
372	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
373	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
374	#103	—	2	1	no	—
375	#103	—	2	1	no	—
376	#103	—	2	1	no	—
377	#103	—	2	1	no	—
378	#103	—	2	1	no	—
379	#103	—	2	1	no	—
380	#103	—	4	1	no	—
381	#103	C₇×D₄	1	1	yes	characteristic
382	#103	C₇×D₄	1	1	yes	characteristic
383	#102	—	2	1	no	—
384	#102	—	2	1	no	—
385	#102	—	2	1	no	—
386	#102	—	2	1	no	—
387	#102	C₇×D₄	1	1	yes	characteristic
388	#102	C₇×D₄	1	1	yes	characteristic
389	C₂²⋊C₈	—	2	1	no	—
390	C₂²⋊C₈	—	2	1	no	—
391	#97	—	2	1	no	—
392	C₄×C₈	—	2	1	no	—
393	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
394	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
395	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
396	C₂²×C₇	—	2	2	no	—
397	C₂²×C₇	—	2	2	no	—
398	C₂²×C₇	—	4	1	no	—
399	C₂²×C₇	—	4	1	no	—
400	C₂²×C₇	—	4	1	no	—
401	C₂²×C₇	—	4	1	no	—
402	C₂²×C₇	D₄²	1	1	yes	characteristic
403	C₄×C₇	—	2	1	no	—
404	C₄×C₇	—	4	1	no	—
405	C₄×C₇	—	4	1	no	—
406	C₄×C₇	—	4	1	no	—
407	C₄×C₇	—	4	1	no	—
408	C₄×C₇	—	4	1	no	—
409	C₄×C₇	—	4	1	no	—
410	C₄×C₇	—	8	1	no	—
411	C₄×C₇	—	8	1	no	—
412	C₄×C₇	—	8	1	no	—
413	C₄×C₇	—	8	1	no	—
414	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
415	C₄×C₇	#575	1	1	yes	characteristic
416	C₄×C₇	#577	1	1	yes	characteristic
417	C₄×C₇	#578	1	1	yes	characteristic
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	4	1	no	—
423	C₂×Q₈	—	4	2	no	—
424	C₂×Q₈	—	8	1	no	—
425	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
426	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
427	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
428	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	4	1	no	—
432	C₂²×C₄	—	4	1	no	—
433	C₂²×C₄	—	4	1	no	—
434	C₂²×C₄	—	4	1	no	—
435	Q₁₆	—	4	1	no	—
436	Q₁₆	—	8	1	no	—
437	Q₁₆	—	8	2	no	—
438	D₈	C₇×Q₁₆	1	4	yes	complemented
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	4	2	no	—
445	C₂×C₈	—	8	1	no	—
446	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
447	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
448	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
449	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	4	1	no	—
461	C₄⋊C₄	—	4	1	no	—
462	C₄⋊C₄	—	4	1	no	—
463	C₄⋊C₄	—	4	1	no	—
464	C₄⋊C₄	—	4	1	no	—
465	C₄⋊C₄	—	4	1	no	—
466	C₄⋊C₄	—	4	1	no	—
467	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
468	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
469	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
470	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
471	C₂²⋊C₄	—	2	1	no	—
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	4	1	no	—
474	C₂²⋊C₄	—	4	1	no	—
475	C₂²⋊C₄	—	4	1	no	—
476	C₂²⋊C₄	—	4	1	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	C₂⁴×C₇	1	1	yes	characteristic
480	C₂×C₇	—	4	2	no	—
481	C₂×C₇	—	4	2	no	—
482	C₂×C₇	#3267	1	1	yes	characteristic
483	C₂×C₇	D₄×Q₁₆	1	1	yes	characteristic
484	C₂×C₇	#3288	1	1	yes	characteristic
485	C₂³	—	2	1	no	—
486	C₂³	—	2	1	no	—
487	Q₈	—	2	1	no	—
488	Q₈	—	2	1	no	—
489	Q₈	—	2	1	no	—
490	Q₈	—	2	1	no	—
491	Q₈	—	4	1	no	—
492	Q₈	—	4	1	no	—
493	Q₈	—	8	1	no	—
494	Q₈	—	8	1	no	—
495	Q₈	—	8	1	no	—
496	Q₈	—	8	2	no	—
497	D₄	—	2	1	no	—
498	D₄	—	2	1	no	—
499	D₄	C₂×C₇×Q₁₆	1	2	yes	—
500	D₄	C₂×C₇×Q₁₆	1	2	yes	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	4	1	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	8	2	no	—
516	C₂×C₄	—	16	1	no	—
517	C₂×C₄	—	16	1	no	—
518	C₂×C₄	—	16	1	no	—
519	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
520	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
521	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
522	C₈	—	2	1	no	—
523	C₈	—	2	1	no	—
524	C₈	—	2	1	no	—
525	C₈	—	4	1	no	—
526	C₈	—	4	1	no	—
527	C₈	—	4	1	no	—
528	C₈	—	4	1	no	—
529	C₈	—	4	1	no	—
530	C₈	C₂×C₇×Q₁₆	1	2	yes	—
531	C₇	#19814	1	1	yes	characteristic, direct product factor, Sylow
532	C₂²	—	2	2	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	—	4	1	no	—
537	C₂²	—	4	1	no	—
538	C₂²	C₇×D₄²	1	1	yes	characteristic
539	C₄	—	2	1	no	—
540	C₄	—	4	1	no	—
541	C₄	—	4	1	no	—
542	C₄	—	4	1	no	—
543	C₄	—	4	1	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	1	no	—
546	C₄	—	8	1	no	—
547	C₄	—	8	1	no	—
548	C₄	—	8	1	no	—
549	C₄	—	8	1	no	—
550	C₄	C₂²×C₇×Q₁₆	1	1	yes	characteristic
551	C₄	C₇×#575	1	1	yes	characteristic
552	C₄	C₇×#577	1	1	yes	characteristic
553	C₄	C₇×#578	1	1	yes	characteristic
554	C₂	—	4	2	no	—
555	C₂	—	4	2	no	—
556	C₂	C₇×#3267	1	1	yes	characteristic
557	C₂	C₇×D₄×Q₁₆	1	1	yes	characteristic
558	C₂	C₇×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×D₈	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#3338	C₂	semidirect
4. C₇×#3299	C₂	semidirect
5. C₇×#1723	C₂	semidirect
6. C₇×#1696	C₂	semidirect
7. C₇×#1683	C₂	semidirect
8. C₇×#1675	C₂	semidirect
9. C₇×#1672	C₂	semidirect
10. C₇×#1644	C₂	semidirect
11. C₇×#1637	C₂	semidirect
12. C₇×#1632	C₂	semidirect
13. C₇×#1632	C₂	semidirect
14. C₇×#1584	C₂	semidirect
67. #19814	C₇	direct (N × H)
147. #3338	C₂×C₇	semidirect
148. #3299	C₂×C₇	semidirect
149. #1723	C₂×C₇	semidirect
150. #1696	C₂×C₇	semidirect
151. #1683	C₂×C₇	semidirect
152. #1675	C₂×C₇	semidirect
153. #1672	C₂×C₇	semidirect
154. #1644	C₂×C₇	semidirect
155. #1637	C₂×C₇	semidirect
156. #1632	C₂×C₇	semidirect
157. #1632	C₂×C₇	semidirect
158. #1584	C₂×C₇	semidirect
180. C₇×D₈	Q₁₆	semidirect
438. D₈	C₇×Q₁₆	semidirect
531. C₇	#19814	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3384	2
C₇×#3380	2
C₇×#3338	2
C₇×#3299	2
C₇×#1723	2
C₇×#1696	2
C₇×#1683	2
C₇×#1675	2
C₇×#1672	2
C₇×#1644	2
C₇×#1637	2
C₇×#1632	2
C₇×#1584	2
C₇×#1580	2
#19814	7

Subgroups — C7×#19814 #13197030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19814 #13197030