Subgroups — C₇×#19808 #13197024

Order 1792.

624 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3381	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3359	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3335	C₂	1	1	yes	maximal, characteristic
4	C₇×#3276	C₂	1	1	yes	maximal, characteristic
5	C₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1691	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1679	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1670	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1654	C₂	1	1	yes	maximal, characteristic
11	C₇×#1653	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1643	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1637	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1583	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×#137	—	2	1	no	—
17	C₇×#557	C₂²	1	1	yes	characteristic
18	C₇×#554	C₂²	1	1	yes	characteristic
19	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
20	C₇×#547	C₂²	1	1	yes	characteristic
21	C₇×#499	—	2	1	no	—
22	C₇×#498	C₂²	1	1	yes	characteristic
23	C₇×#498	C₂²	1	1	yes	characteristic
24	C₇×#498	C₂²	1	1	yes	characteristic
25	C₇×#494	C₂²	1	1	yes	characteristic
26	C₇×#487	—	2	1	no	—
27	C₇×#487	C₂²	1	1	yes	characteristic
28	C₇×#487	C₂²	1	1	yes	characteristic
29	C₇×#483	—	2	1	no	—
30	C₇×#481	—	2	1	no	—
31	C₇×#479	—	2	1	no	—
32	C₇×#478	—	2	1	no	—
33	C₇×#477	C₂²	1	1	yes	characteristic
34	C₇×#476	C₂²	1	1	yes	characteristic
35	C₇×#475	—	2	1	no	—
36	C₇×#475	C₂²	1	1	yes	characteristic
37	C₇×#475	C₂²	1	1	yes	characteristic
38	C₇×#474	C₂²	1	1	yes	characteristic
39	C₇×#470	—	2	1	no	—
40	C₇×#465	—	2	1	no	—
41	C₇×#462	C₂²	1	1	yes	characteristic
42	C₇×#461	—	2	1	no	—
43	C₇×#461	C₂²	1	1	yes	characteristic
44	C₇×#461	C₂²	1	1	yes	characteristic
45	C₇×#452	—	2	1	no	—
46	C₇×#451	—	2	1	no	—
47	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
48	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
49	C₄×C₇×SD₁₆	—	2	1	no	—
50	C₄×C₇×SD₁₆	—	2	1	no	—
51	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
52	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
53	C₇×#436	C₂²	1	1	yes	characteristic
54	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
55	C₂×C₇×#106	—	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×#332	C₂²	1	1	yes	characteristic
60	C₇×#332	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	#19808	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₇×#41	—	2	1	no	—
69	C₂²×C₇×Q₈	—	2	1	no	—
70	C₇×#137	—	4	4	no	—
71	C₂×C₇×Q₁₆	—	2	1	no	—
72	C₂×C₇×Q₁₆	—	2	1	no	—
73	C₂×C₇×Q₁₆	—	2	1	no	—
74	C₂×C₇×SD₁₆	—	2	1	no	—
75	C₂×C₇×SD₁₆	—	2	1	no	—
76	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
77	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
78	C₂×C₇×M₁₆	—	2	1	no	—
79	C₂²×C₇×C₈	—	2	1	no	—
80	C₇×C₄⋊Q₈	—	2	1	no	—
81	C₇×C₄⋊Q₈	—	2	1	no	—
82	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₇×#125	—	2	1	no	—
87	C₇×#125	—	2	1	no	—
88	C₇×#125	—	4	1	no	—
89	C₇×#123	—	4	1	no	—
90	C₇×C₂²⋊Q₈	—	2	1	no	—
91	C₇×C₂²⋊Q₈	—	2	1	no	—
92	C₇×C₂²⋊Q₈	—	2	1	no	—
93	C₇×C₂²⋊Q₈	—	4	1	no	—
94	C₇×C₂²⋊Q₈	—	4	1	no	—
95	C₇×#121	—	2	1	no	—
96	C₄×C₇×Q₈	—	2	1	no	—
97	C₄×C₇×Q₈	—	2	1	no	—
98	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
99	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
100	C₄×C₇×D₄	—	2	1	no	—
101	C₄×C₇×D₄	—	2	1	no	—
102	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
103	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
104	C₂×C₇×C₄⋊C₄	—	2	1	no	—
105	C₂×C₇×C₄⋊C₄	—	2	1	no	—
106	C₇×#107	—	2	1	no	—
107	C₇×#107	—	2	1	no	—
108	C₇×#106	—	2	1	no	—
109	C₇×#106	—	2	1	no	—
110	C₇×#106	—	2	1	no	—
111	C₇×#106	—	2	1	no	—
112	C₇×#106	—	2	1	no	—
113	C₇×#106	—	2	1	no	—
114	C₇×#106	—	2	1	no	—
115	C₇×#106	—	4	1	no	—
116	C₇×#106	D₄	1	1	yes	characteristic
117	C₇×#106	D₄	1	1	yes	characteristic
118	C₇×C₄⋊C₈	—	2	1	no	—
119	C₇×C₄⋊C₈	—	2	1	no	—
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×C₄⋊C₈	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
130	C₇×#103	—	2	1	no	—
131	C₇×#103	—	2	1	no	—
132	C₇×#103	—	4	1	no	—
133	C₇×#103	D₄	1	1	yes	characteristic
134	C₇×#103	D₄	1	1	yes	characteristic
135	C₇×#102	—	2	1	no	—
136	C₇×#102	—	2	1	no	—
137	C₇×#102	D₄	1	1	yes	characteristic
138	C₇×#102	D₄	1	1	yes	characteristic
139	C₇×C₂²⋊C₈	—	2	1	no	—
140	C₇×C₂²⋊C₈	—	2	1	no	—
141	C₄×C₇×C₈	—	2	1	no	—
142	C₄×C₇×C₈	—	2	1	no	—
143	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
146	#3381	C₂×C₇	1	1	yes	characteristic, complemented
147	#3359	C₂×C₇	1	1	yes	characteristic, complemented
148	#3335	C₂×C₇	1	1	yes	characteristic
149	#3276	C₂×C₇	1	1	yes	characteristic
150	#1716	C₂×C₇	1	1	yes	characteristic, complemented
151	#1691	C₂×C₇	1	1	yes	characteristic, complemented
152	#1679	C₂×C₇	1	1	yes	characteristic, complemented
153	#1672	C₂×C₇	1	1	yes	characteristic, complemented
154	#1670	C₂×C₇	1	1	yes	characteristic, complemented
155	#1654	C₂×C₇	1	1	yes	characteristic
156	#1653	C₂×C₇	1	1	yes	characteristic, complemented
157	#1643	C₂×C₇	1	1	yes	characteristic, complemented
158	#1637	C₂×C₇	1	1	yes	characteristic, complemented
159	#1583	C₂×C₇	1	1	yes	characteristic, complemented
160	SD₁₆⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
161	C₇×#41	—	4	2	no	—
162	C₇×#41	—	8	1	no	—
163	C₂×C₇×Q₈	—	2	1	no	—
164	C₂×C₇×Q₈	—	2	1	no	—
165	C₂×C₇×Q₈	—	2	1	no	—
166	C₂×C₇×Q₈	—	2	1	no	—
167	C₂×C₇×Q₈	—	4	1	no	—
168	C₂×C₇×Q₈	—	4	2	no	—
169	C₂×C₇×Q₈	—	8	1	no	—
170	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
171	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₇×D₄	—	4	1	no	—
173	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
174	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₇	—	2	1	no	—
176	C₂²×C₄×C₇	—	2	1	no	—
177	C₂²×C₄×C₇	—	4	1	no	—
178	C₂²×C₄×C₇	—	4	1	no	—
179	C₂²×C₄×C₇	—	4	1	no	—
180	C₇×Q₁₆	—	4	2	no	—
181	C₇×Q₁₆	—	4	2	no	—
182	C₇×Q₁₆	—	8	1	no	—
183	C₇×SD₁₆	—	4	1	no	—
184	C₇×SD₁₆	—	4	2	no	—
185	C₇×SD₁₆	—	4	2	no	—
186	C₇×SD₁₆	SD₁₆	1	4	yes	complemented
187	C₇×M₁₆	—	4	2	no	—
188	C₂×C₇×C₈	—	2	1	no	—
189	C₂×C₇×C₈	—	2	1	no	—
190	C₂×C₇×C₈	—	2	1	no	—
191	C₂×C₇×C₈	—	2	1	no	—
192	C₂×C₇×C₈	—	2	1	no	—
193	C₂×C₇×C₈	—	4	2	no	—
194	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
195	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
196	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
197	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
203	C₇×C₄⋊C₄	—	2	1	no	—
204	C₇×C₄⋊C₄	—	2	1	no	—
205	C₇×C₄⋊C₄	—	2	1	no	—
206	C₇×C₄⋊C₄	—	2	1	no	—
207	C₇×C₄⋊C₄	—	2	1	no	—
208	C₇×C₄⋊C₄	—	4	1	no	—
209	C₇×C₄⋊C₄	—	4	1	no	—
210	C₇×C₄⋊C₄	—	4	1	no	—
211	C₇×C₄⋊C₄	—	4	1	no	—
212	C₇×C₄⋊C₄	—	4	1	no	—
213	C₇×C₄⋊C₄	—	4	1	no	—
214	C₇×C₄⋊C₄	—	4	1	no	—
215	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
216	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
217	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
218	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
219	C₇×C₂²⋊C₄	—	2	1	no	—
220	C₇×C₂²⋊C₄	—	2	1	no	—
221	C₇×C₂²⋊C₄	—	4	1	no	—
222	C₇×C₂²⋊C₄	—	4	1	no	—
223	C₇×C₂²⋊C₄	—	4	1	no	—
224	C₄²×C₇	—	2	1	no	—
225	C₄²×C₇	—	2	1	no	—
226	C₄²×C₇	—	4	1	no	—
227	C₄²×C₇	C₂⁴	1	1	yes	characteristic
228	C₂×#137	—	2	1	no	—
229	#557	C₂²×C₇	1	1	yes	characteristic
230	#554	C₂²×C₇	1	1	yes	characteristic
231	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
232	#547	C₂²×C₇	1	1	yes	characteristic
233	#499	—	2	1	no	—
234	#498	C₂²×C₇	1	1	yes	characteristic
235	#498	C₂²×C₇	1	1	yes	characteristic
236	#498	C₂²×C₇	1	1	yes	characteristic
237	#494	C₂²×C₇	1	1	yes	characteristic
238	#487	—	2	1	no	—
239	#487	C₂²×C₇	1	1	yes	characteristic
240	#487	C₂²×C₇	1	1	yes	characteristic
241	#483	—	2	1	no	—
242	#481	—	2	1	no	—
243	#479	—	2	1	no	—
244	#478	—	2	1	no	—
245	#477	C₂²×C₇	1	1	yes	characteristic
246	#476	C₂²×C₇	1	1	yes	characteristic
247	#475	—	2	1	no	—
248	#475	C₂²×C₇	1	1	yes	characteristic
249	#475	C₂²×C₇	1	1	yes	characteristic
250	#474	C₂²×C₇	1	1	yes	characteristic
251	#470	—	2	1	no	—
252	#465	—	2	1	no	—
253	#462	C₂²×C₇	1	1	yes	characteristic
254	#461	—	2	1	no	—
255	#461	C₂²×C₇	1	1	yes	characteristic
256	#461	C₂²×C₇	1	1	yes	characteristic
257	#452	—	2	1	no	—
258	#451	—	2	1	no	—
259	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
260	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
261	C₄×SD₁₆	—	2	1	no	—
262	C₄×SD₁₆	—	2	1	no	—
263	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
264	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
265	#436	C₂²×C₇	1	1	yes	characteristic
266	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
267	C₂×#106	—	2	1	no	—
268	#335	C₂²×C₇	1	1	yes	characteristic
269	#334	C₂²×C₇	1	1	yes	characteristic
270	#333	C₂²×C₇	1	1	yes	characteristic
271	#332	C₂²×C₇	1	1	yes	characteristic
272	#332	C₂²×C₇	1	1	yes	characteristic
273	#332	C₂²×C₇	1	1	yes	characteristic
274	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
275	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
276	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
277	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
278	#322	C₂²×C₇	1	1	yes	characteristic
279	C₂³×C₇	—	2	1	no	—
280	C₂³×C₇	—	2	1	no	—
281	C₇×Q₈	—	2	1	no	—
282	C₇×Q₈	—	2	1	no	—
283	C₇×Q₈	—	2	1	no	—
284	C₇×Q₈	—	4	1	no	—
285	C₇×Q₈	—	4	1	no	—
286	C₇×Q₈	—	4	1	no	—
287	C₇×Q₈	—	4	1	no	—
288	C₇×Q₈	—	4	1	no	—
289	C₇×Q₈	—	4	1	no	—
290	C₇×Q₈	—	8	2	no	—
291	C₇×Q₈	—	8	1	no	—
292	C₇×Q₈	C₂×SD₁₆	1	2	yes	—
293	C₇×D₄	—	2	1	no	—
294	C₇×D₄	—	2	1	no	—
295	C₇×D₄	—	2	1	no	—
296	C₇×D₄	—	8	2	no	—
297	C₇×D₄	C₂×SD₁₆	1	2	yes	—
298	C₂×C₄×C₇	—	2	1	no	—
299	C₂×C₄×C₇	—	2	1	no	—
300	C₂×C₄×C₇	—	2	1	no	—
301	C₂×C₄×C₇	—	2	1	no	—
302	C₂×C₄×C₇	—	2	1	no	—
303	C₂×C₄×C₇	—	2	1	no	—
304	C₂×C₄×C₇	—	4	1	no	—
305	C₂×C₄×C₇	—	4	1	no	—
306	C₂×C₄×C₇	—	4	1	no	—
307	C₂×C₄×C₇	—	4	1	no	—
308	C₂×C₄×C₇	—	4	1	no	—
309	C₂×C₄×C₇	—	4	1	no	—
310	C₂×C₄×C₇	—	4	1	no	—
311	C₂×C₄×C₇	—	4	1	no	—
312	C₂×C₄×C₇	—	8	2	no	—
313	C₂×C₄×C₇	—	8	2	no	—
314	C₂×C₄×C₇	—	16	1	no	—
315	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
317	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
318	C₇×C₈	—	2	1	no	—
319	C₇×C₈	—	2	1	no	—
320	C₇×C₈	—	2	1	no	—
321	C₇×C₈	—	4	1	no	—
322	C₇×C₈	—	4	1	no	—
323	C₇×C₈	—	4	1	no	—
324	C₇×C₈	—	4	1	no	—
325	C₇×C₈	—	4	1	no	—
326	C₇×C₈	C₂×SD₁₆	1	2	yes	—
327	C₂×#41	—	2	1	no	—
328	C₂²×Q₈	—	2	1	no	—
329	#137	—	4	4	no	—
330	C₂×Q₁₆	—	2	1	no	—
331	C₂×Q₁₆	—	2	1	no	—
332	C₂×Q₁₆	—	2	1	no	—
333	C₂×SD₁₆	—	2	1	no	—
334	C₂×SD₁₆	—	2	1	no	—
335	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
336	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
337	C₂×M₁₆	—	2	1	no	—
338	C₂²×C₈	—	2	1	no	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	—	2	1	no	—
341	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
342	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
343	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
344	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
345	#125	—	2	1	no	—
346	#125	—	2	1	no	—
347	#125	—	4	1	no	—
348	#123	—	4	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	2	1	no	—
352	C₂²⋊Q₈	—	4	1	no	—
353	C₂²⋊Q₈	—	4	1	no	—
354	#121	—	2	1	no	—
355	C₄×Q₈	—	2	1	no	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
358	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
362	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	#107	—	2	1	no	—
366	#107	—	2	1	no	—
367	#106	—	2	1	no	—
368	#106	—	2	1	no	—
369	#106	—	2	1	no	—
370	#106	—	2	1	no	—
371	#106	—	2	1	no	—
372	#106	—	2	1	no	—
373	#106	—	2	1	no	—
374	#106	—	4	1	no	—
375	#106	C₇×D₄	1	1	yes	characteristic
376	#106	C₇×D₄	1	1	yes	characteristic
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
380	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
381	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
382	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
383	#103	—	2	1	no	—
384	#103	—	2	1	no	—
385	#103	—	2	1	no	—
386	#103	—	2	1	no	—
387	#103	—	2	1	no	—
388	#103	—	2	1	no	—
389	#103	—	2	1	no	—
390	#103	—	2	1	no	—
391	#103	—	4	1	no	—
392	#103	C₇×D₄	1	1	yes	characteristic
393	#103	C₇×D₄	1	1	yes	characteristic
394	#102	—	2	1	no	—
395	#102	—	2	1	no	—
396	#102	C₇×D₄	1	1	yes	characteristic
397	#102	C₇×D₄	1	1	yes	characteristic
398	C₂²⋊C₈	—	2	1	no	—
399	C₂²⋊C₈	—	2	1	no	—
400	C₄×C₈	—	2	1	no	—
401	C₄×C₈	—	2	1	no	—
402	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
403	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
404	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
405	C₂²×C₇	—	2	2	no	—
406	C₂²×C₇	—	4	1	no	—
407	C₂²×C₇	—	4	1	no	—
408	C₂²×C₇	—	4	1	no	—
409	C₂²×C₇	—	4	1	no	—
410	C₂²×C₇	—	4	1	no	—
411	C₂²×C₇	D₄²	1	1	yes	characteristic
412	C₄×C₇	—	2	2	no	—
413	C₄×C₇	—	2	1	no	—
414	C₄×C₇	—	4	1	no	—
415	C₄×C₇	—	4	1	no	—
416	C₄×C₇	—	4	1	no	—
417	C₄×C₇	—	4	1	no	—
418	C₄×C₇	—	4	1	no	—
419	C₄×C₇	—	8	1	no	—
420	C₄×C₇	—	8	1	no	—
421	C₄×C₇	—	8	1	no	—
422	C₄×C₇	—	8	1	no	—
423	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
424	C₄×C₇	#575	1	1	yes	characteristic
425	C₄×C₇	#577	1	1	yes	characteristic
426	C₄×C₇	#578	1	1	yes	characteristic
427	#41	—	4	2	no	—
428	#41	—	8	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	4	1	no	—
434	C₂×Q₈	—	4	2	no	—
435	C₂×Q₈	—	8	1	no	—
436	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
437	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
438	C₂×D₄	—	4	1	no	—
439	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
440	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	4	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	Q₁₆	—	4	2	no	—
447	Q₁₆	—	4	2	no	—
448	Q₁₆	—	8	1	no	—
449	SD₁₆	—	4	1	no	—
450	SD₁₆	—	4	2	no	—
451	SD₁₆	—	4	2	no	—
452	SD₁₆	C₇×SD₁₆	1	4	yes	complemented
453	M₁₆	—	4	2	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	4	2	no	—
460	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
461	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
462	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
463	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	—	4	1	no	—
476	C₄⋊C₄	—	4	1	no	—
477	C₄⋊C₄	—	4	1	no	—
478	C₄⋊C₄	—	4	1	no	—
479	C₄⋊C₄	—	4	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
482	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
483	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
484	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
485	C₂²⋊C₄	—	2	1	no	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	4	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	4	1	no	—
493	C₄²	C₂⁴×C₇	1	1	yes	characteristic
494	C₂×C₇	—	4	2	no	—
495	C₂×C₇	—	8	1	no	—
496	C₂×C₇	#3269	1	1	yes	characteristic
497	C₂×C₇	D₄×SD₁₆	1	1	yes	characteristic
498	C₂×C₇	#3289	1	1	yes	characteristic
499	C₂³	—	2	1	no	—
500	C₂³	—	2	1	no	—
501	Q₈	—	2	1	no	—
502	Q₈	—	2	1	no	—
503	Q₈	—	2	1	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	4	1	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	—	4	1	no	—
509	Q₈	—	4	1	no	—
510	Q₈	—	8	2	no	—
511	Q₈	—	8	1	no	—
512	Q₈	C₂×C₇×SD₁₆	1	2	yes	—
513	D₄	—	2	1	no	—
514	D₄	—	2	1	no	—
515	D₄	—	2	1	no	—
516	D₄	—	8	2	no	—
517	D₄	C₂×C₇×SD₁₆	1	2	yes	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	8	2	no	—
533	C₂×C₄	—	8	2	no	—
534	C₂×C₄	—	16	1	no	—
535	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
536	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
537	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
538	C₈	—	2	1	no	—
539	C₈	—	2	1	no	—
540	C₈	—	2	1	no	—
541	C₈	—	4	1	no	—
542	C₈	—	4	1	no	—
543	C₈	—	4	1	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	C₂×C₇×SD₁₆	1	2	yes	—
547	C₇	#19808	1	1	yes	characteristic, direct product factor, Sylow
548	C₂²	—	2	2	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	C₇×D₄²	1	1	yes	characteristic
555	C₄	—	2	2	no	—
556	C₄	—	2	1	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	4	1	no	—
560	C₄	—	4	1	no	—
561	C₄	—	4	1	no	—
562	C₄	—	8	1	no	—
563	C₄	—	8	1	no	—
564	C₄	—	8	1	no	—
565	C₄	—	8	1	no	—
566	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
567	C₄	C₇×#575	1	1	yes	characteristic
568	C₄	C₇×#577	1	1	yes	characteristic
569	C₄	C₇×#578	1	1	yes	characteristic
570	C₂	—	4	2	no	—
571	C₂	—	8	1	no	—
572	C₂	C₇×#3269	1	1	yes	characteristic
573	C₂	C₇×D₄×SD₁₆	1	1	yes	characteristic
574	C₂	C₇×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#547	C₂²
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
1. C₇×#3381	C₂	semidirect
2. C₇×#3359	C₂	semidirect
5. C₇×#1716	C₂	semidirect
6. C₇×#1691	C₂	semidirect
7. C₇×#1679	C₂	semidirect
8. C₇×#1672	C₂	semidirect
9. C₇×#1670	C₂	semidirect
11. C₇×#1653	C₂	semidirect
12. C₇×#1643	C₂	semidirect
13. C₇×#1637	C₂	semidirect
14. C₇×#1583	C₂	semidirect
15. C₇×SD₁₆⋊C₈	C₂	semidirect
67. #19808	C₇	direct (N × H)
146. #3381	C₂×C₇	semidirect
147. #3359	C₂×C₇	semidirect
150. #1716	C₂×C₇	semidirect
151. #1691	C₂×C₇	semidirect
152. #1679	C₂×C₇	semidirect
153. #1672	C₂×C₇	semidirect
154. #1670	C₂×C₇	semidirect
156. #1653	C₂×C₇	semidirect
157. #1643	C₂×C₇	semidirect
158. #1637	C₂×C₇	semidirect
159. #1583	C₂×C₇	semidirect
160. SD₁₆⋊C₈	C₂×C₇	semidirect
186. C₇×SD₁₆	SD₁₆	semidirect
452. SD₁₆	C₇×SD₁₆	semidirect
547. C₇	#19808	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×#3381	2
C₇×#3359	2
C₇×#3335	2
C₇×#3276	2
C₇×#1716	2
C₇×#1691	2
C₇×#1679	2
C₇×#1672	2
C₇×#1670	2
C₇×#1654	2
C₇×#1653	2
C₇×#1643	2
C₇×#1637	2
C₇×#1583	2
C₇×SD₁₆⋊C₈	2
#19808	7

Subgroups — C7×#19808 #13197024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19808 #13197024