Subgroups — C₇×#17966 #13195182

Order 1792.

662 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3171	C₂	1	1	yes	maximal, characteristic
2	C₇×#2966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1590	C₂	1	2	yes	maximal, complemented
5	C₇×#1589	C₂	1	2	yes	maximal, complemented
6	C₇×#1580	C₂	1	2	yes	maximal, complemented
7	C₇×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₇×#1569	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1496	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1495	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#131	—	2	1	no	—
13	C₇×#532	C₂²	1	1	yes	characteristic
14	C₇×#529	C₂²	1	1	yes	characteristic
15	C₇×#501	C₄	1	1	yes	characteristic
16	C₇×#496	C₄	1	1	yes	characteristic
17	C₇×#490	C₄	1	1	yes	characteristic
18	C₇×#489	C₄	1	1	yes	characteristic
19	C₇×#469	C₄	1	2	yes	—
20	C₇×#468	C₄	1	2	yes	—
21	C₇×#446	C₂²	1	1	yes	characteristic
22	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
23	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
24	C₄×C₇×D₈	C₂²	1	2	yes	—
25	C₇×#436	—	2	1	no	—
26	C₇×#436	C₂²	1	2	yes	—
27	C₇×#436	C₂²	1	1	yes	characteristic
28	C₇×#435	—	2	1	no	—
29	C₇×C₈×D₄	—	2	1	no	—
30	C₇×C₈×D₄	—	2	1	no	—
31	C₇×C₈×D₄	C₂²	1	2	yes	—
32	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
33	C₇×#433	—	2	1	no	—
34	C₇×#432	C₂²	1	1	yes	characteristic
35	C₇×#431	C₂²	1	1	yes	characteristic
36	C₇×#423	C₂²	1	1	yes	characteristic
37	C₇×#408	—	2	1	no	—
38	C₇×#405	—	2	1	no	—
39	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
41	C₇×#335	C₂²	1	2	yes	—
42	C₇×#334	C₂²	1	2	yes	—
43	C₇×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₇×D₄⋊C₈	C₂²	1	2	yes	—
45	C₇×D₄⋊C₈	C₂²	1	2	yes	—
46	C₇×D₄⋊C₈	C₂²	1	2	yes	—
47	C₇×#322	C₂²	1	2	yes	—
48	C₇×C₈²	C₂²	1	2	yes	—
49	#17966	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₇×#41	—	2	1	no	—
51	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
52	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
53	C₇×#131	—	4	4	no	—
54	C₂×C₇×M₁₆	—	2	1	no	—
55	C₂×C₇×M₁₆	—	2	1	no	—
56	C₂×C₇×M₁₆	—	2	2	no	—
57	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
58	C₂²×C₇×C₈	—	2	1	no	—
59	C₂²×C₇×C₈	—	2	2	no	—
60	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
61	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
62	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₇×#127	C₂×C₄	1	1	yes	characteristic
64	C₇×#126	—	2	1	no	—
65	C₇×#125	C₂×C₄	1	1	yes	characteristic
66	C₇×#124	—	2	1	no	—
67	C₇×#124	C₂×C₄	1	1	yes	characteristic
68	C₇×#123	—	2	1	no	—
69	C₇×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
70	C₇×#121	—	2	1	no	—
71	C₇×#121	C₂×C₄	1	2	yes	—
72	C₇×#121	C₂×C₄	1	1	yes	characteristic
73	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
74	C₄×C₇×D₄	—	2	1	no	—
75	C₄×C₇×D₄	—	2	1	no	—
76	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
77	C₄×C₇×D₄	C₂³	1	2	yes	—
78	C₇×#117	—	2	1	no	—
79	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
80	C₇×#107	C₂×C₄	1	2	yes	—
81	C₇×#106	C₂×C₄	1	2	yes	—
82	C₇×C₄⋊C₈	—	2	1	no	—
83	C₇×C₄⋊C₈	—	2	1	no	—
84	C₇×C₄⋊C₈	—	2	1	no	—
85	C₇×C₄⋊C₈	—	2	1	no	—
86	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₇×C₄⋊C₈	C₂³	1	2	yes	—
89	C₇×#103	C₂×C₄	1	2	yes	—
90	C₇×#102	C₂×C₄	1	2	yes	—
91	C₇×#102	C₂×C₄	1	2	yes	—
92	C₇×#102	C₂×C₄	1	2	yes	—
93	C₇×C₂²⋊C₈	—	2	1	no	—
94	C₇×C₂²⋊C₈	—	2	1	no	—
95	C₇×C₂²⋊C₈	—	2	2	no	—
96	C₇×C₂²⋊C₈	—	2	2	no	—
97	C₇×C₂²⋊C₈	—	2	1	no	—
98	C₇×C₂²⋊C₈	—	2	1	no	—
99	C₇×#97	—	2	1	no	—
100	C₇×#97	—	2	1	no	—
101	C₇×#97	C₂×C₄	1	1	yes	characteristic
102	C₇×#97	C₂×C₄	1	1	yes	characteristic
103	C₄×C₇×C₈	—	2	1	no	—
104	C₄×C₇×C₈	D₄	1	1	yes	characteristic
105	C₄×C₇×C₈	D₄	1	1	yes	characteristic
106	C₄×C₇×C₈	C₂³	1	2	yes	—
107	C₄×C₇×C₈	C₂³	1	2	yes	—
108	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
109	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
110	#3171	C₂×C₇	1	1	yes	characteristic
111	#2966	C₂×C₇	1	1	yes	characteristic, complemented
112	#2931	C₂×C₇	1	1	yes	characteristic, complemented
113	#1590	C₂×C₇	1	2	yes	complemented
114	#1589	C₂×C₇	1	2	yes	complemented
115	#1580	C₂×C₇	1	2	yes	complemented
116	SD₁₆⋊C₈	C₂×C₇	1	2	yes	complemented
117	#1569	C₂×C₇	1	1	yes	characteristic, complemented
118	#1496	C₂×C₇	1	1	yes	characteristic, complemented
119	#1495	C₂×C₇	1	1	yes	characteristic, complemented
120	C₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
121	C₇×#41	—	4	2	no	—
122	C₇×#41	—	4	2	no	—
123	C₂×C₇×Q₈	—	2	1	no	—
124	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₇×D₄	—	2	1	no	—
126	C₂×C₇×D₄	—	2	2	no	—
127	C₂×C₇×D₄	—	2	1	no	—
128	C₂×C₇×D₄	—	2	1	no	—
129	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₇	—	2	1	no	—
132	C₂²×C₄×C₇	—	2	2	no	—
133	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₇	#41	1	1	yes	characteristic
136	C₇×SD₁₆	—	2	4	no	—
137	C₇×D₈	—	2	4	no	—
138	C₇×M₁₆	—	2	4	no	—
139	C₇×M₁₆	—	4	2	no	—
140	C₇×M₁₆	—	4	2	no	—
141	C₇×M₁₆	—	4	4	no	—
142	C₂×C₇×C₈	—	2	2	no	—
143	C₂×C₇×C₈	—	2	1	no	—
144	C₂×C₇×C₈	—	2	1	no	—
145	C₂×C₇×C₈	—	2	2	no	—
146	C₂×C₇×C₈	—	2	2	no	—
147	C₂×C₇×C₈	—	4	1	no	—
148	C₂×C₇×C₈	—	4	2	no	—
149	C₂×C₇×C₈	—	4	4	no	—
150	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
151	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
152	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
153	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
154	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
155	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
156	C₇×C₄⋊C₄	—	2	1	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	1	no	—
161	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
162	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
163	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
164	C₇×C₂²⋊C₄	—	2	1	no	—
165	C₇×C₂²⋊C₄	—	2	1	no	—
166	C₇×C₂²⋊C₄	—	2	1	no	—
167	C₇×C₂²⋊C₄	—	2	2	no	—
168	C₇×C₂²⋊C₄	—	2	1	no	—
169	C₇×C₂²⋊C₄	—	2	2	no	—
170	C₇×C₂²⋊C₄	—	2	1	no	—
171	C₄²×C₇	—	2	1	no	—
172	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₇	#41	1	1	yes	characteristic
175	C₄²×C₇	C₂⁴	1	1	yes	characteristic
176	C₂×#131	—	2	1	no	—
177	#532	C₂²×C₇	1	1	yes	characteristic
178	#529	C₂²×C₇	1	1	yes	characteristic
179	#501	C₄×C₇	1	1	yes	characteristic
180	#496	C₄×C₇	1	1	yes	characteristic
181	#490	C₄×C₇	1	1	yes	characteristic
182	#489	C₄×C₇	1	1	yes	characteristic
183	#469	C₄×C₇	1	2	yes	—
184	#468	C₄×C₇	1	2	yes	—
185	#446	C₂²×C₇	1	1	yes	characteristic
186	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
187	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
188	C₄×D₈	C₂²×C₇	1	2	yes	—
189	#436	—	2	1	no	—
190	#436	C₂²×C₇	1	1	yes	characteristic
191	#436	C₂²×C₇	1	2	yes	—
192	#435	—	2	1	no	—
193	C₈×D₄	—	2	1	no	—
194	C₈×D₄	—	2	1	no	—
195	C₈×D₄	C₂²×C₇	1	2	yes	—
196	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
197	#433	—	2	1	no	—
198	#432	C₂²×C₇	1	1	yes	characteristic
199	#431	C₂²×C₇	1	1	yes	characteristic
200	#423	C₂²×C₇	1	1	yes	characteristic
201	#408	—	2	1	no	—
202	#405	—	2	1	no	—
203	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
205	#335	C₂²×C₇	1	2	yes	—
206	#334	C₂²×C₇	1	2	yes	—
207	Q₈⋊C₈	C₂²×C₇	1	2	yes	—
208	D₄⋊C₈	C₂²×C₇	1	2	yes	—
209	D₄⋊C₈	C₂²×C₇	1	2	yes	—
210	D₄⋊C₈	C₂²×C₇	1	2	yes	—
211	#322	C₂²×C₇	1	2	yes	—
212	C₈²	C₂²×C₇	1	2	yes	—
213	C₂³×C₇	—	2	1	no	—
214	C₂³×C₇	—	2	2	no	—
215	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
216	C₇×Q₈	—	2	2	no	—
217	C₇×Q₈	—	4	2	no	—
218	C₇×D₄	—	2	2	no	—
219	C₇×D₄	—	2	2	no	—
220	C₇×D₄	—	2	2	no	—
221	C₇×D₄	—	4	2	no	—
222	C₇×D₄	—	4	2	no	—
223	C₇×D₄	—	4	2	no	—
224	C₇×D₄	—	4	4	no	—
225	C₂×C₄×C₇	—	2	2	no	—
226	C₂×C₄×C₇	—	2	1	no	—
227	C₂×C₄×C₇	—	2	1	no	—
228	C₂×C₄×C₇	—	2	1	no	—
229	C₂×C₄×C₇	—	2	1	no	—
230	C₂×C₄×C₇	—	2	1	no	—
231	C₂×C₄×C₇	—	2	2	no	—
232	C₂×C₄×C₇	—	4	1	no	—
233	C₂×C₄×C₇	—	4	2	no	—
234	C₂×C₄×C₇	—	4	1	no	—
235	C₂×C₄×C₇	—	4	1	no	—
236	C₂×C₄×C₇	—	4	2	no	—
237	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
238	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
239	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
240	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
241	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
243	C₇×C₈	—	2	2	no	—
244	C₇×C₈	—	2	1	no	—
245	C₇×C₈	—	2	1	no	—
246	C₇×C₈	—	2	1	no	—
247	C₇×C₈	—	4	2	no	—
248	C₇×C₈	—	4	2	no	—
249	C₇×C₈	—	4	1	no	—
250	C₇×C₈	—	4	1	no	—
251	C₇×C₈	#131	1	4	yes	complemented
252	C₇×C₈	AGL(1,ℤ/8)	1	2	yes	—
253	C₂×#41	—	2	1	no	—
254	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
255	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
256	#131	—	4	4	no	—
257	C₂×M₁₆	—	2	1	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	1	no	—
260	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
264	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
265	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
266	#127	C₂×C₄×C₇	1	1	yes	characteristic
267	#126	—	2	1	no	—
268	#125	C₂×C₄×C₇	1	1	yes	characteristic
269	#124	—	2	1	no	—
270	#124	C₂×C₄×C₇	1	1	yes	characteristic
271	#123	—	2	1	no	—
272	C₂²⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
273	#121	—	2	1	no	—
274	#121	C₂×C₄×C₇	1	1	yes	characteristic
275	#121	C₂×C₄×C₇	1	2	yes	—
276	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
277	C₄×D₄	—	2	1	no	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
280	C₄×D₄	C₂³×C₇	1	2	yes	—
281	#117	—	2	1	no	—
282	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
283	#107	C₂×C₄×C₇	1	2	yes	—
284	#106	C₂×C₄×C₇	1	2	yes	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	—	2	1	no	—
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
290	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
291	C₄⋊C₈	C₂³×C₇	1	2	yes	—
292	#103	C₂×C₄×C₇	1	2	yes	—
293	#102	C₂×C₄×C₇	1	2	yes	—
294	#102	C₂×C₄×C₇	1	2	yes	—
295	#102	C₂×C₄×C₇	1	2	yes	—
296	C₂²⋊C₈	—	2	1	no	—
297	C₂²⋊C₈	—	2	1	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	—	2	2	no	—
300	C₂²⋊C₈	—	2	1	no	—
301	C₂²⋊C₈	—	2	1	no	—
302	#97	—	2	1	no	—
303	#97	—	2	1	no	—
304	#97	C₂×C₄×C₇	1	1	yes	characteristic
305	#97	C₂×C₄×C₇	1	1	yes	characteristic
306	C₄×C₈	—	2	1	no	—
307	C₄×C₈	C₇×D₄	1	1	yes	characteristic
308	C₄×C₈	C₇×D₄	1	1	yes	characteristic
309	C₄×C₈	C₂³×C₇	1	2	yes	—
310	C₄×C₈	C₂³×C₇	1	2	yes	—
311	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
313	C₂²×C₇	—	2	1	no	—
314	C₂²×C₇	—	2	1	no	—
315	C₂²×C₇	—	2	1	no	—
316	C₂²×C₇	—	4	1	no	—
317	C₂²×C₇	—	4	2	no	—
318	C₂²×C₇	—	4	2	no	—
319	C₂²×C₇	—	4	1	no	—
320	C₂²×C₇	—	4	2	no	—
321	C₂²×C₇	—	4	1	no	—
322	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
323	C₄×C₇	—	2	1	no	—
324	C₄×C₇	—	2	1	no	—
325	C₄×C₇	—	2	1	no	—
326	C₄×C₇	—	2	1	no	—
327	C₄×C₇	—	4	1	no	—
328	C₄×C₇	—	4	1	no	—
329	C₄×C₇	—	4	2	no	—
330	C₄×C₇	—	4	1	no	—
331	C₄×C₇	C₂×#131	1	2	yes	—
332	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₄×C₇	#575	1	1	yes	characteristic
334	#41	—	4	2	no	—
335	#41	—	4	2	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
343	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
347	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
348	C₂²×C₄	C₇×#41	1	1	yes	characteristic
349	SD₁₆	—	2	4	no	—
350	D₈	—	2	4	no	—
351	M₁₆	—	2	4	no	—
352	M₁₆	—	4	2	no	—
353	M₁₆	—	4	2	no	—
354	M₁₆	—	4	4	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	4	1	no	—
361	C₂×C₈	—	4	2	no	—
362	C₂×C₈	—	4	4	no	—
363	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
364	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
365	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
366	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
367	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
368	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
375	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
376	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
377	C₂²⋊C₄	—	2	1	no	—
378	C₂²⋊C₄	—	2	1	no	—
379	C₂²⋊C₄	—	2	1	no	—
380	C₂²⋊C₄	—	2	2	no	—
381	C₂²⋊C₄	—	2	1	no	—
382	C₂²⋊C₄	—	2	2	no	—
383	C₂²⋊C₄	—	2	1	no	—
384	C₄²	—	2	1	no	—
385	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
386	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
387	C₄²	C₇×#41	1	1	yes	characteristic
388	C₄²	C₂⁴×C₇	1	1	yes	characteristic
389	C₂×C₇	—	4	1	no	—
390	C₂×C₇	—	8	2	no	—
391	C₂×C₇	—	8	1	no	—
392	C₂×C₇	#2931	1	1	yes	characteristic
393	C₂×C₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
394	C₂×C₇	#2955	1	1	yes	characteristic
395	C₂³	—	2	1	no	—
396	C₂³	—	2	2	no	—
397	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
398	Q₈	—	2	2	no	—
399	Q₈	—	4	2	no	—
400	D₄	—	2	2	no	—
401	D₄	—	2	2	no	—
402	D₄	—	2	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	2	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
420	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
421	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
422	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
423	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
424	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
425	C₈	—	2	2	no	—
426	C₈	—	2	1	no	—
427	C₈	—	2	1	no	—
428	C₈	—	2	1	no	—
429	C₈	—	4	2	no	—
430	C₈	—	4	2	no	—
431	C₈	—	4	1	no	—
432	C₈	—	4	1	no	—
433	C₈	C₇×#131	1	4	yes	complemented
434	C₈	C₇×AGL(1,ℤ/8)	1	2	yes	—
435	C₇	#17966	1	1	yes	characteristic, direct product factor, Sylow
436	C₂²	—	2	1	no	—
437	C₂²	—	2	1	no	—
438	C₂²	—	2	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	1	no	—
445	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	4	1	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	1	no	—
454	C₄	C₂×C₇×#131	1	2	yes	—
455	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
456	C₄	C₇×#575	1	1	yes	characteristic
457	C₂	—	4	1	no	—
458	C₂	—	8	2	no	—
459	C₂	—	8	1	no	—
460	C₂	C₇×#2931	1	1	yes	characteristic
461	C₂	C₄×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
462	C₂	C₇×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#3171	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2966	C₂	semidirect
3. C₇×#2931	C₂	semidirect
4. C₇×#1590	C₂	semidirect
5. C₇×#1589	C₂	semidirect
6. C₇×#1580	C₂	semidirect
7. C₇×SD₁₆⋊C₈	C₂	semidirect
8. C₇×#1569	C₂	semidirect
9. C₇×#1496	C₂	semidirect
10. C₇×#1495	C₂	semidirect
11. C₇×C₈×M₁₆	C₂	semidirect
49. #17966	C₇	direct (N × H)
111. #2966	C₂×C₇	semidirect
112. #2931	C₂×C₇	semidirect
113. #1590	C₂×C₇	semidirect
114. #1589	C₂×C₇	semidirect
115. #1580	C₂×C₇	semidirect
116. SD₁₆⋊C₈	C₂×C₇	semidirect
117. #1569	C₂×C₇	semidirect
118. #1496	C₂×C₇	semidirect
119. #1495	C₂×C₇	semidirect
120. C₈×M₁₆	C₂×C₇	semidirect
251. C₇×C₈	#131	semidirect
433. C₈	C₇×#131	semidirect
435. C₇	#17966	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×#3171	2
C₇×#2966	2
C₇×#2931	2
C₇×#1590	2
C₇×#1589	2
C₇×#1580	2
C₇×SD₁₆⋊C₈	2
C₇×#1569	2
C₇×#1496	2
C₇×#1495	2
C₇×C₈×M₁₆	2
#17966	7

Subgroups — C7×#17966 #13195182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17966 #13195182