Subgroups — C₇×#22529 #13199745

Order 1792.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×#41	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1995	C₂	1	2	yes	maximal, complemented
5	C₇×#1994	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1986	C₂	1	2	yes	maximal, complemented
7	C₇×#1986	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1806	C₂	1	2	yes	maximal
9	C₇×#1806	C₂	1	1	yes	maximal, characteristic
10	C₇×#1759	C₂	1	2	yes	maximal
11	C₄×C₇×#104	C₂	1	1	yes	maximal, characteristic
12	C₇×Q₈²	C₄	1	1	yes	characteristic, complemented
13	C₇×#557	C₄	1	2	yes	complemented
14	C₇×#557	C₄	1	1	yes	characteristic, complemented
15	C₇×#554	—	2	2	no	—
16	C₇×#554	—	2	1	no	—
17	C₇×#554	C₄	1	1	yes	characteristic, complemented
18	C₇×#554	C₄	1	2	yes	complemented
19	C₇×D₄×Q₈	—	2	1	no	—
20	C₇×D₄×Q₈	C₄	1	1	yes	characteristic, complemented
21	C₇×#533	—	2	1	no	—
22	C₇×#533	—	2	2	no	—
23	C₇×#533	C₂²	1	1	yes	characteristic
24	C₇×#533	C₂²	1	2	yes	—
25	C₄×C₇×#41	C₂²	1	1	yes	characteristic
26	C₄×C₇×#41	C₂²	1	2	yes	—
27	C₂×C₄×C₇×Q₈	—	2	1	no	—
28	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
29	C₇×#446	—	2	1	no	—
30	C₇×#446	—	2	2	no	—
31	C₇×C₈×Q₈	—	2	1	no	—
32	C₇×#423	C₂²	1	1	yes	characteristic
33	C₇×#423	C₂²	1	2	yes	—
34	C₂×C₇×#104	C₂²	1	1	yes	characteristic
35	C₂×C₇×#104	C₂²	1	2	yes	—
36	C₂×C₇×#104	C₂²	1	1	yes	characteristic
37	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₇×M₁₆	C₂²	1	2	yes	—
39	C₇×#391	—	2	1	no	—
40	C₇×#389	—	2	1	no	—
41	C₇×#387	—	2	2	no	—
42	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
43	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
44	C₇×#376	C₂²	1	2	yes	—
45	C₄³×C₇	C₂²	1	1	yes	characteristic
46	C₇×#339	C₂²	1	1	yes	characteristic
47	C₇×#339	C₂²	1	2	yes	—
48	C₇×#339	C₂²	1	2	yes	—
49	C₇×#339	C₂²	1	2	yes	—
50	C₇×#339	C₂²	1	2	yes	—
51	C₇×#339	C₂²	1	2	yes	—
52	C₇×#339	C₂²	1	1	yes	characteristic
53	#22529	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₇×#41	C₂³	1	1	yes	characteristic
55	C₂²×C₇×Q₈	—	2	1	no	—
56	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
57	C₂×C₇×M₁₆	C₂³	1	2	yes	—
58	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
60	C₇×C₄⋊Q₈	—	2	1	no	—
61	C₇×C₄⋊Q₈	—	2	2	no	—
62	C₇×C₄⋊Q₈	—	2	2	no	—
63	C₇×C₄⋊Q₈	—	2	1	no	—
64	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
65	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
66	C₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
67	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
68	C₇×#125	—	2	2	no	—
69	C₇×#125	—	2	2	no	—
70	C₇×#125	—	2	2	no	—
71	C₇×#125	C₂×C₄	1	2	yes	—
72	C₇×#125	C₂×C₄	1	2	yes	—
73	C₇×#125	C₂×C₄	1	2	yes	—
74	C₇×C₂²⋊Q₈	—	2	1	no	—
75	C₇×C₂²⋊Q₈	—	2	2	no	—
76	C₇×C₂²⋊Q₈	—	2	1	no	—
77	C₇×C₂²⋊Q₈	—	2	2	no	—
78	C₇×C₂²⋊Q₈	—	2	1	no	—
79	C₇×C₂²⋊Q₈	—	2	2	no	—
80	C₇×C₂²⋊Q₈	—	2	2	no	—
81	C₇×C₂²⋊Q₈	—	2	1	no	—
82	C₇×C₂²⋊Q₈	—	2	2	no	—
83	C₇×C₂²⋊Q₈	—	2	2	no	—
84	C₇×C₂²⋊Q₈	—	2	2	no	—
85	C₄×C₇×Q₈	—	2	1	no	—
86	C₄×C₇×Q₈	—	2	2	no	—
87	C₄×C₇×Q₈	—	2	1	no	—
88	C₄×C₇×Q₈	—	2	1	no	—
89	C₄×C₇×Q₈	—	2	2	no	—
90	C₄×C₇×Q₈	—	2	2	no	—
91	C₄×C₇×Q₈	—	2	1	no	—
92	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
93	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
94	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
95	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
96	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
97	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
98	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
99	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
100	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
101	C₄×C₇×Q₈	D₄	1	2	yes	—
102	C₄×C₇×D₄	—	2	1	no	—
103	C₄×C₇×D₄	—	2	2	no	—
104	C₄×C₇×D₄	—	2	1	no	—
105	C₄×C₇×D₄	—	2	2	no	—
106	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
107	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
108	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
109	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
110	C₇×#117	—	2	2	no	—
111	C₇×#117	—	2	1	no	—
112	C₇×#117	C₂³	1	1	yes	characteristic
113	C₇×#117	C₂³	1	2	yes	—
114	C₂×C₇×C₄⋊C₄	—	2	2	no	—
115	C₂×C₇×C₄⋊C₄	—	2	1	no	—
116	C₂×C₇×C₄⋊C₄	—	2	2	no	—
117	C₂×C₇×C₄⋊C₄	—	2	1	no	—
118	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
119	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₇	—	2	1	no	—
121	C₂×C₄²×C₇	—	2	2	no	—
122	C₂×C₄²×C₇	—	2	1	no	—
123	C₂×C₄²×C₇	—	2	1	no	—
124	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₇	C₂³	1	2	yes	—
126	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₇	C₂³	1	2	yes	—
129	C₇×C₄⋊C₈	—	2	1	no	—
130	C₇×C₄⋊C₈	—	2	2	no	—
131	C₇×C₄⋊C₈	—	2	2	no	—
132	C₇×C₄⋊C₈	—	2	1	no	—
133	C₇×#104	—	2	2	no	—
134	C₇×#104	—	2	4	no	—
135	C₇×#104	Q₈	1	4	yes	complemented
136	C₇×#97	—	2	2	no	—
137	C₇×#97	—	2	1	no	—
138	C₄×C₇×C₈	—	2	1	no	—
139	C₄×C₇×C₈	—	2	2	no	—
140	C₇×#95	—	2	2	no	—
141	C₇×#95	—	2	2	no	—
142	C₇×#95	—	2	1	no	—
143	C₇×#95	—	2	2	no	—
144	Q₈×#41	C₂×C₇	1	1	yes	characteristic
145	Q₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
146	#2295	C₂×C₇	1	1	yes	characteristic, complemented
147	#1995	C₂×C₇	1	2	yes	complemented
148	#1994	C₂×C₇	1	1	yes	characteristic, complemented
149	#1986	C₂×C₇	1	1	yes	characteristic, complemented
150	#1986	C₂×C₇	1	2	yes	complemented
151	#1806	C₂×C₇	1	1	yes	characteristic
152	#1806	C₂×C₇	1	2	yes	—
153	#1759	C₂×C₇	1	2	yes	—
154	C₄×#104	C₂×C₇	1	1	yes	characteristic
155	C₇×#41	—	2	1	no	—
156	C₇×#41	—	2	1	no	—
157	C₇×#41	C₂×Q₈	1	2	yes	—
158	C₇×#41	#41	1	2	yes	—
159	C₂×C₇×Q₈	—	2	2	no	—
160	C₂×C₇×Q₈	—	2	1	no	—
161	C₂×C₇×Q₈	—	2	2	no	—
162	C₂×C₇×Q₈	—	2	1	no	—
163	C₂×C₇×Q₈	—	4	2	no	—
164	C₂×C₇×Q₈	—	4	4	no	—
165	C₂×C₇×Q₈	C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₇×Q₈	C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₇×Q₈	C₂²⋊C₄	1	2	yes	—
168	C₂×C₇×Q₈	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₇×D₄	—	2	1	no	—
170	C₂×C₇×D₄	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₇	—	2	1	no	—
172	C₂²×C₄×C₇	—	2	1	no	—
173	C₂²×C₄×C₇	—	2	1	no	—
174	C₂²×C₄×C₇	—	2	2	no	—
175	C₂²×C₄×C₇	—	2	1	no	—
176	C₂²×C₄×C₇	—	2	1	no	—
177	C₂²×C₄×C₇	—	2	2	no	—
178	C₂²×C₄×C₇	—	2	2	no	—
179	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
184	C₇×M₁₆	—	2	1	no	—
185	C₇×M₁₆	—	2	1	no	—
186	C₇×M₁₆	—	2	2	no	—
187	C₇×M₁₆	—	2	2	no	—
188	C₇×M₁₆	—	2	1	no	—
189	C₇×M₁₆	C₂×Q₈	1	2	yes	complemented
190	C₂×C₇×C₈	—	2	1	no	—
191	C₂×C₇×C₈	—	2	1	no	—
192	C₂×C₇×C₈	—	2	2	no	—
193	C₇×C₄⋊C₄	—	2	2	no	—
194	C₇×C₄⋊C₄	—	2	2	no	—
195	C₇×C₄⋊C₄	—	2	2	no	—
196	C₇×C₄⋊C₄	—	2	2	no	—
197	C₇×C₄⋊C₄	—	2	1	no	—
198	C₇×C₄⋊C₄	—	2	1	no	—
199	C₇×C₄⋊C₄	—	2	2	no	—
200	C₇×C₄⋊C₄	—	2	2	no	—
201	C₇×C₄⋊C₄	—	2	1	no	—
202	C₇×C₄⋊C₄	—	2	2	no	—
203	C₇×C₄⋊C₄	—	2	1	no	—
204	C₇×C₄⋊C₄	—	2	2	no	—
205	C₇×C₄⋊C₄	—	2	2	no	—
206	C₇×C₄⋊C₄	—	2	2	no	—
207	C₇×C₄⋊C₄	—	4	2	no	—
208	C₇×C₄⋊C₄	—	4	4	no	—
209	C₇×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
210	C₇×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
211	C₇×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
212	C₇×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
213	C₇×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
214	C₇×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
215	C₇×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
216	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
217	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
218	C₇×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
219	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
220	C₇×C₂²⋊C₄	—	2	1	no	—
221	C₇×C₂²⋊C₄	—	2	2	no	—
222	C₇×C₂²⋊C₄	—	2	1	no	—
223	C₇×C₂²⋊C₄	—	2	2	no	—
224	C₇×C₂²⋊C₄	—	2	2	no	—
225	C₇×C₂²⋊C₄	—	2	1	no	—
226	C₄²×C₇	—	2	1	no	—
227	C₄²×C₇	—	2	2	no	—
228	C₄²×C₇	—	2	2	no	—
229	C₄²×C₇	—	2	1	no	—
230	C₄²×C₇	—	2	1	no	—
231	C₄²×C₇	—	2	2	no	—
232	C₄²×C₇	—	2	1	no	—
233	C₄²×C₇	—	2	1	no	—
234	C₄²×C₇	—	2	2	no	—
235	C₄²×C₇	—	2	1	no	—
236	C₄²×C₇	—	2	2	no	—
237	C₄²×C₇	—	4	2	no	—
238	C₄²×C₇	—	4	2	no	—
239	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
240	C₄²×C₇	C₂²×C₄	1	2	yes	—
241	C₄²×C₇	C₂²×C₄	1	2	yes	—
242	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
243	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
244	C₄²×C₇	C₂×D₄	1	2	yes	—
245	C₄²×C₇	C₂×D₄	1	2	yes	—
246	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
247	C₄²×C₇	C₂×Q₈	1	2	yes	complemented
248	Q₈²	C₄×C₇	1	1	yes	characteristic, complemented
249	#557	C₄×C₇	1	2	yes	complemented
250	#557	C₄×C₇	1	1	yes	characteristic, complemented
251	#554	—	2	2	no	—
252	#554	—	2	1	no	—
253	#554	C₄×C₇	1	1	yes	characteristic, complemented
254	#554	C₄×C₇	1	2	yes	complemented
255	D₄×Q₈	—	2	1	no	—
256	D₄×Q₈	C₄×C₇	1	1	yes	characteristic, complemented
257	#533	—	2	1	no	—
258	#533	—	2	2	no	—
259	#533	C₂²×C₇	1	1	yes	characteristic
260	#533	C₂²×C₇	1	2	yes	—
261	C₄×#41	C₂²×C₇	1	1	yes	characteristic
262	C₄×#41	C₂²×C₇	1	2	yes	—
263	C₂×C₄×Q₈	—	2	1	no	—
264	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
265	#446	—	2	1	no	—
266	#446	—	2	2	no	—
267	C₈×Q₈	—	2	1	no	—
268	#423	C₂²×C₇	1	1	yes	characteristic
269	#423	C₂²×C₇	1	2	yes	—
270	C₂×#104	C₂²×C₇	1	1	yes	characteristic
271	C₂×#104	C₂²×C₇	1	2	yes	—
272	C₂×#104	C₂²×C₇	1	1	yes	characteristic
273	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
274	C₄×M₁₆	C₂²×C₇	1	2	yes	—
275	#391	—	2	1	no	—
276	#389	—	2	1	no	—
277	#387	—	2	2	no	—
278	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
279	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
280	#376	C₂²×C₇	1	2	yes	—
281	C₄³	C₂²×C₇	1	1	yes	characteristic
282	#339	C₂²×C₇	1	2	yes	—
283	#339	C₂²×C₇	1	2	yes	—
284	#339	C₂²×C₇	1	2	yes	—
285	#339	C₂²×C₇	1	1	yes	characteristic
286	#339	C₂²×C₇	1	2	yes	—
287	#339	C₂²×C₇	1	2	yes	—
288	#339	C₂²×C₇	1	1	yes	characteristic
289	C₂³×C₇	—	2	1	no	—
290	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
291	C₇×Q₈	—	2	1	no	—
292	C₇×Q₈	—	2	2	no	—
293	C₇×Q₈	—	2	2	no	—
294	C₇×Q₈	—	4	2	no	—
295	C₇×Q₈	—	4	4	no	—
296	C₇×Q₈	—	4	2	no	—
297	C₇×Q₈	—	8	1	no	—
298	C₇×Q₈	—	8	2	no	—
299	C₇×Q₈	C₄×Q₈	1	2	yes	—
300	C₇×D₄	—	2	2	no	—
301	C₇×D₄	—	2	2	no	—
302	C₇×D₄	—	2	1	no	—
303	C₇×D₄	C₄×Q₈	1	2	yes	—
304	C₂×C₄×C₇	—	2	1	no	—
305	C₂×C₄×C₇	—	2	2	no	—
306	C₂×C₄×C₇	—	2	2	no	—
307	C₂×C₄×C₇	—	2	2	no	—
308	C₂×C₄×C₇	—	2	2	no	—
309	C₂×C₄×C₇	—	2	2	no	—
310	C₂×C₄×C₇	—	2	1	no	—
311	C₂×C₄×C₇	—	2	2	no	—
312	C₂×C₄×C₇	—	2	1	no	—
313	C₂×C₄×C₇	—	2	2	no	—
314	C₂×C₄×C₇	—	4	1	no	—
315	C₂×C₄×C₇	—	4	2	no	—
316	C₂×C₄×C₇	—	4	2	no	—
317	C₂×C₄×C₇	—	4	4	no	—
318	C₂×C₄×C₇	—	4	1	no	—
319	C₂×C₄×C₇	—	4	1	no	—
320	C₂×C₄×C₇	—	4	1	no	—
321	C₂×C₄×C₇	—	4	1	no	—
322	C₂×C₄×C₇	—	4	2	no	—
323	C₂×C₄×C₇	—	4	2	no	—
324	C₂×C₄×C₇	—	4	2	no	—
325	C₂×C₄×C₇	—	4	2	no	—
326	C₂×C₄×C₇	—	8	1	no	—
327	C₂×C₄×C₇	—	8	2	no	—
328	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
329	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
330	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
331	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
332	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
333	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
334	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
335	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
336	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
337	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
338	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
339	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
340	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
341	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
342	C₇×C₈	—	2	2	no	—
343	C₇×C₈	—	4	1	no	—
344	C₇×C₈	—	4	2	no	—
345	C₂×#41	C₂³×C₇	1	1	yes	characteristic
346	C₂²×Q₈	—	2	1	no	—
347	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
348	C₂×M₁₆	C₂³×C₇	1	2	yes	—
349	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
350	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
351	C₄⋊Q₈	—	2	1	no	—
352	C₄⋊Q₈	—	2	2	no	—
353	C₄⋊Q₈	—	2	2	no	—
354	C₄⋊Q₈	—	2	1	no	—
355	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
356	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
357	C₄⋊Q₈	C₂×C₄×C₇	1	1	yes	characteristic
358	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
359	#125	—	2	2	no	—
360	#125	—	2	2	no	—
361	#125	—	2	2	no	—
362	#125	C₂×C₄×C₇	1	2	yes	—
363	#125	C₂×C₄×C₇	1	2	yes	—
364	#125	C₂×C₄×C₇	1	2	yes	—
365	C₂²⋊Q₈	—	2	1	no	—
366	C₂²⋊Q₈	—	2	2	no	—
367	C₂²⋊Q₈	—	2	1	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	C₂²⋊Q₈	—	2	1	no	—
370	C₂²⋊Q₈	—	2	2	no	—
371	C₂²⋊Q₈	—	2	2	no	—
372	C₂²⋊Q₈	—	2	1	no	—
373	C₂²⋊Q₈	—	2	2	no	—
374	C₂²⋊Q₈	—	2	2	no	—
375	C₂²⋊Q₈	—	2	2	no	—
376	C₄×Q₈	—	2	1	no	—
377	C₄×Q₈	—	2	2	no	—
378	C₄×Q₈	—	2	1	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	—	2	1	no	—
383	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
384	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
385	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
386	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
387	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
388	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
389	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
390	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
391	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
392	C₄×Q₈	C₇×D₄	1	2	yes	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
398	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
399	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
400	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
401	#117	—	2	2	no	—
402	#117	—	2	1	no	—
403	#117	C₂³×C₇	1	1	yes	characteristic
404	#117	C₂³×C₇	1	2	yes	—
405	C₂×C₄⋊C₄	—	2	2	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	—	2	2	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
410	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	—	2	2	no	—
413	C₂×C₄²	—	2	1	no	—
414	C₂×C₄²	—	2	1	no	—
415	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₇	1	2	yes	—
417	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₇	1	2	yes	—
420	C₄⋊C₈	—	2	1	no	—
421	C₄⋊C₈	—	2	2	no	—
422	C₄⋊C₈	—	2	2	no	—
423	C₄⋊C₈	—	2	1	no	—
424	#104	—	2	2	no	—
425	#104	—	2	4	no	—
426	#104	C₇×Q₈	1	4	yes	complemented
427	#97	—	2	2	no	—
428	#97	—	2	1	no	—
429	C₄×C₈	—	2	1	no	—
430	C₄×C₈	—	2	2	no	—
431	#95	—	2	2	no	—
432	#95	—	2	2	no	—
433	#95	—	2	1	no	—
434	#95	—	2	2	no	—
435	C₂²×C₇	—	2	1	no	—
436	C₂²×C₇	—	2	1	no	—
437	C₂²×C₇	—	2	2	no	—
438	C₂²×C₇	—	4	1	no	—
439	C₂²×C₇	—	4	1	no	—
440	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
441	C₂²×C₇	#548	1	1	yes	characteristic
442	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
443	C₄×C₇	—	2	2	no	—
444	C₄×C₇	—	2	1	no	—
445	C₄×C₇	—	2	2	no	—
446	C₄×C₇	—	2	2	no	—
447	C₄×C₇	—	2	1	no	—
448	C₄×C₇	—	2	2	no	—
449	C₄×C₇	—	2	1	no	—
450	C₄×C₇	—	2	2	no	—
451	C₄×C₇	—	4	2	no	—
452	C₄×C₇	—	4	2	no	—
453	C₄×C₇	—	4	2	no	—
454	C₄×C₇	—	4	1	no	—
455	C₄×C₇	—	4	2	no	—
456	C₄×C₇	—	4	1	no	—
457	C₄×C₇	—	8	2	no	—
458	C₄×C₇	#421	1	2	yes	—
459	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
460	C₄×C₇	#519	1	1	yes	characteristic
461	C₄×C₇	#548	1	1	yes	characteristic
462	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
463	#41	—	2	1	no	—
464	#41	—	2	1	no	—
465	#41	C₂×C₇×Q₈	1	2	yes	—
466	#41	C₇×#41	1	2	yes	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	2	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	4	2	no	—
472	C₂×Q₈	—	4	4	no	—
473	C₂×Q₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
474	C₂×Q₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
475	C₂×Q₈	C₇×C₂²⋊C₄	1	2	yes	—
476	C₂×Q₈	C₂²×C₄×C₇	1	1	yes	characteristic
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	C₂²×C₄×C₇	1	1	yes	characteristic
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
488	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
489	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
490	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
492	M₁₆	—	2	1	no	—
493	M₁₆	—	2	1	no	—
494	M₁₆	—	2	2	no	—
495	M₁₆	—	2	2	no	—
496	M₁₆	—	2	1	no	—
497	M₁₆	C₂×C₇×Q₈	1	2	yes	complemented
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	1	no	—
500	C₂×C₈	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	4	2	no	—
516	C₄⋊C₄	—	4	4	no	—
517	C₄⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
518	C₄⋊C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
519	C₄⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
520	C₄⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
521	C₄⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
522	C₄⋊C₄	C₇×C₂²⋊C₄	1	2	yes	—
523	C₄⋊C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
524	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
525	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
526	C₄⋊C₄	C₂²×C₄×C₇	1	1	yes	characteristic
527	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
528	C₂²⋊C₄	—	2	1	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	2	1	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	2	no	—
533	C₂²⋊C₄	—	2	1	no	—
534	C₄²	—	2	1	no	—
535	C₄²	—	2	2	no	—
536	C₄²	—	2	2	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	2	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	2	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	2	no	—
545	C₄²	—	4	2	no	—
546	C₄²	—	4	2	no	—
547	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
548	C₄²	C₂²×C₄×C₇	1	2	yes	—
549	C₄²	C₂²×C₄×C₇	1	2	yes	—
550	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
551	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
552	C₄²	C₂×C₇×D₄	1	2	yes	—
553	C₄²	C₂×C₇×D₄	1	2	yes	—
554	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
555	C₄²	C₂×C₇×Q₈	1	2	yes	complemented
556	C₂×C₇	—	2	1	no	—
557	C₂×C₇	—	2	1	no	—
558	C₂×C₇	—	4	2	no	—
559	C₂×C₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
560	C₂×C₇	C₂×#421	1	1	yes	characteristic
561	C₂×C₇	#2901	1	1	yes	characteristic
562	C₂³	—	2	1	no	—
563	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
564	Q₈	—	2	1	no	—
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	4	2	no	—
568	Q₈	—	4	4	no	—
569	Q₈	—	4	2	no	—
570	Q₈	—	8	1	no	—
571	Q₈	—	8	2	no	—
572	Q₈	C₄×C₇×Q₈	1	2	yes	—
573	D₄	—	2	2	no	—
574	D₄	—	2	2	no	—
575	D₄	—	2	1	no	—
576	D₄	C₄×C₇×Q₈	1	2	yes	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	8	1	no	—
600	C₂×C₄	—	8	2	no	—
601	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
602	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
603	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
604	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
605	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
606	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	2	yes	—
607	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
608	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
609	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
610	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
611	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
612	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
613	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
614	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
615	C₈	—	2	2	no	—
616	C₈	—	4	1	no	—
617	C₈	—	4	2	no	—
618	C₇	#22529	1	1	yes	characteristic, direct product factor, Sylow
619	C₂²	—	2	1	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	1	no	—
624	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
625	C₂²	C₇×#548	1	1	yes	characteristic
626	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
627	C₄	—	2	1	no	—
628	C₄	—	2	2	no	—
629	C₄	—	2	2	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	1	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	4	2	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	1	no	—
641	C₄	—	8	2	no	—
642	C₄	C₇×#421	1	2	yes	—
643	C₄	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
644	C₄	C₇×#519	1	1	yes	characteristic
645	C₄	C₇×#548	1	1	yes	characteristic
646	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
647	C₂	—	2	1	no	—
648	C₂	—	2	1	no	—
649	C₂	—	4	2	no	—
650	C₂	C₇×Q₈×C₂²⋊C₄	1	1	yes	characteristic
651	C₂	C₂×C₇×#421	1	1	yes	characteristic
652	C₂	C₇×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₂²⋊C₄
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×Q₈×M₁₆	C₂	semidirect
3. C₇×#2295	C₂	semidirect
4. C₇×#1995	C₂	semidirect
5. C₇×#1994	C₂	semidirect
6. C₇×#1986	C₂	semidirect
7. C₇×#1986	C₂	semidirect
12. C₇×Q₈²	C₄	semidirect
13. C₇×#557	C₄	semidirect
14. C₇×#557	C₄	semidirect
17. C₇×#554	C₄	semidirect
18. C₇×#554	C₄	semidirect
20. C₇×D₄×Q₈	C₄	semidirect
53. #22529	C₇	direct (N × H)
135. C₇×#104	Q₈	semidirect
145. Q₈×M₁₆	C₂×C₇	semidirect
146. #2295	C₂×C₇	semidirect
147. #1995	C₂×C₇	semidirect
148. #1994	C₂×C₇	semidirect
149. #1986	C₂×C₇	semidirect
150. #1986	C₂×C₇	semidirect
189. C₇×M₁₆	C₂×Q₈	semidirect
247. C₄²×C₇	C₂×Q₈	semidirect
248. Q₈²	C₄×C₇	semidirect
249. #557	C₄×C₇	semidirect
250. #557	C₄×C₇	semidirect
253. #554	C₄×C₇	semidirect
254. #554	C₄×C₇	semidirect
256. D₄×Q₈	C₄×C₇	semidirect
426. #104	C₇×Q₈	semidirect
497. M₁₆	C₂×C₇×Q₈	semidirect
555. C₄²	C₂×C₇×Q₈	semidirect
618. C₇	#22529	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×Q₈×#41	2
C₇×Q₈×M₁₆	2
C₇×#2295	2
C₇×#1995	2
C₇×#1994	2
C₇×#1986	2
C₇×#1806	2
C₇×#1759	2
C₄×C₇×#104	2
#22529	7

Subgroups — C7×#22529 #13199745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22529 #13199745