Subgroups — C₇×#22540 #13199756

Order 1792.

1430 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#550	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Q₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1982	C₂	1	3	yes	maximal, complemented
5	C₇×#1957	C₂	1	3	yes	maximal, complemented
6	C₇×#1802	C₂	1	3	yes	maximal, complemented
7	C₄×C₇×#102	C₂	1	3	yes	maximal, complemented
8	C₂²×C₇×#41	—	2	1	no	—
9	C₇×#550	—	2	4	no	—
10	C₇×#550	C₄	1	8	yes	complemented
11	C₂×C₇×#127	C₂²	1	3	yes	—
12	C₂×C₇×#121	—	2	3	no	—
13	C₂×C₄×C₇×Q₈	—	2	1	no	—
14	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×D₄	C₂²	1	3	yes	—
16	C₇×C₈×Q₈	—	2	4	no	—
17	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
18	C₂×C₇×#102	C₂²	1	3	yes	—
19	C₂×C₇×#102	C₂²	1	3	yes	—
20	C₂×C₇×#102	C₂²	1	6	yes	—
21	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
22	C₇×#389	—	2	3	no	—
23	C₇×#384	C₂²	1	3	yes	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	3	yes	—
25	C₇×#340	C₂²	1	1	yes	characteristic
26	C₇×#340	C₂²	1	3	yes	—
27	#22540	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₇×#41	—	2	8	no	—
29	C₂×C₇×#41	—	4	4	no	—
30	C₂×C₇×#41	—	4	4	no	—
31	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×D₄	—	2	3	no	—
33	C₂²×C₇×D₄	—	2	3	no	—
34	C₂²×C₇×D₄	C₂³	1	3	yes	—
35	C₂³×C₄×C₇	—	2	3	no	—
36	C₂²×C₇×C₈	C₂³	1	3	yes	—
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₇×#127	—	2	6	no	—
39	C₇×#127	C₂×C₄	1	12	yes	—
40	C₇×#121	—	2	24	no	—
41	C₄×C₇×Q₈	—	4	4	no	—
42	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
43	C₄×C₇×Q₈	D₄	1	4	yes	—
44	C₄×C₇×D₄	—	2	6	no	—
45	C₄×C₇×D₄	C₂×C₄	1	12	yes	—
46	C₂×C₇×C₄⋊C₄	—	2	3	no	—
47	C₂×C₇×C₄⋊C₄	—	2	3	no	—
48	C₂×C₇×C₄⋊C₄	D₄	1	3	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
51	C₂×C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
52	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
53	C₂×C₄²×C₇	—	2	3	no	—
54	C₂×C₄²×C₇	C₂³	1	3	yes	—
55	C₇×C₄⋊C₈	—	2	6	no	—
56	C₇×#102	—	2	6	no	—
57	C₇×#102	—	2	6	no	—
58	C₇×#102	—	4	6	no	—
59	C₄×C₇×C₈	—	2	6	no	—
60	C₇×#95	—	2	3	no	—
61	C₂×#550	C₂×C₇	1	1	yes	characteristic
62	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
63	Q₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
64	#1982	C₂×C₇	1	3	yes	complemented
65	#1957	C₂×C₇	1	3	yes	complemented
66	#1802	C₂×C₇	1	3	yes	complemented
67	C₄×#102	C₂×C₇	1	3	yes	complemented
68	C₂⁴×C₇	—	2	3	no	—
69	C₇×#41	—	4	32	no	—
70	C₂×C₇×Q₈	C₂²⋊C₄	1	4	yes	—
71	C₂×C₇×Q₈	D₈	1	4	yes	—
72	C₂×C₇×Q₈	SD₁₆	1	4	yes	—
73	C₂×C₇×D₄	—	2	12	no	—
74	C₂×C₇×D₄	—	2	12	no	—
75	C₂×C₇×D₄	—	2	3	no	—
76	C₂×C₇×D₄	—	2	3	no	—
77	C₂×C₇×D₄	—	2	3	no	—
78	C₂×C₇×D₄	—	2	3	no	—
79	C₂×C₇×D₄	—	2	3	no	—
80	C₂×C₇×D₄	—	4	6	no	—
81	C₂×C₇×D₄	—	4	6	no	—
82	C₂×C₇×D₄	—	8	3	no	—
83	C₂×C₇×D₄	—	8	3	no	—
84	C₂×C₇×D₄	C₂²×C₄	1	6	yes	—
85	C₂²×C₄×C₇	—	2	1	no	—
86	C₂²×C₄×C₇	—	2	1	no	—
87	C₂²×C₄×C₇	—	2	12	no	—
88	C₂²×C₄×C₇	—	2	3	no	—
89	C₂²×C₄×C₇	—	4	6	no	—
90	C₂²×C₄×C₇	—	4	6	no	—
91	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
92	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
93	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
94	C₂×C₇×C₈	—	2	1	no	—
95	C₂×C₇×C₈	—	2	1	no	—
96	C₂×C₇×C₈	—	2	3	no	—
97	C₂×C₇×C₈	—	2	3	no	—
98	C₂×C₇×C₈	—	2	3	no	—
99	C₂×C₇×C₈	#41	1	2	yes	complemented
100	C₇×C₄⋊C₄	—	2	1	no	—
101	C₇×C₄⋊C₄	—	2	1	no	—
102	C₇×C₄⋊C₄	—	2	3	no	—
103	C₇×C₄⋊C₄	—	2	3	no	—
104	C₇×C₄⋊C₄	—	4	6	no	—
105	C₇×C₄⋊C₄	—	8	3	no	—
106	C₇×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
107	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
108	C₇×C₄⋊C₄	#41	1	2	yes	complemented
109	C₇×C₂²⋊C₄	—	2	12	no	—
110	C₄²×C₇	—	4	6	no	—
111	C₄²×C₇	C₂²×C₄	1	6	yes	—
112	C₄²×C₇	C₂×D₄	1	6	yes	—
113	C₂²×#41	—	2	1	no	—
114	#550	—	2	4	no	—
115	#550	C₄×C₇	1	8	yes	complemented
116	C₂×#127	C₂²×C₇	1	3	yes	—
117	C₂×#121	—	2	3	no	—
118	C₂×C₄×Q₈	—	2	1	no	—
119	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄×D₄	C₂²×C₇	1	3	yes	—
121	C₈×Q₈	—	2	4	no	—
122	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
123	C₂×#102	C₂²×C₇	1	3	yes	—
124	C₂×#102	C₂²×C₇	1	3	yes	—
125	C₂×#102	C₂²×C₇	1	6	yes	—
126	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
127	#389	—	2	3	no	—
128	#384	C₂²×C₇	1	3	yes	—
129	C₄×C₄⋊C₄	C₂²×C₇	1	3	yes	—
130	#340	C₂²×C₇	1	3	yes	—
131	#340	C₂²×C₇	1	1	yes	characteristic
132	C₂³×C₇	—	2	6	no	—
133	C₂³×C₇	—	4	3	no	—
134	C₂³×C₇	—	4	3	no	—
135	C₂³×C₇	—	4	3	no	—
136	C₂³×C₇	—	4	3	no	—
137	C₂³×C₇	—	4	3	no	—
138	C₂³×C₇	—	4	3	no	—
139	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
140	C₇×Q₈	#102	1	16	yes	complemented
141	C₇×D₄	—	2	6	no	—
142	C₇×D₄	—	2	6	no	—
143	C₇×D₄	—	4	3	no	—
144	C₇×D₄	—	4	3	no	—
145	C₇×D₄	—	4	24	no	—
146	C₇×D₄	—	8	12	no	—
147	C₂×C₄×C₇	—	2	3	no	—
148	C₂×C₄×C₇	—	2	3	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	4	1	no	—
152	C₂×C₄×C₇	—	4	1	no	—
153	C₂×C₄×C₇	—	4	1	no	—
154	C₂×C₄×C₇	—	4	1	no	—
155	C₂×C₄×C₇	—	4	24	no	—
156	C₂×C₄×C₇	—	4	3	no	—
157	C₂×C₄×C₇	—	4	3	no	—
158	C₂×C₄×C₇	—	4	3	no	—
159	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
160	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	6	yes	—
161	C₂×C₄×C₇	C₂×D₈	1	6	yes	—
162	C₂×C₄×C₇	C₂×SD₁₆	1	6	yes	—
163	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
169	C₇×C₈	—	2	2	no	—
170	C₇×C₈	—	4	3	no	—
171	C₂×#41	—	2	8	no	—
172	C₂×#41	—	4	4	no	—
173	C₂×#41	—	4	4	no	—
174	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
175	C₂²×D₄	—	2	3	no	—
176	C₂²×D₄	—	2	3	no	—
177	C₂²×D₄	C₂³×C₇	1	3	yes	—
178	C₂³×C₄	—	2	3	no	—
179	C₂²×C₈	C₂³×C₇	1	3	yes	—
180	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
181	#127	—	2	6	no	—
182	#127	C₂×C₄×C₇	1	12	yes	—
183	#121	—	2	24	no	—
184	C₄×Q₈	—	4	4	no	—
185	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
186	C₄×Q₈	C₇×D₄	1	4	yes	—
187	C₄×D₄	—	2	6	no	—
188	C₄×D₄	C₂×C₄×C₇	1	12	yes	—
189	C₂×C₄⋊C₄	—	2	3	no	—
190	C₂×C₄⋊C₄	—	2	3	no	—
191	C₂×C₄⋊C₄	C₇×D₄	1	3	yes	—
192	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
193	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
194	C₂×C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	2	3	no	—
196	C₂×C₄²	—	2	3	no	—
197	C₂×C₄²	C₂³×C₇	1	3	yes	—
198	C₄⋊C₈	—	2	6	no	—
199	#102	—	2	6	no	—
200	#102	—	2	6	no	—
201	#102	—	4	6	no	—
202	C₄×C₈	—	2	6	no	—
203	#95	—	2	3	no	—
204	C₂²×C₇	—	4	6	no	—
205	C₂²×C₇	—	4	6	no	—
206	C₂²×C₇	—	4	6	no	—
207	C₂²×C₇	—	8	3	no	—
208	C₂²×C₇	—	8	3	no	—
209	C₂²×C₇	—	8	3	no	—
210	C₂²×C₇	—	8	3	no	—
211	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₇	#548	1	1	yes	characteristic
213	C₂²×C₇	#550	1	1	yes	characteristic
214	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
215	C₂²×C₇	C₂²×SD₁₆	1	1	yes	characteristic
216	C₂²×C₇	#576	1	1	yes	characteristic
217	C₂²×C₇	#577	1	1	yes	characteristic
218	C₄×C₇	—	2	6	no	—
219	C₄×C₇	—	4	2	no	—
220	C₄×C₇	—	4	2	no	—
221	C₄×C₇	—	8	3	no	—
222	C₄×C₇	C₂×#102	1	12	yes	—
223	C₄×C₇	C₄×#41	1	1	yes	characteristic
224	C₄×C₇	#520	1	1	yes	characteristic
225	C₄×C₇	#548	1	1	yes	characteristic
226	C₄×C₇	#550	1	1	yes	characteristic
227	C₂⁴	—	2	3	no	—
228	#41	—	4	32	no	—
229	C₂×Q₈	C₇×C₂²⋊C₄	1	4	yes	—
230	C₂×Q₈	C₇×D₈	1	4	yes	—
231	C₂×Q₈	C₇×SD₁₆	1	4	yes	—
232	C₂×D₄	—	2	12	no	—
233	C₂×D₄	—	2	12	no	—
234	C₂×D₄	—	2	3	no	—
235	C₂×D₄	—	2	3	no	—
236	C₂×D₄	—	2	3	no	—
237	C₂×D₄	—	2	3	no	—
238	C₂×D₄	—	2	3	no	—
239	C₂×D₄	—	4	6	no	—
240	C₂×D₄	—	4	6	no	—
241	C₂×D₄	—	8	3	no	—
242	C₂×D₄	—	8	3	no	—
243	C₂×D₄	C₂²×C₄×C₇	1	6	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	12	no	—
247	C₂²×C₄	—	2	3	no	—
248	C₂²×C₄	—	4	6	no	—
249	C₂²×C₄	—	4	6	no	—
250	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
251	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
252	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	3	no	—
256	C₂×C₈	—	2	3	no	—
257	C₂×C₈	—	2	3	no	—
258	C₂×C₈	C₇×#41	1	2	yes	complemented
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	3	no	—
262	C₄⋊C₄	—	2	3	no	—
263	C₄⋊C₄	—	4	6	no	—
264	C₄⋊C₄	—	8	3	no	—
265	C₄⋊C₄	C₇×C₂²⋊C₄	1	12	yes	—
266	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
267	C₄⋊C₄	C₇×#41	1	2	yes	complemented
268	C₂²⋊C₄	—	2	12	no	—
269	C₄²	—	4	6	no	—
270	C₄²	C₂²×C₄×C₇	1	6	yes	—
271	C₄²	C₂×C₇×D₄	1	6	yes	—
272	C₂×C₇	—	8	6	no	—
273	C₂×C₇	#2341	1	1	yes	characteristic
274	C₂×C₇	C₂²×#102	1	1	yes	characteristic
275	C₂×C₇	#2897	1	1	yes	characteristic
276	C₂×C₇	#3358	1	1	yes	characteristic
277	C₂×C₇	#3362	1	1	yes	characteristic
278	C₂×C₇	#3391	1	1	yes	characteristic
279	C₂×C₇	#3392	1	1	yes	characteristic
280	C₂³	—	2	6	no	—
281	C₂³	—	4	3	no	—
282	C₂³	—	4	3	no	—
283	C₂³	—	4	3	no	—
284	C₂³	—	4	3	no	—
285	C₂³	—	4	3	no	—
286	C₂³	—	4	3	no	—
287	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
288	Q₈	C₇×#102	1	16	yes	complemented
289	D₄	—	2	6	no	—
290	D₄	—	2	6	no	—
291	D₄	—	4	3	no	—
292	D₄	—	4	3	no	—
293	D₄	—	4	24	no	—
294	D₄	—	8	12	no	—
295	C₂×C₄	—	2	3	no	—
296	C₂×C₄	—	2	3	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	24	no	—
304	C₂×C₄	—	4	3	no	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	3	no	—
307	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
308	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	6	yes	—
309	C₂×C₄	C₂×C₇×D₈	1	6	yes	—
310	C₂×C₄	C₂×C₇×SD₁₆	1	6	yes	—
311	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
313	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
314	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
315	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
316	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	4	3	no	—
319	C₇	#22540	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²	—	4	6	no	—
321	C₂²	—	4	6	no	—
322	C₂²	—	4	6	no	—
323	C₂²	—	8	3	no	—
324	C₂²	—	8	3	no	—
325	C₂²	—	8	3	no	—
326	C₂²	—	8	3	no	—
327	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
328	C₂²	C₇×#548	1	1	yes	characteristic
329	C₂²	C₇×#550	1	1	yes	characteristic
330	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
331	C₂²	C₂²×C₇×SD₁₆	1	1	yes	characteristic
332	C₂²	C₇×#576	1	1	yes	characteristic
333	C₂²	C₇×#577	1	1	yes	characteristic
334	C₄	—	2	6	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	3	no	—
338	C₄	C₂×C₇×#102	1	12	yes	—
339	C₄	C₄×C₇×#41	1	1	yes	characteristic
340	C₄	C₇×#520	1	1	yes	characteristic
341	C₄	C₇×#548	1	1	yes	characteristic
342	C₄	C₇×#550	1	1	yes	characteristic
343	C₂	—	8	6	no	—
344	C₂	C₇×#2341	1	1	yes	characteristic
345	C₂	C₂²×C₇×#102	1	1	yes	characteristic
346	C₂	C₇×#2897	1	1	yes	characteristic
347	C₂	C₇×#3358	1	1	yes	characteristic
348	C₂	C₇×#3362	1	1	yes	characteristic
349	C₂	C₇×#3391	1	1	yes	characteristic
350	C₂	C₇×#3392	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₈×Q₈	C₂	semidirect
3. C₇×Q₈×C₄⋊C₄	C₂	semidirect
4. C₇×#1982	C₂	semidirect
5. C₇×#1957	C₂	semidirect
6. C₇×#1802	C₂	semidirect
7. C₄×C₇×#102	C₂	semidirect
10. C₇×#550	C₄	semidirect
27. #22540	C₇	direct (N × H)
62. C₂×C₈×Q₈	C₂×C₇	semidirect
63. Q₈×C₄⋊C₄	C₂×C₇	semidirect
64. #1982	C₂×C₇	semidirect
65. #1957	C₂×C₇	semidirect
66. #1802	C₂×C₇	semidirect
67. C₄×#102	C₂×C₇	semidirect
99. C₂×C₇×C₈	#41	semidirect
108. C₇×C₄⋊C₄	#41	semidirect
115. #550	C₄×C₇	semidirect
140. C₇×Q₈	#102	semidirect
258. C₂×C₈	C₇×#41	semidirect
267. C₄⋊C₄	C₇×#41	semidirect
288. Q₈	C₇×#102	semidirect
319. C₇	#22540	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#550	2
C₂×C₇×C₈×Q₈	2
C₇×Q₈×C₄⋊C₄	2
C₇×#1982	2
C₇×#1957	2
C₇×#1802	2
C₄×C₇×#102	2
#22540	7

Subgroups — C7×#22540 #13199756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22540 #13199756