Subgroups — C₇×#22705 #13199921

Order 1792.

1054 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₇×D₄×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₇×#1947	C₂	1	2	yes	maximal, complemented
4	C₇×#1852	C₂	1	4	yes	maximal, complemented
5	C₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1808	C₂	1	2	yes	maximal, complemented
7	C₄×C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#340	C₂	1	2	yes	maximal, complemented
9	C₂³×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇×D₄	—	2	2	no	—
11	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
13	C₇×C₈×D₄	C₄	1	8	yes	complemented
14	C₂×C₇×#107	C₂²	1	2	yes	—
15	C₂×C₇×#107	C₂²	1	4	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×#389	—	2	2	no	—
20	C₇×#385	—	2	4	no	—
21	C₇×#384	C₂²	1	2	yes	—
22	C₇×#380	C₂²	1	4	yes	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
24	C₇×#340	C₂²	1	1	yes	characteristic
25	C₇×#340	C₂²	1	2	yes	—
26	C₇×#340	C₂²	1	4	yes	—
27	C₇×#340	C₂²	1	1	yes	characteristic
28	C₇×#340	C₂²	1	2	yes	—
29	#22705	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₇	—	2	4	no	—
32	C₂³×C₄×C₇	C₂³	1	2	yes	—
33	C₂²×C₇×C₈	—	4	4	no	—
34	C₂²×C₇×C₈	C₂×C₄	1	8	yes	complemented
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	2	yes	—
38	C₄×C₇×D₄	—	4	8	no	—
39	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
40	C₄×C₇×D₄	Q₈	1	4	yes	—
41	C₂×C₇×C₄⋊C₄	—	2	8	no	—
42	C₂×C₇×C₄⋊C₄	—	2	4	no	—
43	C₂×C₇×C₄⋊C₄	—	2	2	no	—
44	C₂×C₇×C₄⋊C₄	—	2	2	no	—
45	C₂×C₇×C₄⋊C₄	—	4	4	no	—
46	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
51	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
52	C₂×C₄²×C₇	—	2	2	no	—
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₇×#107	—	2	4	no	—
55	C₇×#107	—	4	4	no	—
56	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
57	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
58	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
59	C₇×#95	—	2	2	no	—
60	C₇×#95	—	2	4	no	—
61	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
62	D₄×C₄⋊C₄	C₂×C₇	1	2	yes	—
63	#1947	C₂×C₇	1	2	yes	complemented
64	#1852	C₂×C₇	1	4	yes	complemented
65	#1834	C₂×C₇	1	1	yes	characteristic, complemented
66	#1808	C₂×C₇	1	2	yes	complemented
67	C₄×#107	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂×#340	C₂×C₇	1	2	yes	complemented
69	C₂⁴×C₇	C₂×D₄	1	2	yes	—
70	C₂×C₇×D₄	C₄⋊C₄	1	4	yes	—
71	C₂×C₇×D₄	D₈	1	4	yes	—
72	C₂×C₇×D₄	Q₁₆	1	4	yes	—
73	C₂²×C₄×C₇	—	2	4	no	—
74	C₂²×C₄×C₇	—	2	4	no	—
75	C₂²×C₄×C₇	—	2	2	no	—
76	C₂²×C₄×C₇	—	2	2	no	—
77	C₂²×C₄×C₇	—	2	4	no	—
78	C₂²×C₄×C₇	—	4	8	no	—
79	C₂²×C₄×C₇	—	4	8	no	—
80	C₂²×C₄×C₇	—	8	4	no	—
81	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
86	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
87	C₂×C₇×C₈	—	2	2	no	—
88	C₂×C₇×C₈	—	2	4	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	4	8	no	—
91	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
94	C₇×C₄⋊C₄	—	2	2	no	—
95	C₇×C₄⋊C₄	—	2	2	no	—
96	C₇×C₄⋊C₄	—	2	4	no	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₄⋊C₄	—	4	8	no	—
100	C₇×C₄⋊C₄	—	4	4	no	—
101	C₇×C₄⋊C₄	—	8	2	no	—
102	C₇×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
103	C₇×C₄⋊C₄	#41	1	4	yes	—
104	C₇×C₂²⋊C₄	—	4	8	no	—
105	C₇×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
106	C₄²×C₇	—	4	4	no	—
107	C₄²×C₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₇	C₂×Q₈	1	2	yes	—
109	C₂³×C₈	C₂²×C₇	1	2	yes	—
110	C₂×C₄×D₄	—	2	2	no	—
111	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
112	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
113	C₈×D₄	C₄×C₇	1	8	yes	complemented
114	C₂×#107	C₂²×C₇	1	2	yes	—
115	C₂×#107	C₂²×C₇	1	4	yes	—
116	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
119	#389	—	2	2	no	—
120	#385	—	2	4	no	—
121	#384	C₂²×C₇	1	2	yes	—
122	#380	C₂²×C₇	1	4	yes	—
123	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
124	#340	C₂²×C₇	1	1	yes	characteristic
125	#340	C₂²×C₇	1	2	yes	—
126	#340	C₂²×C₇	1	2	yes	—
127	#340	C₂²×C₇	1	4	yes	—
128	#340	C₂²×C₇	1	1	yes	characteristic
129	C₂³×C₇	—	2	4	no	—
130	C₂³×C₇	—	2	4	no	—
131	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
132	C₂³×C₇	C₂×D₈	1	4	yes	—
133	C₂³×C₇	C₂×Q₁₆	1	4	yes	—
134	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
135	C₇×D₄	#107	1	16	yes	complemented
136	C₂×C₄×C₇	—	2	4	no	—
137	C₂×C₄×C₇	—	2	2	no	—
138	C₂×C₄×C₇	—	2	4	no	—
139	C₂×C₄×C₇	—	2	4	no	—
140	C₂×C₄×C₇	—	2	8	no	—
141	C₂×C₄×C₇	—	2	4	no	—
142	C₂×C₄×C₇	—	4	2	no	—
143	C₂×C₄×C₇	—	4	2	no	—
144	C₂×C₄×C₇	—	4	4	no	—
145	C₂×C₄×C₇	—	4	2	no	—
146	C₂×C₄×C₇	—	4	2	no	—
147	C₂×C₄×C₇	—	4	4	no	—
148	C₂×C₄×C₇	—	4	4	no	—
149	C₂×C₄×C₇	—	4	2	no	—
150	C₂×C₄×C₇	—	8	16	no	—
151	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
153	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
156	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×#41	1	2	yes	—
159	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
160	C₇×C₈	—	2	2	no	—
161	C₇×C₈	—	2	2	no	—
162	C₇×C₈	—	2	4	no	—
163	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
164	C₂³×C₄	—	2	4	no	—
165	C₂³×C₄	C₂³×C₇	1	2	yes	—
166	C₂²×C₈	—	4	4	no	—
167	C₂²×C₈	C₂×C₄×C₇	1	8	yes	complemented
168	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₇	1	2	yes	—
170	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
171	C₄×D₄	—	4	8	no	—
172	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
173	C₄×D₄	C₇×Q₈	1	4	yes	—
174	C₂×C₄⋊C₄	—	2	8	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	4	4	no	—
179	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
181	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
183	C₂×C₂²⋊C₄	—	2	4	no	—
184	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
187	#107	—	2	4	no	—
188	#107	—	4	4	no	—
189	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	complemented
190	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	complemented
191	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
192	#95	—	2	2	no	—
193	#95	—	2	4	no	—
194	C₂²×C₇	—	2	4	no	—
195	C₂²×C₇	—	2	4	no	—
196	C₂²×C₇	—	2	8	no	—
197	C₂²×C₇	—	2	8	no	—
198	C₂²×C₇	C₂×#107	1	8	yes	—
199	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
200	C₂²×C₇	#546	1	2	yes	—
201	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
202	C₂²×C₇	C₂²×Q₁₆	1	1	yes	characteristic
203	C₂²×C₇	#577	1	2	yes	—
204	C₄×C₇	—	2	4	no	—
205	C₄×C₇	—	2	2	no	—
206	C₄×C₇	—	4	4	no	—
207	C₄×C₇	—	8	2	no	—
208	C₄×C₇	—	8	4	no	—
209	C₄×C₇	C₂×#107	1	4	yes	—
210	C₄×C₇	C₄×#41	1	1	yes	characteristic
211	C₄×C₇	#518	1	1	yes	characteristic
212	C₄×C₇	#554	1	2	yes	—
213	C₂⁴	C₂×C₇×D₄	1	2	yes	—
214	C₂×D₄	C₇×C₄⋊C₄	1	4	yes	—
215	C₂×D₄	C₇×D₈	1	4	yes	—
216	C₂×D₄	C₇×Q₁₆	1	4	yes	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	8	no	—
223	C₂²×C₄	—	4	8	no	—
224	C₂²×C₄	—	8	4	no	—
225	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
226	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
228	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	8	no	—
235	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
237	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	8	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	8	2	no	—
246	C₄⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
247	C₄⋊C₄	C₇×#41	1	4	yes	—
248	C₂²⋊C₄	—	4	8	no	—
249	C₂²⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
250	C₄²	—	4	4	no	—
251	C₄²	C₂²×C₄×C₇	1	2	yes	—
252	C₄²	C₂×C₇×Q₈	1	2	yes	—
253	C₂×C₇	—	2	8	no	—
254	C₂×C₇	#2349	1	1	yes	characteristic
255	C₂×C₇	C₂²×#107	1	1	yes	characteristic
256	C₂×C₇	#2912	1	1	yes	characteristic
257	C₂×C₇	#3294	1	2	yes	—
258	C₂×C₇	#3299	1	2	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	C₂×C₇×C₄⋊C₄	1	4	yes	—
262	C₂³	C₂×C₇×D₈	1	4	yes	—
263	C₂³	C₂×C₇×Q₁₆	1	4	yes	—
264	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
265	D₄	C₇×#107	1	16	yes	complemented
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	16	no	—
281	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
282	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
283	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
285	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
286	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
288	C₂×C₄	C₂×C₇×#41	1	2	yes	—
289	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	2	4	no	—
293	C₇	#22705	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	2	4	no	—
295	C₂²	—	2	8	no	—
296	C₂²	—	2	8	no	—
297	C₂²	—	2	4	no	—
298	C₂²	C₂×C₇×#107	1	8	yes	—
299	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
300	C₂²	C₇×#546	1	2	yes	—
301	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
302	C₂²	C₂²×C₇×Q₁₆	1	1	yes	characteristic
303	C₂²	C₇×#577	1	2	yes	—
304	C₄	—	2	2	no	—
305	C₄	—	2	4	no	—
306	C₄	—	4	4	no	—
307	C₄	—	8	4	no	—
308	C₄	—	8	2	no	—
309	C₄	C₂×C₇×#107	1	4	yes	—
310	C₄	C₄×C₇×#41	1	1	yes	characteristic
311	C₄	C₇×#518	1	1	yes	characteristic
312	C₄	C₇×#554	1	2	yes	—
313	C₂	—	2	8	no	—
314	C₂	C₇×#2349	1	1	yes	characteristic
315	C₂	C₂²×C₇×#107	1	1	yes	characteristic
316	C₂	C₇×#2912	1	1	yes	characteristic
317	C₂	C₇×#3294	1	2	yes	—
318	C₂	C₇×#3299	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₇×#1947	C₂	semidirect
4. C₇×#1852	C₂	semidirect
5. C₇×#1834	C₂	semidirect
6. C₇×#1808	C₂	semidirect
7. C₄×C₇×#107	C₂	semidirect
8. C₂×C₇×#340	C₂	semidirect
13. C₇×C₈×D₄	C₄	semidirect
29. #22705	C₇	direct (N × H)
34. C₂²×C₇×C₈	C₂×C₄	semidirect
56. C₇×C₄⋊C₈	C₂×C₄	semidirect
57. C₇×C₂²⋊C₈	C₂×C₄	semidirect
58. C₄×C₇×C₈	C₂×C₄	semidirect
63. #1947	C₂×C₇	semidirect
64. #1852	C₂×C₇	semidirect
65. #1834	C₂×C₇	semidirect
66. #1808	C₂×C₇	semidirect
67. C₄×#107	C₂×C₇	semidirect
68. C₂×#340	C₂×C₇	semidirect
113. C₈×D₄	C₄×C₇	semidirect
135. C₇×D₄	#107	semidirect
167. C₂²×C₈	C₂×C₄×C₇	semidirect
189. C₄⋊C₈	C₂×C₄×C₇	semidirect
190. C₂²⋊C₈	C₂×C₄×C₇	semidirect
191. C₄×C₈	C₂×C₄×C₇	semidirect
265. D₄	C₇×#107	semidirect
293. C₇	#22705	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×C₈×D₄	2
C₇×D₄×C₄⋊C₄	2
C₇×#1947	2
C₇×#1852	2
C₇×#1834	2
C₇×#1808	2
C₄×C₇×#107	2
C₂×C₇×#340	2
#22705	7

Subgroups — C7×#22705 #13199921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22705 #13199921