Subgroups — C₇×#22669 #13199885

Order 1792.

862 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₄×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₇×#436	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#430	C₂	1	2	yes	maximal, complemented
4	C₇×#1950	C₂	1	2	yes	maximal, complemented
5	C₇×#1855	C₂	1	4	yes	maximal, complemented
6	C₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1811	C₂	1	2	yes	maximal, complemented
8	C₄×C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₇×#436	—	2	2	no	—
13	C₇×#436	C₄	1	4	yes	—
14	C₇×#436	C₄	1	4	yes	—
15	C₇×#435	—	2	4	no	—
16	C₇×C₈×D₄	—	2	2	no	—
17	C₇×#430	—	2	8	no	—
18	C₂×C₇×#108	C₂²	1	2	yes	—
19	C₂×C₇×#108	C₂²	1	4	yes	—
20	C₂×C₇×#108	C₂²	1	1	yes	characteristic
21	C₂×C₇×#108	C₂²	1	2	yes	—
22	C₂×C₇×#108	C₂²	1	1	yes	characteristic
23	C₇×#423	C₂²	1	2	yes	—
24	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₇×#407	C₂²	1	4	yes	—
26	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₄×C₇×M₁₆	C₂²	1	2	yes	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#341	C₂²	1	2	yes	—
30	C₇×#341	C₂²	1	4	yes	—
31	#22669	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₇×D₄	Q₈	1	1	yes	characteristic
33	C₂³×C₄×C₇	C₂³	1	2	yes	—
34	C₂×C₇×M₁₆	—	2	4	no	—
35	C₂×C₇×M₁₆	—	2	4	no	—
36	C₂×C₇×M₁₆	—	2	4	no	—
37	C₂×C₇×M₁₆	—	2	8	no	—
38	C₂×C₇×M₁₆	—	4	4	no	—
39	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
40	C₂×C₇×M₁₆	C₂³	1	2	yes	—
41	C₂×C₇×M₁₆	C₂³	1	4	yes	—
42	C₂×C₇×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₇×C₈	—	2	4	no	—
44	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₇×C₈	C₂³	1	2	yes	—
46	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
47	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
48	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
49	C₄×C₇×D₄	D₄	1	2	yes	—
50	C₄×C₇×D₄	D₄	1	2	yes	—
51	C₂×C₇×C₄⋊C₄	Q₈	1	1	yes	characteristic
52	C₂×C₇×C₂²⋊C₄	Q₈	1	2	yes	—
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₇×#108	—	2	8	no	—
55	C₇×#108	—	2	4	no	—
56	C₇×#108	—	2	2	no	—
57	C₇×#108	—	4	2	no	—
58	C₇×#108	D₄	1	4	yes	complemented
59	C₇×C₄⋊C₈	—	2	2	no	—
60	C₇×C₄⋊C₈	—	2	2	no	—
61	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
63	C₇×C₂²⋊C₈	—	2	4	no	—
64	C₇×C₂²⋊C₈	—	2	4	no	—
65	C₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
66	C₇×#97	—	2	2	no	—
67	C₄×C₇×C₈	—	2	2	no	—
68	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
69	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
70	D₄×M₁₆	C₂×C₇	1	2	yes	—
71	C₂×#436	C₂×C₇	1	1	yes	characteristic
72	C₂×#430	C₂×C₇	1	2	yes	complemented
73	#1950	C₂×C₇	1	2	yes	complemented
74	#1855	C₂×C₇	1	4	yes	complemented
75	#1848	C₂×C₇	1	1	yes	characteristic, complemented
76	#1811	C₂×C₇	1	2	yes	complemented
77	C₄×#108	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂⁴×C₇	C₂×Q₈	1	2	yes	—
79	C₂×C₇×D₄	—	4	2	no	—
80	C₂×C₇×D₄	C₄⋊C₄	1	2	yes	—
81	C₂×C₇×D₄	C₄⋊C₄	1	2	yes	—
82	C₂²×C₄×C₇	—	2	2	no	—
83	C₂²×C₄×C₇	—	2	2	no	—
84	C₂²×C₄×C₇	—	2	2	no	—
85	C₂²×C₄×C₇	—	2	2	no	—
86	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
91	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
92	C₇×M₁₆	—	2	2	no	—
93	C₇×M₁₆	—	2	16	no	—
94	C₇×M₁₆	—	2	4	no	—
95	C₇×M₁₆	—	2	2	no	—
96	C₇×M₁₆	—	2	4	no	—
97	C₇×M₁₆	—	2	2	no	—
98	C₇×M₁₆	—	4	8	no	—
99	C₇×M₁₆	C₂×D₄	1	4	yes	—
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	—	2	2	no	—
103	C₂×C₇×C₈	—	2	1	no	—
104	C₂×C₇×C₈	—	2	1	no	—
105	C₂×C₇×C₈	—	2	2	no	—
106	C₂×C₇×C₈	—	2	4	no	—
107	C₂×C₇×C₈	—	4	8	no	—
108	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
110	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₇×C₈	#41	1	2	yes	—
113	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
114	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
115	C₇×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
116	C₇×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
117	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×M₁₆	C₂²×C₇	1	4	yes	—
122	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
123	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
124	#436	—	2	2	no	—
125	#436	C₄×C₇	1	4	yes	—
126	#436	C₄×C₇	1	4	yes	—
127	#435	—	2	4	no	—
128	C₈×D₄	—	2	2	no	—
129	#430	—	2	8	no	—
130	C₂×#108	C₂²×C₇	1	1	yes	characteristic
131	C₂×#108	C₂²×C₇	1	2	yes	—
132	C₂×#108	C₂²×C₇	1	4	yes	—
133	C₂×#108	C₂²×C₇	1	2	yes	—
134	C₂×#108	C₂²×C₇	1	1	yes	characteristic
135	#423	C₂²×C₇	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
137	#407	C₂²×C₇	1	4	yes	—
138	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
139	C₄×M₁₆	C₂²×C₇	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
141	#341	C₂²×C₇	1	4	yes	—
142	#341	C₂²×C₇	1	2	yes	—
143	C₂³×C₇	—	2	2	no	—
144	C₂³×C₇	—	2	2	no	—
145	C₂³×C₇	—	4	2	no	—
146	C₂³×C₇	—	4	2	no	—
147	C₂³×C₇	C₂×C₄⋊C₄	1	4	yes	—
148	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
149	C₇×D₄	—	2	4	no	—
150	C₇×D₄	—	4	2	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	4	no	—
154	C₂×C₄×C₇	—	2	4	no	—
155	C₂×C₄×C₇	—	2	4	no	—
156	C₂×C₄×C₇	—	2	4	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	4	1	no	—
159	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
163	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
170	C₇×C₈	—	2	4	no	—
171	C₇×C₈	—	2	2	no	—
172	C₇×C₈	—	2	4	no	—
173	C₇×C₈	—	4	4	no	—
174	C₇×C₈	—	4	2	no	—
175	C₇×C₈	C₄×D₄	1	4	yes	—
176	C₂²×D₄	C₇×Q₈	1	1	yes	characteristic
177	C₂³×C₄	C₂³×C₇	1	2	yes	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	8	no	—
182	C₂×M₁₆	—	4	4	no	—
183	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	—
184	C₂×M₁₆	C₂³×C₇	1	4	yes	—
185	C₂×M₁₆	C₂³×C₇	1	2	yes	—
186	C₂×M₁₆	C₂³×C₇	1	2	yes	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	C₂³×C₇	1	2	yes	—
189	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
191	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
192	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
193	C₄×D₄	C₇×D₄	1	2	yes	—
194	C₄×D₄	C₇×D₄	1	2	yes	—
195	C₂×C₄⋊C₄	C₇×Q₈	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	C₇×Q₈	1	2	yes	—
197	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
198	#108	—	2	8	no	—
199	#108	—	2	4	no	—
200	#108	—	2	2	no	—
201	#108	—	4	2	no	—
202	#108	C₇×D₄	1	4	yes	complemented
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
206	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
207	C₂²⋊C₈	—	2	4	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	C₂²⋊C₈	C₂×C₄×C₇	1	8	yes	—
210	#97	—	2	2	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
213	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
214	C₂²×C₇	—	2	1	no	—
215	C₂²×C₇	—	2	1	no	—
216	C₂²×C₇	—	2	4	no	—
217	C₂²×C₇	—	2	4	no	—
218	C₂²×C₇	—	2	4	no	—
219	C₂²×C₇	—	8	2	no	—
220	C₂²×C₇	—	8	2	no	—
221	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
223	C₂²×C₇	#554	1	1	yes	characteristic
224	C₄×C₇	—	2	4	no	—
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	#430	1	2	yes	complemented
229	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₄×C₇	#520	1	1	yes	characteristic
231	C₄×C₇	D₄²	1	1	yes	characteristic
232	C₄×C₇	#547	1	1	yes	characteristic
233	C₂⁴	C₂×C₇×Q₈	1	2	yes	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	C₇×C₄⋊C₄	1	2	yes	—
236	C₂×D₄	C₇×C₄⋊C₄	1	2	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
242	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
243	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
245	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
246	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
247	M₁₆	—	2	2	no	—
248	M₁₆	—	2	16	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	—	2	4	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	8	no	—
254	M₁₆	C₂×C₇×D₄	1	4	yes	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	4	8	no	—
263	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
264	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
265	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
266	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
267	C₂×C₈	C₇×#41	1	2	yes	—
268	C₄⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
269	C₄⋊C₄	C₇×C₄⋊C₄	1	2	yes	—
270	C₂²⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
271	C₂²⋊C₄	C₇×C₄⋊C₄	1	4	yes	—
272	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
273	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
274	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
275	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
276	C₂×C₇	—	2	1	no	—
277	C₂×C₇	—	2	1	no	—
278	C₂×C₇	—	4	4	no	—
279	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₇	C₂×#430	1	1	yes	characteristic
281	C₂×C₇	#2916	1	1	yes	characteristic
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	C₂×C₇×C₄⋊C₄	1	4	yes	—
287	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
288	D₄	—	2	4	no	—
289	D₄	—	4	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
302	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
306	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
307	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
308	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
309	C₈	—	2	4	no	—
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	—	4	4	no	—
313	C₈	—	4	2	no	—
314	C₈	C₄×C₇×D₄	1	4	yes	—
315	C₇	#22669	1	1	yes	characteristic, direct product factor, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
324	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
325	C₂²	C₇×#554	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	4	no	—
330	C₄	C₇×#430	1	2	yes	complemented
331	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
332	C₄	C₇×#520	1	1	yes	characteristic
333	C₄	C₇×D₄²	1	1	yes	characteristic
334	C₄	C₇×#547	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	4	4	no	—
338	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
339	C₂	C₂×C₇×#430	1	1	yes	characteristic
340	C₂	C₇×#2916	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×C₄⋊C₄
prime radical	C₂²×C₇×D₄	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×#430	C₂	semidirect
4. C₇×#1950	C₂	semidirect
5. C₇×#1855	C₂	semidirect
6. C₇×#1848	C₂	semidirect
7. C₇×#1811	C₂	semidirect
8. C₄×C₇×#108	C₂	semidirect
31. #22669	C₇	direct (N × H)
58. C₇×#108	D₄	semidirect
72. C₂×#430	C₂×C₇	semidirect
73. #1950	C₂×C₇	semidirect
74. #1855	C₂×C₇	semidirect
75. #1848	C₂×C₇	semidirect
76. #1811	C₂×C₇	semidirect
77. C₄×#108	C₂×C₇	semidirect
202. #108	C₇×D₄	semidirect
228. C₄×C₇	#430	semidirect
315. C₇	#22669	direct (N × H)
330. C₄	C₇×#430	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₄×M₁₆	2
C₂×C₇×#436	2
C₂×C₇×#430	2
C₇×#1950	2
C₇×#1855	2
C₇×#1848	2
C₇×#1811	2
C₄×C₇×#108	2
#22669	7

Subgroups — C7×#22669 #13199885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22669 #13199885