Subgroups — C₂×#11729179 #13506453

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#89873	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	#11729179	C₂	1	8	yes	maximal, direct product factor
5	C₂×#89307	C₂	1	4	yes	maximal, complemented
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×#6489	C₄	1	1	yes	characteristic
8	C₂²×#1136	C₄	1	1	yes	characteristic
9	C₂×C₇×M₃₂	C₄	1	2	yes	complemented
10	C₂×C₇×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₇×M₃₂	C₂²	1	2	yes	—
12	C₂×C₇×M₃₂	C₂²	1	4	yes	—
13	C₂²×C₇×C₁₆	C₄	1	2	yes	complemented
14	#89873	C₄	1	4	yes	—
15	#89873	C₂²	1	4	yes	—
16	C₂×#6389	C₂²	1	2	yes	—
17	C₂×C₈×Dic₇	C₄	1	2	yes	—
18	C₂×#6386	C₄	1	2	yes	complemented
19	C₂×#6386	C₄	1	4	yes	complemented
20	C₂×#6386	C₂²	1	2	yes	—
21	C₂×#6386	C₂²	1	4	yes	—
22	C₂²×C₇⋊C₁₆	C₄	1	2	yes	complemented
23	#89307	C₂²	1	16	yes	complemented
24	C₂×#2110	—	7	1	no	maximal, Sylow
25	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂×C₂²⋊Dic₇	—	2	1	no	—
37	#6489	—	2	2	no	—
38	#6489	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊Dic₇	—	2	1	no	—
40	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
41	C₂×#1136	—	2	1	no	—
42	C₂×#1136	—	2	1	no	—
43	C₂×#1136	—	2	2	no	—
44	C₂×#1136	C₂×C₄	1	4	yes	—
45	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
46	C₇×M₃₂	C₂×C₄	1	4	yes	—
47	C₇×M₃₂	C₂×C₄	1	8	yes	complemented
48	C₇×M₃₂	C₂³	1	4	yes	—
49	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
50	C₂×C₇×C₁₆	C₂×C₄	1	8	yes	complemented
51	#6389	C₂×C₄	1	4	yes	—
52	#6389	C₂³	1	4	yes	—
53	C₈×Dic₇	C₂×C₄	1	8	yes	—
54	#6386	C₂×C₄	1	4	yes	—
55	#6386	C₂×C₄	1	8	yes	complemented
56	#6386	C₂³	1	4	yes	—
57	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
58	C₂×C₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
59	C₂²×M₃₂	—	7	1	no	—
60	C₂²×M₃₂	D₇	1	1	yes	characteristic
61	C₂×#405	—	7	1	no	—
62	#2110	—	7	8	no	—
63	C₂×#347	—	7	4	no	—
64	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₄²	1	2	yes	—
68	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₇	M₁₆	1	2	yes	—
70	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₇	M₁₆	1	2	yes	—
73	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
75	C₂²×Dic₇	—	2	2	no	—
76	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
77	C₂×C₇×C₈	C₄²	1	2	yes	—
78	C₂×C₇×C₈	C₄²	1	2	yes	—
79	C₂×C₇×C₈	C₄²	1	2	yes	—
80	C₂×C₇×C₈	C₄²	1	2	yes	—
81	C₂×C₇×C₈	C₄²	1	2	yes	—
82	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
94	C₂²⋊Dic₇	—	2	4	no	—
95	C₄⋊Dic₇	—	2	4	no	—
96	C₄×Dic₇	—	2	2	no	—
97	C₄×Dic₇	C₂²×C₄	1	4	yes	—
98	#1136	—	2	4	no	—
99	#1136	—	2	4	no	—
100	C₂×C₇⋊C₈	—	2	2	no	—
101	C₂×C₇⋊C₈	—	2	2	no	—
102	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
103	C₇×C₁₆	C₂²×C₄	1	8	yes	—
104	C₇⋊C₁₆	C₂²×C₄	1	8	yes	—
105	C₂²×M₁₆	—	7	1	no	—
106	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
107	C₂×#117	—	7	1	no	—
108	C₂×M₃₂	—	7	2	no	—
109	C₂×M₃₂	—	7	4	no	—
110	C₂×M₃₂	—	7	2	no	—
111	C₂×M₃₂	—	7	4	no	—
112	C₂×M₃₂	Dic₇	1	2	yes	complemented
113	C₂×M₃₂	Dic₇	1	4	yes	complemented
114	C₂×M₃₂	C₂×D₇	1	2	yes	—
115	C₂×M₃₂	C₂×D₇	1	4	yes	—
116	C₂²×C₁₆	—	7	2	no	—
117	C₂²×C₁₆	Dic₇	1	2	yes	complemented
118	#405	—	7	4	no	—
119	#405	—	7	4	no	—
120	C₂×#97	—	7	2	no	—
121	C₂×C₄×C₈	—	7	2	no	—
122	#347	—	7	16	no	—
123	C₂³×C₇	—	4	2	no	—
124	C₂³×C₇	#97	1	1	yes	characteristic
125	C₂³×C₇	#97	1	1	yes	characteristic
126	C₂³×C₇	#97	1	2	yes	—
127	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂³×C₇	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₇	#97	1	2	yes	—
130	C₂×C₄×C₇	#97	1	2	yes	—
131	C₂×C₄×C₇	#97	1	2	yes	—
132	C₂×C₄×C₇	#97	1	2	yes	—
133	C₂×C₄×C₇	#97	1	2	yes	—
134	C₂×C₄×C₇	#97	1	1	yes	characteristic
135	C₂×C₄×C₇	#97	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
143	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
144	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×Dic₇	—	2	4	no	—
148	C₂×Dic₇	—	4	4	no	—
149	C₇×C₈	C₂×C₄²	1	2	yes	—
150	C₇×C₈	C₂×C₄²	1	1	yes	characteristic
151	C₇×C₈	C₂×C₄²	1	1	yes	characteristic
152	C₇×C₈	C₂×C₄²	1	2	yes	—
153	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
154	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
155	C₇⋊C₈	—	2	4	no	—
156	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
157	C₂×M₁₆	—	7	4	no	—
158	C₂×M₁₆	—	14	1	no	—
159	C₂×M₁₆	—	14	1	no	—
160	C₂×M₁₆	—	14	2	no	—
161	C₂²×C₈	—	7	2	no	—
162	C₂²×C₈	C₄×D₇	1	2	yes	—
163	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
165	C₂²×C₈	C₄×D₇	1	2	yes	—
166	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
168	C₂²×C₈	C₂×Dic₇	1	2	yes	—
169	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₇	1	2	yes	—
172	#117	—	7	4	no	—
173	#117	—	14	2	no	—
174	C₂×C₄⋊C₄	—	14	1	no	—
175	C₂×C₂²⋊C₄	—	14	1	no	—
176	C₂×C₄²	—	7	2	no	—
177	M₃₂	—	7	4	no	—
178	M₃₂	—	7	8	no	—
179	M₃₂	—	7	4	no	—
180	M₃₂	C₂×Dic₇	1	4	yes	—
181	M₃₂	C₂×Dic₇	1	8	yes	complemented
182	M₃₂	C₂²×D₇	1	4	yes	—
183	C₂×C₁₆	—	7	4	no	—
184	C₂×C₁₆	—	7	8	no	—
185	C₂×C₁₆	C₂×Dic₇	1	4	yes	—
186	C₂×C₁₆	C₂×Dic₇	1	8	yes	complemented
187	#97	—	7	4	no	—
188	#97	—	7	4	no	—
189	C₄×C₈	—	7	8	no	—
190	C₂²×C₇	—	2	2	no	—
191	C₂²×C₇	—	2	2	no	—
192	C₂²×C₇	—	2	2	no	—
193	C₂²×C₇	—	4	2	no	—
194	C₂²×C₇	—	4	2	no	—
195	C₂²×C₇	C₂×#97	1	1	yes	characteristic
196	C₂²×C₇	C₂×#97	1	2	yes	—
197	C₂²×C₇	C₂×#97	1	2	yes	—
198	C₂²×C₇	C₂×#97	1	1	yes	characteristic
199	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
200	C₄×C₇	C₂×#97	1	1	yes	characteristic
201	C₄×C₇	C₂×#97	1	1	yes	characteristic
202	C₄×C₇	C₂×#97	1	2	yes	—
203	C₄×C₇	C₂×#97	1	2	yes	—
204	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
205	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
206	Dic₇	—	4	4	no	—
207	C₂⁴	#1131	1	1	yes	characteristic
208	C₂²×C₄	—	14	2	no	—
209	C₂²×C₄	#1131	1	1	yes	characteristic
210	C₂²×C₄	#1131	1	1	yes	characteristic
211	C₂²×C₄	#1131	1	2	yes	—
212	C₂²×C₄	#1131	1	1	yes	characteristic
213	C₂²×C₄	#1131	1	2	yes	—
214	C₂²×C₄	C₄×Dic₇	1	2	yes	—
215	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
216	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
219	M₁₆	—	14	4	no	—
220	M₁₆	—	14	4	no	—
221	C₂×C₈	—	7	4	no	—
222	C₂×C₈	—	14	2	no	—
223	C₂×C₈	—	14	2	no	—
224	C₂×C₈	C₄×Dic₇	1	2	yes	—
225	C₂×C₈	C₄×Dic₇	1	2	yes	—
226	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
227	C₂×C₈	C₄×Dic₇	1	2	yes	—
228	C₂×C₈	C₄×Dic₇	1	2	yes	—
229	C₂×C₈	C₄×Dic₇	1	2	yes	—
230	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
237	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×Dic₇	1	2	yes	—
239	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₇	1	2	yes	—
241	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
242	C₄⋊C₄	—	14	4	no	—
243	C₂²⋊C₄	—	14	4	no	—
244	C₄²	—	7	4	no	—
245	C₄²	—	14	2	no	—
246	C₁₆	—	7	8	no	—
247	C₁₆	C₂²×Dic₇	1	8	yes	—
248	C₂×C₇	—	2	1	no	—
249	C₂×C₇	—	2	1	no	—
250	C₂×C₇	—	2	2	no	—
251	C₂×C₇	—	2	2	no	—
252	C₂×C₇	#2110	1	2	yes	complemented
253	C₂×C₇	C₂²×#97	1	1	yes	characteristic
254	C₂³	—	4	2	no	—
255	C₂³	#6389	1	1	yes	characteristic
256	C₂³	#6389	1	1	yes	characteristic
257	C₂³	#6389	1	2	yes	—
258	C₂³	C₂×#1131	1	1	yes	characteristic
259	C₂³	C₂×#1131	1	2	yes	—
260	C₂×C₄	—	14	4	no	—
261	C₂×C₄	—	28	4	no	—
262	C₂×C₄	#6389	1	2	yes	—
263	C₂×C₄	#6389	1	2	yes	—
264	C₂×C₄	#6389	1	2	yes	—
265	C₂×C₄	#6389	1	2	yes	—
266	C₂×C₄	#6389	1	2	yes	—
267	C₂×C₄	#6389	1	1	yes	characteristic
268	C₂×C₄	#6389	1	1	yes	characteristic
269	C₂×C₄	C₂×#1131	1	1	yes	characteristic
270	C₂×C₄	C₂×#1131	1	2	yes	—
271	C₂×C₄	C₂×#1131	1	1	yes	characteristic
272	C₂×C₄	C₂×#1131	1	1	yes	characteristic
273	C₂×C₄	C₂×#1131	1	2	yes	—
274	C₂×C₄	C₂×#1131	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
280	C₈	—	14	4	no	—
281	C₈	C₂×C₄×Dic₇	1	2	yes	—
282	C₈	C₂×C₄×Dic₇	1	1	yes	characteristic
283	C₈	C₂×C₄×Dic₇	1	1	yes	characteristic
284	C₈	C₂×C₄×Dic₇	1	2	yes	—
285	C₈	C₂²×C₄×D₇	1	1	yes	characteristic
286	C₈	C₂³×Dic₇	1	1	yes	characteristic
287	C₇	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×#6389	1	1	yes	characteristic
294	C₂²	C₂×#6389	1	2	yes	—
295	C₂²	C₂×#6389	1	2	yes	—
296	C₂²	C₂×#6389	1	1	yes	characteristic
297	C₂²	C₂²×#1131	1	1	yes	characteristic
298	C₄	—	28	4	no	—
299	C₄	C₂×#6389	1	1	yes	characteristic
300	C₄	C₂×#6389	1	1	yes	characteristic
301	C₄	C₂×#6389	1	2	yes	—
302	C₄	C₂×#6389	1	2	yes	—
303	C₄	C₂²×#1131	1	1	yes	characteristic
304	C₄	C₂²×C₄×Dic₇	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#11729179	1	2	yes	direct product factor
310	C₂	C₂²×#6389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇
commutator	C₄×C₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂²×C₇×M₃₂	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂²×M₃₂	D₇
hypocenter	C₇	C₂×#2110
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. #11729179	C₂	direct (N × H)
5. C₂×#89307	C₂	semidirect
9. C₂×C₇×M₃₂	C₄	semidirect
10. C₂×C₇×M₃₂	C₄	semidirect
13. C₂²×C₇×C₁₆	C₄	semidirect
18. C₂×#6386	C₄	semidirect
19. C₂×#6386	C₄	semidirect
22. C₂²×C₇⋊C₁₆	C₄	semidirect
23. #89307	C₂²	semidirect
47. C₇×M₃₂	C₂×C₄	semidirect
50. C₂×C₇×C₁₆	C₂×C₄	semidirect
55. #6386	C₂×C₄	semidirect
58. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
112. C₂×M₃₂	Dic₇	semidirect
113. C₂×M₃₂	Dic₇	semidirect
117. C₂²×C₁₆	Dic₇	semidirect
181. M₃₂	C₂×Dic₇	semidirect
186. C₂×C₁₆	C₂×Dic₇	semidirect
252. C₂×C₇	#2110	semidirect
287. C₇	C₂×#2110	semidirect
309. C₂	#11729179	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×M₃₂	2
C₂×#89873	2
C₂²×#6386	2
#11729179	2
C₂×#89307	2
C₂×#2110	7

Subgroups — C2×#11729179 #13506453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729179 #13506453