Subgroups — #14907187

Order 2816.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13081752	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622313	C₂	1	2	yes	maximal, complemented
3	#13070503	C₂	1	4	yes	maximal, complemented
4	#13070498	C₂	1	4	yes	maximal, complemented
5	#13070411	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070345	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#66957	C₂²	1	2	yes	—
9	C₂×#66954	—	2	2	no	—
10	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66883	C₂²	1	2	yes	—
14	C₂×#66869	—	2	2	no	—
15	#10622522	C₄	1	4	yes	complemented
16	#10622521	C₄	1	4	yes	complemented
17	#10622517	C₂²	1	4	yes	—
18	#10622313	—	2	2	no	—
19	#10622313	—	2	2	no	—
20	#10622313	C₂²	1	4	yes	complemented
21	#10622313	C₂²	1	4	yes	complemented
22	C₁₁×#346	C₂²	1	2	yes	complemented
23	C₄×C₁₁×C₁₆	C₂²	1	2	yes	complemented
24	#10622148	C₂²	1	8	yes	complemented
25	#10622144	C₂²	1	4	yes	complemented
26	C₂×#4339	—	2	2	no	—
27	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
30	C₂×#4318	—	2	2	no	—
31	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
32	#66957	—	2	2	no	—
33	#66957	—	2	2	no	—
34	#66957	C₂×C₄	1	4	yes	—
35	#66957	C₂³	1	4	yes	—
36	C₂×D₈₈	C₂×C₄	1	2	yes	—
37	C₂×#4314	—	2	2	no	—
38	#66954	—	2	4	no	—
39	#66954	—	2	4	no	—
40	#66954	—	4	2	no	—
41	#66954	—	4	2	no	—
42	C₂×#4313	—	4	2	no	—
43	C₂×C₈×D₁₁	—	4	2	no	—
44	C₁₁×M₃₂	—	2	2	no	—
45	C₁₁×M₃₂	C₂×C₄	1	4	yes	—
46	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
48	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
52	#66883	—	2	2	no	—
53	#66883	C₂³	1	4	yes	—
54	#66869	—	2	4	no	—
55	#66869	—	2	4	no	—
56	Dic₈₈	C₂×C₄	1	4	yes	complemented
57	#66864	C₂×C₄	1	8	yes	complemented
58	D₁₇₆	C₂×C₄	1	4	yes	complemented
59	#33463	—	11	1	no	maximal, Sylow
60	C₂²×C₄×C₁₁	—	2	1	no	—
61	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₁	—	4	2	no	—
63	#4339	—	2	4	no	—
64	#4339	—	2	4	no	—
65	#4339	—	4	2	no	—
66	#4339	—	4	2	no	—
67	C₂×D₄₄	—	2	2	no	—
68	C₂×C₄×D₁₁	—	4	2	no	—
69	C₂×Dic₂₂	—	2	2	no	—
70	C₂×C₈×C₁₁	D₈	1	2	yes	—
71	C₂×C₈×C₁₁	D₈	1	2	yes	—
72	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₁	#41	1	2	yes	—
77	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
78	C₄²×C₁₁	—	2	1	no	—
79	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
81	#4318	—	2	4	no	—
82	#4318	—	4	2	no	—
83	C₂×C₁₁⋊C₈	—	4	2	no	—
84	Dic₄₄	—	2	2	no	—
85	Dic₄₄	C₂²×C₄	1	4	yes	—
86	D₈₈	—	2	2	no	—
87	D₈₈	C₂²×C₄	1	4	yes	—
88	#4314	—	2	4	no	—
89	#4314	—	2	4	no	—
90	#4313	—	4	4	no	—
91	#4313	—	4	4	no	—
92	C₈×D₁₁	—	4	4	no	—
93	C₈×D₁₁	—	4	4	no	—
94	C₁₁×C₁₆	—	2	2	no	—
95	C₁₁×C₁₆	C₂²×C₄	1	4	yes	—
96	#3414	—	11	1	no	—
97	C₂×#443	—	11	2	no	—
98	#2179	—	11	4	no	—
99	#2178	—	11	4	no	—
100	#2161	—	11	1	no	—
101	#2143	—	11	2	no	—
102	C₄×M₃₂	D₁₁	1	1	yes	characteristic, complemented
103	C₂³×C₁₁	#135	1	1	yes	characteristic
104	C₂²×D₁₁	—	4	2	no	—
105	C₂×C₄×C₁₁	—	2	1	no	—
106	C₂×C₄×C₁₁	—	2	1	no	—
107	C₂×C₄×C₁₁	—	4	2	no	—
108	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
112	C₂²⋊D₁₁	—	4	4	no	—
113	C₂²⋊D₁₁	—	4	4	no	—
114	C₂×Dic₁₁	—	4	2	no	—
115	D₄₄	—	2	4	no	—
116	D₄₄	—	4	2	no	—
117	C₄×D₁₁	—	4	4	no	—
118	C₄×D₁₁	—	4	4	no	—
119	Dic₂₂	—	2	4	no	—
120	Dic₂₂	—	4	2	no	—
121	C₈×C₁₁	C₄×D₄	1	2	yes	—
122	C₈×C₁₁	C₂×D₈	1	2	yes	—
123	C₈×C₁₁	C₂×D₈	1	1	yes	characteristic
124	C₈×C₁₁	C₂×D₈	1	1	yes	characteristic
125	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
127	C₁₁⋊C₈	—	4	4	no	—
128	C₂×#135	—	11	2	no	—
129	C₂×#131	—	22	2	no	—
130	#510	—	11	4	no	—
131	#509	—	11	4	no	—
132	#508	—	11	4	no	—
133	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
134	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
135	#443	—	11	4	no	—
136	#443	—	11	4	no	—
137	#443	—	22	2	no	—
138	#443	—	22	2	no	—
139	C₂×#108	—	11	2	no	—
140	C₂×#104	—	22	2	no	—
141	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
142	#368	—	11	4	no	—
143	#359	—	11	8	no	—
144	#346	C₂×D₁₁	1	2	yes	complemented
145	C₄×C₁₆	C₂×D₁₁	1	2	yes	complemented
146	C₂²×C₁₁	—	4	1	no	—
147	C₂²×C₁₁	C₄×D₈	1	2	yes	—
148	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
149	C₂×D₁₁	—	4	4	no	—
150	C₂×D₁₁	—	8	4	no	—
151	C₄×C₁₁	—	2	2	no	—
152	C₄×C₁₁	—	4	1	no	—
153	C₄×C₁₁	C₄×D₈	1	2	yes	—
154	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
156	Dic₁₁	—	4	4	no	—
157	C₂×#41	—	22	2	no	—
158	#135	—	11	4	no	—
159	#135	—	11	4	no	—
160	#135	—	22	2	no	—
161	#135	—	22	2	no	—
162	C₂×Q₁₆	—	11	2	no	—
163	C₂×SD₁₆	—	22	2	no	—
164	C₂×D₈	—	11	2	no	—
165	#131	—	22	4	no	—
166	#131	—	22	4	no	—
167	#131	—	44	2	no	—
168	#131	—	44	2	no	—
169	C₂×M₁₆	—	22	2	no	—
170	C₂×M₁₆	—	44	2	no	—
171	C₂²×C₈	—	44	2	no	—
172	C₂²×C₈	D₄₄	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
174	C₂×C₄²	D₄₄	1	1	yes	characteristic
175	Q₃₂	—	11	4	no	—
176	SD₃₂	—	11	8	no	—
177	D₁₆	—	11	4	no	—
178	M₃₂	—	2	2	no	—
179	M₃₂	C₄×D₁₁	1	4	yes	—
180	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
181	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
182	#108	—	11	4	no	—
183	#108	—	22	2	no	—
184	#104	—	22	4	no	—
185	#104	—	22	4	no	—
186	C₄×C₈	D₄₄	1	1	yes	characteristic
187	C₄×C₈	D₄₄	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
190	C₂×C₁₁	—	2	1	no	—
191	C₂×C₁₁	—	2	2	no	—
192	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
193	D₁₁	—	8	4	no	—
194	#41	—	22	4	no	—
195	#41	—	22	4	no	—
196	#41	—	44	2	no	—
197	#41	—	44	2	no	—
198	C₂×Q₈	—	22	2	no	—
199	C₂×D₄	—	22	2	no	—
200	C₂×D₄	—	44	2	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	44	2	no	—
203	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
204	Q₁₆	—	11	4	no	—
205	Q₁₆	—	22	2	no	—
206	SD₁₆	—	22	4	no	—
207	SD₁₆	—	22	4	no	—
208	D₈	—	11	4	no	—
209	D₈	—	22	2	no	—
210	M₁₆	—	22	4	no	—
211	M₁₆	—	44	2	no	—
212	M₁₆	—	44	4	no	—
213	M₁₆	—	44	4	no	—
214	C₂×C₈	—	44	2	no	—
215	C₂×C₈	—	44	4	no	—
216	C₂×C₈	—	44	4	no	—
217	C₂×C₈	D₈₈	1	2	yes	—
218	C₂×C₈	D₈₈	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
220	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
223	C₂×C₈	#4339	1	2	yes	—
224	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
225	C₄²	—	2	1	no	—
226	C₄²	C₂×D₄₄	1	1	yes	characteristic
227	C₄²	C₂×D₄₄	1	1	yes	characteristic
228	C₁₆	—	2	2	no	—
229	C₁₆	C₂×C₄×D₁₁	1	4	yes	—
230	C₁₁	#33463	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	44	2	no	—
232	C₂³	#66957	1	1	yes	characteristic
233	Q₈	—	22	4	no	—
234	Q₈	—	44	2	no	—
235	D₄	—	22	4	no	—
236	D₄	—	44	4	no	—
237	D₄	—	44	4	no	—
238	D₄	—	44	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	44	2	no	—
243	C₂×C₄	—	44	4	no	—
244	C₂×C₄	—	44	4	no	—
245	C₂×C₄	C₄×D₄₄	1	2	yes	—
246	C₂×C₄	C₂×D₈₈	1	2	yes	—
247	C₂×C₄	#66957	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
249	C₈	—	44	4	no	—
250	C₈	C₄×D₄₄	1	2	yes	—
251	C₈	C₂×D₈₈	1	2	yes	—
252	C₈	C₂×D₈₈	1	1	yes	characteristic
253	C₈	C₂×D₈₈	1	1	yes	characteristic
254	C₈	C₂²×C₄×D₁₁	1	1	yes	characteristic
255	C₈	C₂×#4339	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	44	4	no	—
258	C₂²	—	88	4	no	—
259	C₂²	C₄×D₈₈	1	2	yes	—
260	C₂²	C₂×#66957	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	44	4	no	—
264	C₄	C₄×D₈₈	1	2	yes	—
265	C₄	C₂×C₄×D₄₄	1	1	yes	characteristic
266	C₄	C₂²×D₈₈	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	88	4	no	—
270	C₂	C₂×C₄×D₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₈₈
commutator	C₈×C₁₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₄×C₁₁×M₃₂	C₂
socle	C₂×C₁₁	C₂×C₄×D₈
prime radical	#13081752	C₂
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₄×M₃₂	D₁₁
hypocenter	C₁₁	#33463
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#10622313	C₂	semidirect
3. #13070503	C₂	semidirect
4. #13070498	C₂	semidirect
5. #13070411	C₂	semidirect
6. #13070345	C₂	semidirect
7. C₄×C₁₁×M₃₂	C₂	semidirect
15. #10622522	C₄	semidirect
16. #10622521	C₄	semidirect
20. #10622313	C₂²	semidirect
21. #10622313	C₂²	semidirect
22. C₁₁×#346	C₂²	semidirect
23. C₄×C₁₁×C₁₆	C₂²	semidirect
24. #10622148	C₂²	semidirect
25. #10622144	C₂²	semidirect
56. Dic₈₈	C₂×C₄	semidirect
57. #66864	C₂×C₄	semidirect
58. D₁₇₆	C₂×C₄	semidirect
102. C₄×M₃₂	D₁₁	semidirect
144. #346	C₂×D₁₁	semidirect
145. C₄×C₁₆	C₂×D₁₁	semidirect
230. C₁₁	#33463	semidirect

Maximal subgroups

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

subgroup	index
#13081752	2
C₂×#10622313	2
#13070503	2
#13070498	2
#13070411	2
#13070345	2
C₄×C₁₁×M₃₂	2
#33463	11