Subgroups — #14672312

Order 2816.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622292	C₂	1	1	yes	maximal, characteristic
2	#13070372	C₂	1	1	yes	maximal, characteristic
3	#13066699	C₂	1	4	yes	maximal
4	C₄×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₂₂	C₄	1	1	yes	characteristic
6	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×#66879	C₂²	1	2	yes	—
9	C₂×Dic₁₁⋊C₈	C₄	1	2	yes	—
10	C₂×#66876	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	1	yes	characteristic
12	#10622292	—	2	1	no	—
13	#10622292	—	2	1	no	—
14	#10622292	—	2	1	no	—
15	#10622292	—	2	1	no	—
16	C₁₁×#346	—	2	1	no	—
17	C₄×C₁₁×C₁₆	—	2	1	no	—
18	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
19	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
20	C₂²×Dic₂₂	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₁	C₈	1	1	yes	characteristic
25	C₂×Dic₁₁⋊C₄	C₈	1	2	yes	—
26	C₂×C₄×Dic₁₁	C₂×C₄	1	2	yes	—
27	C₂²×C₁₁⋊C₈	C₂×C₄	1	2	yes	—
28	C₄×Dic₂₂	—	2	1	no	—
29	C₄×Dic₂₂	—	2	1	no	—
30	C₄×Dic₂₂	—	2	1	no	—
31	C₄×Dic₂₂	—	2	1	no	—
32	C₁₁×M₃₂	—	2	2	no	—
33	C₁₁×M₃₂	Q₈	1	4	yes	complemented
34	C₂×C₁₁×C₁₆	—	2	2	no	—
35	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
39	#66879	—	2	2	no	—
40	#66879	—	2	2	no	—
41	Dic₁₁⋊C₈	—	2	2	no	—
42	Dic₁₁⋊C₈	—	2	2	no	—
43	#66876	—	2	4	no	—
44	C₂×C₁₁⋊C₁₆	—	2	2	no	—
45	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
46	C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
47	#13268	—	11	1	no	maximal, Sylow
48	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
52	C₂²×Dic₁₁	C₂×C₈	1	2	yes	—
53	C₂×Dic₂₂	—	2	1	no	—
54	C₂×Dic₂₂	—	2	1	no	—
55	C₂×Dic₂₂	—	4	2	no	—
56	C₂×C₈×C₁₁	—	2	2	no	—
57	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₁₁	#41	1	2	yes	—
63	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
66	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
67	C₄⋊Dic₁₁	—	2	1	no	—
68	C₄⋊Dic₁₁	—	2	1	no	—
69	Dic₁₁⋊C₄	—	2	2	no	—
70	Dic₁₁⋊C₄	—	2	2	no	—
71	C₄×Dic₁₁	—	2	2	no	—
72	C₄×Dic₁₁	—	2	2	no	—
73	C₂×C₁₁⋊C₈	—	2	2	no	—
74	C₂×C₁₁⋊C₈	—	2	4	no	—
75	C₁₁×C₁₆	—	2	4	no	—
76	C₁₁⋊C₁₆	—	4	2	no	—
77	C₂×#446	—	11	1	no	—
78	#2150	—	11	1	no	—
79	C₄×M₃₂	D₁₁	1	1	yes	characteristic
80	#1377	—	11	4	no	—
81	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₁	—	2	2	no	—
83	C₂×C₄×C₁₁	—	4	1	no	—
84	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
90	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₁	#131	1	2	yes	—
92	C₂×Dic₁₁	—	2	2	no	—
93	C₂×Dic₁₁	—	2	2	no	—
94	C₂×Dic₁₁	—	4	2	no	—
95	C₂×Dic₁₁	—	4	2	no	—
96	Dic₂₂	—	4	2	no	—
97	Dic₂₂	—	4	2	no	—
98	C₈×C₁₁	—	2	1	no	—
99	C₈×C₁₁	—	2	1	no	—
100	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
101	C₈×C₁₁	#124	1	1	yes	characteristic
102	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
103	C₁₁⋊C₈	—	4	2	no	—
104	C₂×C₄×Q₈	—	11	1	no	—
105	C₂×M₃₂	—	11	2	no	—
106	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
107	#446	—	22	1	no	—
108	#446	—	22	1	no	—
109	#446	—	22	1	no	—
110	#446	—	22	1	no	—
111	C₂×C₄⋊C₈	—	11	2	no	—
112	C₂×C₄⋊C₈	—	11	1	no	—
113	C₂×#97	—	11	2	no	—
114	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
115	C₄⋊C₁₆	—	22	1	no	—
116	C₄⋊C₁₆	—	22	1	no	—
117	#346	—	2	1	no	—
118	C₄×C₁₆	—	2	1	no	—
119	C₂²×C₁₁	—	2	1	no	—
120	C₂²×C₁₁	—	2	1	no	—
121	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₁₁	C₈×Q₈	1	2	yes	—
123	C₄×C₁₁	—	2	1	no	—
124	C₄×C₁₁	—	2	1	no	—
125	C₄×C₁₁	—	2	1	no	—
126	C₄×C₁₁	#349	1	2	yes	—
127	C₄×C₁₁	#391	1	1	yes	characteristic
128	C₄×C₁₁	#408	1	1	yes	characteristic
129	C₄×C₁₁	#446	1	2	yes	—
130	Dic₁₁	—	4	2	no	—
131	Dic₁₁	—	4	2	no	—
132	C₂²×Q₈	—	11	1	no	—
133	C₂²×C₈	—	11	2	no	—
134	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
136	C₄×Q₈	—	22	1	no	—
137	C₄×Q₈	—	22	1	no	—
138	C₄×Q₈	—	22	1	no	—
139	C₄×Q₈	—	22	1	no	—
140	C₂×C₄⋊C₄	—	11	2	no	—
141	C₂×C₄⋊C₄	—	11	1	no	—
142	C₂×C₄²	—	11	2	no	—
143	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
144	M₃₂	—	2	2	no	—
145	M₃₂	—	22	4	no	—
146	M₃₂	Dic₂₂	1	4	yes	complemented
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	22	2	no	—
149	C₄⋊C₈	—	22	2	no	—
150	C₄⋊C₈	—	22	2	no	—
151	C₄⋊C₈	—	22	1	no	—
152	C₄⋊C₈	—	22	1	no	—
153	#97	—	22	2	no	—
154	#97	—	22	2	no	—
155	C₄×C₈	D₄₄	1	1	yes	characteristic
156	C₄×C₈	D₄₄	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
159	C₂×C₁₁	—	2	2	no	—
160	C₂×C₁₁	#1984	1	1	yes	characteristic
161	C₂×C₁₁	C₂×#349	1	1	yes	characteristic
162	C₂×C₁₁	#2115	1	1	yes	characteristic
163	C₂×Q₈	—	22	1	no	—
164	C₂×Q₈	—	22	1	no	—
165	C₂×Q₈	—	44	2	no	—
166	C₂²×C₄	—	11	2	no	—
167	C₂²×C₄	—	11	2	no	—
168	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
169	C₂²×C₄	C₈×D₁₁	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	22	4	no	—
173	C₂×C₈	—	22	2	no	—
174	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
176	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
177	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
178	C₂×C₈	#4339	1	2	yes	—
179	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
180	C₄⋊C₄	—	22	2	no	—
181	C₄⋊C₄	—	22	2	no	—
182	C₄⋊C₄	—	22	1	no	—
183	C₄⋊C₄	—	22	1	no	—
184	C₄²	—	22	2	no	—
185	C₄²	—	22	2	no	—
186	C₄²	#4313	1	1	yes	characteristic
187	C₄²	#4313	1	1	yes	characteristic
188	C₄²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
189	C₄²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
190	C₁₆	—	2	4	no	—
191	C₁₆	—	44	2	no	—
192	C₁₁	#13268	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂×C₈×D₁₁	1	1	yes	characteristic
194	Q₈	—	44	2	no	—
195	Q₈	—	44	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	22	2	no	—
199	C₂×C₄	—	22	2	no	—
200	C₂×C₄	—	44	2	no	—
201	C₂×C₄	—	44	2	no	—
202	C₂×C₄	(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
203	C₂×C₄	(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
204	C₂×C₄	(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
205	C₂×C₄	(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
206	C₂×C₄	C₄×Dic₂₂	1	2	yes	—
207	C₂×C₄	C₂×#4313	1	1	yes	characteristic
208	C₂×C₄	#66954	1	2	yes	—
209	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	44	2	no	—
213	C₈	#66922	1	1	yes	characteristic
214	C₈	#66928	1	1	yes	characteristic
215	C₈	#66977	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₈×Dic₂₂	1	2	yes	—
219	C₂²	C₂×(C₂×D₁₁)⋊C₈	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	44	2	no	—
224	C₄	—	44	2	no	—
225	C₄	#10622152	1	2	yes	—
226	C₄	#10622292	1	2	yes	—
227	C₄	#10622542	1	1	yes	characteristic
228	C₄	#10622723	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	#13069552	1	1	yes	characteristic
231	C₂	C₂×#10622152	1	1	yes	characteristic
232	C₂	#13070236	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄
commutator, prime radical	C₂³×C₁₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₄×C₁₁×M₃₂	C₂
socle	C₂²×C₁₁	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₄×M₃₂	D₁₁
hypocenter	C₁₁	#13268
LCS	G ▷ C₂³×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
33. C₁₁×M₃₂	Q₈	semidirect
146. M₃₂	Dic₂₂	semidirect
192. C₁₁	#13268	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622292	2
#13070372	2
#13066699	2
C₄×C₁₁×M₃₂	2
#13268	11