Subgroups — #13077558

Order 1408.

356 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	#10623109	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#10622637	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622452	C₂	1	2	yes	maximal
5	D₁₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#10622443	C₂	1	1	yes	maximal, characteristic, complemented
7	#10622441	C₂	1	2	yes	maximal, complemented
8	#10622302	C₂	1	2	yes	maximal
9	C₄×#4313	C₂	1	1	yes	maximal, characteristic
10	#10622292	C₂	1	1	yes	maximal, characteristic, complemented
11	#10622292	C₂	1	2	yes	maximal, complemented
12	C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
13	#66999	C₄	1	1	yes	characteristic
14	#66997	C₄	1	1	yes	characteristic
15	C₂×#4313	C₂²	1	1	yes	characteristic
16	C₂×#4313	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₁	C₂²	1	2	yes	—
18	#66950	C₄	1	1	yes	characteristic
19	#66949	C₄	1	2	yes	—
20	#66942	C₄	1	2	yes	—
21	#66941	C₄	1	1	yes	characteristic
22	#66925	C₂²	1	2	yes	—
23	C₄²×D₁₁	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₂	C₂²	1	2	yes	—
25	C₄×Dic₂₂	C₂²	1	1	yes	characteristic
26	C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
27	C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₁	C₂²	1	2	yes	—
30	#66879	C₂²	1	1	yes	characteristic
31	#66879	C₂²	1	2	yes	—
32	#66879	C₂²	1	1	yes	characteristic
33	Dic₁₁⋊C₈	C₂²	1	2	yes	—
34	Dic₁₁⋊C₈	C₂²	1	2	yes	—
35	Dic₁₁⋊C₈	C₂²	1	2	yes	—
36	C₈×Dic₁₁	C₂²	1	2	yes	—
37	C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
39	#66867	C₂²	1	2	yes	—
40	C₄×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₂	C₂×C₄	1	1	yes	characteristic
46	C₂×Dic₂₂	C₂×C₄	1	2	yes	—
47	C₂×C₈×C₁₁	C₂³	1	2	yes	—
48	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₄²×C₁₁	C₂³	1	1	yes	characteristic
53	C₄²×C₁₁	C₂³	1	2	yes	—
54	(C₂×D₁₁)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₁₁)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₁₁	C₂×C₄	1	2	yes	—
58	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₁₁⋊C₄	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₁	C₂³	1	2	yes	—
66	C₄×Dic₁₁	C₂³	1	1	yes	characteristic
67	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₁⋊C₈	C₂³	1	2	yes	—
69	C₂×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
70	#4313	—	2	2	no	—
71	#4313	Q₈	1	4	yes	complemented
72	C₈×D₁₁	—	2	4	no	—
73	#2962	—	11	1	no	maximal, Sylow
74	C₂²×D₁₁	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₁₁	—	2	2	no	—
76	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₁	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₄×D₁₁	—	2	2	no	—
87	C₄×D₁₁	—	4	1	no	—
88	C₄×D₁₁	C₂×Q₈	1	2	yes	—
89	C₄×D₁₁	#41	1	2	yes	—
90	Dic₂₂	—	2	2	no	—
91	Dic₂₂	—	4	2	no	—
92	C₈×C₁₁	—	2	2	no	—
93	C₈×C₁₁	—	2	1	no	—
94	C₈×C₁₁	C₂×Q₈	1	2	yes	—
95	C₁₁⋊C₈	—	2	2	no	—
96	C₁₁⋊C₈	—	2	1	no	—
97	C₁₁⋊C₈	C₂×Q₈	1	2	yes	—
98	#533	—	11	1	no	—
99	#446	—	11	1	no	—
100	#446	—	11	2	no	—
101	#446	—	11	1	no	—
102	#446	—	11	2	no	—
103	C₈×Q₈	—	11	1	no	—
104	C₈×Q₈	D₁₁	1	1	yes	characteristic, complemented
105	#424	—	11	2	no	—
106	C₂×C₄⋊C₈	—	11	1	no	—
107	#405	—	11	2	no	—
108	C₄×M₁₆	—	11	1	no	—
109	C₂²×C₁₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×D₁₁	—	2	1	no	—
111	C₂×D₁₁	—	2	1	no	—
112	C₂×D₁₁	C₄×Q₈	1	2	yes	—
113	C₄×C₁₁	—	2	2	no	—
114	C₄×C₁₁	—	2	2	no	—
115	C₄×C₁₁	—	2	1	no	—
116	C₄×C₁₁	#131	1	2	yes	—
117	C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
118	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
119	Dic₁₁	—	2	2	no	—
120	Dic₁₁	—	2	1	no	—
121	Dic₁₁	—	2	1	no	—
122	Dic₁₁	—	2	2	no	—
123	Dic₁₁	C₄×Q₈	1	2	yes	—
124	C₂×M₁₆	—	11	1	no	—
125	C₂×M₁₆	—	11	1	no	—
126	C₂²×C₈	—	11	2	no	—
127	C₄⋊Q₈	—	11	1	no	—
128	C₄⋊Q₈	—	11	1	no	—
129	#125	—	11	2	no	—
130	C₂²⋊Q₈	—	11	1	no	—
131	C₂²⋊Q₈	—	11	2	no	—
132	C₂²⋊Q₈	—	11	1	no	—
133	C₄×Q₈	—	11	1	no	—
134	C₄×Q₈	—	11	2	no	—
135	C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
136	#117	—	11	2	no	—
137	C₂×C₄²	—	11	1	no	—
138	C₄⋊C₈	—	11	2	no	—
139	C₄⋊C₈	—	11	2	no	—
140	C₄⋊C₈	—	11	2	no	—
141	C₄⋊C₈	—	11	1	no	—
142	C₄⋊C₈	—	11	2	no	—
143	C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₁₁	1	2	yes	—
145	#97	—	11	1	no	—
146	#97	—	11	1	no	—
147	#97	—	11	2	no	—
148	#97	—	11	2	no	—
149	C₄×C₈	—	11	2	no	—
150	C₄×C₈	—	11	1	no	—
151	C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
152	C₄×C₈	C₂×D₁₁	1	2	yes	—
153	C₂×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
154	C₂×C₁₁	C₂×#131	1	1	yes	characteristic
155	C₂×C₁₁	#568	1	1	yes	characteristic
156	D₁₁	—	2	2	no	—
157	C₂×Q₈	—	11	1	no	—
158	C₂×Q₈	—	11	2	no	—
159	C₂×Q₈	C₄×D₁₁	1	1	yes	characteristic
160	C₂²×C₄	—	11	1	no	—
161	C₂²×C₄	—	11	1	no	—
162	C₂²×C₄	—	11	1	no	—
163	M₁₆	—	11	4	no	—
164	M₁₆	—	22	2	no	—
165	C₂×C₈	—	11	1	no	—
166	C₂×C₈	—	11	1	no	—
167	C₂×C₈	—	11	2	no	—
168	C₂×C₈	—	22	4	no	—
169	C₂×C₈	C₂²×D₁₁	1	2	yes	—
170	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₁	1	1	yes	characteristic
172	C₄⋊C₄	—	11	2	no	—
173	C₄⋊C₄	—	11	2	no	—
174	C₄⋊C₄	—	11	2	no	—
175	C₄⋊C₄	—	11	2	no	—
176	C₄⋊C₄	—	11	2	no	—
177	C₄⋊C₄	—	11	2	no	—
178	C₄⋊C₄	—	11	1	no	—
179	C₄⋊C₄	C₄×D₁₁	1	2	yes	—
180	C₄⋊C₄	C₄×D₁₁	1	1	yes	characteristic
181	C₂²⋊C₄	—	11	2	no	—
182	C₂²⋊C₄	—	11	2	no	—
183	C₄²	—	11	1	no	—
184	C₄²	—	11	1	no	—
185	C₄²	—	11	2	no	—
186	C₄²	—	11	1	no	—
187	C₄²	C₂²×D₁₁	1	1	yes	characteristic
188	C₄²	C₂²×D₁₁	1	2	yes	—
189	C₁₁	#2962	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	11	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	22	2	no	—
193	Q₈	—	44	2	no	—
194	C₂×C₄	—	11	1	no	—
195	C₂×C₄	—	11	2	no	—
196	C₂×C₄	—	11	2	no	—
197	C₂×C₄	—	11	1	no	—
198	C₂×C₄	—	11	1	no	—
199	C₂×C₄	—	11	2	no	—
200	C₂×C₄	—	11	2	no	—
201	C₂×C₄	—	22	2	no	—
202	C₂×C₄	—	44	1	no	—
203	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₁	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₁	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	11	2	no	—
211	C₈	—	22	2	no	—
212	C₈	—	22	1	no	—
213	C₈	Q₈×D₁₁	1	2	yes	—
214	C₂²	—	11	2	no	—
215	C₂²	—	22	1	no	—
216	C₂²	—	22	1	no	—
217	C₂²	C₂²×C₄×D₁₁	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	11	2	no	—
222	C₄	—	22	1	no	—
223	C₄	—	22	2	no	—
224	C₄	—	22	1	no	—
225	C₄	—	22	2	no	—
226	C₄	#66954	1	2	yes	—
227	C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
228	C₄	#41×D₁₁	1	1	yes	characteristic
229	C₂	—	22	2	no	—
230	C₂	C₄×Q₈×D₁₁	1	1	yes	characteristic
231	C₂	C₂×#66954	1	1	yes	characteristic
232	C₂	#10623281	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₄
Fitting	C₈×Q₈×C₁₁	C₂
prime radical	C₂²×D₁₁	C₂²×C₄
cosocle	C₂×C₄×C₁₁	C₂⁴
hypercenter	C₈×Q₈	D₁₁
hypocenter	C₁₁	#2962
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₁₁	C₂	semidirect
3. #10622637	C₂	semidirect
6. #10622443	C₂	semidirect
7. #10622441	C₂	semidirect
10. #10622292	C₂	semidirect
11. #10622292	C₂	semidirect
71. #4313	Q₈	semidirect
104. C₈×Q₈	D₁₁	semidirect
189. C₁₁	#2962	semidirect

Maximal subgroups

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

subgroup	index
#10623109	2
C₈×Q₈×C₁₁	2
#10622637	2
#10622452	2
D₁₁×C₄⋊C₈	2
#10622443	2
#10622441	2
#10622302	2
C₄×#4313	2
#10622292	2
#2962	11