Subgroups — #14772778

Order 2816.

642 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#4318	C₂	1	2	yes	maximal
2	#13069392	C₂	1	6	yes	maximal
3	#13068754	C₂	1	3	yes	maximal
4	C₄×#66883	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₁	C₄	1	6	yes	—
8	C₈×Q₈×C₁₁	C₄	1	2	yes	—
9	C₂×C₁₁×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
11	C₂×#66883	C₂²	1	6	yes	—
12	#10622637	—	2	6	no	—
13	Q₈×C₁₁⋊C₈	—	2	2	no	—
14	#10622537	C₂²	1	6	yes	—
15	C₄×#4318	C₂²	1	6	yes	—
16	#10622190	C₂²	1	6	yes	—
17	#10622190	C₂²	1	3	yes	—
18	#10622190	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₁₁	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₁₁	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₁	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₁	D₄	1	3	yes	—
26	C₂×C₁₁×C₄⋊C₄	Q₈	1	3	yes	—
27	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
28	C₂×#4318	C₂³	1	6	yes	—
29	C₂×#4318	C₂³	1	2	yes	—
30	C₁₁×C₄⋊C₈	C₂×C₄	1	6	yes	—
31	C₁₁×C₄⋊C₈	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₁₁	C₂×C₄	1	6	yes	—
33	C₄×C₈×C₁₁	C₂×C₄	1	6	yes	—
34	#66883	—	2	12	no	—
35	C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
36	C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
37	#66867	—	2	6	no	—
38	C₄×C₁₁⋊C₈	—	2	6	no	—
39	#22746	—	11	1	no	maximal, Sylow
40	C₂×Q₈×C₁₁	—	2	4	no	—
41	C₂×Q₈×C₁₁	C₄⋊C₄	1	3	yes	—
42	C₂×Q₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
44	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
46	C₂×C₈×C₁₁	—	2	6	no	—
47	C₂×C₈×C₁₁	—	2	2	no	—
48	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₁₁	C₂²×C₄	1	6	yes	—
50	C₁₁×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
51	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
52	C₁₁×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
53	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
54	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
55	C₄²×C₁₁	C₂×D₄	1	3	yes	—
56	C₄²×C₁₁	C₂×D₄	1	3	yes	—
57	#4318	—	2	6	no	—
58	#4318	—	2	2	no	—
59	#4318	—	2	6	no	—
60	#4318	#41	1	4	yes	—
61	C₂×C₁₁⋊C₈	—	2	2	no	—
62	C₂×C₁₁⋊C₈	—	2	6	no	—
63	Q₈×M₁₆	—	11	2	no	—
64	C₂×C₈×Q₈	D₁₁	1	1	yes	characteristic
65	#1950	—	11	6	no	—
66	#1837	—	11	3	no	—
67	C₄×#108	—	11	3	no	—
68	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
69	Q₈×C₁₁	—	2	6	no	—
70	Q₈×C₁₁	#108	1	4	yes	complemented
71	C₂×C₄×C₁₁	—	2	6	no	—
72	C₂×C₄×C₁₁	—	4	3	no	—
73	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
80	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
81	C₈×C₁₁	—	2	6	no	—
82	C₈×C₁₁	—	2	2	no	—
83	C₁₁⋊C₈	—	2	4	no	—
84	C₁₁⋊C₈	—	4	6	no	—
85	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
86	#446	—	22	6	no	—
87	C₈×Q₈	—	22	2	no	—
88	C₈×Q₈	Dic₁₁	1	6	yes	—
89	C₈×Q₈	Dic₁₁	1	2	yes	—
90	C₂×#108	—	11	6	no	—
91	#423	—	11	6	no	—
92	C₂×C₄⋊C₈	C₂×D₁₁	1	3	yes	—
93	C₄×M₁₆	—	11	6	no	—
94	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
95	#341	—	11	3	no	—
96	#341	—	11	6	no	—
97	#341	—	11	1	no	—
98	C₂²×C₁₁	—	2	1	no	—
99	C₂²×C₁₁	—	2	1	no	—
100	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₁	#557	1	2	yes	—
102	C₄×C₁₁	—	2	3	no	—
103	C₄×C₁₁	—	2	3	no	—
104	C₄×C₁₁	—	2	3	no	—
105	C₄×C₁₁	C₂×#108	1	6	yes	—
106	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
107	C₄×C₁₁	#519	1	1	yes	characteristic
108	C₄×C₁₁	#548	1	2	yes	—
109	C₂²×Q₈	Dic₂₂	1	1	yes	characteristic
110	C₂×M₁₆	—	11	2	no	—
111	C₂×M₁₆	—	11	6	no	—
112	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
114	C₄×Q₈	D₄₄	1	1	yes	characteristic
115	C₄×Q₈	D₄₄	1	3	yes	—
116	C₄×Q₈	C₂×Dic₁₁	1	3	yes	—
117	C₄×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
118	C₂×C₄⋊C₄	Dic₂₂	1	3	yes	—
119	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
120	#108	—	22	12	no	—
121	C₄⋊C₈	—	22	6	no	—
122	C₄⋊C₈	—	22	6	no	—
123	C₄⋊C₈	C₂×Dic₁₁	1	6	yes	—
124	C₄⋊C₈	C₂×Dic₁₁	1	6	yes	—
125	#97	—	22	6	no	—
126	C₄×C₈	—	22	6	no	—
127	C₄×C₈	C₂×Dic₁₁	1	6	yes	—
128	C₄×C₈	C₂×Dic₁₁	1	6	yes	—
129	C₂×C₁₁	—	2	2	no	—
130	C₂×C₁₁	#2351	1	1	yes	characteristic
131	C₂×C₁₁	C₂²×#108	1	1	yes	characteristic
132	C₂×C₁₁	#2916	1	1	yes	characteristic
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	C₄⋊Dic₁₁	1	3	yes	—
135	C₂×Q₈	C₄⋊Dic₁₁	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
137	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
138	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
139	M₁₆	—	11	4	no	—
140	M₁₆	—	22	2	no	—
141	M₁₆	—	22	6	no	—
142	M₁₆	—	22	6	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	6	no	—
145	C₂×C₈	—	22	2	no	—
146	C₂×C₈	—	22	6	no	—
147	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
148	C₂×C₈	C₂²×Dic₁₁	1	6	yes	—
149	C₄⋊C₄	C₄⋊Dic₁₁	1	6	yes	—
150	C₄⋊C₄	C₄⋊Dic₁₁	1	3	yes	—
151	C₄⋊C₄	C₄⋊Dic₁₁	1	3	yes	—
152	C₄²	C₂×D₄₄	1	3	yes	—
153	C₄²	C₂×D₄₄	1	3	yes	—
154	C₄²	C₂²×Dic₁₁	1	3	yes	—
155	C₄²	C₂²×Dic₁₁	1	3	yes	—
156	C₁₁	#22746	1	1	yes	characteristic, complemented, Sylow
157	C₂³	C₂²×Dic₂₂	1	1	yes	characteristic
158	Q₈	—	2	6	no	—
159	Q₈	#66883	1	4	yes	complemented
160	C₂×C₄	—	2	6	no	—
161	C₂×C₄	—	4	3	no	—
162	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
166	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
167	C₂×C₄	#67040	1	2	yes	—
168	C₂×C₄	#41×D₁₁	1	2	yes	—
169	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	6	no	—
172	C₈	—	22	4	no	—
173	C₈	—	44	6	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#10623100	1	2	yes	—
177	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₂×#66883	1	6	yes	—
182	C₄	#10623107	1	2	yes	—
183	C₄	#10623343	1	1	yes	characteristic
184	C₄	#41×Dic₁₁	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#13071537	1	1	yes	characteristic
187	C₂	C₂²×#66883	1	1	yes	characteristic
188	C₂	#13077227	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₈×Q₈×C₁₁	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₁	C₂²×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	#22746
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
70. Q₈×C₁₁	#108	semidirect
156. C₁₁	#22746	semidirect
159. Q₈	#66883	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#4318	2
#13069392	2
#13068754	2
C₄×#66883	2
C₂×C₈×Q₈×C₁₁	2
#22746	11