Subgroups — #14772712

Order 2816.

822 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₁₁	C₂	1	2	yes	maximal
2	#13069380	C₂	1	6	yes	maximal
3	#13068744	C₂	1	3	yes	maximal
4	C₄×#66881	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	C₂×Q₈×Dic₁₁	—	2	2	no	—
8	C₈×Q₈×C₁₁	C₄	1	8	yes	complemented
9	C₂×C₁₁×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
11	C₂×#66881	C₂²	1	6	yes	—
12	#10622598	—	2	6	no	—
13	#10622548	C₂²	1	6	yes	—
14	C₄×C₄⋊Dic₁₁	C₂²	1	6	yes	—
15	#10622189	C₂²	1	6	yes	—
16	#10622189	C₂²	1	3	yes	—
17	#10622189	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
21	C₄×Q₈×C₁₁	C₂×C₄	1	4	yes	—
22	C₄×Q₈×C₁₁	Q₈	1	4	yes	—
23	C₂×C₁₁×C₄⋊C₄	D₄	1	3	yes	—
24	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
25	Q₈×Dic₁₁	—	4	8	no	—
26	C₂×C₄⋊Dic₁₁	—	2	6	no	—
27	C₂×C₄⋊Dic₁₁	—	2	6	no	—
28	C₂×C₄⋊Dic₁₁	C₂³	1	6	yes	—
29	C₂×C₄⋊Dic₁₁	C₂³	1	2	yes	—
30	C₂×C₄×Dic₁₁	—	2	6	no	—
31	C₁₁×C₄⋊C₈	C₂×C₄	1	12	yes	—
32	C₄×C₈×C₁₁	C₂×C₄	1	12	yes	—
33	C₁₁⋊#95	—	2	6	no	—
34	#66881	—	2	12	no	—
35	#22738	—	11	1	no	maximal, Sylow
36	C₂×Q₈×C₁₁	C₄⋊C₄	1	4	yes	—
37	C₂×Q₈×C₁₁	SD₁₆	1	8	yes	—
38	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₁₁	—	2	2	no	—
42	C₂²×Dic₁₁	—	2	6	no	—
43	C₂×C₈×C₁₁	—	2	6	no	—
44	C₂×C₈×C₁₁	—	2	2	no	—
45	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₁₁	C₂²×C₄	1	6	yes	—
47	C₁₁×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
48	C₄²×C₁₁	C₂²×C₄	1	6	yes	—
49	C₄²×C₁₁	C₂×Q₈	1	6	yes	—
50	C₄⋊Dic₁₁	—	2	6	no	—
51	C₄⋊Dic₁₁	—	2	2	no	—
52	C₄⋊Dic₁₁	—	2	6	no	—
53	C₄⋊Dic₁₁	—	4	12	no	—
54	C₄⋊Dic₁₁	—	8	6	no	—
55	C₄⋊Dic₁₁	#41	1	4	yes	—
56	C₄×Dic₁₁	—	4	12	no	—
57	C₂×C₈×Q₈	D₁₁	1	1	yes	characteristic
58	Q₈×C₄⋊C₄	—	11	2	no	—
59	#1948	—	11	6	no	—
60	#1834	—	11	3	no	—
61	C₄×#106	—	11	3	no	—
62	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₁₁	#106	1	16	yes	complemented
64	C₂×C₄×C₁₁	—	2	3	no	—
65	C₂×C₄×C₁₁	—	2	3	no	—
66	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
67	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
68	C₂×C₄×C₁₁	C₂×SD₁₆	1	12	yes	—
69	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
72	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
73	C₂×Dic₁₁	—	2	4	no	—
74	C₂×Dic₁₁	—	4	2	no	—
75	C₂×Dic₁₁	—	4	2	no	—
76	C₂×Dic₁₁	—	4	6	no	—
77	C₂×Dic₁₁	—	4	6	no	—
78	C₂×Dic₁₁	—	4	6	no	—
79	C₈×C₁₁	—	2	6	no	—
80	C₈×C₁₁	—	2	2	no	—
81	C₂×C₄×Q₈	—	22	2	no	—
82	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
83	C₈×Q₈	Dic₁₁	1	8	yes	complemented
84	C₂×#106	—	11	6	no	—
85	C₂×C₄⋊C₈	C₂×D₁₁	1	3	yes	—
86	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
87	#389	—	22	6	no	—
88	#384	—	11	6	no	—
89	C₄×C₄⋊C₄	—	11	6	no	—
90	#340	—	11	3	no	—
91	#340	—	11	6	no	—
92	#340	—	11	1	no	—
93	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₁₁	#548	1	2	yes	—
95	C₂²×C₁₁	C₂²×SD₁₆	1	2	yes	—
96	C₂²×C₁₁	#576	1	1	yes	characteristic
97	C₂²×C₁₁	#578	1	1	yes	characteristic
98	C₄×C₁₁	—	2	6	no	—
99	C₄×C₁₁	C₂×#106	1	12	yes	—
100	C₄×C₁₁	C₄×#41	1	1	yes	characteristic
101	C₄×C₁₁	#519	1	1	yes	characteristic
102	C₄×C₁₁	#557	1	2	yes	—
103	Dic₁₁	—	4	4	no	—
104	Dic₁₁	—	8	6	no	—
105	C₂²×Q₈	D₄₄	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
108	C₄×Q₈	—	44	8	no	—
109	C₄×Q₈	Dic₂₂	1	4	yes	—
110	C₄×Q₈	C₂×Dic₁₁	1	4	yes	—
111	C₂×C₄⋊C₄	—	11	2	no	—
112	C₂×C₄⋊C₄	—	11	6	no	—
113	C₂×C₄⋊C₄	—	22	6	no	—
114	C₂×C₄⋊C₄	—	22	6	no	—
115	C₂×C₄⋊C₄	D₄₄	1	3	yes	—
116	C₂×C₄²	—	22	6	no	—
117	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
118	#106	—	22	12	no	—
119	C₄⋊C₈	C₂×Dic₁₁	1	12	yes	—
120	C₄×C₈	C₂×Dic₁₁	1	12	yes	—
121	#95	—	22	6	no	—
122	C₂×C₁₁	#2351	1	1	yes	characteristic
123	C₂×C₁₁	C₂²×#106	1	1	yes	characteristic
124	C₂×C₁₁	#2913	1	1	yes	characteristic
125	C₂×C₁₁	#3359	1	2	yes	—
126	C₂×C₁₁	#3362	1	2	yes	—
127	C₂×Q₈	#4314	1	8	yes	—
128	C₂×Q₈	C₄⋊Dic₁₁	1	4	yes	—
129	C₂²×C₄	—	22	2	no	—
130	C₂²×C₄	—	22	6	no	—
131	C₂²×C₄	C₂×D₄₄	1	3	yes	—
132	C₂²×C₄	C₂×D₄₄	1	3	yes	—
133	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
137	C₂×C₈	C₂²×Dic₁₁	1	6	yes	—
138	C₄⋊C₄	—	11	4	no	—
139	C₄⋊C₄	—	22	2	no	—
140	C₄⋊C₄	—	22	6	no	—
141	C₄⋊C₄	—	22	6	no	—
142	C₄⋊C₄	—	44	12	no	—
143	C₄⋊C₄	—	88	6	no	—
144	C₄⋊C₄	C₄⋊Dic₁₁	1	12	yes	—
145	C₄²	—	44	12	no	—
146	C₄²	C₂×Dic₂₂	1	6	yes	—
147	C₄²	C₂²×Dic₁₁	1	6	yes	—
148	C₁₁	#22738	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₂²×D₄₄	1	1	yes	characteristic
150	Q₈	#66881	1	16	yes	complemented
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	22	4	no	—
154	C₂×C₄	—	44	2	no	—
155	C₂×C₄	—	44	2	no	—
156	C₂×C₄	—	44	6	no	—
157	C₂×C₄	—	44	6	no	—
158	C₂×C₄	—	44	6	no	—
159	C₂×C₄	C₂×#4314	1	12	yes	—
160	C₂×C₄	C₂×C₄⋊Dic₁₁	1	6	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₁₁	1	6	yes	—
162	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
163	C₂×C₄	#67040	1	2	yes	—
164	C₂×C₄	#41×D₁₁	1	2	yes	—
165	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	2	6	no	—
168	C₂²	#10623107	1	2	yes	—
169	C₂²	C₂²×#4314	1	2	yes	—
170	C₂²	#10623283	1	1	yes	characteristic
171	C₂²	#10623284	1	1	yes	characteristic
172	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
173	C₄	—	2	6	no	—
174	C₄	—	44	4	no	—
175	C₄	—	88	6	no	—
176	C₄	C₂×#66881	1	12	yes	—
177	C₄	#10623100	1	2	yes	—
178	C₄	#10623343	1	1	yes	characteristic
179	C₄	#41×Dic₁₁	1	1	yes	characteristic
180	C₂	#13071537	1	1	yes	characteristic
181	C₂	C₂²×#66881	1	1	yes	characteristic
182	C₂	#13077216	1	1	yes	characteristic
183	C₂	#13081259	1	2	yes	—
184	C₂	#13081288	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₈×Q₈×C₁₁	C₂
socle, prime radical	C₂³×C₁₁	C₂²×D₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	#22738
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₁₁	C₄	semidirect
63. Q₈×C₁₁	#106	semidirect
83. C₈×Q₈	Dic₁₁	semidirect
148. C₁₁	#22738	semidirect
150. Q₈	#66881	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₁₁	2
#13069380	2
#13068744	2
C₄×#66881	2
C₂×C₈×Q₈×C₁₁	2
#22738	11