Subgroups — C₂³×#66886 #15617461

Order 2816.

2846 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#4318	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×#66886	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×C₁₁×M₁₆	—	2	7	no	—
7	C₂²×C₁₁×M₁₆	C₂²	1	14	yes	—
8	C₂³×C₈×C₁₁	—	2	1	no	—
9	C₂²×#4318	—	2	7	no	—
10	C₂²×#4318	C₂²	1	14	yes	—
11	C₂³×C₁₁⋊C₈	—	2	1	no	—
12	C₂³×C₁₁⋊C₈	C₄	1	14	yes	—
13	C₂³×C₁₁⋊C₈	C₄	1	2	yes	—
14	C₂³×C₁₁⋊C₈	C₂²	1	14	yes	—
15	C₂×#66886	C₂²	1	112	yes	direct product factor
16	C₂⁵×C₁₁	Q₈	1	1	yes	characteristic
17	C₂³×C₄×C₁₁	C₂×C₄	1	7	yes	—
18	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₁	D₄	1	7	yes	—
20	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₁	Q₈	1	7	yes	—
22	C₂³×C₄×C₁₁	C₂³	1	7	yes	—
23	C₂×C₁₁×M₁₆	—	2	42	no	—
24	C₂×C₁₁×M₁₆	C₂³	1	28	yes	—
25	C₂²×C₈×C₁₁	—	2	14	no	—
26	C₂×#4318	—	2	42	no	—
27	C₂×#4318	C₂³	1	28	yes	—
28	C₂²×C₁₁⋊C₈	—	2	14	no	—
29	C₂²×C₁₁⋊C₈	C₂×C₄	1	84	yes	—
30	C₂²×C₁₁⋊C₈	C₂×C₄	1	28	yes	—
31	C₂²×C₁₁⋊C₈	C₂³	1	28	yes	—
32	#66886	C₂³	1	64	yes	direct product factor
33	C₂³×#108	—	11	1	no	maximal, Sylow
34	C₂⁴×C₁₁	—	2	8	no	—
35	C₂⁴×C₁₁	C₄⋊C₄	1	7	yes	—
36	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂⁴×C₁₁	C₂×Q₈	1	7	yes	—
38	C₂²×C₄×C₁₁	C₄⋊C₄	1	42	yes	—
39	C₂²×C₄×C₁₁	C₄⋊C₄	1	7	yes	—
40	C₂²×C₄×C₁₁	C₄⋊C₄	1	7	yes	—
41	C₂²×C₄×C₁₁	C₂²×C₄	1	21	yes	—
42	C₂²×C₄×C₁₁	C₂²×C₄	1	7	yes	—
43	C₂²×C₄×C₁₁	C₂×D₄	1	7	yes	—
44	C₂²×C₄×C₁₁	C₂×D₄	1	21	yes	—
45	C₂²×C₄×C₁₁	C₂×Q₈	1	21	yes	—
46	C₂²×C₄×C₁₁	C₂⁴	1	7	yes	—
47	C₁₁×M₁₆	—	2	28	no	—
48	C₁₁×M₁₆	C₂⁴	1	8	yes	—
49	C₂×C₈×C₁₁	—	2	28	no	—
50	#4318	—	2	28	no	—
51	#4318	C₂⁴	1	8	yes	—
52	C₂×C₁₁⋊C₈	—	2	28	no	—
53	C₂×C₁₁⋊C₈	C₂²×C₄	1	56	yes	—
54	C₂×C₁₁⋊C₈	C₂²×C₄	1	56	yes	—
55	C₂×C₁₁⋊C₈	C₂⁴	1	8	yes	—
56	C₂³×M₁₆	—	11	1	no	—
57	C₂³×M₁₆	D₁₁	1	1	yes	characteristic
58	C₂⁴×C₈	—	11	1	no	—
59	C₂²×#108	—	11	28	no	—
60	C₂³×C₁₁	—	2	28	no	—
61	C₂³×C₁₁	—	2	28	no	—
62	C₂³×C₁₁	#108	1	8	yes	complemented
63	C₂³×C₁₁	C₂×C₄⋊C₄	1	21	yes	—
64	C₂³×C₁₁	C₂×C₄⋊C₄	1	7	yes	—
65	C₂³×C₁₁	C₂²×Q₈	1	7	yes	—
66	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	42	yes	—
67	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	21	yes	—
68	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	21	yes	—
69	C₂×C₄×C₁₁	C₂³×C₄	1	7	yes	—
70	C₂×C₄×C₁₁	C₂³×C₄	1	7	yes	—
71	C₂×C₄×C₁₁	C₂²×D₄	1	7	yes	—
72	C₂×C₄×C₁₁	C₂²×D₄	1	7	yes	—
73	C₂×C₄×C₁₁	C₂²×Q₈	1	7	yes	—
74	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
75	C₈×C₁₁	—	2	8	no	—
76	C₁₁⋊C₈	—	2	8	no	—
77	C₁₁⋊C₈	C₂³×C₄	1	16	yes	—
78	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
79	C₂²×M₁₆	—	2	7	no	—
80	C₂²×M₁₆	—	11	14	no	—
81	C₂²×M₁₆	—	22	7	no	—
82	C₂²×M₁₆	C₂×D₁₁	1	14	yes	—
83	C₂³×C₈	—	2	1	no	—
84	C₂³×C₈	—	11	14	no	—
85	C₂³×C₈	—	11	2	no	—
86	C₂³×C₈	—	11	14	no	—
87	C₂³×C₈	—	22	1	no	—
88	C₂×#108	—	11	112	no	—
89	C₂²×C₁₁	—	2	42	no	—
90	C₂²×C₁₁	—	2	14	no	—
91	C₂²×C₁₁	C₂×#108	1	28	yes	complemented
92	C₂²×C₁₁	C₂²×C₄⋊C₄	1	7	yes	—
93	C₂²×C₁₁	C₂²×C₄⋊C₄	1	7	yes	—
94	C₂²×C₁₁	C₂³×Q₈	1	1	yes	characteristic
95	C₄×C₁₁	C₂²×C₄⋊C₄	1	7	yes	—
96	C₄×C₁₁	C₂²×C₄⋊C₄	1	7	yes	—
97	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
98	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
99	C₂⁵	Dic₂₂	1	1	yes	characteristic
100	C₂³×C₄	Dic₂₂	1	7	yes	—
101	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
102	C₂³×C₄	C₄×D₁₁	1	7	yes	—
103	C₂³×C₄	C₂²⋊D₁₁	1	7	yes	—
104	C₂³×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
105	C₂³×C₄	C₂²×D₁₁	1	7	yes	—
106	C₂×M₁₆	—	2	42	no	—
107	C₂×M₁₆	—	11	28	no	—
108	C₂×M₁₆	—	22	42	no	—
109	C₂×M₁₆	C₂²×D₁₁	1	28	yes	—
110	C₂²×C₈	—	2	14	no	—
111	C₂²×C₈	—	11	84	no	—
112	C₂²×C₈	—	11	28	no	—
113	C₂²×C₈	—	11	28	no	—
114	C₂²×C₈	—	22	14	no	—
115	#108	—	11	64	no	—
116	C₂×C₁₁	—	2	7	no	—
117	C₂×C₁₁	—	2	1	no	—
118	C₂×C₁₁	C₂²×#108	1	14	yes	complemented
119	C₂×C₁₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
120	C₂⁴	—	2	8	no	—
121	C₂⁴	Dic₁₁⋊C₄	1	7	yes	—
122	C₂⁴	Dic₁₁⋊C₄	1	1	yes	characteristic
123	C₂⁴	C₂×Dic₂₂	1	7	yes	—
124	C₂²×C₄	Dic₁₁⋊C₄	1	42	yes	—
125	C₂²×C₄	Dic₁₁⋊C₄	1	7	yes	—
126	C₂²×C₄	Dic₁₁⋊C₄	1	7	yes	—
127	C₂²×C₄	C₂×Dic₂₂	1	21	yes	—
128	C₂²×C₄	C₂×C₄×D₁₁	1	21	yes	—
129	C₂²×C₄	C₂×C₄×D₁₁	1	7	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₁₁	1	7	yes	—
131	C₂²×C₄	C₂×C₂²⋊D₁₁	1	21	yes	—
132	C₂²×C₄	C₂³×D₁₁	1	7	yes	—
133	M₁₆	—	2	28	no	—
134	M₁₆	—	11	8	no	—
135	M₁₆	—	22	28	no	—
136	M₁₆	C₂³×D₁₁	1	8	yes	—
137	C₂×C₈	—	2	28	no	—
138	C₂×C₈	—	11	56	no	—
139	C₂×C₈	—	11	56	no	—
140	C₂×C₈	—	11	8	no	—
141	C₂×C₈	—	22	28	no	—
142	C₁₁	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	2	28	no	—
144	C₂³	—	2	28	no	—
145	C₂³	#66886	1	8	yes	direct product factor
146	C₂³	C₂×Dic₁₁⋊C₄	1	21	yes	—
147	C₂³	C₂×Dic₁₁⋊C₄	1	7	yes	—
148	C₂³	C₂²×Dic₂₂	1	7	yes	—
149	C₂×C₄	C₂×Dic₁₁⋊C₄	1	42	yes	—
150	C₂×C₄	C₂×Dic₁₁⋊C₄	1	21	yes	—
151	C₂×C₄	C₂×Dic₁₁⋊C₄	1	21	yes	—
152	C₂×C₄	C₂²×Dic₂₂	1	7	yes	—
153	C₂×C₄	C₂²×C₄×D₁₁	1	7	yes	—
154	C₂×C₄	C₂²×C₄×D₁₁	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₁₁	1	7	yes	—
156	C₂×C₄	C₂²×C₂²⋊D₁₁	1	7	yes	—
157	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
158	C₈	—	2	8	no	—
159	C₈	—	11	16	no	—
160	C₈	—	22	8	no	—
161	C₂²	—	2	42	no	—
162	C₂²	—	2	14	no	—
163	C₂²	C₂×#66886	1	28	yes	direct product factor
164	C₂²	C₂²×Dic₁₁⋊C₄	1	7	yes	—
165	C₂²	C₂²×Dic₁₁⋊C₄	1	7	yes	—
166	C₂²	C₂³×Dic₂₂	1	1	yes	characteristic
167	C₄	C₂²×Dic₁₁⋊C₄	1	7	yes	—
168	C₄	C₂²×Dic₁₁⋊C₄	1	7	yes	—
169	C₄	C₂³×C₄×D₁₁	1	1	yes	characteristic
170	C₄	C₂³×C₂²⋊D₁₁	1	1	yes	characteristic
171	C₂	—	2	7	no	—
172	C₂	—	2	1	no	—
173	C₂	C₂²×#66886	1	14	yes	direct product factor
174	C₂	C₂³×Dic₁₁⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₁₁
commutator	C₄×C₁₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂³×C₁₁×M₁₆	C₂
socle	C₂⁴×C₁₁	C₄⋊C₄
prime radical	C₂⁵×C₁₁	Q₈
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂³×M₁₆	D₁₁
hypocenter	C₁₁	C₂³×#108
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂²×#66886	C₂	direct (N × H)
15. C₂×#66886	C₂²	direct (N × H)
32. #66886	C₂³	direct (N × H)
62. C₂³×C₁₁	#108	semidirect
91. C₂²×C₁₁	C₂×#108	semidirect
118. C₂×C₁₁	C₂²×#108	semidirect
142. C₁₁	C₂³×#108	semidirect
145. C₂³	#66886	direct (N × H)
163. C₂²	C₂×#66886	direct (N × H)
173. C₂	C₂²×#66886	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#4318	2
C₂⁴×C₁₁⋊C₈	2
C₂²×#66886	2
C₂³×C₁₁×M₁₆	2
C₂³×#108	11

Subgroups — C23×#66886 #15617461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66886 #15617461