Subgroups — #41014117

Order 27008.

260 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₂₁₁⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#27212901	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₂₁₁⋊C₈)	—	2	4	no	—
10	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
11	C₄×C₂₁₁⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₁₁	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₁₁	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	#95×C₂₁₁	C₄	1	2	yes	—
17	C₂×C₂₁₁⋊C₈	—	2	1	no	—
18	C₂×C₂₁₁⋊C₈	—	2	1	no	—
19	C₂×C₂₁₁⋊C₈	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₂₁₁⋊C₈	—	2	1	no	—
22	C₂×C₂₁₁⋊C₈	—	2	1	no	—
23	C₂×C₂₁₁⋊C₈	—	2	2	no	—
24	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₂₁₁⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₂₁₁	—	2	2	no	—
32	C₄⋊C₄×C₂₁₁	C₈	1	4	yes	complemented
33	C₄²×C₂₁₁	—	2	2	no	—
34	C₄²×C₂₁₁	—	2	2	no	—
35	C₄²×C₂₁₁	D₄	1	2	yes	—
36	C₄²×C₂₁₁	Q₈	1	2	yes	—
37	C₂₁₁⋊C₈	—	2	2	no	—
38	C₂₁₁⋊C₈	—	2	2	no	—
39	C₂₁₁⋊C₈	—	4	2	no	—
40	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	—	2	2	no	—
42	C₂×C₄×C₂₁₁	—	2	1	no	—
43	C₂×C₄×C₂₁₁	—	2	1	no	—
44	C₂×C₄×C₂₁₁	—	2	2	no	—
45	C₂×C₄×C₂₁₁	—	2	2	no	—
46	C₂×C₄×C₂₁₁	—	2	2	no	—
47	C₂×C₄×C₂₁₁	—	2	2	no	—
48	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₁₁	C₂×C₈	1	4	yes	—
51	C₂×C₄×C₂₁₁	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₂₁₁	M₁₆	1	2	yes	—
53	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
58	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
59	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₁₁	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₂₁₁	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₂₁₁	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₂₁₁	#131	1	1	yes	characteristic
65	C₂²×C₂₁₁	#131	1	1	yes	characteristic
66	C₄×C₂₁₁	—	2	2	no	—
67	C₄×C₂₁₁	—	2	4	no	—
68	C₄×C₂₁₁	—	4	2	no	—
69	C₄×C₂₁₁	C₄⋊C₈	1	4	yes	complemented
70	C₄×C₂₁₁	#121	1	1	yes	characteristic
71	C₄×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
72	C₄×C₂₁₁	#125	1	1	yes	characteristic
73	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₂₁₁	#389	1	1	yes	characteristic
75	C₂×C₂₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₂₁₁	#424	1	1	yes	characteristic
77	C₂×C₂₁₁	C₈×D₄	1	1	yes	characteristic
78	C₂×C₂₁₁	#436	1	1	yes	characteristic
79	C₂×C₂₁₁	C₈×Q₈	1	1	yes	characteristic
80	C₂×C₂₁₁	#446	1	1	yes	characteristic
81	C₂₁₁	#1939	1	1	yes	characteristic, complemented, Sylow
82	#1939	—	211	1	no	maximal, Sylow
83	C₂×C₄⋊C₈	—	211	2	no	—
84	C₂×C₄⋊C₈	—	211	1	no	—
85	C₂×C₄×C₈	—	211	1	no	—
86	C₄×C₄⋊C₄	D₂₁₁	1	1	yes	characteristic
87	#336	—	211	2	no	—
88	C₂²×C₈	—	211	2	no	—
89	C₂²×C₈	—	211	1	no	—
90	C₂²×C₈	—	211	1	no	—
91	C₂×C₄⋊C₄	Dic₂₁₁	1	1	yes	characteristic
92	C₂×C₄⋊C₄	Dic₂₁₁	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₂₁₁	1	2	yes	—
94	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
95	C₄⋊C₈	—	422	4	no	—
96	C₄⋊C₈	—	422	2	no	—
97	C₄×C₈	—	422	2	no	—
98	#95	Dic₂₁₁	1	2	yes	—
99	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
100	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
101	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
102	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
104	C₂×C₈	—	422	2	no	—
105	C₂×C₈	—	422	2	no	—
106	C₂×C₈	—	422	2	no	—
107	C₂×C₈	—	422	1	no	—
108	C₂×C₈	—	422	1	no	—
109	C₂×C₈	—	422	1	no	—
110	C₂×C₈	—	422	1	no	—
111	C₂×C₈	—	211	2	no	—
112	C₂×C₈	—	211	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₂₁₁⋊C₈	1	4	yes	complemented
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	D₈₄₄	1	2	yes	—
118	C₄²	Dic₄₂₂	1	2	yes	—
119	C₂³	C₂²×Dic₂₁₁	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
128	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
129	C₂×C₄	C₂×C₂₁₁⋊C₈	1	4	yes	—
130	C₂×C₄	C₂×C₂₁₁⋊C₈	1	2	yes	—
131	C₂×C₄	#16809946	1	2	yes	—
132	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
133	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
136	C₂×C₄	#16809961	1	1	yes	characteristic
137	C₂×C₄	#16809963	1	1	yes	characteristic
138	C₈	—	844	2	no	—
139	C₈	—	422	2	no	—
140	C₈	—	422	2	no	—
141	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
142	C₂²	D₄×Dic₂₁₁	1	1	yes	characteristic
143	C₂²	Q₈×Dic₂₁₁	1	1	yes	characteristic
144	C₂²	C₂²×C₂₁₁⋊C₈	1	1	yes	characteristic
145	C₂²	C₂×#16809946	1	1	yes	characteristic
146	C₂²	#20269984	1	1	yes	characteristic
147	C₂²	#20269984	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	2	no	—
151	C₄	C₄⋊(C₂₁₁⋊C₈)	1	4	yes	complemented
152	C₄	#20269952	1	1	yes	characteristic
153	C₄	#20269957	1	1	yes	characteristic
154	C₄	#20269968	1	1	yes	characteristic
155	C₄	#20269976	1	1	yes	characteristic
156	C₂	#27213045	1	1	yes	characteristic
157	C₂	C₂×C₄⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
158	C₂	#27213173	1	1	yes	characteristic
159	C₂	D₄×C₂₁₁⋊C₈	1	1	yes	characteristic
160	C₂	#27213224	1	1	yes	characteristic
161	C₂	Q₈×C₂₁₁⋊C₈	1	1	yes	characteristic
162	C₂	#27213277	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator	C₂²×C₂₁₁	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₁₁
hypocenter	C₂₁₁	#1939
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
32. C₄⋊C₄×C₂₁₁	C₈	semidirect
69. C₄×C₂₁₁	C₄⋊C₈	semidirect
81. C₂₁₁	#1939	semidirect
114. C₄⋊C₄	C₂₁₁⋊C₈	semidirect
151. C₄	C₄⋊(C₂₁₁⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₂₁₁⋊C₈)	2
C₂×C₄×C₂₁₁⋊C₈	2
#27212901	2
C₄×C₄⋊C₄×C₂₁₁	2
#1939	211