Subgroups — #26110016

Order 12032.

462 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	2	yes	maximal
2	C₂×#15688874	C₂	1	4	yes	maximal
3	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
5	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
6	#15688874	—	2	8	no	—
7	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
11	C₄³×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄₇⋊C₈	—	2	4	no	—
13	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
14	C₄⋊(C₄₇⋊C₈)	—	4	8	no	—
15	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	D₄	1	2	yes	—
20	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₄₇	—	2	4	no	—
23	C₄×C₈×C₄₇	—	2	2	no	—
24	C₄×C₈×C₄₇	C₈	1	4	yes	complemented
25	C₄×C₈×C₄₇	D₄	1	4	yes	—
26	C₄×C₈×C₄₇	Q₈	1	4	yes	—
27	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
38	C₄²×C₄₇	—	2	4	no	—
39	C₄²×C₄₇	—	2	2	no	—
40	C₄²×C₄₇	C₂×C₈	1	2	yes	—
41	C₄²×C₄₇	M₁₆	1	2	yes	—
42	C₄²×C₄₇	SD₁₆	1	8	yes	—
43	C₄²×C₄₇	C₂×D₄	1	2	yes	—
44	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄₇⋊C₈	—	4	4	no	—
46	C₂×C₄₇⋊C₈	—	4	8	no	—
47	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₄₇	#106	1	4	yes	—
53	C₂×C₄×C₄₇	#106	1	4	yes	—
54	C₂×C₄×C₄₇	#108	1	4	yes	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
58	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
63	C₈×C₄₇	—	2	4	no	—
64	C₈×C₄₇	C₄⋊C₈	1	8	yes	complemented
65	C₄₇⋊C₈	—	8	4	no	—
66	#8805	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	#384	1	1	yes	characteristic
68	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₄₇	#423	1	1	yes	characteristic
70	C₂²×C₄₇	C₂×#106	1	2	yes	—
71	C₂²×C₄₇	C₂×#108	1	2	yes	—
72	C₄×C₄₇	—	2	4	no	—
73	C₄×C₄₇	—	2	2	no	—
74	C₄×C₄₇	#334	1	8	yes	complemented
75	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
76	C₄×C₄₇	#423	1	2	yes	—
77	C₄×C₄₇	#492	1	2	yes	—
78	C₄×C₄₇	#498	1	2	yes	—
79	#1842	—	47	2	no	—
80	C₄²×C₈	D₄₇	1	1	yes	characteristic
81	C₂×#334	—	47	4	no	—
82	C₂×C₄₇	C₂×#334	1	2	yes	—
83	C₂×C₄₇	#1566	1	2	yes	—
84	C₂×C₄₇	#1842	1	1	yes	characteristic
85	C₂×C₄₇	#1844	1	1	yes	characteristic
86	C₂×C₄₇	#1848	1	1	yes	characteristic
87	C₂×C₄⋊C₈	—	47	4	no	—
88	C₂×C₄⋊C₈	—	94	4	no	—
89	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
90	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
91	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
93	C₄³	C₂×D₄₇	1	1	yes	characteristic
94	#334	—	94	8	no	—
95	C₄₇	#8805	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	94	4	no	—
97	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
98	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
99	C₂×C₄²	Dic₉₄	1	2	yes	—
100	C₂×C₄²	D₁₈₈	1	2	yes	—
101	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
104	C₄⋊C₈	—	94	8	no	—
105	C₄⋊C₈	—	188	8	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	C₄₇⋊C₈	1	4	yes	complemented
109	C₄×C₈	Dic₉₄	1	4	yes	—
110	C₄×C₈	D₁₈₈	1	4	yes	—
111	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
112	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
113	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
115	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	188	4	no	—
119	C₂×C₈	—	188	8	no	—
120	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
121	C₂×C₈	#10625329	1	4	yes	—
122	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
123	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	#10625325	1	8	yes	—
127	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
128	C₄²	#10625329	1	2	yes	—
129	C₄²	C₂×Dic₉₄	1	2	yes	—
130	C₄²	C₂×D₁₈₈	1	2	yes	—
131	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
136	C₂×C₄	#13093956	1	4	yes	—
137	C₂×C₄	#13093956	1	4	yes	—
138	C₂×C₄	#13093960	1	4	yes	—
139	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
140	C₂×C₄	#13094021	1	1	yes	characteristic
141	C₂×C₄	#13094024	1	2	yes	—
142	C₂×C₄	#13094024	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
144	C₂×C₄	C₂×#10625329	1	2	yes	—
145	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
146	C₂×C₄	C₂×#10625325	1	4	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	376	4	no	—
149	C₈	C₄⋊(C₄₇⋊C₈)	1	8	yes	complemented
150	C₂²	#15688993	1	1	yes	characteristic
151	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
152	C₂²	#15689147	1	1	yes	characteristic
153	C₂²	C₂×#13093956	1	2	yes	—
154	C₂²	C₂×#13093960	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	2	4	no	—
157	C₄	#15688874	1	8	yes	complemented
158	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
159	C₄	#15689147	1	2	yes	—
160	C₄	#15689496	1	2	yes	—
161	C₄	#15689521	1	2	yes	—
162	C₂	C₂×#15688874	1	2	yes	—
163	C₂	#20242144	1	2	yes	—
164	C₂	#20243540	1	1	yes	characteristic
165	C₂	#20243545	1	1	yes	characteristic
166	C₂	#20243560	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#8805
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
24. C₄×C₈×C₄₇	C₈	semidirect
64. C₈×C₄₇	C₄⋊C₈	semidirect
74. C₄×C₄₇	#334	semidirect
95. C₄₇	#8805	semidirect
108. C₄×C₈	C₄₇⋊C₈	semidirect
149. C₈	C₄⋊(C₄₇⋊C₈)	semidirect
157. C₄	#15688874	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
C₂×#15688874	2
C₄²×C₈×C₄₇	2
#8805	47