Finite Group Databasebeta

Subgroups — #26137168

Order 12032.

176 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245662	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241590	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241589	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₄₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241566	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1415	C₂	1	1	yes	maximal, characteristic
8	#15689527	—	2	1	no	—
9	D₈×Dic₄₇	—	2	1	no	—
10	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	#15688949	C₂²	1	1	yes	characteristic
12	#15688949	C₂²	1	1	yes	characteristic
13	#15688948	—	2	1	no	—
14	#15688924	—	2	1	no	—
15	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#366	C₂²	1	1	yes	characteristic
19	#13094023	—	4	1	no	—
20	D₄×Dic₄₇	—	4	1	no	—
21	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
22	#13093958	—	2	1	no	—
23	#13093958	—	2	1	no	—
24	#13093957	—	4	1	no	—
25	#13093945	—	4	1	no	—
26	C₈×Dic₄₇	D₄	1	1	yes	characteristic
27	C₄₇⋊C₃₂	—	2	1	no	—
28	C₂×D₈×C₄₇	—	2	1	no	—
29	D₁₆×C₄₇	—	2	1	no	—
30	D₁₆×C₄₇	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
32	#107×C₄₇	—	2	1	no	—
33	C₃₂×C₄₇	—	2	1	no	—
34	C₂×D₄×C₄₇	—	4	1	no	—
35	C₂²×Dic₄₇	—	8	1	no	—
36	C₂×Dic₉₄	—	8	1	no	—
37	D₈×C₄₇	—	2	2	no	—
38	D₈×C₄₇	—	4	1	no	—
39	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₄₇	—	4	1	no	—
41	C₂²⋊Dic₄₇	—	8	1	no	—
42	C₄⋊Dic₄₇	—	4	1	no	—
43	C₄⋊Dic₄₇	—	4	1	no	—
44	Dic₄₇⋊C₄	—	8	1	no	—
45	C₄×Dic₄₇	D₈	1	1	yes	characteristic
46	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
47	C₁₆×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₄₇	#41	1	1	yes	characteristic
49	C₄₇⋊C₁₆	—	2	1	no	—
50	C₂³×C₄₇	—	8	1	no	—
51	D₄×C₄₇	—	4	2	no	—
52	D₄×C₄₇	—	8	1	no	—
53	C₂×C₄×C₄₇	—	8	1	no	—
54	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₄₇	—	8	1	no	—
56	C₂×Dic₄₇	—	8	1	no	—
57	C₂×Dic₄₇	—	16	2	no	—
58	C₂×Dic₄₇	D₁₆	1	1	yes	characteristic
59	C₂×Dic₄₇	D₁₆	1	1	yes	characteristic
60	Dic₉₄	—	16	2	no	—
61	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₄₇	#137	1	1	yes	characteristic
63	C₄₇⋊C₈	—	2	1	no	—
64	#13705	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	8	2	no	—
66	C₂²×C₄₇	—	16	1	no	—
67	C₂²×C₄₇	—	16	1	no	—
68	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₄₇	—	16	1	no	—
70	C₄×C₄₇	#474	1	1	yes	characteristic
71	C₄×C₄₇	#510	1	1	yes	characteristic
72	Dic₄₇	—	2	1	no	—
73	Dic₄₇	—	16	1	no	—
74	Dic₄₇	—	16	1	no	—
75	Dic₄₇	SD₆₄	1	2	yes	—
76	#2252	—	47	1	no	—
77	C₄×D₁₆	—	47	1	no	—
78	#1424	—	47	1	no	—
79	#1424	—	47	1	no	—
80	C₄⋊C₃₂	—	47	1	no	—
81	#1415	—	47	1	no	—
82	#1415	D₄₇	1	1	yes	characteristic
83	C₂×C₄₇	—	16	2	no	—
84	C₂×C₄₇	#2224	1	1	yes	characteristic
85	C₂×C₄₇	C₂×SD₆₄	1	1	yes	characteristic
86	C₂×C₄₇	#2263	1	1	yes	characteristic
87	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
88	#500	—	94	1	no	—
89	C₄×D₈	—	94	1	no	—
90	C₂×C₃₂	—	47	1	no	—
91	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
92	#366	—	47	1	no	—
93	#366	—	47	1	no	—
94	#366	—	94	1	no	—
95	#366	C₂×D₄₇	1	1	yes	characteristic
96	#357	—	94	1	no	—
97	C₄×C₁₆	—	47	1	no	—
98	C₄₇	#13705	1	1	yes	characteristic, complemented, Sylow
99	C₂×D₈	—	2	1	no	—
100	C₄⋊Q₈	—	188	1	no	—
101	C₄×D₄	—	188	1	no	—
102	D₁₆	—	2	1	no	—
103	D₁₆	Dic₉₄	1	2	yes	complemented
104	C₂×C₁₆	—	47	1	no	—
105	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
106	#107	—	2	1	no	—
107	#107	—	94	1	no	—
108	#107	—	94	1	no	—
109	#107	—	188	1	no	—
110	#102	—	188	1	no	—
111	C₄×C₈	—	47	1	no	—
112	C₃₂	—	2	1	no	—
113	C₃₂	—	94	1	no	—
114	C₂×Q₈	—	376	1	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂²×C₄	—	376	1	no	—
117	D₈	—	2	2	no	—
118	D₈	—	4	1	no	—
119	C₂×C₈	—	47	1	no	—
120	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	376	1	no	—
123	C₄⋊C₄	—	188	1	no	—
124	C₄⋊C₄	—	188	1	no	—
125	C₂²⋊C₄	—	376	1	no	—
126	C₄²	—	47	1	no	—
127	C₁₆	—	94	1	no	—
128	C₁₆	C₂×Dic₉₄	1	1	yes	characteristic
129	C₁₆	#10625352	1	1	yes	characteristic
130	C₂³	—	8	1	no	—
131	Q₈	—	752	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	47	1	no	—
136	C₂×C₄	—	47	1	no	—
137	C₂×C₄	—	376	1	no	—
138	C₂×C₄	—	376	1	no	—
139	C₂×C₄	—	752	2	no	—
140	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
141	C₈	—	94	1	no	—
142	C₈	#13094001	1	1	yes	characteristic
143	C₈	#13094053	1	1	yes	characteristic
144	C₂²	—	8	2	no	—
145	C₂²	—	16	1	no	—
146	C₂²	—	16	1	no	—
147	C₂²	D₁₆×D₄₇	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	47	2	no	—
150	C₄	—	94	1	no	—
151	C₄	—	752	1	no	—
152	C₄	—	752	1	no	—
153	C₄	#15689401	1	1	yes	characteristic
154	C₄	#15689562	1	1	yes	characteristic
155	C₂	—	16	2	no	—
156	C₂	#20245523	1	1	yes	characteristic
157	C₂	SD₆₄×D₄₇	1	1	yes	characteristic
158	C₂	#20245699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₄₇
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1415	C₂
socle	C₂²×C₄₇	C₂×D₁₆
prime radical	C₂×D₁₆×C₄₇	C₂²
hypercenter	#1415	D₄₇
hypocenter	C₄₇	#13705
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1415

Complements

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

N	H ≅ G/N	type
1. #20245662	C₂	semidirect
3. #20241590	C₂	semidirect
4. #20241589	C₂	semidirect
5. Dic₄₇⋊C₃₂	C₂	semidirect
30. D₁₆×C₄₇	Q₈	semidirect
98. C₄₇	#13705	semidirect
103. D₁₆	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20245662	2
D₁₆×Dic₄₇	2
#20241590	2
#20241589	2
Dic₄₇⋊C₃₂	2
#20241566	2
C₄₇×#1415	2
#13705	47