Finite Group Databasebeta

Subgroups — #26136782

Order 12032.

194 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#15688956	C₂	1	1	yes	maximal, characteristic
2	#20245169	C₂	1	1	yes	maximal, characteristic
3	#366×D₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241590	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241589	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241587	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
9	#15689160	—	2	1	no	—
10	#107×D₄₇	—	2	1	no	—
11	#15688956	C₄	1	4	yes	complemented
12	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
13	#15688950	—	2	1	no	—
14	#15688949	C₂²	1	1	yes	characteristic
15	#15688949	C₂²	1	1	yes	characteristic
16	#15688948	—	2	1	no	—
17	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#366	C₂²	1	1	yes	characteristic
19	C₄₇×#366	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
21	#13093982	—	4	1	no	—
22	C₄⋊C₄×D₄₇	—	4	1	no	—
23	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₄₇	C₂×C₄	1	2	yes	—
25	C₁₆×D₄₇	Q₈	1	2	yes	—
26	#13093958	—	2	1	no	—
27	#13093958	—	2	1	no	—
28	#13093957	—	4	1	no	—
29	#13093955	—	4	1	no	—
30	C₄₇⋊C₃₂	C₂×C₄	1	2	yes	complemented
31	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
32	#107×C₄₇	—	2	1	no	—
33	#107×C₄₇	—	2	1	no	—
34	C₃₂×C₄₇	C₂×C₄	1	2	yes	complemented
35	C₂×C₄×D₄₇	—	8	1	no	—
36	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₄₇	—	4	1	no	—
39	C₄⋊C₄×C₄₇	—	4	1	no	—
40	(C₂×D₄₇)⋊C₄	—	8	1	no	—
41	C₄⋊Dic₄₇	—	4	1	no	—
42	C₄⋊Dic₄₇	—	4	1	no	—
43	Dic₄₇⋊C₄	—	8	1	no	—
44	C₄×Dic₄₇	—	8	1	no	—
45	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
46	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
47	C₈×D₄₇	Q₁₆	1	2	yes	—
48	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₄₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
51	C₂²×D₄₇	D₁₆	1	1	yes	characteristic
52	C₂×C₄×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	D₁₆	1	1	yes	characteristic
58	C₄×D₄₇	—	16	2	no	—
59	C₄×D₄₇	#107	1	2	yes	—
60	C₄×D₄₇	Q₃₂	1	2	yes	—
61	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₄₇⋊C₈	#107	1	2	yes	—
64	#13634	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
66	C₂×D₄₇	—	2	1	no	—
67	C₂×D₄₇	—	2	1	no	—
68	C₂×D₄₇	#366	1	2	yes	—
69	C₄×C₄₇	—	16	1	no	—
70	C₄×C₄₇	—	16	1	no	—
71	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
72	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
73	Dic₄₇	—	16	1	no	—
74	Dic₄₇	—	16	1	no	—
75	Dic₄₇	#366	1	2	yes	—
76	C₂×M₆₄	—	47	1	no	—
77	#2158	—	47	1	no	—
78	C₂×#366	—	47	1	no	—
79	#1424	—	47	1	no	—
80	#1424	—	47	1	no	—
81	#1423	—	47	1	no	—
82	#1423	D₄₇	1	1	yes	characteristic, complemented
83	C₂×C₄₇	C₂×#366	1	1	yes	characteristic
84	C₂×C₄₇	#2263	1	1	yes	characteristic
85	C₂×C₄₇	#2264	1	1	yes	characteristic
86	D₄₇	—	2	2	no	—
87	C₂²×C₁₆	—	47	1	no	—
88	#427	—	94	1	no	—
89	C₂×#107	—	94	1	no	—
90	M₆₄	—	47	4	no	—
91	C₂×C₃₂	—	47	1	no	—
92	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
93	#367	—	94	1	no	—
94	#366	—	47	1	no	—
95	#366	—	47	1	no	—
96	#366	—	94	1	no	—
97	#366	C₂×D₄₇	1	1	yes	characteristic
98	#366	C₂×D₄₇	1	1	yes	characteristic
99	C₄₇	#13634	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	47	1	no	—
101	#117	—	188	1	no	—
102	C₂×C₄⋊C₄	—	188	1	no	—
103	C₂×C₁₆	—	47	1	no	—
104	C₂×C₁₆	—	47	2	no	—
105	C₂×C₁₆	—	47	2	no	—
106	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
107	#107	—	2	1	no	—
108	#107	—	2	1	no	—
109	#107	—	94	1	no	—
110	#107	—	94	1	no	—
111	#107	—	188	1	no	—
112	#106	—	188	1	no	—
113	C₃₂	—	47	2	no	—
114	C₃₂	C₄×D₄₇	1	2	yes	complemented
115	C₂²×C₄	—	47	1	no	—
116	C₂²×C₄	—	376	1	no	—
117	C₂×C₈	—	47	1	no	—
118	C₂×C₈	—	47	2	no	—
119	C₂×C₈	—	47	2	no	—
120	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	376	1	no	—
124	C₄⋊C₄	—	188	1	no	—
125	C₄⋊C₄	—	188	1	no	—
126	C₂²⋊C₄	—	376	1	no	—
127	C₄²	—	376	1	no	—
128	C₁₆	—	47	2	no	—
129	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₁₆	Q₈×D₄₇	1	1	yes	characteristic
131	C₂³	—	47	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	47	1	no	—
135	C₂×C₄	—	47	2	no	—
136	C₂×C₄	—	47	2	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	—	47	2	no	—
142	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
143	C₈	Q₁₆×D₄₇	1	1	yes	characteristic
144	C₂²	—	47	2	no	—
145	C₂²	—	94	1	no	—
146	C₂²	—	94	1	no	—
147	C₂²	D₁₆×D₄₇	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	47	2	no	—
151	C₄	—	752	1	no	—
152	C₄	—	752	1	no	—
153	C₄	#107×D₄₇	1	1	yes	characteristic
154	C₄	Q₃₂×D₄₇	1	1	yes	characteristic
155	C₂	—	94	2	no	—
156	C₂	#366×D₄₇	1	1	yes	characteristic
157	C₂	#20245699	1	1	yes	characteristic
158	C₂	#20245707	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₄₇
commutator	C₁₆×C₄₇	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1423	C₂
socle	C₂²×C₄₇	C₂×D₁₆
prime radical	C₂²×D₄₇	D₁₆
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	#1423	D₄₇
hypocenter	C₄₇	#13634
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

Complements

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

N	H ≅ G/N	type
4. #20241590	C₂	semidirect
5. #20241589	C₂	semidirect
6. #20241587	C₂	semidirect
7. C₄₇×#1423	C₂	semidirect
11. #15688956	C₄	semidirect
30. C₄₇⋊C₃₂	C₂×C₄	semidirect
34. C₃₂×C₄₇	C₂×C₄	semidirect
82. #1423	D₄₇	semidirect
99. C₄₇	#13634	semidirect
114. C₃₂	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688956	2
#20245169	2
#366×D₄₇	2
#20241590	2
#20241589	2
#20241587	2
C₄₇×#1423	2
#13634	47