Finite Group Databasebeta

Subgroups — #26136900

Order 12032.

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

normal subgroups only

This group has 194 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	#20245700	C₂	1	1	yes	maximal, characteristic
2	#20245274	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245263	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241593	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241579	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241523	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
9	#15689240	—	2	1	no	—
10	#15689205	—	2	1	no	—
11	#15688960	C₄	1	2	yes	complemented
12	#15688958	C₄	1	2	yes	complemented
13	#15688957	C₂²	1	1	yes	characteristic
14	#15688953	C₂²	1	1	yes	characteristic
15	#15688930	—	2	1	no	—
16	#15688923	—	2	1	no	—
17	#15688898	C₂²	1	1	yes	characteristic
18	C₄₇×#508	C₂²	1	1	yes	characteristic
19	C₄₇×M₆₄	C₂²	1	1	yes	characteristic
20	C₄₇×#367	C₂²	1	1	yes	characteristic
21	C₂×D₃₇₆	—	2	1	no	—
22	#13094031	—	4	1	no	—
23	#13093986	—	4	1	no	—
24	D₇₅₂	—	2	1	no	—
25	D₇₅₂	C₂×C₄	1	2	yes	—
26	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
27	#13093960	—	2	1	no	—
28	#13093951	—	4	1	no	—
29	#13093944	—	4	1	no	—
30	C₈×Dic₄₇	D₄	1	1	yes	characteristic
31	C₄₇⋊C₃₂	C₂×C₄	1	2	yes	complemented
32	#135×C₄₇	—	2	1	no	—
33	Q₃₂×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
34	SD₃₂×C₄₇	—	2	1	no	—
35	D₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
37	#107×C₄₇	—	2	1	no	—
38	C₃₂×C₄₇	—	2	1	no	—
39	#41×C₄₇	—	4	1	no	—
40	C₂×D₁₈₈	—	4	1	no	—
41	C₂×C₄×D₄₇	—	8	1	no	—
42	Q₁₆×C₄₇	—	2	1	no	—
43	SD₁₆×C₄₇	—	4	1	no	—
44	D₈×C₄₇	—	2	1	no	—
45	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₄₇	—	4	1	no	—
47	(C₂×D₄₇)⋊C₄	—	8	1	no	—
48	C₄×Dic₄₇	D₈	1	1	yes	characteristic
49	#10625329	—	4	1	no	—
50	#10625329	—	8	1	no	—
51	C₂×C₄₇⋊C₈	—	8	1	no	—
52	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
53	D₃₇₆	—	2	2	no	—
54	D₃₇₆	—	4	1	no	—
55	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₄₇	#41	1	1	yes	characteristic
57	C₄₇⋊C₁₆	—	2	1	no	—
58	C₂²×D₄₇	—	8	1	no	—
59	Q₈×C₄₇	—	4	1	no	—
60	D₄×C₄₇	—	4	1	no	—
61	D₄×C₄₇	—	8	1	no	—
62	C₂×C₄×C₄₇	—	8	1	no	—
63	C₂×C₄×C₄₇	—	8	1	no	—
64	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
65	C₂×Dic₄₇	—	2	1	no	—
66	D₁₈₈	—	4	2	no	—
67	D₁₈₈	—	8	1	no	—
68	C₄×D₄₇	—	16	2	no	—
69	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
70	C₈×C₄₇	#135	1	1	yes	characteristic
71	C₄₇⋊C₈	—	2	1	no	—
72	C₄₇⋊C₈	—	8	1	no	—
73	#13658	—	47	1	no	maximal, Sylow
74	C₂²×C₄₇	—	8	1	no	—
75	C₂²×C₄₇	#509	1	1	yes	characteristic
76	C₂×D₄₇	—	8	2	no	—
77	C₂×D₄₇	—	16	1	no	—
78	C₂×D₄₇	—	16	1	no	—
79	C₄×C₄₇	—	8	1	no	—
80	C₄×C₄₇	—	16	1	no	—
81	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
82	C₄×C₄₇	#509	1	1	yes	characteristic
83	Dic₄₇	—	4	1	no	—
84	#2263	—	47	1	no	—
85	#2179	—	47	1	no	—
86	#2177	—	47	1	no	—
87	#1426	—	47	1	no	—
88	#1419	—	47	1	no	—
89	#1418	D₄₇	1	1	yes	characteristic, complemented
90	#1398	—	47	1	no	—
91	C₂×C₄₇	—	2	1	no	—
92	C₂×C₄₇	—	16	1	no	—
93	C₂×C₄₇	#2177	1	1	yes	characteristic
94	D₄₇	—	16	2	no	—
95	#508	C₂×D₄₇	1	1	yes	characteristic
96	C₂×D₁₆	—	47	1	no	—
97	#443	—	94	1	no	—
98	C₄×D₈	—	94	1	no	—
99	SD₆₄	—	47	2	no	—
100	D₃₂	—	47	2	no	—
101	M₆₄	—	47	1	no	—
102	M₆₄	C₂×D₄₇	1	1	yes	characteristic
103	#368	—	47	1	no	—
104	#367	C₂×D₄₇	1	1	yes	characteristic
105	#359	—	94	1	no	—
106	#357	—	94	1	no	—
107	#346	—	47	1	no	—
108	C₄₇	#13658	1	1	yes	characteristic, complemented, Sylow
109	#135	—	2	1	no	—
110	C₂×D₈	—	94	1	no	—
111	#131	—	188	1	no	—
112	C₄×D₄	—	188	1	no	—
113	Q₃₂	C₄×D₄₇	1	1	yes	characteristic, complemented
114	SD₃₂	—	2	1	no	—
115	D₁₆	—	47	2	no	—
116	D₁₆	—	94	1	no	—
117	D₁₆	C₄×D₄₇	1	1	yes	characteristic, complemented
118	C₂×C₁₆	—	47	1	no	—
119	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
120	#108	—	94	1	no	—
121	#107	—	2	1	no	—
122	#104	—	188	1	no	—
123	#102	—	188	1	no	—
124	C₄×C₈	—	47	1	no	—
125	C₃₂	—	2	1	no	—
126	C₃₂	—	47	2	no	—
127	#41	—	4	1	no	—
128	C₂×D₄	—	188	1	no	—
129	C₂²×C₄	—	376	1	no	—
130	Q₁₆	—	2	1	no	—
131	SD₁₆	—	4	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	94	2	no	—
134	D₈	—	188	1	no	—
135	M₁₆	—	376	1	no	—
136	M₁₆	—	188	1	no	—
137	C₂×C₈	—	47	1	no	—
138	C₂×C₈	—	376	1	no	—
139	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
140	C₄⋊C₄	—	4	1	no	—
141	C₂²⋊C₄	—	376	1	no	—
142	C₄²	—	47	1	no	—
143	C₁₆	—	94	1	no	—
144	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
145	C₁₆	#10625352	1	1	yes	characteristic
146	C₂³	—	376	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	1	no	—
149	D₄	—	8	1	no	—
150	D₄	—	376	1	no	—
151	D₄	—	188	2	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	94	1	no	—
155	C₂×C₄	—	752	2	no	—
156	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
157	C₈	—	94	1	no	—
158	C₈	—	376	1	no	—
159	C₈	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
160	C₈	#13094045	1	1	yes	characteristic
161	C₂²	—	8	1	no	—
162	C₂²	—	376	2	no	—
163	C₂²	—	752	1	no	—
164	C₂²	—	752	1	no	—
165	C₂²	#15689557	1	1	yes	characteristic
166	C₄	—	8	1	no	—
167	C₄	—	16	1	no	—
168	C₄	—	188	1	no	—
169	C₄	#15689203	1	1	yes	characteristic
170	C₄	#15689560	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	1	no	—
173	C₂	—	752	2	no	—
174	C₂	#20245261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20245261
commutator	C₁₆×C₄₇	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1418	C₂
socle	C₂×C₄₇	#2177
prime radical	#20245700	C₂
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	#1418	D₄₇
hypocenter	C₄₇	#13658
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
2. #20245274	C₂	semidirect
3. #20245263	C₂	semidirect
4. #20241593	C₂	semidirect
5. #20241579	C₂	semidirect
6. #20241523	C₂	semidirect
7. C₄₇×#1418	C₂	semidirect
11. #15688960	C₄	semidirect
12. #15688958	C₄	semidirect
31. C₄₇⋊C₃₂	C₂×C₄	semidirect
33. Q₃₂×C₄₇	C₂×C₄	semidirect
35. D₁₆×C₄₇	C₂×C₄	semidirect
89. #1418	D₄₇	semidirect
108. C₄₇	#13658	semidirect
113. Q₃₂	C₄×D₄₇	semidirect
117. D₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20245700	2
#20245274	2
#20245263	2
#20241593	2
#20241579	2
#20241523	2
C₄₇×#1418	2
#13658	47