Subgroups — #26102806

Order 12032.

414 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	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
4	C₂×C₄×C₄₇⋊C₈	—	2	2	no	—
5	#15688878	—	2	2	no	—
6	C₂×C₄×C₈×C₄₇	—	2	2	no	—
7	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
8	C₄³×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₈)	C₈	1	8	yes	complemented
12	C₄×C₄₇⋊C₈	—	4	2	no	—
13	C₄×C₄₇⋊C₈	—	4	2	no	—
14	C₂²×C₈×C₄₇	—	2	2	no	—
15	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	D₄	1	2	yes	—
18	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
20	C₄×C₈×C₄₇	—	2	8	no	—
21	C₄×C₈×C₄₇	—	2	2	no	—
22	C₄×C₈×C₄₇	C₈	1	4	yes	complemented
23	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	—	2	8	no	—
29	C₂×C₈×C₄₇	—	2	4	no	—
30	C₄²×C₄₇	—	2	4	no	—
31	C₄²×C₄₇	—	2	2	no	—
32	C₄²×C₄₇	C₂×C₈	1	4	yes	—
33	C₄²×C₄₇	C₂×C₈	1	2	yes	—
34	C₄²×C₄₇	M₁₆	1	4	yes	—
35	C₄²×C₄₇	M₁₆	1	2	yes	—
36	C₄²×C₄₇	D₈	1	1	yes	characteristic
37	C₄²×C₄₇	SD₁₆	1	2	yes	—
38	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	—	2	8	no	—
40	C₂×C₄₇⋊C₈	—	4	4	no	—
41	C₂×C₄₇⋊C₈	—	4	4	no	—
42	C₂³×C₄₇	#95	1	1	yes	characteristic
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#97	1	2	yes	—
49	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
51	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
52	C₂×C₄×C₄₇	#102	1	2	yes	—
53	C₂×C₄×C₄₇	#103	1	2	yes	—
54	C₂×C₄×C₄₇	#104	1	4	yes	—
55	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
56	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#108	1	2	yes	—
60	C₈×C₄₇	—	2	8	no	—
61	C₄₇⋊C₈	—	4	8	no	—
62	#7015	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#336	1	2	yes	—
64	C₂²×C₄₇	#336	1	1	yes	characteristic
65	C₂²×C₄₇	#339	1	2	yes	—
66	C₂²×C₄₇	#340	1	1	yes	characteristic
67	C₂²×C₄₇	#341	1	1	yes	characteristic
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	—	2	4	no	—
70	C₄×C₄₇	C₈²	1	1	yes	characteristic
71	C₄×C₄₇	#322	1	1	yes	characteristic
72	C₄×C₄₇	#322	1	2	yes	—
73	C₄×C₄₇	D₄⋊C₈	1	4	yes	—
74	C₄×C₄₇	Q₈⋊C₈	1	4	yes	—
75	C₄×C₄₇	#334	1	2	yes	—
76	C₄×C₄₇	#335	1	2	yes	—
77	C₄×C₄⋊C₈	—	47	2	no	—
78	C₄²×C₈	D₄₇	1	1	yes	characteristic
79	C₂×C₄₇	#1273	1	1	yes	characteristic
80	C₂×C₄₇	#1276	1	1	yes	characteristic
81	C₂×C₄₇	#1277	1	1	yes	characteristic
82	C₂×C₄₇	#1278	1	2	yes	—
83	C₂×C₄₇	#1279	1	2	yes	—
84	C₂×C₄⋊C₈	—	47	4	no	—
85	C₂×C₄×C₈	—	2	2	no	—
86	C₂×C₄×C₈	—	94	2	no	—
87	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
88	C₄³	C₂×D₄₇	1	1	yes	characteristic
89	#336	—	94	2	no	—
90	C₄₇	#7015	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	94	4	no	—
93	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₄₇	1	2	yes	—
95	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
98	C₄⋊C₈	—	47	8	no	—
99	C₄⋊C₈	—	94	4	no	—
100	C₄×C₈	—	2	8	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	188	2	no	—
103	C₄×C₈	—	188	2	no	—
104	C₄×C₈	C₄₇⋊C₈	1	4	yes	complemented
105	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
106	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
107	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
108	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
109	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	94	8	no	—
113	C₂×C₈	—	188	4	no	—
114	C₂×C₈	—	188	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	2	no	—
117	C₄²	C₈×D₄₇	1	4	yes	—
118	C₄²	#10625324	1	4	yes	—
119	C₄²	#10625325	1	2	yes	—
120	C₄²	D₃₇₆	1	1	yes	characteristic
121	C₄²	Dic₁₈₈	1	1	yes	characteristic
122	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
123	C₄²	#10625329	1	2	yes	—
124	C₂³	C₄₇⋊#95	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
129	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
130	C₂×C₄	#13093934	1	2	yes	—
131	C₂×C₄	#13093935	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
133	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
134	C₂×C₄	#13093946	1	2	yes	—
135	C₂×C₄	#13093949	1	2	yes	—
136	C₂×C₄	#13093952	1	4	yes	—
137	C₂×C₄	Dic₄₇⋊C₈	1	4	yes	—
138	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
139	C₂×C₄	#13093956	1	1	yes	characteristic
140	C₂×C₄	#13093958	1	1	yes	characteristic
141	C₂×C₄	#13093960	1	2	yes	—
142	C₈	—	2	8	no	—
143	C₈	—	188	8	no	—
144	C₂²	#15688877	1	2	yes	—
145	C₂²	#15688878	1	1	yes	characteristic
146	C₂²	#15688883	1	2	yes	—
147	C₂²	#15688886	1	1	yes	characteristic
148	C₂²	#15688888	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	4	no	—
151	C₄	C₈×C₄₇⋊C₈	1	1	yes	characteristic
152	C₄	#15688844	1	1	yes	characteristic
153	C₄	#15688845	1	2	yes	—
154	C₄	#15688852	1	4	yes	—
155	C₄	#15688855	1	4	yes	—
156	C₄	#15688874	1	2	yes	—
157	C₄	#15688876	1	2	yes	—
158	C₂	C₄₇⋊#1273	1	1	yes	characteristic
159	C₂	#20241221	1	1	yes	characteristic
160	C₂	#20241223	1	1	yes	characteristic
161	C₂	#20241225	1	2	yes	—
162	C₂	#20241228	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₁₈₈
commutator	C₄×C₄₇	C₈²
Frattini	C₄³	C₂×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#7015
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
11. C₄⋊(C₄₇⋊C₈)	C₈	semidirect
22. C₄×C₈×C₄₇	C₈	semidirect
90. C₄₇	#7015	semidirect
104. C₄×C₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
C₄²×C₈×C₄₇	2
#7015	47