Subgroups — #26235230

Order 12032.

822 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
2	#20244147	C₂	1	6	yes	maximal
3	#20243516	C₂	1	3	yes	maximal
4	C₄×#13093958	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₄₇	—	2	2	no	—
7	C₂×#13093958	C₂²	1	6	yes	—
8	#15689022	—	2	6	no	—
9	#15688993	C₂²	1	6	yes	—
10	C₄×C₄⋊Dic₄₇	C₂²	1	6	yes	—
11	#15688886	C₂²	1	3	yes	—
12	#15688886	C₂²	1	6	yes	—
13	#15688886	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₄₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
18	Q₈×Dic₄₇	—	4	8	no	—
19	C₂×C₄⋊Dic₄₇	—	2	6	no	—
20	C₂×C₄⋊Dic₄₇	—	2	6	no	—
21	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
23	C₂×C₄×Dic₄₇	—	2	6	no	—
24	#13093958	—	2	12	no	—
25	C₄₇⋊#95	—	2	6	no	—
26	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
29	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
30	C₄×Q₈×C₄₇	Q₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
33	C₄⋊C₈×C₄₇	C₂×C₄	1	12	yes	—
34	C₄×C₈×C₄₇	C₂×C₄	1	12	yes	—
35	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
36	C₂×Q₈×C₄₇	D₈	1	4	yes	—
37	C₂×Q₈×C₄₇	Q₁₆	1	4	yes	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₄₇	—	2	2	no	—
42	C₂²×Dic₄₇	—	2	6	no	—
43	C₂×C₈×C₄₇	—	2	6	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
46	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
47	C₄⋊C₄×C₄₇	C₄⋊C₄	1	12	yes	—
48	C₄²×C₄₇	C₂²×C₄	1	6	yes	—
49	C₄²×C₄₇	C₂×Q₈	1	6	yes	—
50	C₄⋊Dic₄₇	—	2	2	no	—
51	C₄⋊Dic₄₇	—	2	6	no	—
52	C₄⋊Dic₄₇	—	2	6	no	—
53	C₄⋊Dic₄₇	—	4	12	no	—
54	C₄⋊Dic₄₇	—	8	6	no	—
55	C₄⋊Dic₄₇	#41	1	4	yes	—
56	C₄×Dic₄₇	—	4	12	no	—
57	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
58	Q₈×C₄₇	#107	1	16	yes	complemented
59	C₂×C₄×C₄₇	—	2	6	no	—
60	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
61	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
62	C₂×C₄×C₄₇	C₂×D₈	1	6	yes	—
63	C₂×C₄×C₄₇	C₂×Q₁₆	1	6	yes	—
64	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
67	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
68	C₂×Dic₄₇	—	2	4	no	—
69	C₂×Dic₄₇	—	4	2	no	—
70	C₂×Dic₄₇	—	4	2	no	—
71	C₂×Dic₄₇	—	4	6	no	—
72	C₂×Dic₄₇	—	4	6	no	—
73	C₂×Dic₄₇	—	4	6	no	—
74	C₈×C₄₇	—	2	6	no	—
75	C₈×C₄₇	—	2	2	no	—
76	#22740	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄₇	#548	1	2	yes	—
79	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
80	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
81	C₂²×C₄₇	#577	1	2	yes	—
82	C₄×C₄₇	—	2	6	no	—
83	C₄×C₄₇	C₂×#107	1	12	yes	—
84	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
85	C₄×C₄₇	#519	1	1	yes	characteristic
86	C₄×C₄₇	#557	1	2	yes	—
87	Dic₄₇	—	4	4	no	—
88	Dic₄₇	—	8	6	no	—
89	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
90	Q₈×C₄⋊C₄	—	47	2	no	—
91	#1947	—	47	6	no	—
92	#1834	—	47	3	no	—
93	C₄×#107	—	47	3	no	—
94	C₂×C₄₇	#2351	1	1	yes	characteristic
95	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
96	C₂×C₄₇	#2912	1	1	yes	characteristic
97	C₂×C₄₇	#3358	1	2	yes	—
98	C₂×C₄₇	#3363	1	2	yes	—
99	C₂×C₄×Q₈	—	94	2	no	—
100	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
101	C₈×Q₈	Dic₄₇	1	8	yes	complemented
102	C₂×#107	—	47	6	no	—
103	C₂×C₄⋊C₈	C₂×D₄₇	1	3	yes	—
104	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
105	#389	—	94	6	no	—
106	#384	—	47	6	no	—
107	C₄×C₄⋊C₄	—	47	6	no	—
108	#340	—	47	3	no	—
109	#340	—	47	6	no	—
110	#340	—	47	1	no	—
111	C₄₇	#22740	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
115	C₄×Q₈	—	188	8	no	—
116	C₄×Q₈	Dic₉₄	1	4	yes	—
117	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
118	C₂×C₄⋊C₄	—	47	2	no	—
119	C₂×C₄⋊C₄	—	47	6	no	—
120	C₂×C₄⋊C₄	—	94	6	no	—
121	C₂×C₄⋊C₄	—	94	6	no	—
122	C₂×C₄⋊C₄	D₁₈₈	1	3	yes	—
123	C₂×C₄²	—	94	6	no	—
124	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
125	#107	—	94	12	no	—
126	C₄⋊C₈	C₂×Dic₄₇	1	12	yes	—
127	C₄×C₈	C₂×Dic₄₇	1	12	yes	—
128	#95	—	94	6	no	—
129	C₂×Q₈	D₃₇₆	1	4	yes	—
130	C₂×Q₈	Dic₁₈₈	1	4	yes	—
131	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
132	C₂²×C₄	—	94	2	no	—
133	C₂²×C₄	—	94	6	no	—
134	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
135	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
136	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
140	C₂×C₈	C₂²×Dic₄₇	1	6	yes	—
141	C₄⋊C₄	—	47	4	no	—
142	C₄⋊C₄	—	94	2	no	—
143	C₄⋊C₄	—	94	6	no	—
144	C₄⋊C₄	—	94	6	no	—
145	C₄⋊C₄	—	376	6	no	—
146	C₄⋊C₄	—	188	12	no	—
147	C₄⋊C₄	C₄⋊Dic₄₇	1	12	yes	—
148	C₄²	—	188	12	no	—
149	C₄²	C₂×Dic₉₄	1	6	yes	—
150	C₄²	C₂²×Dic₄₇	1	6	yes	—
151	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
152	Q₈	#13093958	1	16	yes	complemented
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	94	4	no	—
155	C₂×C₄	—	188	6	no	—
156	C₂×C₄	—	188	6	no	—
157	C₂×C₄	—	188	6	no	—
158	C₂×C₄	—	188	2	no	—
159	C₂×C₄	—	188	2	no	—
160	C₂×C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
162	C₂×C₄	C₂×D₃₇₆	1	6	yes	—
163	C₂×C₄	C₂×Dic₁₈₈	1	6	yes	—
164	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
165	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
166	C₂×C₄	#41×D₄₇	1	2	yes	—
167	C₂×C₄	#13094072	1	2	yes	—
168	C₈	—	2	6	no	—
169	C₈	—	2	2	no	—
170	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
171	C₂²	#15689785	1	2	yes	—
172	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
173	C₂²	C₂²×Dic₁₈₈	1	1	yes	characteristic
174	C₂²	#15689935	1	2	yes	—
175	C₄	—	2	6	no	—
176	C₄	—	376	6	no	—
177	C₄	—	188	4	no	—
178	C₄	C₂×#13093958	1	12	yes	—
179	C₄	#41×Dic₄₇	1	1	yes	characteristic
180	C₄	#15689601	1	1	yes	characteristic
181	C₄	#15689849	1	2	yes	—
182	C₂	#20246309	1	1	yes	characteristic
183	C₂	C₂²×#13093958	1	1	yes	characteristic
184	C₂	#20251983	1	1	yes	characteristic
185	C₂	#20256024	1	2	yes	—
186	C₂	#20256067	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	#22740
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₄₇	C₄	semidirect
58. Q₈×C₄₇	#107	semidirect
101. C₈×Q₈	Dic₄₇	semidirect
111. C₄₇	#22740	semidirect
152. Q₈	#13093958	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₄₇	2
#20244147	2
#20243516	2
C₄×#13093958	2
C₂×C₈×Q₈×C₄₇	2
#22740	47