Subgroups — #26369970

Order 12032.

532 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×C₄₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
2	#20245220	C₂	1	6	yes	maximal, complemented
3	#20245187	C₂	1	3	yes	maximal, complemented
4	#20245144	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	6	yes	—
7	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	6	yes	—
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	3	yes	—
9	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	6	yes	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	6	yes	—
12	C₄×#41×C₄₇	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₄₇	C₄	1	1	yes	characteristic
14	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₄₇	C₄	1	3	yes	—
16	C₈×D₄×C₄₇	C₂²	1	3	yes	—
17	C₄₇×#431	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
19	#13093964	—	4	6	no	—
20	C₂×C₄₇⋊C₁₆	C₂³	1	6	yes	—
21	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
22	C₂×#41×C₄₇	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
25	C₄×Q₈×C₄₇	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₄₇	C₈	1	3	yes	—
28	C₄×D₄×C₄₇	C₂×C₄	1	3	yes	—
29	#117×C₄₇	C₈	1	3	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₄₇	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₄₇	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₄₇	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₄₇	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₄₇	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₄₇	C₂³	1	3	yes	—
37	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
38	#41×C₄₇	—	2	4	no	—
39	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
40	C₂×D₄×C₄₇	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₄₇	—	2	3	no	—
45	C₂×C₈×C₄₇	—	2	3	no	—
46	C₂×C₈×C₄₇	—	4	3	no	—
47	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₄₇	C₂×C₈	1	3	yes	—
52	C₄⋊C₄×C₄₇	C₂×C₈	1	3	yes	—
53	C₂²⋊C₄×C₄₇	C₂×C₈	1	6	yes	—
54	C₄²×C₄₇	C₂×C₈	1	3	yes	—
55	C₄²×C₄₇	C₂×C₈	1	3	yes	—
56	C₄²×C₄₇	C₂²×C₄	1	3	yes	—
57	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₄₇⋊C₁₆	—	2	6	no	—
59	C₄₇⋊C₁₆	—	2	2	no	—
60	C₂³×C₄₇	C₂²×C₈	1	3	yes	—
61	Q₈×C₄₇	M₃₂	1	4	yes	complemented
62	D₄×C₄₇	—	2	6	no	—
63	C₂×C₄×C₄₇	—	2	3	no	—
64	C₂×C₄×C₄₇	—	2	3	no	—
65	C₂×C₄×C₄₇	—	2	6	no	—
66	C₂×C₄×C₄₇	—	2	6	no	—
67	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₄₇	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₄₇	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
73	C₈×C₄₇	—	2	3	no	—
74	C₈×C₄₇	—	2	1	no	—
75	C₈×C₄₇	—	2	3	no	—
76	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
77	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
78	#33505	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	3	no	—
80	C₂²×C₄₇	—	2	3	no	—
81	C₂²×C₄₇	—	2	3	no	—
82	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	—	2	3	no	—
85	C₄×C₄₇	—	2	3	no	—
86	C₄×C₄₇	C₂×M₃₂	1	6	yes	—
87	C₄×C₄₇	#504	1	2	yes	—
88	C₄×C₄₇	#520	1	1	yes	characteristic
89	C₄×C₄₇	#568	1	1	yes	characteristic
90	C₈×#41	D₄₇	1	1	yes	characteristic
91	Q₈×C₁₆	—	47	2	no	—
92	#2169	—	47	6	no	—
93	#2162	—	47	3	no	—
94	#2150	—	47	3	no	—
95	C₂×C₄₇	—	4	3	no	—
96	C₂×C₄₇	#2988	1	1	yes	characteristic
97	C₂×C₄₇	C₂²×M₃₂	1	1	yes	characteristic
98	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
99	C₄×#41	Dic₄₇	1	1	yes	characteristic
100	C₂×M₃₂	—	47	6	no	—
101	C₈×Q₈	Dic₄₇	1	1	yes	characteristic
102	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
103	C₈×D₄	Dic₄₇	1	3	yes	—
104	C₈×D₄	C₂×D₄₇	1	3	yes	—
105	#431	Dic₄₇	1	3	yes	—
106	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
107	C₄⋊C₁₆	—	47	3	no	—
108	C₄⋊C₁₆	—	47	6	no	—
109	C₄⋊C₁₆	—	47	1	no	—
110	C₂²⋊C₁₆	—	47	6	no	—
111	C₄×C₁₆	—	47	6	no	—
112	C₄₇	#33505	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	C₄₇⋊C₈	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
115	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
116	C₄×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
117	C₄×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
118	C₄×D₄	C₄₇⋊C₈	1	3	yes	—
119	C₄×D₄	C₂×Dic₄₇	1	3	yes	—
120	#117	C₄₇⋊C₈	1	3	yes	—
121	C₂×C₄²	C₂×Dic₄₇	1	3	yes	—
122	M₃₂	—	188	6	no	—
123	C₂×C₁₆	—	47	2	no	—
124	C₂×C₁₆	—	47	6	no	—
125	C₄⋊C₈	C₂×Dic₄₇	1	3	yes	—
126	C₄⋊C₈	C₂×Dic₄₇	1	3	yes	—
127	C₂²⋊C₈	C₂×Dic₄₇	1	6	yes	—
128	C₄×C₈	C₂×Dic₄₇	1	3	yes	—
129	C₄×C₈	C₂×Dic₄₇	1	3	yes	—
130	C₄×C₈	C₂²×D₄₇	1	3	yes	—
131	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
132	#41	—	2	4	no	—
133	C₂×Q₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
134	C₂×D₄	C₂×C₄₇⋊C₈	1	3	yes	—
135	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
136	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
137	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
138	C₂×C₈	—	2	3	no	—
139	C₂×C₈	—	2	3	no	—
140	C₂×C₈	—	4	3	no	—
141	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
143	C₂×C₈	C₂²×Dic₄₇	1	3	yes	—
144	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
145	C₄⋊C₄	C₂×C₄₇⋊C₈	1	3	yes	—
146	C₄⋊C₄	C₂×C₄₇⋊C₈	1	3	yes	—
147	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	6	yes	—
148	C₄²	C₂×C₄₇⋊C₈	1	3	yes	—
149	C₄²	C₂×C₄₇⋊C₈	1	3	yes	—
150	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
151	C₄²	C₂²×Dic₄₇	1	3	yes	—
152	C₁₆	—	94	6	no	—
153	C₁₆	—	94	2	no	—
154	C₂³	C₂²×C₄₇⋊C₈	1	3	yes	—
155	Q₈	#13093964	1	4	yes	complemented
156	D₄	—	2	6	no	—
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	3	no	—
160	C₂×C₄	—	2	3	no	—
161	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₄₇⋊C₈	1	3	yes	—
163	C₂×C₄	C₂²×C₄₇⋊C₈	1	3	yes	—
164	C₂×C₄	C₂²×C₄₇⋊C₈	1	3	yes	—
165	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
167	C₈	—	2	3	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	3	no	—
170	C₈	#13094070	1	1	yes	characteristic
171	C₈	#13094071	1	1	yes	characteristic
172	C₂²	—	2	3	no	—
173	C₂²	—	2	3	no	—
174	C₂²	—	2	3	no	—
175	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
176	C₄	—	2	3	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	3	no	—
179	C₄	C₂×#13093964	1	6	yes	—
180	C₄	#15689540	1	2	yes	—
181	C₄	#15689606	1	1	yes	characteristic
182	C₄	#15689890	1	1	yes	characteristic
183	C₂	—	4	3	no	—
184	C₂	#20252499	1	1	yes	characteristic
185	C₂	C₂²×#13093964	1	1	yes	characteristic
186	C₂	C₂×#15689540	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₈×#41×C₄₇	C₂
prime radical	C₂×#41×C₄₇	C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₈×#41	D₄₇
hypocenter	C₄₇	#33505
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×C₄₇⋊C₁₆	C₂	semidirect
2. #20245220	C₂	semidirect
3. #20245187	C₂	semidirect
4. #20245144	C₂	semidirect
61. Q₈×C₄₇	M₃₂	semidirect
112. C₄₇	#33505	semidirect
155. Q₈	#13093964	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄₇⋊C₁₆	2
#20245220	2
#20245187	2
#20245144	2
C₈×#41×C₄₇	2
#33505	47