Subgroups — #26125295

Order 12032.

290 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245187	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20241306	C₂	1	2	yes	maximal, complemented
4	#20241305	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	—
7	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
8	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
9	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄₇×#431	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#322	C₄	1	2	yes	—
17	C₈²×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄₇⋊C₁₆	—	2	4	no	—
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₄₇	C₈	1	2	yes	—
23	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₄₇	C₈	1	2	yes	—
27	C₂²⋊C₈×C₄₇	C₈	1	2	yes	—
28	#97×C₄₇	C₈	1	2	yes	—
29	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
38	M₁₆×C₄₇	—	2	4	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₄²	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₄₇⋊C₁₆	—	2	2	no	—
52	C₄₇⋊C₁₆	—	2	4	no	—
53	C₄₇⋊C₁₆	M₁₆	1	4	yes	complemented
54	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	M₃₂	1	4	yes	—
62	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₄₇	—	2	1	no	—
66	C₈×C₄₇	—	2	1	no	—
67	C₈×C₄₇	—	2	4	no	—
68	C₈×C₄₇	—	2	4	no	—
69	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
70	C₈×C₄₇	#131	1	2	yes	—
71	#11408	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	2	1	no	—
73	C₂²×C₄₇	—	2	1	no	—
74	C₂²×C₄₇	#346	1	2	yes	—
75	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₄₇	—	2	1	no	—
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	#346	1	2	yes	—
79	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
80	C₄×C₄₇	#405	1	1	yes	characteristic
81	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
82	C₄×C₄₇	#504	1	2	yes	—
83	#2162	—	47	1	no	—
84	C₂×C₄×C₁₆	—	47	1	no	—
85	C₈×M₁₆	D₄₇	1	1	yes	characteristic
86	#1312	—	47	2	no	—
87	#1311	—	47	2	no	—
88	C₂×C₄₇	—	2	2	no	—
89	C₂×C₄₇	C₈×M₁₆	1	1	yes	characteristic
90	C₂×C₄₇	C₂×#346	1	1	yes	characteristic
91	C₂×C₄₇	#2108	1	1	yes	characteristic
92	C₂²×C₁₆	—	47	2	no	—
93	#431	Dic₄₇	1	1	yes	characteristic
94	C₄×M₁₆	Dic₄₇	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
96	C₄⋊C₁₆	—	47	2	no	—
97	C₂²⋊C₁₆	—	47	2	no	—
98	C₄×C₁₆	—	47	2	no	—
99	C₄×C₁₆	—	47	1	no	—
100	C₄×C₁₆	—	47	2	no	—
101	C₄×C₁₆	—	47	1	no	—
102	#322	Dic₄₇	1	2	yes	—
103	C₈²	C₂×D₄₇	1	2	yes	—
104	C₄₇	#11408	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	C₄₇⋊C₈	1	2	yes	—
106	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
107	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
109	C₂×C₁₆	—	47	2	no	—
110	C₂×C₁₆	—	47	2	no	—
111	C₂×C₁₆	—	47	4	no	—
112	C₂×C₁₆	—	94	4	no	—
113	C₄⋊C₈	C₄₇⋊C₈	1	2	yes	—
114	C₂²⋊C₈	C₄₇⋊C₈	1	2	yes	—
115	#97	C₄₇⋊C₈	1	2	yes	—
116	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
117	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
119	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
120	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
125	M₁₆	—	2	4	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
129	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
130	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
134	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
135	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
136	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
137	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
138	C₁₆	—	47	4	no	—
139	C₁₆	—	94	2	no	—
140	C₁₆	—	94	4	no	—
141	C₂³	C₄×C₄₇⋊C₈	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
146	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
147	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
148	C₂×C₄	#13093964	1	4	yes	—
149	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
150	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
151	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	2	4	no	—
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	M₁₆×D₄₇	1	2	yes	—
157	C₈	#13094032	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#15688899	1	2	yes	—
161	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#15688899	1	2	yes	—
165	C₄	M₁₆×Dic₄₇	1	1	yes	characteristic
166	C₄	#15689081	1	1	yes	characteristic
167	C₄	C₂×#13093964	1	2	yes	—
168	C₄	#15689540	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	M₁₆×C₄₇⋊C₈	1	1	yes	characteristic
171	C₂	C₂×#15688899	1	1	yes	characteristic
172	C₂	#20244985	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₈×M₁₆×C₄₇	C₂
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#11408
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20241306	C₂	semidirect
4. #20241305	C₂	semidirect
53. C₄₇⋊C₁₆	M₁₆	semidirect
104. C₄₇	#11408	semidirect

Maximal subgroups

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

subgroup	index
#20245187	2
C₂×C₄×C₄₇⋊C₁₆	2
#20241306	2
#20241305	2
C₈×M₁₆×C₄₇	2
#11408	47