Subgroups — #26125308

Order 12032.

306 conjugacy classes of subgroups.

The subgroup lattice has 184 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×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241307	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2162	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
9	C₂²⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
11	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
12	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×C₄⋊C₁₆	C₄	1	2	yes	—
17	C₄₇×C₂²⋊C₁₆	C₄	1	2	yes	—
18	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄₇⋊C₈	C₈	1	2	yes	—
20	C₂²×C₄₇⋊C₈	C₈	1	2	yes	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
23	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
24	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₄²	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄₇⋊C₈	—	2	4	no	—
50	C₂×C₄₇⋊C₈	C₁₆	1	8	yes	complemented
51	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
52	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
53	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
54	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
55	C₁₆×C₄₇	—	2	4	no	—
56	C₄₇⋊C₁₆	—	2	4	no	—
57	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×C₁₆	1	4	yes	—
65	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
68	C₈×C₄₇	—	2	1	no	—
69	C₈×C₄₇	—	2	1	no	—
70	C₈×C₄₇	#131	1	2	yes	—
71	C₈×C₄₇	#131	1	2	yes	—
72	C₄₇⋊C₈	—	2	2	no	—
73	C₄₇⋊C₈	C₂×C₁₆	1	4	yes	—
74	C₄₇⋊C₈	C₂×C₁₆	1	4	yes	—
75	C₄₇⋊C₈	M₃₂	1	4	yes	—
76	#11410	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	1	no	—
78	C₂²×C₄₇	—	2	1	no	—
79	C₂²×C₄₇	C₄×C₁₆	1	2	yes	—
80	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	C₄×C₁₆	1	2	yes	—
84	C₄×C₄₇	#405	1	1	yes	characteristic
85	C₄×C₄₇	#405	1	1	yes	characteristic
86	C₄×C₄₇	C₂²×C₁₆	1	2	yes	—
87	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
88	#2162	—	47	1	no	—
89	#2162	D₄₇	1	1	yes	characteristic
90	C₂×C₈²	—	47	1	no	—
91	#1312	—	47	2	no	—
92	C₈×C₁₆	—	47	2	no	—
93	C₂×C₄₇	—	2	2	no	—
94	C₂×C₄₇	#1450	1	1	yes	characteristic
95	C₂×C₄₇	C₂×C₄×C₁₆	1	1	yes	characteristic
96	C₂×C₄₇	C₄×M₃₂	1	1	yes	characteristic
97	C₂×C₄×C₈	—	47	1	no	—
98	C₂×C₄×C₈	—	47	1	no	—
99	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
100	C₄⋊C₁₆	—	47	2	no	—
101	C₄⋊C₁₆	Dic₄₇	1	2	yes	—
102	C₂²⋊C₁₆	—	47	2	no	—
103	C₂²⋊C₁₆	Dic₄₇	1	2	yes	—
104	C₄×C₁₆	—	47	2	no	—
105	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
106	C₈²	—	47	1	no	—
107	C₈²	—	47	1	no	—
108	C₈²	—	47	1	no	—
109	C₈²	—	47	1	no	—
110	C₄₇	#11410	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	47	2	no	—
112	C₂²×C₈	—	47	2	no	—
113	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
116	C₂×C₁₆	—	47	4	no	—
117	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
118	C₄×C₈	—	47	2	no	—
119	C₄×C₈	—	47	2	no	—
120	C₄×C₈	—	47	1	no	—
121	C₄×C₈	—	47	1	no	—
122	C₄×C₈	—	47	1	no	—
123	C₄×C₈	—	47	1	no	—
124	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
125	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
128	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
129	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	47	8	no	—
134	C₂×C₈	—	47	2	no	—
135	C₂×C₈	—	47	2	no	—
136	C₂×C₈	—	47	2	no	—
137	C₂×C₈	—	47	2	no	—
138	C₂×C₈	—	94	4	no	—
139	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
140	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
143	C₄²	C₈×D₄₇	1	1	yes	characteristic
144	C₄²	C₈×D₄₇	1	1	yes	characteristic
145	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
147	C₁₆	—	2	4	no	—
148	C₁₆	—	94	4	no	—
149	C₂³	C₈×Dic₄₇	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
154	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
155	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
156	C₂×C₄	C₁₆×D₄₇	1	4	yes	—
157	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
158	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
159	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	47	4	no	—
163	C₈	—	47	4	no	—
164	C₈	—	47	4	no	—
165	C₈	—	94	2	no	—
166	C₈	#13094031	1	2	yes	—
167	C₈	#13094032	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	C₁₆×Dic₄₇	1	2	yes	—
171	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	C₁₆×Dic₄₇	1	2	yes	—
175	C₄	#15689079	1	1	yes	characteristic
176	C₄	#15689081	1	1	yes	characteristic
177	C₄	C₂×C₁₆×D₄₇	1	2	yes	—
178	C₄	M₃₂×D₄₇	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#20241672	1	1	yes	characteristic
181	C₂	C₂×C₁₆×Dic₄₇	1	1	yes	characteristic
182	C₂	M₃₂×Dic₄₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator	C₂×C₄₇	C₂×C₄×C₁₆
Fitting	C₄₇×#2162	C₂
socle	C₂²×C₄₇	C₂×C₄×C₈
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#2162	D₄₇
hypocenter	C₄₇	#11410
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #20241307	C₂	semidirect
4. C₁₆×C₄₇⋊C₈	C₂	semidirect
50. C₂×C₄₇⋊C₈	C₁₆	semidirect
110. C₄₇	#11410	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×C₈×C₄₇⋊C₈	2
#20241307	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#2162	2
#11410	47