Subgroups — #26127896

Order 12032.

238 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245191	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#20241458	C₂	1	2	yes	maximal, complemented
4	#20241452	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
10	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
12	#15688899	—	2	1	no	—
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	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	1	no	—
21	C₂×C₄₇⋊C₁₆	—	4	2	no	—
22	C₂×M₁₆×C₄₇	—	2	1	no	—
23	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	Q₈	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	—	2	1	no	—
29	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
38	M₁₆×C₄₇	—	2	4	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	1	no	—
41	C₂×C₈×C₄₇	—	2	1	no	—
42	C₂×C₈×C₄₇	—	4	1	no	—
43	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
51	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₄₇⋊C₁₆	—	4	1	no	—
53	C₄₇⋊C₁₆	—	4	2	no	—
54	C₄₇⋊C₁₆	—	4	1	no	—
55	C₂³×C₄₇	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₄₇	—	2	4	no	—
67	C₈×C₄₇	—	2	1	no	—
68	C₈×C₄₇	—	2	1	no	—
69	C₈×C₄₇	—	2	1	no	—
70	C₈×C₄₇	—	2	4	no	—
71	C₈×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₈×C₄₇	#137	1	1	yes	characteristic
73	#12016	—	47	1	no	maximal, Sylow
74	C₂²×C₄₇	—	2	1	no	—
75	C₂²×C₄₇	—	2	1	no	—
76	C₂²×C₄₇	#364	1	2	yes	—
77	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	—	2	1	no	—
80	C₄×C₄₇	—	2	1	no	—
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	#428	1	1	yes	characteristic
83	C₄×C₄₇	#430	1	1	yes	characteristic
84	C₄×C₄₇	#504	1	2	yes	—
85	#2163	—	47	1	no	—
86	C₂×C₄⋊C₁₆	—	47	1	no	—
87	C₈×M₁₆	D₄₇	1	1	yes	characteristic
88	#1370	—	47	2	no	—
89	#1367	—	47	2	no	—
90	C₂×C₄₇	—	2	2	no	—
91	C₂×C₄₇	#1565	1	1	yes	characteristic
92	C₂×C₄₇	#2151	1	1	yes	characteristic
93	C₂×C₄₇	C₂×#364	1	1	yes	characteristic
94	C₂²×C₁₆	—	94	1	no	—
95	#431	Dic₄₇	1	1	yes	characteristic
96	C₄×M₁₆	Dic₄₇	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
98	C₄⋊C₁₆	—	47	1	no	—
99	C₄⋊C₁₆	—	47	2	no	—
100	C₄⋊C₁₆	—	47	1	no	—
101	C₄⋊C₁₆	—	94	1	no	—
102	C₂²⋊C₁₆	—	94	1	no	—
103	#346	—	94	1	no	—
104	#322	Dic₄₇	1	2	yes	—
105	C₈²	C₂×D₄₇	1	2	yes	—
106	C₄₇	#12016	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₁₆	—	2	1	no	—
108	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
109	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
111	C₂×C₁₆	—	94	1	no	—
112	C₂×C₁₆	—	94	1	no	—
113	C₂×C₁₆	—	94	2	no	—
114	C₂×C₁₆	—	188	2	no	—
115	C₄⋊C₈	C₄₇⋊C₈	1	2	yes	—
116	C₂²⋊C₈	C₄₇⋊C₈	1	2	yes	—
117	#97	—	2	1	no	—
118	C₄×C₈	Dic₉₄	1	1	yes	characteristic
119	C₄×C₈	D₁₈₈	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
121	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
122	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
125	C₂²×C₄	#10625329	1	1	yes	characteristic
126	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
127	M₁₆	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
133	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
135	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
136	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
137	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
138	C₄²	#10625329	1	1	yes	characteristic
139	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
140	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
141	C₁₆	—	188	1	no	—
142	C₁₆	—	188	2	no	—
143	C₁₆	—	188	1	no	—
144	C₂³	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
150	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
151	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
154	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	4	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	#13094052	1	1	yes	characteristic
161	C₈	#13094057	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#15688944	1	2	yes	—
165	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#15689165	1	1	yes	characteristic
171	C₄	#15689172	1	1	yes	characteristic
172	C₄	#15689540	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	#20242140	1	1	yes	characteristic
175	C₂	#20245147	1	1	yes	characteristic
176	C₂	C₂×#15688944	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₈
prime radical	C₂³×C₄₇	C₄⋊C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#12016
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20241458	C₂	semidirect
4. #20241452	C₂	semidirect
106. C₄₇	#12016	semidirect

Maximal subgroups

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

subgroup	index
#20245191	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20241458	2
#20241452	2
C₈×M₁₆×C₄₇	2
#12016	47