Subgroups — #26103170

Order 12032.

238 conjugacy classes of subgroups.

The subgroup lattice has 170 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	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₄₇⋊C₁₆)	C₄	1	2	yes	—
6	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
7	C₄×C₄₇⋊C₁₆	—	2	1	no	—
8	C₄×C₄₇⋊C₁₆	—	2	1	no	—
9	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
10	C₄₇×#431	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#322	—	2	1	no	—
14	C₈²×C₄₇	—	2	1	no	—
15	C₂×C₄₇⋊C₁₆	—	2	4	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×M₁₆×C₄₇	C₈	1	2	yes	—
19	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	C₈	1	2	yes	—
23	C₂²⋊C₈×C₄₇	—	2	1	no	—
24	#97×C₄₇	—	2	1	no	—
25	C₄×C₈×C₄₇	—	2	1	no	—
26	C₄×C₈×C₄₇	—	2	1	no	—
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
34	M₁₆×C₄₇	—	2	2	no	—
35	M₁₆×C₄₇	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	2	2	no	—
38	C₂×C₈×C₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₄₇	C₄²	1	1	yes	characteristic
45	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
48	C₄₇⋊C₁₆	—	2	4	no	—
49	C₄₇⋊C₁₆	—	4	2	no	—
50	C₂³×C₄₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
62	C₈×C₄₇	—	2	1	no	—
63	C₈×C₄₇	—	2	1	no	—
64	C₈×C₄₇	—	2	4	no	—
65	C₈×C₄₇	—	4	2	no	—
66	C₈×C₄₇	#104	1	2	yes	—
67	C₈×C₄₇	#108	1	2	yes	—
68	#7163	—	47	1	no	maximal, Sylow
69	C₂²×C₄₇	—	2	1	no	—
70	C₂²×C₄₇	—	2	1	no	—
71	C₂²×C₄₇	#336	1	1	yes	characteristic
72	C₂²×C₄₇	C₄⋊C₁₆	1	2	yes	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	—	2	1	no	—
75	C₄×C₄₇	#339	1	1	yes	characteristic
76	C₄×C₄₇	#341	1	1	yes	characteristic
77	C₄×C₄₇	C₄×C₁₆	1	1	yes	characteristic
78	C₄×C₄₇	#346	1	1	yes	characteristic
79	C₄×C₄₇	C₂²⋊C₁₆	1	2	yes	—
80	C₄×C₄₇	#350	1	2	yes	—
81	#2162	—	47	1	no	—
82	C₂×C₄×C₁₆	—	47	1	no	—
83	C₈×M₁₆	D₄₇	1	1	yes	characteristic
84	C₂×C₄₇	—	2	2	no	—
85	C₂×C₄₇	#1278	1	1	yes	characteristic
86	C₂×C₄₇	#1374	1	1	yes	characteristic
87	C₂×C₄₇	#1379	1	1	yes	characteristic
88	C₂²×C₁₆	—	94	1	no	—
89	#431	Dic₄₇	1	1	yes	characteristic
90	C₄×M₁₆	Dic₄₇	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
92	C₄⋊C₁₆	—	47	2	no	—
93	C₂²⋊C₁₆	—	94	1	no	—
94	C₄×C₁₆	—	47	2	no	—
95	C₄×C₁₆	—	94	1	no	—
96	C₄×C₁₆	—	94	1	no	—
97	#322	—	2	1	no	—
98	C₈²	—	2	1	no	—
99	C₄₇	#7163	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	C₄₇⋊C₈	1	2	yes	—
101	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
102	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
103	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
104	C₂×C₁₆	—	94	4	no	—
105	C₂×C₁₆	—	94	2	no	—
106	C₂×C₁₆	—	94	2	no	—
107	C₄⋊C₈	C₄₇⋊C₈	1	2	yes	—
108	C₂²⋊C₈	—	2	1	no	—
109	#97	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
113	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
117	C₂²×C₄	#10625329	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	C₄₇⋊C₁₆	1	4	yes	complemented
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
126	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
127	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
128	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
129	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
130	C₄²	#10625329	1	1	yes	characteristic
131	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
132	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
133	C₁₆	—	94	4	no	—
134	C₁₆	—	188	2	no	—
135	C₂³	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
140	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
141	C₂×C₄	#13093935	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
144	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
146	C₂×C₄	#13093964	1	2	yes	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	4	no	—
150	C₈	—	4	2	no	—
151	C₈	#13093950	1	2	yes	—
152	C₈	#13093959	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#15688878	1	1	yes	characteristic
156	C₂²	C₄⋊(C₄₇⋊C₁₆)	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#15688884	1	1	yes	characteristic
160	C₄	#15688887	1	1	yes	characteristic
161	C₄	C₄×C₄₇⋊C₁₆	1	1	yes	characteristic
162	C₄	#15688899	1	1	yes	characteristic
163	C₄	C₂²⋊(C₄₇⋊C₁₆)	1	2	yes	—
164	C₄	#15688907	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#20241226	1	1	yes	characteristic
167	C₂	#20241466	1	1	yes	characteristic
168	C₂	#20241477	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₄₇
commutator	C₄×C₄₇	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#336
prime radical	C₂³×C₄₇	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#7163
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
35. M₁₆×C₄₇	C₁₆	semidirect
99. C₄₇	#7163	semidirect
120. M₁₆	C₄₇⋊C₁₆	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
C₈×M₁₆×C₄₇	2
#7163	47