Subgroups — #26103105

Order 12032.

306 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
6	C₄×C₄₇⋊C₁₆	—	2	1	no	—
7	C₄×C₄₇⋊C₁₆	—	2	1	no	—
8	C₈×C₄₇⋊C₈	—	2	1	no	—
9	C₈×C₄₇⋊C₈	—	2	1	no	—
10	C₂²×C₁₆×C₄₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₄₇	—	2	1	no	—
13	C₄×C₁₆×C₄₇	—	2	1	no	—
14	C₂²×C₄₇⋊C₈	C₈	1	4	yes	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	C₈	1	4	yes	complemented
18	C₄×C₄₇⋊C₈	—	2	2	no	—
19	C₄×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₇	—	2	2	no	—
24	C₂×C₁₆×C₄₇	—	2	2	no	—
25	C₂×C₁₆×C₄₇	C₈	1	4	yes	complemented
26	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
39	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
40	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₄₇	M₁₆	1	2	yes	—
43	C₂×C₄₇⋊C₈	—	2	4	no	—
44	C₂×C₄₇⋊C₈	—	2	4	no	—
45	C₂×C₄₇⋊C₈	C₁₆	1	8	yes	complemented
46	C₁₆×C₄₇	—	2	4	no	—
47	C₄₇⋊C₁₆	—	2	4	no	—
48	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×C₁₆	1	4	yes	—
58	C₂×C₄×C₄₇	M₃₂	1	4	yes	—
59	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
60	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
61	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
62	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
63	C₄₇⋊C₈	—	2	8	no	—
64	#7136	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	C₈²	1	1	yes	characteristic
66	C₂²×C₄₇	#322	1	1	yes	characteristic
67	C₂²×C₄₇	#336	1	1	yes	characteristic
68	C₂²×C₄₇	C₄×C₁₆	1	2	yes	—
69	C₂²×C₄₇	#346	1	2	yes	—
70	C₄×C₄₇	#322	1	1	yes	characteristic
71	C₄×C₄₇	#322	1	1	yes	characteristic
72	C₄×C₄₇	#336	1	1	yes	characteristic
73	C₄×C₄₇	#336	1	1	yes	characteristic
74	C₄×C₄₇	C₂²⋊C₁₆	1	4	yes	—
75	C₄×C₄₇	C₄⋊C₁₆	1	4	yes	—
76	C₂×C₄×C₁₆	—	47	1	no	—
77	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
78	C₂×C₈²	—	47	1	no	—
79	C₂×C₄₇	#1273	1	1	yes	characteristic
80	C₂×C₄₇	C₈×C₁₆	1	1	yes	characteristic
81	C₂×C₄₇	#1311	1	1	yes	characteristic
82	C₂×C₄₇	#1312	1	2	yes	—
83	C₂×C₄₇	#1374	1	2	yes	—
84	C₂²×C₁₆	—	47	2	no	—
85	C₂²×C₁₆	Dic₄₇	1	2	yes	—
86	C₂×C₄×C₈	—	47	2	no	—
87	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	—	2	1	no	—
90	C₄×C₁₆	—	94	1	no	—
91	C₄×C₁₆	—	94	1	no	—
92	C₈²	—	94	1	no	—
93	C₈²	—	94	1	no	—
94	C₄₇	#7136	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	47	4	no	—
96	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
97	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
98	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	47	4	no	—
102	C₂×C₁₆	—	94	2	no	—
103	C₂×C₁₆	—	94	2	no	—
104	C₂×C₁₆	C₄₇⋊C₈	1	4	yes	complemented
105	C₄×C₈	—	94	2	no	—
106	C₄×C₈	—	94	2	no	—
107	C₄×C₈	Dic₉₄	1	1	yes	characteristic
108	C₄×C₈	D₁₈₈	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
111	C₂²×C₄	C₈×D₄₇	1	2	yes	—
112	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
113	C₂×C₈	—	47	8	no	—
114	C₂×C₈	—	94	4	no	—
115	C₂×C₈	—	94	4	no	—
116	C₂×C₈	C₈×D₄₇	1	2	yes	—
117	C₂×C₈	#10625324	1	2	yes	—
118	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
119	C₂×C₈	#10625329	1	2	yes	—
120	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
122	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
124	C₄²	#10625324	1	2	yes	—
125	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
126	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
127	C₁₆	—	2	4	no	—
128	C₁₆	—	94	4	no	—
129	C₂³	C₈×Dic₄₇	1	1	yes	characteristic
130	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
131	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
132	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
133	C₂×C₄	#13093934	1	1	yes	characteristic
134	C₂×C₄	#13093934	1	1	yes	characteristic
135	C₂×C₄	#13093935	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
137	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
138	C₂×C₄	C₁₆×D₄₇	1	4	yes	—
139	C₂×C₄	#13093963	1	4	yes	—
140	C₈	—	94	8	no	—
141	C₈	(C₂×D₄₇)⋊C₈	1	2	yes	—
142	C₈	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
143	C₈	Dic₄₇⋊C₈	1	2	yes	—
144	C₈	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
145	C₂²	C₈×C₄₇⋊C₈	1	1	yes	characteristic
146	C₂²	#15688845	1	1	yes	characteristic
147	C₂²	#15688877	1	1	yes	characteristic
148	C₂²	C₁₆×Dic₄₇	1	2	yes	—
149	C₂²	#15688898	1	2	yes	—
150	C₄	#15688844	1	1	yes	characteristic
151	C₄	#15688845	1	1	yes	characteristic
152	C₄	#15688877	1	1	yes	characteristic
153	C₄	#15688878	1	1	yes	characteristic
154	C₄	(C₂×D₄₇)⋊C₁₆	1	4	yes	—
155	C₄	Dic₄₇⋊C₁₆	1	4	yes	—
156	C₂	C₄₇⋊#1273	1	1	yes	characteristic
157	C₂	C₁₆×C₄₇⋊C₈	1	1	yes	characteristic
158	C₂	#20241304	1	1	yes	characteristic
159	C₂	#20241307	1	2	yes	—
160	C₂	#20241465	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄×C₈	C₂×D₄₇
commutator	C₂×C₄₇	C₈×C₁₆
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#7136
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
17. C₂×C₄₇⋊C₁₆	C₈	semidirect
25. C₂×C₁₆×C₄₇	C₈	semidirect
45. C₂×C₄₇⋊C₈	C₁₆	semidirect
94. C₄₇	#7136	semidirect
104. C₂×C₁₆	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
C₂×C₈×C₄₇⋊C₈	2
C₂×C₄×C₁₆×C₄₇	2
#7136	47