Subgroups — #26132625

Order 12032.

366 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	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal
2	#20241466	C₂	1	4	yes	maximal
3	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
5	C₄⋊(C₄₇⋊C₁₆)	—	2	4	no	—
6	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₄³×C₄₇	C₄	1	1	yes	characteristic
11	C₂×C₄₇⋊C₁₆	—	2	4	no	—
12	C₂×C₄₇⋊C₁₆	—	2	8	no	—
13	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₄₇	C₈	1	2	yes	—
17	C₂×C₄²×C₄₇	C₈	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	C₈	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₄₇	—	2	4	no	—
22	C₄×C₈×C₄₇	—	2	4	no	—
23	C₄×C₈×C₄₇	—	2	2	no	—
24	C₄×C₈×C₄₇	D₄	1	2	yes	—
25	C₄×C₈×C₄₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
36	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	#41	1	4	yes	—
39	C₄²×C₄₇	—	2	4	no	—
40	C₄²×C₄₇	—	2	2	no	—
41	C₄²×C₄₇	—	2	2	no	—
42	C₄²×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	C₁₆	1	4	yes	complemented
44	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₄²×C₄₇	M₁₆	1	2	yes	—
46	C₄₇⋊C₁₆	—	4	4	no	—
47	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×C₁₆	1	4	yes	—
56	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₄₇	M₃₂	1	4	yes	—
58	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#117	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₄₇	—	2	2	no	—
64	C₈×C₄₇	—	2	4	no	—
65	C₈×C₄₇	#124	1	2	yes	—
66	C₈×C₄₇	#125	1	2	yes	—
67	#12879	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₄₇	#431	1	2	yes	—
70	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
71	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
72	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
73	C₄×C₄₇	—	2	2	no	—
74	C₄×C₄₇	—	2	4	no	—
75	C₄×C₄₇	—	2	4	no	—
76	C₄×C₄₇	C₄⋊C₁₆	1	4	yes	—
77	C₄×C₄₇	#382	1	1	yes	characteristic
78	C₄×C₄₇	#423	1	1	yes	characteristic
79	C₄×C₄₇	#432	1	2	yes	—
80	C₂×C₄⋊C₁₆	—	47	2	no	—
81	C₄²×C₈	D₄₇	1	1	yes	characteristic
82	#1374	—	47	4	no	—
83	C₂×C₄₇	#1831	1	1	yes	characteristic
84	C₂×C₄₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
85	C₂×C₄₇	#2150	1	1	yes	characteristic
86	C₂×C₄₇	#2162	1	2	yes	—
87	C₂×C₄₇	#2163	1	2	yes	—
88	C₂²×C₁₆	—	47	4	no	—
89	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
90	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
92	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
93	C₄³	Dic₄₇	1	1	yes	characteristic
94	C₄⋊C₁₆	—	94	4	no	—
95	C₄₇	#12879	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
97	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×C₄²	C₄₇⋊C₈	1	2	yes	—
100	C₂×C₄²	C₄₇⋊C₈	1	1	yes	characteristic
101	C₂×C₄²	C₄₇⋊C₈	1	1	yes	characteristic
102	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
103	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
104	C₂×C₁₆	—	94	4	no	—
105	C₂×C₁₆	—	94	8	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	4	no	—
108	C₄×C₈	—	2	2	no	—
109	C₄×C₈	Dic₉₄	1	2	yes	—
110	C₄×C₈	D₁₈₈	1	2	yes	—
111	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
113	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
114	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	#10625329	1	4	yes	—
120	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
121	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
122	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
123	C₂×C₈	#10625350	1	4	yes	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	C₄₇⋊C₁₆	1	4	yes	complemented
129	C₄²	#10625329	1	2	yes	—
130	C₄²	C₄⋊Dic₄₇	1	2	yes	—
131	C₁₆	—	188	4	no	—
132	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
139	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
140	C₂×C₄	C₂×C₄₇⋊C₁₆	1	4	yes	—
141	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
142	C₂×C₄	#13093964	1	4	yes	—
143	C₂×C₄	#13093964	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
145	C₂×C₄	#13093983	1	2	yes	—
146	C₂×C₄	C₂×#10625329	1	2	yes	—
147	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	4	no	—
150	C₈	#13094012	1	2	yes	—
151	C₈	#13094016	1	2	yes	—
152	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
153	C₂²	#15689176	1	2	yes	—
154	C₂²	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
155	C₂²	C₂×#13093964	1	1	yes	characteristic
156	C₂²	C₂×#13093964	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	4	no	—
160	C₄	C₄⋊(C₄₇⋊C₁₆)	1	4	yes	—
161	C₄	#15688989	1	1	yes	characteristic
162	C₄	#15689147	1	1	yes	characteristic
163	C₄	#15689180	1	2	yes	—
164	C₂	#20243505	1	1	yes	characteristic
165	C₂	C₂×C₄⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
166	C₂	#20245144	1	1	yes	characteristic
167	C₂	#20245187	1	2	yes	—
168	C₂	#20245191	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₁₆
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#12879
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
43. C₄²×C₄₇	C₁₆	semidirect
95. C₄₇	#12879	semidirect
128. C₄²	C₄₇⋊C₁₆	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
#20241466	2
C₄²×C₈×C₄₇	2
#12879	47