Subgroups — #26107403

Order 12032.

502 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#15688844	C₂	1	2	yes	maximal
3	C₄₇⋊#1273	C₂	1	2	yes	maximal
4	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
6	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
7	#15688878	C₄	1	4	yes	—
8	#15688844	—	2	4	no	—
9	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
15	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
16	C₄×C₄₇⋊C₈	—	2	4	no	—
17	C₄×C₄₇⋊C₈	—	2	4	no	—
18	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₄₇	—	2	4	no	—
26	C₄×C₈×C₄₇	—	2	2	no	—
27	C₄×C₈×C₄₇	C₈	1	8	yes	complemented
28	C₄×C₈×C₄₇	D₄	1	2	yes	—
29	C₄×C₈×C₄₇	Q₈	1	2	yes	—
30	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
32	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
41	C₄²×C₄₇	—	2	2	no	—
42	C₄²×C₄₇	—	2	4	no	—
43	C₄²×C₄₇	C₂×C₈	1	4	yes	—
44	C₄²×C₄₇	M₁₆	1	8	yes	—
45	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₄₇	#41	1	2	yes	—
48	C₂×C₄₇⋊C₈	—	2	8	no	—
49	C₂×C₄₇⋊C₈	—	2	8	no	—
50	C₂×C₄₇⋊C₈	—	2	8	no	—
51	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
56	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₄₇	#97	1	4	yes	—
58	C₂×C₄×C₄₇	#97	1	4	yes	—
59	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
65	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₄₇	#131	1	4	yes	—
67	C₈×C₄₇	—	2	4	no	—
68	C₈×C₄₇	C₄⋊C₈	1	8	yes	complemented
69	C₄₇⋊C₈	—	4	8	no	—
70	#8260	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
73	C₂²×C₄₇	C₂×#97	1	2	yes	—
74	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
75	C₂²×C₄₇	#405	1	2	yes	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	2	4	no	—
78	C₄×C₄₇	#322	1	4	yes	complemented
79	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
80	C₄×C₄₇	#431	1	2	yes	—
81	C₄×C₄₇	#436	1	4	yes	—
82	C₄×C₄₇	#446	1	4	yes	—
83	C₄×C₄⋊C₈	—	47	2	no	—
84	C₄²×C₈	D₄₇	1	1	yes	characteristic
85	C₂×#322	—	47	2	no	—
86	#1273	—	47	2	no	—
87	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
88	C₂×C₄₇	#1450	1	1	yes	characteristic
89	C₂×C₄₇	#1451	1	2	yes	—
90	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₄₇	#1770	1	2	yes	—
92	C₂×C₄⋊C₈	—	47	4	no	—
93	C₂×C₄×C₈	—	47	4	no	—
94	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
95	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
98	C₄³	C₂×D₄₇	1	1	yes	characteristic
99	#336	—	47	4	no	—
100	#322	—	94	4	no	—
101	C₄₇	#8260	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	47	8	no	—
103	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
104	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
105	C₂×C₄²	C₄×D₄₇	1	2	yes	—
106	C₂×C₄²	C₄×D₄₇	1	2	yes	—
107	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
109	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
110	C₄⋊C₈	—	94	8	no	—
111	C₄×C₈	—	2	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	94	4	no	—
114	C₄×C₈	—	94	4	no	—
115	C₄×C₈	C₄₇⋊C₈	1	8	yes	complemented
116	C₄×C₈	Dic₉₄	1	2	yes	—
117	C₄×C₈	D₁₈₈	1	2	yes	—
118	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
119	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
120	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
122	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	94	8	no	—
126	C₂×C₈	—	94	8	no	—
127	C₂×C₈	—	94	8	no	—
128	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
129	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
130	C₂×C₈	#10625329	1	4	yes	—
131	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	4	no	—
134	C₄²	#10625324	1	8	yes	—
135	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
136	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
137	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
138	C₄²	#10625350	1	2	yes	—
139	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
144	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
145	C₂×C₄	#13093934	1	4	yes	—
146	C₂×C₄	#13093934	1	4	yes	—
147	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
148	C₂×C₄	#13093983	1	1	yes	characteristic
149	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
150	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
151	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
152	C₂×C₄	C₂×#10625324	1	4	yes	—
153	C₂×C₄	C₂×#10625329	1	2	yes	—
154	C₂×C₄	#13094030	1	4	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	188	8	no	—
157	C₈	C₄⋊(C₄₇⋊C₈)	1	8	yes	complemented
158	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
159	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
160	C₂²	C₂×#13093934	1	2	yes	—
161	C₂²	C₄×#10625329	1	1	yes	characteristic
162	C₂²	#15689078	1	2	yes	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	#15688844	1	4	yes	complemented
166	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
167	C₄	#15689176	1	2	yes	—
168	C₄	#15689202	1	4	yes	—
169	C₄	#15689253	1	4	yes	—
170	C₂	C₂×#15688844	1	1	yes	characteristic
171	C₂	#20241671	1	1	yes	characteristic
172	C₂	#20241677	1	2	yes	—
173	C₂	C₄×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
174	C₂	#20243255	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#8260
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
27. C₄×C₈×C₄₇	C₈	semidirect
68. C₈×C₄₇	C₄⋊C₈	semidirect
78. C₄×C₄₇	#322	semidirect
101. C₄₇	#8260	semidirect
115. C₄×C₈	C₄₇⋊C₈	semidirect
157. C₈	C₄⋊(C₄₇⋊C₈)	semidirect
165. C₄	#15688844	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688844	2
C₄₇⋊#1273	2
C₄²×C₈×C₄₇	2
#8260	47