Subgroups — C₈×C₄⋊(C₄₇⋊C₈) #26107399

Order 12032.

550 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×C₈×C₄₇⋊C₈	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	C₈×C₄₇⋊C₈	—	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)	C₈	1	16	yes	direct product factor
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	2	yes	—
23	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
31	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
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	8	yes	—
44	C₄²×C₄₇	C₂×C₈	1	4	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	C₂×C₈	1	16	yes	—
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	4	yes	—
56	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
57	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₄₇	C₄×C₈	1	2	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	4	yes	—
64	C₂×C₄×C₄₇	C₂²×C₈	1	2	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	—	2	16	no	—
70	C₈×C₄⋊C₈	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄₇	C₂×C₄×C₈	1	2	yes	—
73	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
74	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
75	C₂²×C₄₇	#405	1	2	yes	—
76	C₄×C₄₇	—	2	4	no	—
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	C₈²	1	4	yes	complemented
79	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
80	C₄×C₄₇	#431	1	2	yes	—
81	C₄×C₄₇	C₈×D₄	1	4	yes	—
82	C₄×C₄₇	C₈×Q₈	1	4	yes	—
83	C₄×C₄⋊C₈	—	47	2	no	—
84	C₄²×C₈	D₄₇	1	1	yes	characteristic
85	C₂×C₈²	—	47	2	no	—
86	#1273	—	47	2	no	—
87	C₂×C₄₇	C₂×C₈²	1	1	yes	characteristic
88	C₂×C₄₇	C₈×M₁₆	1	2	yes	—
89	C₂×C₄₇	#1450	1	1	yes	characteristic
90	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₄₇	C₈×C₄⋊C₄	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	C₈²	—	94	4	no	—
101	C₄₇	C₈×C₄⋊C₈	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	—	47	16	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	2	yes	—
119	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
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	—	47	16	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	C₈×D₄₇	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	C₈×Dic₄₇	1	4	yes	—
144	C₂×C₄	C₈×Dic₄₇	1	4	yes	—
145	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
146	C₂×C₄	C₄×C₄₇⋊C₈	1	2	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	C₂×C₈×D₄₇	1	4	yes	—
152	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
153	C₂×C₄	C₂×#10625329	1	2	yes	—
154	C₂×C₄	#13094030	1	4	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	94	16	no	—
157	C₈	C₄⋊(C₄₇⋊C₈)	1	8	yes	direct product factor
158	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
159	C₂²	C₂×C₈×Dic₄₇	1	2	yes	—
160	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
161	C₂²	C₄×#10625329	1	1	yes	characteristic
162	C₂²	#15689078	1	2	yes	—
163	C₄	—	2	4	no	—
164	C₄	—	2	2	no	—
165	C₄	C₈×C₄₇⋊C₈	1	4	yes	complemented
166	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
167	C₄	#15689176	1	2	yes	—
168	C₄	C₈×D₁₈₈	1	4	yes	—
169	C₄	C₈×Dic₉₄	1	4	yes	—
170	C₂	C₂×C₈×C₄₇⋊C₈	1	1	yes	characteristic
171	C₂	C₈×#10625329	1	2	yes	—
172	C₂	#20241671	1	1	yes	characteristic
173	C₂	C₄×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
174	C₂	C₈×C₄⋊Dic₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₄₇
commutator	C₂×C₄₇	C₂×C₈²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
15. C₄⋊(C₄₇⋊C₈)	C₈	direct (N × H)
27. C₄×C₈×C₄₇	C₈	semidirect
68. C₈×C₄₇	C₄⋊C₈	semidirect
78. C₄×C₄₇	C₈²	semidirect
101. C₄₇	C₈×C₄⋊C₈	semidirect
115. C₄×C₈	C₄₇⋊C₈	semidirect
157. C₈	C₄⋊(C₄₇⋊C₈)	direct (N × H)
165. C₄	C₈×C₄₇⋊C₈	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×C₈×C₄₇⋊C₈	2
C₄₇⋊#1273	2
C₄²×C₈×C₄₇	2
C₈×C₄⋊C₈	47

Subgroups — C8×C4⋊(C47⋊C8) #26107399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₄⋊(C₄₇⋊C₈) #26107399