Subgroups — C₄²×C₂²⋊Dic₄₇ #26137353

Order 12032.

3342 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂²⋊Dic₄₇	C₂	1	6	yes	maximal
2	C₂×C₄²×Dic₄₇	C₂	1	2	yes	maximal, complemented
3	C₄×C₄₇⋊#95	C₂	1	6	yes	maximal, complemented
4	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
6	C₂²×C₄×Dic₄₇	C₂²	1	12	yes	—
7	C₄×C₂²⋊Dic₄₇	C₄	1	48	yes	direct product factor
8	C₂×C₄₇⋊#95	C₂²	1	12	yes	—
9	C₄²×Dic₄₇	—	2	8	no	—
10	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
11	C₂²×C₄²×C₄₇	—	2	8	no	—
12	C₂²×C₄²×C₄₇	C₄	1	8	yes	complemented
13	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×Dic₄₇	C₂³	1	8	yes	—
16	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	48	yes	—
17	C₂×C₄×Dic₄₇	—	2	24	no	—
18	C₂×C₄×Dic₄₇	—	2	24	no	—
19	C₂×C₄×Dic₄₇	C₂×C₄	1	48	yes	complemented
20	C₄₇⋊#95	C₂×C₄	1	48	yes	complemented
21	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	—	2	12	no	—
23	C₂³×C₄×C₄₇	—	2	12	no	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	24	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	12	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
28	C₂×C₄²×C₄₇	—	2	12	no	—
29	C₂×C₄²×C₄₇	—	2	32	no	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	12	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₄₇	D₄	1	8	yes	—
33	C₂⁴×C₄₇	—	2	2	no	—
34	C₂⁴×C₄₇	—	2	6	no	—
35	C₂⁴×C₄₇	C₂²×C₄	1	12	yes	—
36	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
37	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	—	2	6	no	—
39	C₂²×C₄×C₄₇	—	2	6	no	—
40	C₂²×C₄×C₄₇	—	2	24	no	—
41	C₂²×C₄×C₄₇	—	2	24	no	—
42	C₂²×C₄×C₄₇	—	2	96	no	—
43	C₂²×C₄×C₄₇	C₄²	1	48	yes	complemented
44	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
49	C₂²×C₄×C₄₇	#41	1	12	yes	—
50	C₂²×Dic₄₇	—	2	8	no	—
51	C₂²×Dic₄₇	—	2	24	no	—
52	C₂²×Dic₄₇	C₂²×C₄	1	48	yes	—
53	C₄²×C₄₇	—	2	24	no	—
54	C₄²×C₄₇	C₂²⋊C₄	1	16	yes	complemented
55	C₂²⋊Dic₄₇	C₄²	1	64	yes	direct product factor
56	C₄×Dic₄₇	—	2	96	no	—
57	C₂³×C₄₇	—	2	2	no	—
58	C₂³×C₄₇	—	2	6	no	—
59	C₂³×C₄₇	—	2	24	no	—
60	C₂³×C₄₇	—	2	24	no	—
61	C₂³×C₄₇	C₂×C₄²	1	16	yes	—
62	C₂³×C₄₇	C₂×C₄²	1	12	yes	—
63	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
64	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
65	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
66	C₂³×C₄₇	C₂×#41	1	6	yes	—
67	C₂×C₄×C₄₇	—	2	6	no	—
68	C₂×C₄×C₄₇	—	2	6	no	—
69	C₂×C₄×C₄₇	—	2	24	no	—
70	C₂×C₄×C₄₇	—	2	96	no	—
71	C₂×C₄×C₄₇	C₂×C₄²	1	12	yes	—
72	C₂×C₄×C₄₇	C₂×C₄²	1	12	yes	—
73	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
74	C₂×C₄×C₄₇	#117	1	12	yes	—
75	C₂×C₄×C₄₇	C₄×D₄	1	48	yes	—
76	C₂×Dic₄₇	—	2	48	no	—
77	C₂×Dic₄₇	—	2	48	no	—
78	C₂×Dic₄₇	C₂×C₄²	1	32	yes	complemented
79	C₄²×C₂²⋊C₄	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	—	2	12	no	—
81	C₂²×C₄₇	—	2	12	no	—
82	C₂²×C₄₇	—	2	32	no	—
83	C₂²×C₄₇	C₄³	1	8	yes	complemented
84	C₂²×C₄₇	C₂²×C₄²	1	4	yes	—
85	C₂²×C₄₇	C₂²×C₄²	1	3	yes	—
86	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×#117	1	3	yes	—
88	C₂²×C₄₇	C₂×C₄×D₄	1	12	yes	—
89	C₂²×C₄₇	C₄×#41	1	12	yes	—
90	C₄×C₄₇	—	2	12	no	—
91	C₄×C₄₇	C₄×C₂²⋊C₄	1	24	yes	complemented
92	Dic₄₇	—	2	32	no	—
93	C₂³×C₄²	D₄₇	1	1	yes	characteristic
94	C₂×C₄×C₂²⋊C₄	—	47	6	no	—
95	C₂×C₄³	—	47	2	no	—
96	C₄×#95	—	47	6	no	—
97	C₂×C₄₇	—	2	8	no	—
98	C₂×C₄₇	C₂×C₄³	1	1	yes	characteristic
99	C₂×C₄₇	C₂×C₄×C₂²⋊C₄	1	3	yes	—
100	C₂×C₄₇	C₄×#117	1	3	yes	—
101	C₂×C₄₇	C₄²×D₄	1	8	yes	—
102	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
103	C₂²×C₂²⋊C₄	—	47	4	no	—
104	C₂²×C₄²	—	2	8	no	—
105	C₂²×C₄²	—	47	12	no	—
106	C₂²×C₄²	Dic₄₇	1	8	yes	complemented
107	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
108	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
109	C₄×C₂²⋊C₄	—	47	48	no	—
110	C₂×#95	—	47	12	no	—
111	C₄³	—	94	8	no	—
112	C₄₇	C₄²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
113	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
114	C₂³×C₄	—	2	12	no	—
115	C₂³×C₄	—	2	12	no	—
116	C₂³×C₄	—	47	8	no	—
117	C₂³×C₄	C₄×D₄₇	1	24	yes	—
118	C₂³×C₄	C₂×Dic₄₇	1	12	yes	—
119	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
120	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
121	C₂×C₂²⋊C₄	—	47	48	no	—
122	C₂×C₄²	—	2	12	no	—
123	C₂×C₄²	—	2	32	no	—
124	C₂×C₄²	—	47	48	no	—
125	C₂×C₄²	—	94	24	no	—
126	C₂×C₄²	—	94	24	no	—
127	C₂×C₄²	C₂×Dic₄₇	1	12	yes	—
128	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
129	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
130	#95	—	47	48	no	—
131	C₂⁴	—	2	2	no	—
132	C₂⁴	—	2	6	no	—
133	C₂⁴	C₂×C₄×D₄₇	1	12	yes	—
134	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
135	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
136	C₂²×C₄	—	2	6	no	—
137	C₂²×C₄	—	2	6	no	—
138	C₂²×C₄	—	2	24	no	—
139	C₂²×C₄	—	2	24	no	—
140	C₂²×C₄	—	2	96	no	—
141	C₂²×C₄	—	47	48	no	—
142	C₂²×C₄	—	94	8	no	—
143	C₂²×C₄	—	94	24	no	—
144	C₂²×C₄	C₄×Dic₄₇	1	48	yes	complemented
145	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
146	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
147	C₂²×C₄	#10625350	1	12	yes	—
148	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
149	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
151	C₂²⋊C₄	—	47	64	no	—
152	C₄²	—	2	24	no	—
153	C₄²	—	94	96	no	—
154	C₄²	C₂²⋊Dic₄₇	1	16	yes	direct product factor
155	C₂³	—	2	2	no	—
156	C₂³	—	2	6	no	—
157	C₂³	—	2	24	no	—
158	C₂³	—	2	24	no	—
159	C₂³	C₄²×D₄₇	1	16	yes	—
160	C₂³	C₂×C₄×Dic₄₇	1	12	yes	—
161	C₂³	C₂²×C₄×D₄₇	1	6	yes	—
162	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
163	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
164	C₂³	C₂×#10625350	1	6	yes	—
165	C₂×C₄	—	2	6	no	—
166	C₂×C₄	—	2	6	no	—
167	C₂×C₄	—	2	24	no	—
168	C₂×C₄	—	2	96	no	—
169	C₂×C₄	—	47	32	no	—
170	C₂×C₄	—	94	48	no	—
171	C₂×C₄	—	94	48	no	—
172	C₂×C₄	C₂×C₄×Dic₄₇	1	12	yes	—
173	C₂×C₄	C₂×C₄×Dic₄₇	1	12	yes	—
174	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
175	C₂×C₄	#13093983	1	12	yes	—
176	C₂×C₄	C₄×C₂²⋊D₄₇	1	48	yes	—
177	C₂²	—	2	12	no	—
178	C₂²	—	2	12	no	—
179	C₂²	—	2	32	no	—
180	C₂²	C₄²×Dic₄₇	1	8	yes	complemented
181	C₂²	C₂×C₄²×D₄₇	1	4	yes	—
182	C₂²	C₂²×C₄×Dic₄₇	1	3	yes	—
183	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
184	C₂²	C₂×#13093983	1	3	yes	—
185	C₂²	C₂×C₄×C₂²⋊D₄₇	1	12	yes	—
186	C₂²	C₄×#10625350	1	12	yes	—
187	C₄	—	2	12	no	—
188	C₄	—	94	32	no	—
189	C₄	C₄×C₂²⋊Dic₄₇	1	24	yes	direct product factor
190	C₂	—	2	8	no	—
191	C₂	C₂×C₄²×Dic₄₇	1	1	yes	characteristic
192	C₂	C₂×C₄×C₂²⋊Dic₄₇	1	3	yes	—
193	C₂	C₄×#13093983	1	3	yes	—
194	C₂	C₄²×C₂²⋊D₄₇	1	8	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₄₇
commutator	C₂×C₄₇	C₂×C₄³
Frattini	C₂⁴	C₂³×D₄₇
Fitting	C₂³×C₄²×C₄₇	C₂
socle, cosocle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁵×C₄₇	C₂³
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	C₄²×C₂²⋊C₄
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄²×Dic₄₇	C₂	semidirect
3. C₄×C₄₇⋊#95	C₂	semidirect
7. C₄×C₂²⋊Dic₄₇	C₄	direct (N × H)
12. C₂²×C₄²×C₄₇	C₄	semidirect
19. C₂×C₄×Dic₄₇	C₂×C₄	semidirect
20. C₄₇⋊#95	C₂×C₄	semidirect
43. C₂²×C₄×C₄₇	C₄²	semidirect
54. C₄²×C₄₇	C₂²⋊C₄	semidirect
55. C₂²⋊Dic₄₇	C₄²	direct (N × H)
78. C₂×Dic₄₇	C₂×C₄²	semidirect
83. C₂²×C₄₇	C₄³	semidirect
91. C₄×C₄₇	C₄×C₂²⋊C₄	semidirect
106. C₂²×C₄²	Dic₄₇	semidirect
112. C₄₇	C₄²×C₂²⋊C₄	semidirect
144. C₂²×C₄	C₄×Dic₄₇	semidirect
154. C₄²	C₂²⋊Dic₄₇	direct (N × H)
180. C₂²	C₄²×Dic₄₇	semidirect
189. C₄	C₄×C₂²⋊Dic₄₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂²⋊Dic₄₇	2
C₂×C₄²×Dic₄₇	2
C₄×C₄₇⋊#95	2
C₂³×C₄²×C₄₇	2
C₄²×C₂²⋊C₄	47

Subgroups — C42×C22⋊Dic47 #26137353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₂²⋊Dic₄₇ #26137353