Subgroups — C₂×#20246494 #26375950

Order 12032.

2422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246494	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689009	C₂	1	4	yes	maximal
3	C₂×#15689009	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₄×Dic₄₇	C₂	1	4	yes	maximal
5	C₂×#15688969	C₂	1	2	yes	maximal
6	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
10	#15689009	C₂²	1	32	yes	—
11	#15689009	C₂²	1	32	yes	—
12	C₄⋊C₄×Dic₄₇	C₂²	1	32	yes	—
13	#15688969	C₂²	1	16	yes	—
14	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
15	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
17	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
18	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
19	C₂×#125×C₄₇	C₄	1	16	yes	complemented
20	C₂×#125×C₄₇	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₂³×Dic₄₇	C₂³	1	4	yes	—
25	C₂³×Dic₄₇	C₂³	1	4	yes	—
26	C₂×C₄⋊Dic₄₇	—	2	16	no	—
27	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
28	C₂×C₄×Dic₄₇	—	2	16	no	—
29	C₂×C₄×Dic₄₇	—	2	16	no	—
30	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
32	C₄₇⋊#95	C₂³	1	8	yes	—
33	C₄₇⋊#95	C₂³	1	8	yes	—
34	C₄₇⋊#95	C₂³	1	16	yes	—
35	C₄₇⋊#95	C₂³	1	16	yes	—
36	C₄₇⋊#95	C₂³	1	16	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
40	#125×C₄₇	C₂×C₄	1	64	yes	complemented
41	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
47	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	8	no	—
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	8	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
55	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
58	C₂²×Dic₄₇	—	2	8	no	—
59	C₂²×Dic₄₇	—	2	8	no	—
60	C₂²×Dic₄₇	—	2	8	no	—
61	C₂²×Dic₄₇	—	2	8	no	—
62	C₂²×Dic₄₇	—	2	8	no	—
63	C₂²×Dic₄₇	#41	1	16	yes	—
64	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
65	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
66	C₄⋊C₄×C₄₇	C₂²×C₄	1	64	yes	—
67	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
69	C₄⋊Dic₄₇	—	2	32	no	—
70	C₄×Dic₄₇	—	2	32	no	—
71	C₄×Dic₄₇	—	2	32	no	—
72	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
73	C₂³×C₄₇	C₂×#41	1	8	yes	—
74	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
75	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
76	C₂×C₄×C₄₇	—	2	16	no	—
77	C₂×C₄×C₄₇	—	2	32	no	—
78	C₂×C₄×C₄₇	—	2	8	no	—
79	C₂×C₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	2	4	no	—
83	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
84	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
85	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
86	C₂×Dic₄₇	—	2	32	no	—
87	C₂×Dic₄₇	—	2	8	no	—
88	C₂×Dic₄₇	—	2	8	no	—
89	C₂×Dic₄₇	—	2	8	no	—
90	C₂×Dic₄₇	—	2	8	no	—
91	C₂×Dic₄₇	—	2	8	no	—
92	C₂×Dic₄₇	—	4	16	no	—
93	C₂×Dic₄₇	C₂×#41	1	16	yes	—
94	C₂×#2371	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	C₄×#41	1	8	yes	—
96	C₂²×C₄₇	#519	1	4	yes	—
97	C₂²×C₄₇	#552	1	16	yes	—
98	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
99	C₂²×C₄₇	C₂²×#41	1	4	yes	—
100	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
101	C₄×C₄₇	—	2	8	no	—
102	C₄×C₄₇	—	2	16	no	—
103	C₄×C₄₇	—	2	4	no	—
104	Dic₄₇	—	2	16	no	—
105	Dic₄₇	—	4	8	no	—
106	C₂²×#125	D₄₇	1	1	yes	characteristic
107	#2371	—	47	16	no	—
108	C₂×#387	—	47	4	no	—
109	C₂×#387	—	47	4	no	—
110	C₂×C₄×C₄⋊C₄	—	47	4	no	—
111	C₂×#376	—	47	2	no	—
112	C₂×C₄₇	#2371	1	8	yes	complemented
113	C₂×C₄₇	C₂×C₄×#41	1	2	yes	—
114	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
115	C₂×C₄₇	C₂×#552	1	4	yes	—
116	C₂×#125	Dic₄₇	1	16	yes	complemented
117	C₂×#125	C₂×D₄₇	1	8	yes	—
118	C₂²×C₄⋊C₄	—	47	4	no	—
119	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
120	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
121	C₂²×C₄²	—	47	4	no	—
122	C₂²×C₄²	—	47	4	no	—
123	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
124	#387	—	47	32	no	—
125	#387	—	47	32	no	—
126	C₄×C₄⋊C₄	—	47	32	no	—
127	#376	—	47	16	no	—
128	C₂×#95	—	47	2	no	—
129	C₂×#95	—	47	4	no	—
130	C₂×#95	—	47	2	no	—
131	C₂×#95	—	47	4	no	—
132	C₂×#95	—	47	4	no	—
133	C₄₇	C₂×#2371	1	1	yes	characteristic, complemented, Sylow
134	C₂³×C₄	—	47	4	no	—
135	C₂³×C₄	—	47	4	no	—
136	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
137	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
139	#125	C₂×Dic₄₇	1	64	yes	complemented
140	C₂×C₄⋊C₄	—	47	16	no	—
141	C₂×C₄⋊C₄	—	94	16	no	—
142	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
143	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
144	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
145	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
146	C₂×C₄²	—	47	16	no	—
147	C₂×C₄²	—	47	16	no	—
148	C₂×C₄²	—	94	16	no	—
149	C₂×C₄²	—	94	16	no	—
150	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
151	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
152	#95	—	47	8	no	—
153	#95	—	47	16	no	—
154	#95	—	47	8	no	—
155	#95	—	47	16	no	—
156	#95	—	47	16	no	—
157	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	47	16	no	—
163	C₂²×C₄	—	47	8	no	—
164	C₂²×C₄	—	47	8	no	—
165	C₂²×C₄	—	94	8	no	—
166	C₂²×C₄	—	94	8	no	—
167	C₂²×C₄	—	94	8	no	—
168	C₂²×C₄	—	94	8	no	—
169	C₂²×C₄	—	94	8	no	—
170	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
171	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
172	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
173	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
174	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
175	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
176	C₄⋊C₄	—	94	32	no	—
177	C₄⋊C₄	C₂²×Dic₄₇	1	64	yes	—
178	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
179	C₄²	—	94	32	no	—
180	C₄²	—	94	32	no	—
181	C₄²	C₂²×Dic₄₇	1	16	yes	—
182	C₂³	C₂³×Dic₄₇	1	2	yes	—
183	C₂³	#41×D₄₇	1	8	yes	—
184	C₂³	#13094072	1	4	yes	—
185	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
186	C₂×C₄	—	2	16	no	—
187	C₂×C₄	—	2	32	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	47	16	no	—
194	C₂×C₄	—	94	32	no	—
195	C₂×C₄	—	94	8	no	—
196	C₂×C₄	—	94	8	no	—
197	C₂×C₄	—	94	8	no	—
198	C₂×C₄	—	94	8	no	—
199	C₂×C₄	—	94	8	no	—
200	C₂×C₄	—	188	16	no	—
201	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
202	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
203	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
204	C₂²	#41×Dic₄₇	1	8	yes	—
205	C₂²	#15689601	1	4	yes	—
206	C₂²	#15689805	1	16	yes	—
207	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
208	C₂²	C₂×#41×D₄₇	1	4	yes	—
209	C₂²	C₂×#13094072	1	2	yes	—
210	C₄	—	2	8	no	—
211	C₄	—	2	16	no	—
212	C₄	—	2	4	no	—
213	C₄	—	94	16	no	—
214	C₄	—	188	8	no	—
215	C₂	#20246494	1	8	yes	direct product factor
216	C₂	C₂×#41×Dic₄₇	1	2	yes	—
217	C₂	C₂×#15689601	1	1	yes	characteristic
218	C₂	C₂×#15689805	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	C₂×#2371
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
1. #20246494	C₂	direct (N × H)
19. C₂×#125×C₄₇	C₄	semidirect
40. #125×C₄₇	C₂×C₄	semidirect
112. C₂×C₄₇	#2371	semidirect
116. C₂×#125	Dic₄₇	semidirect
133. C₄₇	C₂×#2371	semidirect
139. #125	C₂×Dic₄₇	semidirect
215. C₂	#20246494	direct (N × H)

Maximal subgroups

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

subgroup	index
#20246494	2
C₂×#15689009	2
C₂×C₄⋊C₄×Dic₄₇	2
C₂×#15688969	2
C₂²×#125×C₄₇	2
C₂×#2371	47

Subgroups — C2×#20246494 #26375950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246494 #26375950