Subgroups — C₂×#20257882 #27124114

Order 12032.

6338 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20257882	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₄₇⋊#564	C₂	1	3	yes	maximal, complemented
3	C₂×#15689850	C₂	1	12	yes	maximal, complemented
4	C₂×#15689732	C₂	1	12	yes	maximal
5	C₂×#15689679	C₂	1	3	yes	maximal
6	C₂×C₄₇×#564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇⋊#564	C₂²	1	48	yes	complemented
8	#15689850	C₂²	1	192	yes	complemented
9	#15689732	C₂²	1	192	yes	—
10	#15689679	C₂²	1	48	yes	—
11	C₂×#13094023	C₂²	1	6	yes	—
12	C₂×#13094023	C₂²	1	12	yes	—
13	C₂×#13094019	C₂²	1	12	yes	—
14	C₂×#13094018	C₂²	1	4	yes	—
15	C₂×#13094016	C₂²	1	3	yes	—
16	C₂×#13094015	C₂²	1	12	yes	—
17	C₂×#13094014	C₂²	1	12	yes	—
18	C₂×#13094014	C₂²	1	12	yes	—
19	C₂×#13094014	C₂²	1	12	yes	—
20	C₂×#13094002	C₂²	1	12	yes	—
21	C₂×C₄×Dic₉₄	C₂²	1	12	yes	—
22	C₂×#13093989	C₂²	1	12	yes	—
23	C₂×#13093989	C₂²	1	12	yes	—
24	C₂×#13093982	C₂²	1	6	yes	—
25	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#564	C₂²	1	16	yes	complemented
27	C₂×#125×C₄₇	C₂²	1	3	yes	—
28	C₂×#125×C₄₇	C₂²	1	12	yes	—
29	C₂²×Dic₉₄	C₂³	1	12	yes	—
30	C₂²×C₄×D₄₇	C₂³	1	3	yes	—
31	#13094023	C₂³	1	96	yes	—
32	#13094023	C₂³	1	48	yes	—
33	#13094019	C₂³	1	96	yes	—
34	#13094018	C₂³	1	32	yes	—
35	#13094016	C₂³	1	24	yes	—
36	#13094015	C₂³	1	96	yes	—
37	#13094014	C₂³	1	96	yes	—
38	#13094014	C₂³	1	96	yes	—
39	#13094014	C₂³	1	96	yes	—
40	#13094002	C₂³	1	96	yes	—
41	C₄×Dic₉₄	C₂³	1	96	yes	—
42	#13093989	C₂³	1	96	yes	—
43	#13093989	C₂³	1	96	yes	—
44	#13093982	C₂³	1	48	yes	—
45	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
47	C₂×Dic₄₇⋊C₄	C₂³	1	12	yes	—
48	C₂×Dic₄₇⋊C₄	C₂³	1	12	yes	—
49	C₂×Dic₄₇⋊C₄	C₂³	1	12	yes	—
50	C₂×Dic₄₇⋊C₄	C₂³	1	12	yes	—
51	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	12	yes	—
52	C₂×C₄×Dic₄₇	C₂³	1	3	yes	—
53	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
54	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
55	C₄²×D₄₇	C₂³	1	8	yes	—
56	#125×C₄₇	C₂³	1	96	yes	—
57	#125×C₄₇	C₂³	1	24	yes	—
58	C₂×C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
59	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
60	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
63	C₂³×D₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
66	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
67	C₂²×Dic₄₇	C₂⁴	1	3	yes	—
68	C₂×C₄×D₄₇	—	2	12	no	—
69	C₂×C₄×D₄₇	C₂⁴	1	12	yes	—
70	C₂×Dic₉₄	—	2	48	no	—
71	C₂×Dic₉₄	C₂⁴	1	48	yes	—
72	C₄⋊C₄×C₄₇	C₂⁴	1	24	yes	—
73	C₄⋊C₄×C₄₇	C₂⁴	1	48	yes	—
74	C₄⋊C₄×C₄₇	C₂⁴	1	48	yes	—
75	C₄²×C₄₇	C₂⁴	1	4	yes	—
76	C₄²×C₄₇	C₂⁴	1	16	yes	—
77	(C₂×D₄₇)⋊C₄	C₂⁴	1	48	yes	—
78	C₄⋊Dic₄₇	C₂⁴	1	24	yes	—
79	C₄⋊Dic₄₇	C₂⁴	1	48	yes	—
80	Dic₄₇⋊C₄	C₂⁴	1	48	yes	—
81	Dic₄₇⋊C₄	C₂⁴	1	48	yes	—
82	Dic₄₇⋊C₄	C₂⁴	1	48	yes	—
83	Dic₄₇⋊C₄	C₂⁴	1	48	yes	—
84	C₄×Dic₄₇	C₂⁴	1	12	yes	—
85	C₄×Dic₄₇	C₂⁴	1	48	yes	—
86	C₄×Dic₄₇	C₂⁴	1	48	yes	—
87	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
88	C₂²×D₄₇	—	2	6	no	—
89	C₂²×D₄₇	C₂⁵	1	2	yes	—
90	C₂×C₄×C₄₇	—	2	24	no	—
91	C₂×C₄×C₄₇	—	2	6	no	—
92	C₂×C₄×C₄₇	C₂⁵	1	24	yes	—
93	C₂×C₄×C₄₇	C₂⁵	1	6	yes	—
94	C₂×Dic₄₇	—	2	24	no	—
95	C₂×Dic₄₇	2₋¹⁺⁴	1	12	yes	—
96	C₂×Dic₄₇	C₂⁵	1	24	yes	—
97	C₂×Dic₄₇	C₂⁵	1	6	yes	—
98	C₄×D₄₇	—	2	24	no	—
99	Dic₉₄	—	2	96	no	—
100	C₂×#3556	—	47	1	no	maximal, Sylow
101	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	6	yes	—
102	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
103	C₂×D₄₇	—	2	6	no	—
104	C₂×D₄₇	—	4	4	no	—
105	C₄×C₄₇	—	2	24	no	—
106	C₄×C₄₇	—	2	6	no	—
107	Dic₄₇	—	2	24	no	—
108	Dic₄₇	C₂×2₋¹⁺⁴	1	12	yes	—
109	#3556	—	47	32	no	—
110	C₂×#564	—	47	3	no	—
111	C₂×#564	D₄₇	1	1	yes	characteristic, complemented
112	C₂×#557	—	47	12	no	—
113	C₂×#541	—	47	12	no	—
114	C₂×#533	—	47	3	no	—
115	C₂×C₄₇	#3556	1	4	yes	complemented
116	C₂×C₄₇	C₂²×2₋¹⁺⁴	1	3	yes	—
117	D₄₇	—	4	2	no	—
118	#564	—	47	48	no	—
119	#564	C₂×D₄₇	1	16	yes	complemented
120	#557	—	47	192	no	—
121	#541	—	47	192	no	—
122	#533	—	47	48	no	—
123	C₂×C₄⋊Q₈	—	47	6	no	—
124	C₂×C₄⋊Q₈	—	47	12	no	—
125	C₂×#126	—	47	12	no	—
126	C₂×#126	—	47	4	no	—
127	C₂×#125	—	47	12	no	—
128	C₂×#125	—	47	12	no	—
129	C₂×#125	—	47	12	no	—
130	C₂×#125	—	47	12	no	—
131	C₂×#125	—	47	3	no	—
132	C₂×#125	C₂×D₄₇	1	3	yes	—
133	C₂×#125	C₂×D₄₇	1	12	yes	—
134	C₂×C₂²⋊Q₈	—	47	12	no	—
135	C₂×C₄×Q₈	—	47	12	no	—
136	C₂×C₄×Q₈	—	47	12	no	—
137	C₂×C₄×Q₈	—	47	12	no	—
138	C₂×#117	—	47	6	no	—
139	C₂²×C₄²	—	47	1	no	—
140	C₄₇	C₂×#3556	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	47	12	no	—
142	C₂³×C₄	—	47	3	no	—
143	C₄⋊Q₈	—	47	48	no	—
144	C₄⋊Q₈	—	47	96	no	—
145	#126	—	47	96	no	—
146	#126	—	47	32	no	—
147	#125	—	47	96	no	—
148	#125	—	47	96	no	—
149	#125	—	47	96	no	—
150	#125	—	47	96	no	—
151	#125	—	47	24	no	—
152	#125	C₂²×D₄₇	1	24	yes	—
153	#125	C₂²×D₄₇	1	96	yes	—
154	C₂²⋊Q₈	—	47	96	no	—
155	C₄×Q₈	—	47	96	no	—
156	C₄×Q₈	—	47	96	no	—
157	C₄×Q₈	—	47	96	no	—
158	#117	—	47	48	no	—
159	C₂×C₄⋊C₄	—	47	6	no	—
160	C₂×C₄⋊C₄	—	47	12	no	—
161	C₂×C₄⋊C₄	—	47	12	no	—
162	C₂×C₄⋊C₄	—	47	12	no	—
163	C₂×C₄⋊C₄	—	47	12	no	—
164	C₂×C₄⋊C₄	—	47	12	no	—
165	C₂×C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
166	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
167	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
168	C₂×C₂²⋊C₄	—	47	12	no	—
169	C₂×C₄²	—	47	12	no	—
170	C₂×C₄²	—	47	3	no	—
171	C₂×C₄²	—	47	12	no	—
172	C₂×C₄²	—	47	8	no	—
173	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
174	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
175	C₂⁴	—	47	1	no	—
176	C₂×Q₈	—	47	48	no	—
177	C₂×Q₈	—	94	48	no	—
178	C₂²×C₄	—	47	3	no	—
179	C₂²×C₄	—	47	12	no	—
180	C₂²×C₄	—	47	12	no	—
181	C₂²×C₄	—	94	12	no	—
182	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
183	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
184	C₄⋊C₄	—	47	48	no	—
185	C₄⋊C₄	—	47	48	no	—
186	C₄⋊C₄	—	47	48	no	—
187	C₄⋊C₄	—	47	48	no	—
188	C₄⋊C₄	—	47	24	no	—
189	C₄⋊C₄	—	47	48	no	—
190	C₄⋊C₄	C₂³×D₄₇	1	24	yes	—
191	C₄⋊C₄	C₂³×D₄₇	1	48	yes	—
192	C₄⋊C₄	C₂³×D₄₇	1	48	yes	—
193	C₂²⋊C₄	—	47	48	no	—
194	C₄²	—	47	12	no	—
195	C₄²	—	47	48	no	—
196	C₄²	—	47	48	no	—
197	C₄²	C₂³×D₄₇	1	4	yes	—
198	C₄²	C₂³×D₄₇	1	16	yes	—
199	C₂³	—	47	2	no	—
200	C₂³	—	94	6	no	—
201	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
202	Q₈	—	94	96	no	—
203	C₂×C₄	—	2	6	no	—
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	47	12	no	—
206	C₂×C₄	—	47	6	no	—
207	C₂×C₄	—	47	24	no	—
208	C₂×C₄	—	94	24	no	—
209	C₂×C₄	—	94	24	no	—
210	C₂×C₄	C₂⁴×D₄₇	1	24	yes	—
211	C₂×C₄	C₂⁴×D₄₇	1	6	yes	—
212	C₂²	—	94	6	no	—
213	C₂²	—	188	4	no	—
214	C₂²	2₋¹⁺⁴×D₄₇	1	6	yes	—
215	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
216	C₄	—	2	6	no	—
217	C₄	—	2	24	no	—
218	C₄	—	47	12	no	—
219	C₄	—	94	24	no	—
220	C₂	—	188	2	no	—
221	C₂	#20257882	1	4	yes	direct product factor
222	C₂	C₂×2₋¹⁺⁴×D₄₇	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×#564	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂³×D₄₇	C₂⁴
hypercenter	C₂×#564	D₄₇
hypocenter	C₄₇	C₂×#3556
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#564

Complements

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

N	H ≅ G/N	type
1. #20257882	C₂	direct (N × H)
2. C₂×C₄₇⋊#564	C₂	semidirect
3. C₂×#15689850	C₂	semidirect
6. C₂×C₄₇×#564	C₂	semidirect
7. C₄₇⋊#564	C₂²	semidirect
8. #15689850	C₂²	semidirect
26. C₄₇×#564	C₂²	semidirect
111. C₂×#564	D₄₇	semidirect
115. C₂×C₄₇	#3556	semidirect
119. #564	C₂×D₄₇	semidirect
140. C₄₇	C₂×#3556	semidirect
221. C₂	#20257882	direct (N × H)

Maximal subgroups

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

subgroup	index
#20257882	2
C₂×C₄₇⋊#564	2
C₂×#15689850	2
C₂×#15689732	2
C₂×#15689679	2
C₂×C₄₇×#564	2
C₂×#3556	47

Subgroups — C2×#20257882 #27124114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20257882 #27124114