Subgroups — C₂×#20256541 #27122804

Order 12032.

2022 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689940	C₂	1	2	yes	maximal, complemented
2	#20256541	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689566	C₂	1	6	yes	maximal, complemented
4	C₂×#15689555	C₂	1	3	yes	maximal, complemented
5	C₂²×Dic₃₇₆	C₂	1	3	yes	maximal, complemented
6	C₂×C₄₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13094071	—	2	2	no	—
8	#15689940	—	2	8	no	—
9	#15689940	C₂²	1	16	yes	complemented
10	C₂×#13094057	—	2	6	no	—
11	C₂×#13094046	C₂²	1	6	yes	—
12	C₂²×Dic₁₈₈	C₂²	1	6	yes	—
13	#15689566	C₂²	1	48	yes	complemented
14	#15689555	C₂²	1	24	yes	complemented
15	C₂×Dic₃₇₆	C₂²	1	6	yes	—
16	C₂×Dic₃₇₆	C₂²	1	3	yes	—
17	C₂×Dic₃₇₆	C₂²	1	24	yes	complemented
18	C₂×#13093969	C₂²	1	6	yes	—
19	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
20	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#504	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₄₇	C₂²	1	3	yes	—
23	C₂²×C₁₆×C₄₇	C₂²	1	3	yes	—
24	#13094071	—	2	16	no	—
25	#13094071	—	4	8	no	—
26	C₂×#10625350	—	2	6	no	—
27	C₂×#10625352	—	4	6	no	—
28	C₂²×Dic₉₄	—	2	6	no	—
29	C₂×Q₈×D₄₇	—	4	2	no	—
30	#13094057	—	2	48	no	—
31	#13094046	—	2	12	no	—
32	#13094046	C₂³	1	24	yes	—
33	C₂×Dic₁₈₈	—	2	6	no	—
34	C₂×Dic₁₈₈	—	2	12	no	—
35	C₂×Dic₁₈₈	C₂³	1	6	yes	—
36	C₂×Dic₁₈₈	C₂³	1	24	yes	—
37	C₂×#10625325	—	2	6	no	—
38	C₂×D₃₇₆	C₂³	1	2	yes	—
39	Dic₃₇₆	C₂³	1	12	yes	—
40	Dic₃₇₆	C₂³	1	24	yes	—
41	#13093969	C₂³	1	24	yes	—
42	D₇₅₂	C₂³	1	4	yes	—
43	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
44	#131×C₄₇	D₄	1	4	yes	—
45	#131×C₄₇	C₂³	1	4	yes	—
46	C₂×M₁₆×C₄₇	D₄	1	3	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
48	M₃₂×C₄₇	C₂³	1	12	yes	—
49	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₄₇	C₂³	1	3	yes	—
51	C₂×C₁₆×C₄₇	C₂³	1	12	yes	—
52	#41×C₄₇	D₈	1	4	yes	—
53	#41×C₄₇	C₂×D₄	1	4	yes	—
54	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
55	C₂×D₄×C₄₇	D₈	1	3	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
57	C₂×C₂²⋊D₄₇	—	4	6	no	—
58	C₂²×Dic₄₇	—	4	6	no	—
59	Q₈×D₄₇	—	4	16	no	—
60	#10625352	—	4	48	no	—
61	#10625350	—	2	24	no	—
62	#10625350	—	4	12	no	—
63	C₂×D₁₈₈	—	2	2	no	—
64	C₂×C₄×D₄₇	—	4	6	no	—
65	C₂×Dic₉₄	—	2	6	no	—
66	C₂×Dic₉₄	—	2	24	no	—
67	C₂×Dic₉₄	—	4	6	no	—
68	C₂×Dic₉₄	—	4	12	no	—
69	M₁₆×C₄₇	C₂×D₄	1	12	yes	—
70	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
71	C₂×C₈×C₄₇	C₂×D₄	1	6	yes	—
72	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
74	Dic₁₈₈	—	2	6	no	—
75	Dic₁₈₈	—	2	24	no	—
76	Dic₁₈₈	C₂⁴	1	12	yes	—
77	D₃₇₆	—	2	2	no	—
78	D₃₇₆	C₂⁴	1	4	yes	—
79	#10625325	—	2	24	no	—
80	C₁₆×C₄₇	C₂⁴	1	2	yes	—
81	C₁₆×C₄₇	C₂⁴	1	6	yes	—
82	C₂³×C₄₇	C₂×D₈	1	3	yes	—
83	C₂²×D₄₇	—	4	2	no	—
84	Q₈×C₄₇	C₂×D₈	1	4	yes	—
85	D₄×C₄₇	C₂×D₈	1	12	yes	—
86	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
87	C₂×C₄×C₄₇	C₂×D₈	1	6	yes	—
88	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
90	C₂²⋊D₄₇	—	4	24	no	—
91	C₂×Dic₄₇	—	4	6	no	—
92	C₂×Dic₄₇	—	4	24	no	—
93	D₁₈₈	—	2	4	no	—
94	D₁₈₈	—	4	2	no	—
95	C₄×D₄₇	—	4	24	no	—
96	Dic₉₄	—	2	12	no	—
97	Dic₉₄	—	4	6	no	—
98	Dic₉₄	—	4	24	no	—
99	C₈×C₄₇	C₂²×D₄	1	6	yes	—
100	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₄₇	C₂⁵	1	1	yes	characteristic
102	C₂×#3417	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	6	no	—
104	C₂²×C₄₇	C₂²×D₈	1	6	yes	—
105	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
106	C₂×D₄₇	—	4	4	no	—
107	C₂×D₄₇	—	8	4	no	—
108	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
109	C₄×C₄₇	C₂²×D₈	1	6	yes	—
110	C₄×C₄₇	C₂³×D₄	1	1	yes	characteristic
111	Dic₄₇	—	4	12	no	—
112	C₂×#578	—	47	2	no	—
113	#3417	—	47	16	no	—
114	C₂×#510	—	47	6	no	—
115	C₂×#508	—	47	3	no	—
116	C₂²×Q₃₂	—	47	3	no	—
117	C₂×#504	D₄₇	1	1	yes	characteristic, complemented
118	C₂×C₄₇	—	2	6	no	—
119	C₂×C₄₇	#3417	1	2	yes	complemented
120	C₂×C₄₇	C₂³×D₈	1	1	yes	characteristic
121	D₄₇	—	8	4	no	—
122	C₂×2₋¹⁺⁴	—	94	2	no	—
123	#578	—	47	16	no	—
124	#578	—	94	8	no	—
125	C₂×#137	—	94	6	no	—
126	C₂×#135	—	47	6	no	—
127	C₂²×Q₁₆	—	47	6	no	—
128	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
129	#510	—	47	48	no	—
130	#508	—	47	24	no	—
131	C₂×Q₃₂	—	47	3	no	—
132	C₂×Q₃₂	—	47	6	no	—
133	C₂×Q₃₂	—	47	24	no	—
134	C₂×SD₃₂	—	47	6	no	—
135	C₂×D₁₆	—	47	1	no	—
136	#504	C₂×D₄₇	1	8	yes	complemented
137	C₂×M₃₂	C₂×D₄₇	1	3	yes	—
138	C₂²×C₁₆	C₂×D₄₇	1	3	yes	—
139	C₄₇	C₂×#3417	1	1	yes	characteristic, complemented, Sylow
140	2₋¹⁺⁴	—	94	16	no	—
141	2₋¹⁺⁴	—	188	8	no	—
142	C₂×#41	—	94	6	no	—
143	C₂×#41	—	188	6	no	—
144	C₂×#41	D₁₈₈	1	1	yes	characteristic
145	C₂²×Q₈	—	94	6	no	—
146	C₂²×Q₈	—	188	2	no	—
147	#137	—	94	48	no	—
148	#135	—	47	24	no	—
149	#135	—	94	12	no	—
150	C₂×Q₁₆	—	47	6	no	—
151	C₂×Q₁₆	—	47	24	no	—
152	C₂×Q₁₆	—	94	6	no	—
153	C₂×Q₁₆	—	94	12	no	—
154	C₂×SD₁₆	—	94	6	no	—
155	C₂×D₈	—	47	2	no	—
156	#131	D₁₈₈	1	4	yes	—
157	#131	C₂²×D₄₇	1	4	yes	—
158	C₂×M₁₆	D₁₈₈	1	3	yes	—
159	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
160	Q₃₂	—	47	12	no	—
161	Q₃₂	—	47	24	no	—
162	SD₃₂	—	47	24	no	—
163	D₁₆	—	47	4	no	—
164	M₃₂	C₂²×D₄₇	1	12	yes	—
165	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×C₁₆	C₂²×D₄₇	1	3	yes	—
167	C₂×C₁₆	C₂²×D₄₇	1	12	yes	—
168	#41	—	94	24	no	—
169	#41	—	188	48	no	—
170	#41	—	188	12	no	—
171	#41	D₃₇₆	1	4	yes	—
172	#41	C₂×D₁₈₈	1	4	yes	—
173	C₂×Q₈	—	94	6	no	—
174	C₂×Q₈	—	94	24	no	—
175	C₂×Q₈	—	188	16	no	—
176	C₂×Q₈	—	188	6	no	—
177	C₂×Q₈	—	188	12	no	—
178	C₂×Q₈	D₃₇₆	1	1	yes	characteristic
179	C₂×D₄	—	94	2	no	—
180	C₂×D₄	—	188	6	no	—
181	C₂×D₄	D₃₇₆	1	3	yes	—
182	C₂²×C₄	—	188	6	no	—
183	C₂²×C₄	—	188	6	no	—
184	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
185	Q₁₆	—	47	12	no	—
186	Q₁₆	—	94	6	no	—
187	Q₁₆	—	94	24	no	—
188	SD₁₆	—	94	24	no	—
189	D₈	—	47	4	no	—
190	D₈	—	94	2	no	—
191	M₁₆	C₂×D₁₈₈	1	12	yes	—
192	C₂×C₈	C₂×D₁₈₈	1	3	yes	—
193	C₂×C₈	C₂×D₁₈₈	1	6	yes	—
194	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
195	C₂×C₈	C₂³×D₄₇	1	6	yes	—
196	C₁₆	C₂³×D₄₇	1	2	yes	—
197	C₁₆	C₂³×D₄₇	1	6	yes	—
198	C₂³	—	188	2	no	—
199	C₂³	C₂×D₃₇₆	1	3	yes	—
200	Q₈	—	94	12	no	—
201	Q₈	—	188	24	no	—
202	Q₈	—	188	6	no	—
203	Q₈	C₂×D₃₇₆	1	4	yes	—
204	D₄	—	94	4	no	—
205	D₄	—	188	24	no	—
206	D₄	—	188	2	no	—
207	D₄	C₂×D₃₇₆	1	12	yes	—
208	C₂×C₄	—	188	24	no	—
209	C₂×C₄	—	188	6	no	—
210	C₂×C₄	—	188	24	no	—
211	C₂×C₄	C₂×D₃₇₆	1	6	yes	—
212	C₂×C₄	C₂×D₃₇₆	1	3	yes	—
213	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
214	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
215	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
216	C₈	C₂²×D₁₈₈	1	6	yes	—
217	C₈	C₂⁴×D₄₇	1	1	yes	characteristic
218	C₂²	—	2	6	no	—
219	C₂²	—	376	4	no	—
220	C₂²	—	188	4	no	—
221	C₂²	C₂²×D₃₇₆	1	6	yes	—
222	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
223	C₄	—	188	12	no	—
224	C₄	C₂²×D₃₇₆	1	6	yes	—
225	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
226	C₄	C₂³×D₁₈₈	1	1	yes	characteristic
227	C₂	—	2	6	no	—
228	C₂	—	376	4	no	—
229	C₂	#20256541	1	2	yes	direct product factor
230	C₂	C₂³×D₃₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₇₆
commutator, cosocle	C₈×C₄₇	C₂⁵
Frattini	C₈	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#504	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂×#504	D₄₇
hypocenter	C₄₇	C₂×#3417
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#15689940	C₂	semidirect
2. #20256541	C₂	direct (N × H)
3. C₂×#15689566	C₂	semidirect
4. C₂×#15689555	C₂	semidirect
5. C₂²×Dic₃₇₆	C₂	semidirect
6. C₂×C₄₇×#504	C₂	semidirect
9. #15689940	C₂²	semidirect
13. #15689566	C₂²	semidirect
14. #15689555	C₂²	semidirect
17. C₂×Dic₃₇₆	C₂²	semidirect
21. C₄₇×#504	C₂²	semidirect
117. C₂×#504	D₄₇	semidirect
119. C₂×C₄₇	#3417	semidirect
136. #504	C₂×D₄₇	semidirect
139. C₄₇	C₂×#3417	semidirect
229. C₂	#20256541	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×#15689940	2
#20256541	2
C₂×#15689566	2
C₂×#15689555	2
C₂²×Dic₃₇₆	2
C₂×C₄₇×#504	2
C₂×#3417	47

Subgroups — C2×#20256541 #27122804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256541 #27122804