Subgroups — C₂×#20256363 #27097172

Order 12032.

2582 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689795	C₂	1	2	yes	maximal, complemented
2	#20256363	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689500	C₂	1	3	yes	maximal, complemented
4	C₂×#15689319	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₃₇₆	C₂	1	3	yes	maximal, complemented
6	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625354	—	2	2	no	—
8	C₂²×D₃₇₆	C₂²	1	3	yes	—
9	C₂²×D₃₇₆	C₂²	1	6	yes	—
10	#15689795	—	2	8	no	—
11	#15689795	C₂²	1	16	yes	complemented
12	C₂×#13094021	C₂²	1	6	yes	—
13	C₂×#13093999	—	2	6	no	—
14	C₂×C₄×D₁₈₈	C₂²	1	6	yes	—
15	#15689500	C₂²	1	24	yes	complemented
16	#15689319	C₂²	1	48	yes	complemented
17	C₄×D₃₇₆	C₂²	1	24	yes	complemented
18	C₂×#13093958	C₂²	1	1	yes	characteristic
19	C₂×#13093946	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₄₇	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
24	C₂×#10625354	—	2	16	no	—
25	C₂×#10625354	—	4	16	no	—
26	C₂²×D₁₈₈	—	2	6	no	—
27	C₂²×D₁₈₈	—	2	6	no	—
28	C₂²×D₁₈₈	C₂³	1	6	yes	—
29	C₂²×C₄×D₄₇	—	2	6	no	—
30	C₂×D₃₇₆	—	2	12	no	—
31	C₂×D₃₇₆	—	4	12	no	—
32	C₂×D₃₇₆	C₂³	1	12	yes	—
33	C₂×D₃₇₆	C₂³	1	24	yes	—
34	#13094021	—	2	12	no	—
35	#13094021	C₂³	1	24	yes	—
36	#13093999	—	2	48	no	—
37	C₄×D₁₈₈	—	2	12	no	—
38	C₄×D₁₈₈	C₂³	1	24	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
40	C₂×(C₂×D₄₇)⋊C₄	—	2	6	no	—
41	#13093958	C₂³	1	4	yes	—
42	#13093946	C₂³	1	24	yes	—
43	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₄₇	D₄	1	4	yes	—
47	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
50	C₄⋊C₈×C₄₇	C₂³	1	12	yes	—
51	C₄×C₈×C₄₇	C₂³	1	12	yes	—
52	C₂³×D₄₇	—	2	6	no	—
53	C₂×Q₈×C₄₇	D₈	1	8	yes	complemented
54	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	2	no	—
59	#10625354	—	4	64	no	—
60	C₂×D₁₈₈	—	2	12	no	—
61	C₂×D₁₈₈	—	2	12	no	—
62	C₂×D₁₈₈	—	2	6	no	—
63	C₂×D₁₈₈	—	2	24	no	—
64	C₂×D₁₈₈	—	2	24	no	—
65	C₂×D₁₈₈	—	4	24	no	—
66	C₂×D₁₈₈	—	8	12	no	—
67	C₂×D₁₈₈	C₂⁴	1	12	yes	—
68	C₂×C₄×D₄₇	—	2	24	no	—
69	C₂×C₄×D₄₇	—	4	24	no	—
70	C₂×C₈×C₄₇	—	2	6	no	—
71	C₂×C₈×C₄₇	#41	1	4	yes	complemented
72	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
73	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
74	C₄⋊C₄×C₄₇	C₂×D₄	1	12	yes	—
75	C₄²×C₄₇	C₂×D₄	1	6	yes	—
76	C₄²×C₄₇	C₂⁴	1	6	yes	—
77	(C₂×D₄₇)⋊C₄	—	2	24	no	—
78	C₄⋊Dic₄₇	—	2	2	no	—
79	C₄⋊Dic₄₇	C₂⁴	1	4	yes	—
80	D₃₇₆	—	2	24	no	—
81	D₃₇₆	—	4	24	no	—
82	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂²×D₄₇	—	2	12	no	—
84	C₂²×D₄₇	—	4	12	no	—
85	C₂²×D₄₇	—	4	12	no	—
86	C₂²×D₄₇	—	4	12	no	—
87	Q₈×C₄₇	C₂×D₈	1	16	yes	complemented
88	C₂×C₄×C₄₇	—	2	6	no	—
89	C₂×C₄×C₄₇	C₂×D₈	1	12	yes	—
90	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
91	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
93	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
94	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
95	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
96	C₂×Dic₄₇	—	2	4	no	—
97	C₂×Dic₄₇	—	4	4	no	—
98	D₁₈₈	—	2	12	no	—
99	D₁₈₈	—	2	12	no	—
100	D₁₈₈	—	4	12	no	—
101	D₁₈₈	—	4	48	no	—
102	D₁₈₈	—	8	24	no	—
103	C₄×D₄₇	—	4	48	no	—
104	C₈×C₄₇	—	2	6	no	—
105	C₈×C₄₇	C₂×#41	1	4	yes	complemented
106	C₂×#3391	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	#550	1	2	yes	—
108	C₂²×C₄₇	C₂²×D₈	1	2	yes	—
109	C₂²×C₄₇	#576	1	2	yes	—
110	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
111	C₂×D₄₇	—	4	12	no	—
112	C₂×D₄₇	—	4	12	no	—
113	C₂×D₄₇	—	4	12	no	—
114	C₂×D₄₇	—	8	24	no	—
115	C₄×C₄₇	—	2	6	no	—
116	C₄×C₄₇	#550	1	2	yes	—
117	C₄×C₄₇	C₂²×D₈	1	12	yes	—
118	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
119	C₄×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
120	Dic₄₇	—	4	4	no	—
121	C₂×#550	—	47	2	no	—
122	#3391	—	47	16	no	—
123	C₂×#493	—	47	3	no	—
124	C₂×#459	—	47	6	no	—
125	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
126	C₂×C₄×D₈	—	47	3	no	—
127	C₂×C₄₇	#3391	1	4	yes	complemented
128	C₂×C₄₇	C₂×#550	1	1	yes	characteristic
129	C₂×C₄₇	C₂³×D₈	1	1	yes	characteristic
130	C₂×C₄₇	C₂×#576	1	1	yes	characteristic
131	D₄₇	—	8	12	no	—
132	C₂²×#41	—	94	2	no	—
133	C₂²×D₈	—	47	3	no	—
134	C₂²×D₈	—	47	6	no	—
135	#550	—	47	16	no	—
136	#550	—	94	8	no	—
137	C₂×#127	—	47	6	no	—
138	C₂×#121	—	94	6	no	—
139	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
140	C₂×C₄×D₄	—	47	6	no	—
141	#493	—	47	24	no	—
142	#459	—	47	48	no	—
143	C₈×Q₈	C₂×D₄₇	1	8	yes	complemented
144	C₄×D₈	—	47	24	no	—
145	C₂×#107	—	47	1	no	—
146	C₂×C₄⋊C₈	C₂×D₄₇	1	3	yes	—
147	C₂×#102	—	47	6	no	—
148	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
149	C₄₇	C₂×#3391	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	—	94	16	no	—
151	C₂×#41	—	188	16	no	—
152	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
153	C₂²×D₄	—	47	6	no	—
154	C₂²×D₄	—	94	6	no	—
155	C₂²×D₄	—	94	6	no	—
156	C₂³×C₄	—	94	6	no	—
157	C₂×D₈	—	47	12	no	—
158	C₂×D₈	—	47	24	no	—
159	C₂×D₈	—	94	12	no	—
160	C₂×D₈	—	188	12	no	—
161	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
162	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
163	#127	—	47	24	no	—
164	#127	—	94	12	no	—
165	#121	—	94	48	no	—
166	C₄×Q₈	D₁₈₈	1	4	yes	—
167	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
168	C₄×D₄	—	47	24	no	—
169	C₄×D₄	—	94	12	no	—
170	C₂×C₄⋊C₄	—	47	2	no	—
171	C₂×C₄⋊C₄	D₁₈₈	1	3	yes	—
172	C₂×C₂²⋊C₄	—	94	6	no	—
173	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
174	#107	—	47	4	no	—
175	C₄⋊C₈	C₂²×D₄₇	1	12	yes	—
176	#102	—	47	24	no	—
177	C₄×C₈	C₂²×D₄₇	1	12	yes	—
178	C₂⁴	—	94	6	no	—
179	#41	—	188	64	no	—
180	C₂×Q₈	D₃₇₆	1	8	yes	complemented
181	C₂×Q₈	C₂×D₁₈₈	1	4	yes	—
182	C₂×D₄	—	47	12	no	—
183	C₂×D₄	—	94	24	no	—
184	C₂×D₄	—	94	6	no	—
185	C₂×D₄	—	94	24	no	—
186	C₂×D₄	—	94	12	no	—
187	C₂×D₄	—	94	12	no	—
188	C₂×D₄	—	376	12	no	—
189	C₂×D₄	—	188	24	no	—
190	C₂²×C₄	—	94	2	no	—
191	C₂²×C₄	—	94	24	no	—
192	C₂²×C₄	—	188	24	no	—
193	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
194	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
195	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
196	D₈	—	94	24	no	—
197	D₈	—	188	24	no	—
198	C₂×C₈	—	2	6	no	—
199	C₂×C₈	#10625354	1	4	yes	complemented
200	C₂×C₈	C₂³×D₄₇	1	6	yes	—
201	C₂×C₈	C₂³×D₄₇	1	2	yes	—
202	C₄⋊C₄	—	47	4	no	—
203	C₄⋊C₄	—	94	2	no	—
204	C₄⋊C₄	C₂×D₁₈₈	1	12	yes	—
205	C₂²⋊C₄	—	94	24	no	—
206	C₄²	C₂×D₁₈₈	1	6	yes	—
207	C₄²	C₂³×D₄₇	1	6	yes	—
208	C₂³	—	94	12	no	—
209	C₂³	—	188	12	no	—
210	C₂³	—	188	12	no	—
211	C₂³	—	188	12	no	—
212	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
213	Q₈	C₂×D₃₇₆	1	16	yes	complemented
214	D₄	—	94	12	no	—
215	D₄	—	94	12	no	—
216	D₄	—	376	24	no	—
217	D₄	—	188	12	no	—
218	D₄	—	188	48	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	94	4	no	—
221	C₂×C₄	—	188	4	no	—
222	C₂×C₄	—	188	48	no	—
223	C₂×C₄	C₂×D₃₇₆	1	12	yes	—
224	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
225	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
227	C₂×C₄	C₂×#10625354	1	2	yes	—
228	C₂×C₄	#13094070	1	2	yes	—
229	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
230	C₈	—	2	6	no	—
231	C₈	C₂×#10625354	1	4	yes	complemented
232	C₂²	—	376	24	no	—
233	C₂²	—	188	12	no	—
234	C₂²	—	188	12	no	—
235	C₂²	—	188	12	no	—
236	C₂²	#15689795	1	2	yes	—
237	C₂²	C₂²×D₃₇₆	1	2	yes	—
238	C₂²	#15689928	1	2	yes	—
239	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	188	4	no	—
242	C₄	#15689795	1	2	yes	—
243	C₄	C₂²×D₃₇₆	1	12	yes	—
244	C₄	C₂²×#10625354	1	1	yes	characteristic
245	C₄	C₂×#13094070	1	1	yes	characteristic
246	C₂	—	376	12	no	—
247	C₂	#20256363	1	4	yes	direct product factor
248	C₂	C₂×#15689795	1	1	yes	characteristic
249	C₂	C₂³×D₃₇₆	1	1	yes	characteristic
250	C₂	C₂×#15689928	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	C₂×#3391
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15689795	C₂	semidirect
2. #20256363	C₂	direct (N × H)
3. C₂×#15689500	C₂	semidirect
4. C₂×#15689319	C₂	semidirect
5. C₂×C₄×D₃₇₆	C₂	semidirect
6. C₂×C₈×Q₈×C₄₇	C₂	semidirect
11. #15689795	C₂²	semidirect
15. #15689500	C₂²	semidirect
16. #15689319	C₂²	semidirect
17. C₄×D₃₇₆	C₂²	semidirect
21. C₈×Q₈×C₄₇	C₂²	semidirect
53. C₂×Q₈×C₄₇	D₈	semidirect
71. C₂×C₈×C₄₇	#41	semidirect
87. Q₈×C₄₇	C₂×D₈	semidirect
105. C₈×C₄₇	C₂×#41	semidirect
125. C₂×C₈×Q₈	D₄₇	semidirect
127. C₂×C₄₇	#3391	semidirect
143. C₈×Q₈	C₂×D₄₇	semidirect
149. C₄₇	C₂×#3391	semidirect
180. C₂×Q₈	D₃₇₆	semidirect
199. C₂×C₈	#10625354	semidirect
213. Q₈	C₂×D₃₇₆	semidirect
231. C₈	C₂×#10625354	semidirect
247. C₂	#20256363	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×#15689795	2
#20256363	2
C₂×#15689500	2
C₂×#15689319	2
C₂×C₄×D₃₇₆	2
C₂×C₈×Q₈×C₄₇	2
C₂×#3391	47

Subgroups — C2×#20256363 #27097172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256363 #27097172