Subgroups — C₂×#20245526 #26370882

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
3	#20245526	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688949	C₂	1	2	yes	maximal, complemented
5	C₂×#15688925	C₂	1	2	yes	maximal
6	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
8	C₂×#13094024	—	2	1	no	—
9	C₂×C₄×D₁₈₈	—	2	1	no	—
10	#15689528	—	2	2	no	—
11	#15689528	C₂²	1	4	yes	complemented
12	C₄×D₃₇₆	—	2	2	no	—
13	C₄×D₃₇₆	C₂²	1	4	yes	—
14	C₂×#13093958	—	2	1	no	—
15	C₂×#13093958	C₂²	1	2	yes	—
16	C₂×#13093958	C₂²	1	1	yes	characteristic
17	C₂×#13093946	—	2	1	no	—
18	#15688949	C₂²	1	8	yes	complemented
19	#15688925	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂²×Dic₉₄	—	4	1	no	—
24	C₂²×D₁₈₈	—	2	1	no	—
25	C₂²×C₄×D₄₇	—	4	1	no	—
26	C₂×D₃₇₆	—	2	1	no	—
27	C₂×D₃₇₆	—	2	2	no	—
28	C₂×D₃₇₆	Q₈	1	4	yes	complemented
29	C₂×D₃₇₆	C₂³	1	2	yes	—
30	#13094024	—	2	4	no	—
31	#13094024	—	4	2	no	—
32	C₄×D₁₈₈	—	2	4	no	—
33	C₄×D₁₈₈	—	4	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	1	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄⋊Dic₄₇	—	4	1	no	—
37	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
38	#13093958	—	2	2	no	—
39	#13093958	—	2	4	no	—
40	#13093958	—	2	1	no	—
41	#13093958	C₂³	1	4	yes	—
42	#13093958	C₂³	1	2	yes	—
43	#13093946	—	2	4	no	—
44	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₁₆×C₄₇	—	2	4	no	—
48	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
49	C₄×C₈×C₄₇	D₄	1	2	yes	—
50	C₄×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂³×D₄₇	—	4	1	no	—
52	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	4	1	no	—
56	C₂²×Dic₄₇	—	4	2	no	—
57	C₂×D₁₈₈	—	2	4	no	—
58	C₂×D₁₈₈	—	2	2	no	—
59	C₂×D₁₈₈	—	4	1	no	—
60	C₂×D₁₈₈	—	4	2	no	—
61	C₂×C₄×D₄₇	—	4	4	no	—
62	C₂×C₄×D₄₇	—	8	4	no	—
63	C₂×Dic₉₄	—	4	4	no	—
64	C₂×Dic₉₄	—	8	4	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₄₇	#41	1	2	yes	—
70	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₄²×C₄₇	D₈	1	2	yes	—
72	C₄²×C₄₇	C₂×D₄	1	2	yes	—
73	(C₂×D₄₇)⋊C₄	—	4	4	no	—
74	C₄⋊Dic₄₇	—	2	2	no	—
75	C₄⋊Dic₄₇	—	2	4	no	—
76	C₄⋊Dic₄₇	—	4	1	no	—
77	C₄⋊Dic₄₇	—	4	2	no	—
78	C₄⋊Dic₄₇	—	4	4	no	—
79	D₃₇₆	—	2	4	no	—
80	D₃₇₆	—	2	2	no	—
81	D₃₇₆	C₂×Q₈	1	4	yes	complemented
82	C₁₆×C₄₇	—	2	4	no	—
83	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₄₇	—	4	2	no	—
85	C₂²×D₄₇	—	4	4	no	—
86	C₂²×D₄₇	—	8	2	no	—
87	C₂²×D₄₇	—	8	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	D₁₆	1	4	yes	—
90	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₄₇	#137	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₂×Dic₄₇	—	4	4	no	—
97	C₂×Dic₄₇	—	4	2	no	—
98	C₂×Dic₄₇	—	8	4	no	—
99	C₂×Dic₄₇	—	8	2	no	—
100	D₁₈₈	—	2	4	no	—
101	D₁₈₈	—	4	2	no	—
102	D₁₈₈	—	4	4	no	—
103	C₄×D₄₇	—	8	8	no	—
104	Dic₉₄	—	8	8	no	—
105	C₈×C₄₇	—	2	2	no	—
106	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
107	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₂×#2224	—	47	1	no	maximal, Sylow
110	C₂²×C₄₇	#474	1	2	yes	—
111	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
112	C₂²×C₄₇	#510	1	2	yes	—
113	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
114	C₂×D₄₇	—	4	4	no	—
115	C₂×D₄₇	—	8	2	no	—
116	C₂×D₄₇	—	8	2	no	—
117	C₂×D₄₇	—	8	8	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	#474	1	2	yes	—
120	C₄×C₄₇	C₂×D₁₆	1	4	yes	—
121	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
123	Dic₄₇	—	8	4	no	—
124	Dic₄₇	—	8	2	no	—
125	C₂×#500	—	47	1	no	—
126	C₂×C₄×D₈	—	47	1	no	—
127	#2224	—	47	8	no	—
128	C₂×#366	—	47	2	no	—
129	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
130	C₂×#357	—	47	2	no	—
131	C₂×C₄₇	#2224	1	4	yes	complemented
132	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
133	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
134	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
135	D₄₇	—	8	4	no	—
136	C₂²×D₈	—	47	1	no	—
137	C₂×C₄⋊Q₈	—	94	1	no	—
138	C₂×C₄×D₄	—	94	1	no	—
139	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
140	#500	—	47	4	no	—
141	#500	—	94	2	no	—
142	C₄×D₈	—	47	4	no	—
143	C₄×D₈	—	94	2	no	—
144	C₂×#107	—	47	2	no	—
145	C₂×#107	—	47	1	no	—
146	C₂×#107	—	94	1	no	—
147	C₂×#102	—	94	1	no	—
148	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
149	#366	—	47	8	no	—
150	C₄⋊C₁₆	C₂×D₄₇	1	4	yes	complemented
151	#357	—	47	8	no	—
152	C₄₇	C₂×#2224	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	188	1	no	—
154	C₂²×D₄	—	94	1	no	—
155	C₂³×C₄	—	188	1	no	—
156	C₂×D₈	—	47	4	no	—
157	C₂×D₈	—	47	2	no	—
158	C₂×D₈	—	94	1	no	—
159	C₂×D₈	—	94	2	no	—
160	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
162	C₄⋊Q₈	—	94	4	no	—
163	C₄⋊Q₈	—	188	2	no	—
164	C₄×D₄	—	94	4	no	—
165	C₄×D₄	—	188	2	no	—
166	C₂×C₄⋊C₄	—	94	1	no	—
167	C₂×C₄⋊C₄	—	94	2	no	—
168	C₂×C₄⋊C₄	—	188	1	no	—
169	C₂×C₂²⋊C₄	—	188	1	no	—
170	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
173	#107	—	47	4	no	—
174	#107	—	47	2	no	—
175	#107	—	94	1	no	—
176	#107	—	94	2	no	—
177	#107	—	94	4	no	—
178	#102	—	94	4	no	—
179	C₄×C₈	D₁₈₈	1	2	yes	—
180	C₄×C₈	C₂²×D₄₇	1	2	yes	—
181	C₂⁴	—	188	1	no	—
182	C₂×Q₈	—	376	4	no	—
183	C₂×Q₈	—	188	4	no	—
184	C₂×D₄	—	94	4	no	—
185	C₂×D₄	—	94	2	no	—
186	C₂×D₄	—	188	1	no	—
187	C₂×D₄	—	188	2	no	—
188	C₂²×C₄	—	376	4	no	—
189	C₂²×C₄	—	188	1	no	—
190	C₂²×C₄	—	188	2	no	—
191	C₂²×C₄	—	188	4	no	—
192	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
193	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
195	D₈	—	47	4	no	—
196	D₈	—	94	4	no	—
197	D₈	—	94	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
200	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
201	C₂×C₈	#10625352	1	2	yes	—
202	C₂×C₈	Q₈×D₄₇	1	2	yes	—
203	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
204	C₄⋊C₄	—	94	2	no	—
205	C₄⋊C₄	—	94	4	no	—
206	C₄⋊C₄	—	188	1	no	—
207	C₄⋊C₄	—	188	2	no	—
208	C₄⋊C₄	—	188	4	no	—
209	C₂²⋊C₄	—	188	4	no	—
210	C₄²	D₃₇₆	1	2	yes	—
211	C₄²	C₂×D₁₈₈	1	2	yes	—
212	C₁₆	—	2	4	no	—
213	C₂³	—	376	2	no	—
214	C₂³	—	376	2	no	—
215	C₂³	—	188	2	no	—
216	C₂³	—	188	4	no	—
217	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
218	Q₈	—	376	8	no	—
219	D₄	—	94	4	no	—
220	D₄	—	188	4	no	—
221	D₄	—	188	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	376	2	no	—
224	C₂×C₄	—	376	4	no	—
225	C₂×C₄	—	376	8	no	—
226	C₂×C₄	—	188	2	no	—
227	C₂×C₄	—	188	4	no	—
228	C₂×C₄	D₇₅₂	1	4	yes	—
229	C₂×C₄	#13094004	1	2	yes	—
230	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
231	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
232	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
233	C₂×C₄	#13094057	1	2	yes	—
234	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	#13094004	1	2	yes	—
237	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
238	C₈	C₂×#10625352	1	1	yes	characteristic
239	C₂²	—	376	2	no	—
240	C₂²	—	376	8	no	—
241	C₂²	—	376	2	no	—
242	C₂²	—	188	4	no	—
243	C₂²	#15689404	1	2	yes	—
244	C₂²	C₂×D₇₅₂	1	2	yes	—
245	C₂²	#15689566	1	2	yes	—
246	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	376	2	no	—
249	C₄	—	376	4	no	—
250	C₄	#15689404	1	2	yes	—
251	C₄	C₂×D₇₅₂	1	4	yes	—
252	C₄	C₂×#13094004	1	1	yes	characteristic
253	C₄	C₂×#13094057	1	1	yes	characteristic
254	C₂	—	376	4	no	—
255	C₂	#20245526	1	4	yes	direct product factor
256	C₂	C₂×#15689404	1	1	yes	characteristic
257	C₂	C₂²×D₇₅₂	1	1	yes	characteristic
258	C₂	C₂×#15689566	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₇₆
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×D₃₇₆	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	C₂×#2224
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15689528	C₂	semidirect
3. #20245526	C₂	direct (N × H)
4. C₂×#15688949	C₂	semidirect
6. C₂×C₄₇×C₄⋊C₁₆	C₂	semidirect
11. #15689528	C₂²	semidirect
18. #15688949	C₂²	semidirect
22. C₄₇×C₄⋊C₁₆	C₂²	semidirect
28. C₂×D₃₇₆	Q₈	semidirect
81. D₃₇₆	C₂×Q₈	semidirect
129. C₂×C₄⋊C₁₆	D₄₇	semidirect
131. C₂×C₄₇	#2224	semidirect
150. C₄⋊C₁₆	C₂×D₄₇	semidirect
152. C₄₇	C₂×#2224	semidirect
255. C₂	#20245526	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×#15689528	2
C₂×C₄×D₃₇₆	2
#20245526	2
C₂×#15688949	2
C₂×#15688925	2
C₂×C₄₇×C₄⋊C₁₆	2
C₂×#2224	47

Subgroups — C2×#20245526 #26370882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245526 #26370882