Subgroups — C₂×#20245565 #26371015

Order 12032.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689525	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689241	C₂	1	1	yes	maximal, characteristic
3	#20245565	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688946	C₂	1	2	yes	maximal, complemented
5	C₂×#15688934	C₂	1	2	yes	maximal
6	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13094046	C₂²	1	1	yes	characteristic
8	C₂×#13094030	—	2	1	no	—
9	C₂×#13094016	—	2	1	no	—
10	#15689525	—	2	2	no	—
11	#15689525	C₂²	1	4	yes	complemented
12	#15689241	—	2	2	no	—
13	#15689241	C₂²	1	4	yes	—
14	C₂×#13093960	C₂²	1	1	yes	characteristic
15	C₂×#13093958	—	2	1	no	—
16	C₂×#13093956	C₂²	1	2	yes	—
17	C₂×#13093952	—	2	1	no	—
18	#15688946	C₂²	1	8	yes	complemented
19	#15688934	C₂²	1	8	yes	—
20	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#364	C₂²	1	4	yes	complemented
23	C₂×#10625350	—	2	1	no	—
24	#13094046	—	2	1	no	—
25	#13094046	—	2	1	no	—
26	#13094046	Q₈	1	2	yes	—
27	#13094046	C₂³	1	2	yes	—
28	C₂×Dic₁₈₈	Q₈	1	1	yes	characteristic
29	C₂×#10625325	—	2	1	no	—
30	C₂×D₃₇₆	Q₈	1	1	yes	characteristic
31	#13094030	—	2	4	no	—
32	#13094030	—	4	2	no	—
33	C₂×#10625329	—	2	1	no	—
34	C₂×#10625324	—	4	1	no	—
35	C₂×C₈×D₄₇	—	4	1	no	—
36	#13094016	—	2	4	no	—
37	#13094016	—	4	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×Dic₄₇⋊C₄	—	4	2	no	—
40	#13093960	—	2	1	no	—
41	#13093960	C₂³	1	2	yes	—
42	#13093958	—	2	4	no	—
43	#13093956	—	2	2	no	—
44	#13093956	C₂³	1	4	yes	—
45	#13093952	—	2	4	no	—
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	M₃₂×C₄₇	—	2	2	no	—
50	M₃₂×C₄₇	C₂³	1	4	yes	—
51	C₂×C₁₆×C₄₇	—	2	2	no	—
52	C₄×C₈×C₄₇	D₄	1	2	yes	—
53	C₄×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₂²⋊D₄₇	—	4	1	no	—
57	C₂²×Dic₄₇	—	4	2	no	—
58	#10625350	—	2	2	no	—
59	#10625350	—	2	2	no	—
60	#10625350	—	4	1	no	—
61	#10625350	—	4	1	no	—
62	C₂×D₁₈₈	—	2	1	no	—
63	C₂×C₄×D₄₇	—	4	1	no	—
64	C₂×Dic₉₄	—	2	1	no	—
65	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
67	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
68	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
73	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₄²×C₄₇	—	2	1	no	—
76	C₄²×C₄₇	C₂×D₄	1	2	yes	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	4	2	no	—
79	Dic₄₇⋊C₄	—	4	8	no	—
80	#10625329	—	2	2	no	—
81	#10625329	—	4	1	no	—
82	C₂×C₄₇⋊C₈	—	4	1	no	—
83	Dic₁₈₈	—	2	1	no	—
84	Dic₁₈₈	C₂×Q₈	1	2	yes	—
85	D₃₇₆	—	2	1	no	—
86	D₃₇₆	C₂×Q₈	1	2	yes	—
87	#10625325	—	2	2	no	—
88	#10625325	—	2	2	no	—
89	#10625324	—	4	4	no	—
90	C₈×D₄₇	—	4	4	no	—
91	C₁₆×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂²×D₄₇	—	4	1	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂²⋊D₄₇	—	4	2	no	—
103	C₂²⋊D₄₇	—	4	2	no	—
104	C₂×Dic₄₇	—	4	1	no	—
105	C₂×Dic₄₇	—	4	4	no	—
106	C₂×Dic₄₇	—	8	4	no	—
107	D₁₈₈	—	2	2	no	—
108	D₁₈₈	—	4	1	no	—
109	C₄×D₄₇	—	4	2	no	—
110	C₄×D₄₇	—	4	2	no	—
111	Dic₉₄	—	2	2	no	—
112	Dic₉₄	—	4	1	no	—
113	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
114	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
115	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
116	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
117	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₄₇⋊C₈	—	4	2	no	—
120	C₂×#2231	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	#476	1	1	yes	characteristic
123	C₂²×C₄₇	#476	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×D₄₇	—	4	2	no	—
126	C₂×D₄₇	—	8	2	no	—
127	C₄×C₄₇	—	4	2	no	—
128	C₄×C₄₇	#476	1	1	yes	characteristic
129	C₄×C₄₇	#476	1	1	yes	characteristic
130	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
132	Dic₄₇	—	4	2	no	—
133	Dic₄₇	—	8	4	no	—
134	C₂×#499	—	47	1	no	—
135	C₂×#443	—	47	1	no	—
136	#2231	—	47	8	no	—
137	C₂×#365	—	47	2	no	—
138	C₂×#364	D₄₇	1	1	yes	characteristic, complemented
139	C₂×#360	—	47	2	no	—
140	C₂×C₄₇	—	2	1	no	—
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	#2231	1	2	yes	complemented
143	C₂×C₄₇	C₂×#476	1	1	yes	characteristic
144	D₄₇	—	8	2	no	—
145	C₂×#135	—	47	1	no	—
146	C₂×#131	—	94	1	no	—
147	C₂×#125	—	94	1	no	—
148	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
149	#499	—	47	4	no	—
150	#499	—	94	2	no	—
151	#443	—	47	4	no	—
152	#443	—	94	2	no	—
153	C₂×#108	—	47	1	no	—
154	C₂×#107	—	94	1	no	—
155	C₂×#106	—	47	2	no	—
156	C₂×#104	—	94	1	no	—
157	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
158	#365	—	47	8	no	—
159	#364	C₂×D₄₇	1	4	yes	complemented
160	#360	—	47	8	no	—
161	C₄₇	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	94	1	no	—
163	#135	—	47	2	no	—
164	#135	—	47	2	no	—
165	#135	—	94	1	no	—
166	#135	—	94	1	no	—
167	C₂×Q₁₆	—	47	1	no	—
168	C₂×SD₁₆	—	94	1	no	—
169	C₂×D₈	—	47	1	no	—
170	#131	—	94	4	no	—
171	#131	—	188	2	no	—
172	C₂×M₁₆	—	94	1	no	—
173	C₂×M₁₆	—	188	1	no	—
174	C₂²×C₈	—	188	1	no	—
175	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
177	#125	—	94	4	no	—
178	#125	—	188	2	no	—
179	C₂×C₄⋊C₄	—	94	2	no	—
180	C₂×C₄⋊C₄	—	188	2	no	—
181	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
182	M₃₂	—	2	2	no	—
183	M₃₂	C₂²×D₄₇	1	4	yes	—
184	C₂×C₁₆	—	2	2	no	—
185	#108	—	47	2	no	—
186	#108	—	94	1	no	—
187	#107	—	94	4	no	—
188	#106	—	47	4	no	—
189	#106	—	94	2	no	—
190	#104	—	94	4	no	—
191	C₄×C₈	D₁₈₈	1	2	yes	—
192	C₄×C₈	C₂²×D₄₇	1	2	yes	—
193	#41	—	94	2	no	—
194	#41	—	94	2	no	—
195	#41	—	188	1	no	—
196	#41	—	188	1	no	—
197	C₂×Q₈	—	94	1	no	—
198	C₂×D₄	—	94	1	no	—
199	C₂×D₄	—	188	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	188	2	no	—
202	C₂²×C₄	—	188	1	no	—
203	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
204	Q₁₆	—	47	2	no	—
205	Q₁₆	—	94	1	no	—
206	SD₁₆	—	94	2	no	—
207	SD₁₆	—	94	2	no	—
208	D₈	—	47	2	no	—
209	D₈	—	94	1	no	—
210	M₁₆	—	94	2	no	—
211	M₁₆	—	188	1	no	—
212	M₁₆	—	188	4	no	—
213	C₂×C₈	—	188	1	no	—
214	C₂×C₈	—	188	4	no	—
215	C₂×C₈	#10625325	1	1	yes	characteristic
216	C₂×C₈	#10625325	1	1	yes	characteristic
217	C₂×C₈	#10625325	1	2	yes	—
218	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
219	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
220	C₂×C₈	#10625352	1	1	yes	characteristic
221	C₂×C₈	#10625352	1	1	yes	characteristic
222	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
223	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
224	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
225	C₄⋊C₄	—	94	4	no	—
226	C₄⋊C₄	—	188	8	no	—
227	C₄⋊C₄	—	188	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×D₁₈₈	1	2	yes	—
230	C₁₆	—	2	4	no	—
231	C₂³	—	188	1	no	—
232	C₂³	#13094052	1	1	yes	characteristic
233	Q₈	—	94	2	no	—
234	Q₈	—	188	1	no	—
235	D₄	—	94	2	no	—
236	D₄	—	188	2	no	—
237	D₄	—	188	2	no	—
238	D₄	—	188	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	376	4	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	—	188	1	no	—
244	C₂×C₄	—	188	2	no	—
245	C₂×C₄	—	188	2	no	—
246	C₂×C₄	#13094004	1	1	yes	characteristic
247	C₂×C₄	#13094004	1	1	yes	characteristic
248	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
249	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
250	C₂×C₄	#13094052	1	1	yes	characteristic
251	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
252	C₈	—	188	2	no	—
253	C₈	#13094004	1	1	yes	characteristic
254	C₈	#13094004	1	1	yes	characteristic
255	C₈	C₂×#10625325	1	2	yes	—
256	C₈	C₂×#10625325	1	2	yes	—
257	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
258	C₈	C₂×#10625352	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	376	2	no	—
261	C₂²	—	188	2	no	—
262	C₂²	#15689414	1	1	yes	characteristic
263	C₂²	#15689414	1	1	yes	characteristic
264	C₂²	C₂×#13094052	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	376	4	no	—
267	C₄	—	188	2	no	—
268	C₄	#15689414	1	1	yes	characteristic
269	C₄	#15689414	1	1	yes	characteristic
270	C₄	C₂×#13094004	1	1	yes	characteristic
271	C₄	C₂²×#10625325	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	376	2	no	—
275	C₂	#20245565	1	2	yes	direct product factor
276	C₂	C₂×#15689414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689414
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂²×C₄₇	#476
prime radical	C₂×#13094046	C₂²
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	C₂×#2231
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#15689525	C₂	semidirect
3. #20245565	C₂	direct (N × H)
4. C₂×#15688946	C₂	semidirect
6. C₂×C₄₇×#364	C₂	semidirect
11. #15689525	C₂²	semidirect
18. #15688946	C₂²	semidirect
22. C₄₇×#364	C₂²	semidirect
138. C₂×#364	D₄₇	semidirect
142. C₂×C₄₇	#2231	semidirect
159. #364	C₂×D₄₇	semidirect
161. C₄₇	C₂×#2231	semidirect
275. C₂	#20245565	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×#15689525	2
C₂×#15689241	2
#20245565	2
C₂×#15688946	2
C₂×#15688934	2
C₂×C₄₇×#364	2
C₂×#2231	47

Subgroups — C2×#20245565 #26371015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245565 #26371015