Subgroups — C₂×#20251675 #26422115

Order 12032.

7502 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094003	C₂	1	3	yes	maximal, complemented
2	#20251675	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15688996	C₂	1	6	yes	maximal, complemented
4	C₂×#15688983	C₂	1	3	yes	maximal, complemented
5	C₂×#15688979	C₂	1	2	yes	maximal
6	C₂⁵×C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₉₄	C₂²	1	1	yes	characteristic, complemented
8	C₂×#13094003	—	2	24	no	—
9	C₂×#13094003	C₂²	1	24	yes	complemented
10	C₂²×C₄⋊Dic₄₇	C₂²	1	3	yes	complemented
11	C₂²×Dic₄₇⋊C₄	C₂²	1	6	yes	complemented
12	C₂²×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
13	C₂²×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
14	#15688996	C₂²	1	48	yes	complemented
15	#15688983	C₂²	1	24	yes	complemented
16	#15688979	C₂²	1	16	yes	—
17	C₂×C₄₇⋊#95	C₂²	1	6	yes	complemented
18	C₂⁶×C₄₇	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₄₇	—	2	24	no	—
20	C₂⁴×C₄×C₄₇	C₂²	1	8	yes	—
21	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
22	C₂⁴×C₄×C₄₇	C₂²	1	3	yes	—
23	C₂²×Dic₉₄	—	2	12	no	—
24	C₂²×Dic₉₄	C₂³	1	4	yes	complemented
25	C₂³×Dic₄₇	C₂³	1	6	yes	—
26	C₂³×Dic₄₇	C₂³	1	2	yes	—
27	#13094003	—	2	96	no	—
28	C₂×C₄⋊Dic₄₇	—	2	12	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	complemented
30	C₂×Dic₄₇⋊C₄	—	2	24	no	—
31	C₂×Dic₄₇⋊C₄	C₂³	1	24	yes	complemented
32	C₂×C₂²⋊Dic₄₇	—	2	24	no	—
33	C₂×C₂²⋊Dic₄₇	—	2	24	no	—
34	C₂×C₂²⋊Dic₄₇	C₂³	1	24	yes	—
35	C₂×C₂²⋊Dic₄₇	C₂³	1	24	yes	—
36	C₄₇⋊#95	C₂³	1	24	yes	complemented
37	C₂⁵×C₄₇	—	2	12	no	—
38	C₂⁵×C₄₇	—	2	12	no	—
39	C₂⁵×C₄₇	Q₈	1	8	yes	complemented
40	C₂⁵×C₄₇	C₂³	1	4	yes	—
41	C₂⁵×C₄₇	C₂³	1	3	yes	—
42	C₂³×C₄×C₄₇	—	2	128	no	—
43	C₂³×C₄×C₄₇	—	2	12	no	—
44	C₂³×C₄×C₄₇	—	2	24	no	—
45	C₂³×C₄×C₄₇	—	2	96	no	—
46	C₂³×C₄×C₄₇	—	2	24	no	—
47	C₂³×C₄×C₄₇	D₄	1	24	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
49	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	12	yes	—
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂⁴×C₄₇	—	2	32	no	—
53	C₂⁴×C₄₇	—	2	96	no	—
54	C₂⁴×C₄₇	—	2	96	no	—
55	C₂⁴×C₄₇	—	2	24	no	—
56	C₂⁴×C₄₇	—	2	12	no	—
57	C₂⁴×C₄₇	—	2	24	no	—
58	C₂⁴×C₄₇	—	2	12	no	—
59	C₂⁴×C₄₇	C₂×D₄	1	12	yes	—
60	C₂⁴×C₄₇	C₂×Q₈	1	12	yes	—
61	C₂⁴×C₄₇	C₂×Q₈	1	16	yes	complemented
62	C₂⁴×C₄₇	#41	1	12	yes	—
63	C₂⁴×C₄₇	C₂⁴	1	6	yes	—
64	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	96	no	—
66	C₂²×C₄×C₄₇	—	2	256	no	—
67	C₂²×C₄×C₄₇	—	2	24	no	—
68	C₂²×C₄×C₄₇	—	2	12	no	—
69	C₂²×C₄×C₄₇	—	2	96	no	—
70	C₂²×C₄×C₄₇	—	2	48	no	—
71	C₂²×C₄×C₄₇	—	2	48	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	48	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
77	C₂²×Dic₄₇	—	2	12	no	—
78	C₂²×Dic₄₇	—	2	12	no	—
79	C₂²×Dic₄₇	—	2	24	no	—
80	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
81	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
82	C₂×Dic₉₄	—	2	24	no	—
83	C₂×Dic₉₄	—	4	16	no	—
84	C₂²⋊Dic₄₇	—	2	48	no	—
85	C₂²⋊Dic₄₇	—	2	48	no	—
86	C₄⋊Dic₄₇	—	2	24	no	—
87	Dic₄₇⋊C₄	—	2	48	no	—
88	C₂³×C₄₇	—	2	32	no	—
89	C₂³×C₄₇	—	2	256	no	—
90	C₂³×C₄₇	—	2	96	no	—
91	C₂³×C₄₇	—	2	24	no	—
92	C₂³×C₄₇	—	2	48	no	—
93	C₂³×C₄₇	—	2	96	no	—
94	C₂³×C₄₇	—	2	24	no	—
95	C₂³×C₄₇	—	2	48	no	—
96	C₂³×C₄₇	—	2	12	no	—
97	C₂³×C₄₇	—	2	12	no	—
98	C₂³×C₄₇	C₂²⋊Q₈	1	48	yes	complemented
99	C₂³×C₄₇	C₂²×D₄	1	12	yes	—
100	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
101	C₂³×C₄₇	C₂²×Q₈	1	12	yes	—
102	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
103	C₂³×C₄₇	C₂×#41	1	12	yes	—
104	C₂³×C₄₇	C₂×#41	1	6	yes	—
105	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
106	C₂×C₄×C₄₇	—	2	96	no	—
107	C₂×C₄×C₄₇	—	2	24	no	—
108	C₂×C₄×C₄₇	—	2	12	no	—
109	C₂×C₄×C₄₇	—	2	96	no	—
110	C₂×C₄×C₄₇	#120	1	16	yes	—
111	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
112	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
113	C₂×Dic₄₇	—	2	12	no	—
114	C₂×Dic₄₇	—	2	12	no	—
115	C₂×Dic₄₇	—	2	24	no	—
116	C₂×Dic₄₇	—	4	8	no	—
117	C₂×Dic₄₇	—	4	24	no	—
118	Dic₉₄	—	4	16	no	—
119	C₂×#2848	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	128	no	—
121	C₂²×C₄₇	—	2	96	no	—
122	C₂²×C₄₇	—	2	12	no	—
123	C₂²×C₄₇	—	2	24	no	—
124	C₂²×C₄₇	—	2	12	no	—
125	C₂²×C₄₇	—	2	24	no	—
126	C₂²×C₄₇	C₂×#120	1	4	yes	—
127	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	12	yes	—
128	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	24	yes	complemented
129	C₂²×C₄₇	#525	1	12	yes	—
130	C₂²×C₄₇	C₂³×D₄	1	3	yes	—
131	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×#41	1	3	yes	—
133	C₄×C₄₇	—	2	12	no	—
134	C₄×C₄₇	C₂×#120	1	8	yes	—
135	Dic₄₇	—	4	4	no	—
136	Dic₄₇	—	4	12	no	—
137	C₂⁵×C₄	D₄₇	1	1	yes	characteristic
138	C₂²×C₂²⋊Q₈	—	47	3	no	—
139	#2848	—	47	16	no	—
140	C₂×#385	—	47	6	no	—
141	C₂×#380	—	47	3	no	—
142	C₂×#379	—	47	2	no	—
143	C₂×C₄₇	—	2	24	no	—
144	C₂×C₄₇	#2848	1	8	yes	complemented
145	C₂×C₄₇	C₂²×#120	1	1	yes	characteristic
146	C₂×C₄₇	C₂²×C₂²⋊Q₈	1	3	yes	—
147	C₂×C₄₇	C₂×#525	1	3	yes	—
148	C₂⁶	C₂×D₄₇	1	1	yes	characteristic
149	C₂³×Q₈	—	47	1	no	—
150	C₂⁴×C₄	—	2	24	no	—
151	C₂⁴×C₄	C₂×D₄₇	1	8	yes	—
152	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
153	C₂⁴×C₄	C₂×D₄₇	1	3	yes	—
154	C₂×C₂²⋊Q₈	—	47	24	no	—
155	C₂×C₂²⋊Q₈	—	94	24	no	—
156	C₂²×C₄⋊C₄	—	47	6	no	—
157	C₂²×C₄⋊C₄	—	47	3	no	—
158	C₂²×C₂²⋊C₄	—	47	6	no	—
159	C₂²×C₂²⋊C₄	—	47	6	no	—
160	#385	—	47	48	no	—
161	#380	—	47	24	no	—
162	#379	—	47	16	no	—
163	C₂×#95	—	47	6	no	—
164	C₄₇	C₂×#2848	1	1	yes	characteristic, complemented, Sylow
165	C₂⁵	—	2	12	no	—
166	C₂⁵	—	2	12	no	—
167	C₂⁵	Dic₉₄	1	8	yes	complemented
168	C₂⁵	C₂²×D₄₇	1	4	yes	—
169	C₂⁵	C₂²×D₄₇	1	3	yes	—
170	C₂²×Q₈	—	47	4	no	—
171	C₂²×Q₈	—	94	12	no	—
172	C₂³×C₄	—	2	128	no	—
173	C₂³×C₄	—	2	12	no	—
174	C₂³×C₄	—	2	24	no	—
175	C₂³×C₄	—	2	96	no	—
176	C₂³×C₄	—	2	24	no	—
177	C₂³×C₄	—	47	2	no	—
178	C₂³×C₄	—	47	6	no	—
179	C₂³×C₄	C₂²⋊D₄₇	1	24	yes	—
180	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
181	C₂³×C₄	C₂²×D₄₇	1	12	yes	—
182	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂²⋊Q₈	—	94	96	no	—
185	C₂×C₄⋊C₄	—	47	24	no	—
186	C₂×C₄⋊C₄	—	47	12	no	—
187	C₂×C₄⋊C₄	—	94	24	no	—
188	C₂×C₄⋊C₄	—	94	12	no	—
189	C₂×C₂²⋊C₄	—	47	24	no	—
190	C₂×C₂²⋊C₄	—	47	24	no	—
191	C₂×C₂²⋊C₄	—	94	24	no	—
192	C₂×C₂²⋊C₄	—	94	24	no	—
193	#95	—	47	24	no	—
194	C₂⁴	—	2	32	no	—
195	C₂⁴	—	2	96	no	—
196	C₂⁴	—	2	96	no	—
197	C₂⁴	—	2	24	no	—
198	C₂⁴	—	2	12	no	—
199	C₂⁴	—	2	24	no	—
200	C₂⁴	—	2	12	no	—
201	C₂⁴	C₂×Dic₉₄	1	12	yes	—
202	C₂⁴	C₂×Dic₉₄	1	16	yes	complemented
203	C₂⁴	#10625350	1	12	yes	—
204	C₂⁴	C₂×C₂²⋊D₄₇	1	12	yes	—
205	C₂⁴	C₂³×D₄₇	1	6	yes	—
206	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
207	C₂×Q₈	—	94	24	no	—
208	C₂×Q₈	—	188	16	no	—
209	C₂²×C₄	—	2	96	no	—
210	C₂²×C₄	—	2	256	no	—
211	C₂²×C₄	—	2	24	no	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	—	2	96	no	—
214	C₂²×C₄	—	2	48	no	—
215	C₂²×C₄	—	2	48	no	—
216	C₂²×C₄	—	47	12	no	—
217	C₂²×C₄	—	47	4	no	—
218	C₂²×C₄	—	94	12	no	—
219	C₂²×C₄	—	94	12	no	—
220	C₂²×C₄	—	94	24	no	—
221	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	48	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
224	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
225	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
226	C₄⋊C₄	—	94	48	no	—
227	C₄⋊C₄	—	94	24	no	—
228	C₂²⋊C₄	—	94	48	no	—
229	C₂²⋊C₄	—	94	48	no	—
230	C₂³	—	2	32	no	—
231	C₂³	—	2	256	no	—
232	C₂³	—	2	96	no	—
233	C₂³	—	2	24	no	—
234	C₂³	—	2	48	no	—
235	C₂³	—	2	96	no	—
236	C₂³	—	2	48	no	—
237	C₂³	—	2	24	no	—
238	C₂³	—	2	12	no	—
239	C₂³	—	2	12	no	—
240	C₂³	#13094003	1	48	yes	complemented
241	C₂³	C₂²×C₂²⋊D₄₇	1	12	yes	—
242	C₂³	C₂²×C₂²⋊D₄₇	1	6	yes	—
243	C₂³	C₂²×Dic₉₄	1	12	yes	—
244	C₂³	C₂²×Dic₉₄	1	2	yes	—
245	C₂³	C₂×#10625350	1	12	yes	—
246	C₂³	C₂×#10625350	1	6	yes	—
247	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
248	Q₈	—	188	16	no	—
249	C₂×C₄	—	2	96	no	—
250	C₂×C₄	—	2	24	no	—
251	C₂×C₄	—	2	12	no	—
252	C₂×C₄	—	2	96	no	—
253	C₂×C₄	—	94	12	no	—
254	C₂×C₄	—	94	12	no	—
255	C₂×C₄	—	94	24	no	—
256	C₂×C₄	—	188	8	no	—
257	C₂×C₄	—	188	24	no	—
258	C₂×C₄	#13093993	1	16	yes	—
259	C₂×C₄	C₂²×C₂²⋊D₄₇	1	12	yes	—
260	C₂×C₄	C₂²×C₂²⋊D₄₇	1	12	yes	—
261	C₂²	—	2	128	no	—
262	C₂²	—	2	96	no	—
263	C₂²	—	2	12	no	—
264	C₂²	—	2	24	no	—
265	C₂²	—	2	12	no	—
266	C₂²	—	2	24	no	—
267	C₂²	C₂×#13093993	1	4	yes	—
268	C₂²	C₂×#13094003	1	12	yes	—
269	C₂²	C₂×#13094003	1	24	yes	complemented
270	C₂²	#15689634	1	12	yes	—
271	C₂²	C₂³×C₂²⋊D₄₇	1	3	yes	—
272	C₂²	C₂³×Dic₉₄	1	1	yes	characteristic
273	C₂²	C₂²×#10625350	1	3	yes	—
274	C₄	—	2	12	no	—
275	C₄	—	188	4	no	—
276	C₄	—	188	12	no	—
277	C₄	C₂×#13093993	1	8	yes	—
278	C₂	—	2	24	no	—
279	C₂	#20251675	1	8	yes	direct product factor
280	C₂	C₂²×#13093993	1	1	yes	characteristic
281	C₂	C₂²×#13094003	1	3	yes	—
282	C₂	C₂×#15689634	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂⁵×C₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂⁶×C₄₇	C₂²
hypercenter	C₂⁵×C₄	D₄₇
hypocenter	C₄₇	C₂×#2848
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13094003	C₂	semidirect
2. #20251675	C₂	direct (N × H)
3. C₂×#15688996	C₂	semidirect
4. C₂×#15688983	C₂	semidirect
7. C₂³×Dic₉₄	C₂²	semidirect
9. C₂×#13094003	C₂²	semidirect
10. C₂²×C₄⋊Dic₄₇	C₂²	semidirect
11. C₂²×Dic₄₇⋊C₄	C₂²	semidirect
14. #15688996	C₂²	semidirect
15. #15688983	C₂²	semidirect
17. C₂×C₄₇⋊#95	C₂²	semidirect
24. C₂²×Dic₉₄	C₂³	semidirect
29. C₂×C₄⋊Dic₄₇	C₂³	semidirect
31. C₂×Dic₄₇⋊C₄	C₂³	semidirect
36. C₄₇⋊#95	C₂³	semidirect
39. C₂⁵×C₄₇	Q₈	semidirect
61. C₂⁴×C₄₇	C₂×Q₈	semidirect
98. C₂³×C₄₇	C₂²⋊Q₈	semidirect
128. C₂²×C₄₇	C₂×C₂²⋊Q₈	semidirect
144. C₂×C₄₇	#2848	semidirect
164. C₄₇	C₂×#2848	semidirect
167. C₂⁵	Dic₉₄	semidirect
202. C₂⁴	C₂×Dic₉₄	semidirect
240. C₂³	#13094003	semidirect
269. C₂²	C₂×#13094003	semidirect
279. C₂	#20251675	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13094003	2
#20251675	2
C₂×#15688996	2
C₂×#15688983	2
C₂×#15688979	2
C₂⁵×C₄×C₄₇	2
C₂×#2848	47

Subgroups — C2×#20251675 #26422115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20251675 #26422115