Subgroups — C₂×#20245523 #26370879

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689527	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20245523	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688949	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688924	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#357	C₂	1	1	yes	maximal, characteristic
9	C₂×#13094023	—	2	1	no	—
10	C₂×D₄×Dic₄₇	—	2	1	no	—
11	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#15689527	—	2	2	no	—
13	#15689527	C₂²	1	4	yes	complemented
14	D₈×Dic₄₇	—	2	2	no	—
15	D₈×Dic₄₇	C₂²	1	4	yes	—
16	C₂×#13093958	C₂²	1	1	yes	characteristic
17	C₂×#13093958	C₂²	1	1	yes	characteristic
18	C₂×#13093957	—	2	1	no	—
19	C₂×#13093945	—	2	1	no	—
20	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688949	C₂²	1	4	yes	complemented
22	#15688948	C₂²	1	4	yes	complemented
23	Dic₄₇⋊C₁₆	C₂²	1	4	yes	complemented
24	#15688924	C₂²	1	4	yes	—
25	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#357	C₂²	1	4	yes	—
29	C₂²×Dic₉₄	—	4	1	no	—
30	C₂³×Dic₄₇	—	4	1	no	—
31	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
32	#13094023	—	2	4	no	—
33	#13094023	—	4	2	no	—
34	D₄×Dic₄₇	—	2	4	no	—
35	D₄×Dic₄₇	—	4	2	no	—
36	C₂×C₄⋊Dic₄₇	—	2	1	no	—
37	C₂×C₄⋊Dic₄₇	—	2	1	no	—
38	C₂×Dic₄₇⋊C₄	—	4	1	no	—
39	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
40	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄₇⋊C₁₆	—	2	2	no	—
42	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
43	#13093958	—	2	1	no	—
44	#13093958	—	2	1	no	—
45	#13093958	C₂³	1	2	yes	—
46	#13093958	C₂³	1	2	yes	—
47	#13093957	—	2	4	no	—
48	#13093945	—	2	4	no	—
49	C₈×Dic₄₇	D₄	1	2	yes	—
50	C₈×Dic₄₇	C₂³	1	2	yes	—
51	C₂²×D₄×C₄₇	—	2	1	no	—
52	C₂×D₈×C₄₇	—	2	1	no	—
53	C₂×D₈×C₄₇	—	2	2	no	—
54	C₂×D₈×C₄₇	Q₈	1	4	yes	complemented
55	C₂×D₈×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
58	C₂×C₁₆×C₄₇	—	2	2	no	—
59	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
60	#107×C₄₇	—	2	1	no	—
61	#107×C₄₇	C₂³	1	2	yes	—
62	C₂⁴×C₄₇	—	4	1	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	4	1	no	—
66	C₂×D₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	—	4	1	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	4	1	no	—
70	C₂²×Dic₄₇	—	4	1	no	—
71	C₂²×Dic₄₇	—	4	4	no	—
72	C₂²×Dic₄₇	—	8	4	no	—
73	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
74	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
75	C₂×Dic₉₄	—	4	4	no	—
76	C₂×Dic₉₄	—	8	4	no	—
77	D₈×C₄₇	—	2	4	no	—
78	D₈×C₄₇	—	2	2	no	—
79	D₈×C₄₇	C₂×Q₈	1	4	yes	complemented
80	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₄₇	#41	1	2	yes	—
83	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	4	1	no	—
86	C₂²⋊Dic₄₇	—	4	4	no	—
87	C₄⋊Dic₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	—	2	2	no	—
89	C₄⋊Dic₄₇	—	4	1	no	—
90	C₄⋊Dic₄₇	—	4	1	no	—
91	Dic₄₇⋊C₄	—	4	4	no	—
92	C₄×Dic₄₇	D₈	1	2	yes	—
93	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₁₆×C₄₇	—	2	2	no	—
97	C₄₇⋊C₁₆	—	2	2	no	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	—	4	4	no	—
100	C₂³×C₄₇	—	8	2	no	—
101	C₂³×C₄₇	—	8	2	no	—
102	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
103	D₄×C₄₇	—	2	4	no	—
104	D₄×C₄₇	—	4	2	no	—
105	D₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	8	2	no	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#137	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	4	2	no	—
114	C₂×Dic₄₇	—	4	2	no	—
115	C₂×Dic₄₇	—	8	2	no	—
116	C₂×Dic₄₇	—	8	2	no	—
117	C₂×Dic₄₇	—	8	8	no	—
118	C₂×Dic₄₇	D₁₆	1	4	yes	—
119	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
120	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
121	Dic₉₄	—	8	8	no	—
122	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
123	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
124	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₄₇⋊C₈	—	2	2	no	—
126	C₂×#2224	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	4	4	no	—
128	C₂²×C₄₇	—	8	2	no	—
129	C₂²×C₄₇	—	8	2	no	—
130	C₂²×C₄₇	—	8	8	no	—
131	C₂²×C₄₇	#474	1	2	yes	—
132	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
133	C₂²×C₄₇	#510	1	2	yes	—
134	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₄₇	—	8	2	no	—
136	C₄×C₄₇	#474	1	2	yes	—
137	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
138	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
139	Dic₄₇	—	2	2	no	—
140	Dic₄₇	—	8	2	no	—
141	Dic₄₇	—	8	2	no	—
142	Dic₄₇	C₂×D₁₆	1	4	yes	—
143	C₂×#500	—	47	1	no	—
144	C₂×C₄×D₈	—	47	1	no	—
145	#2224	—	47	8	no	—
146	C₂×#366	—	47	1	no	—
147	C₂×#366	—	47	1	no	—
148	C₂×C₄⋊C₁₆	—	47	1	no	—
149	C₂×#357	—	47	1	no	—
150	C₂×#357	D₄₇	1	1	yes	characteristic
151	C₂×C₄₇	—	8	4	no	—
152	C₂×C₄₇	#2224	1	4	yes	complemented
153	C₂×C₄₇	C₂×#474	1	1	yes	characteristic
154	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
156	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂×C₄⋊Q₈	—	94	1	no	—
158	C₂×C₄×D₄	—	94	1	no	—
159	C₂²×C₁₆	—	47	1	no	—
160	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
161	#500	—	47	4	no	—
162	#500	—	94	2	no	—
163	C₄×D₈	—	47	4	no	—
164	C₄×D₈	—	94	2	no	—
165	C₂×#107	—	47	1	no	—
166	C₂×#107	—	47	1	no	—
167	C₂×#107	—	94	1	no	—
168	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
169	C₂×#102	—	94	1	no	—
170	C₂×C₄×C₈	—	47	1	no	—
171	#366	—	47	4	no	—
172	#366	—	47	4	no	—
173	C₄⋊C₁₆	—	47	4	no	—
174	#357	—	47	4	no	—
175	#357	C₂×D₄₇	1	4	yes	—
176	C₄₇	C₂×#2224	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₈	—	188	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂³×C₄	—	188	1	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	Dic₉₄	1	4	yes	complemented
183	C₂×D₈	C₂²×D₄₇	1	2	yes	—
184	C₂²×C₈	—	47	1	no	—
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	C₄⋊Q₈	—	94	4	no	—
187	C₄⋊Q₈	—	188	2	no	—
188	C₄×D₄	—	94	4	no	—
189	C₄×D₄	—	188	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	94	1	no	—
192	C₂×C₄⋊C₄	—	94	1	no	—
193	C₂×C₄⋊C₄	—	188	1	no	—
194	C₂×C₂²⋊C₄	—	188	1	no	—
195	C₂×C₄²	—	47	1	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	47	2	no	—
198	C₂×C₁₆	—	94	2	no	—
199	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
200	#107	—	2	1	no	—
201	#107	—	47	2	no	—
202	#107	—	47	2	no	—
203	#107	—	94	4	no	—
204	#107	—	94	1	no	—
205	#107	—	94	1	no	—
206	#107	C₂²×D₄₇	1	2	yes	—
207	#102	—	94	4	no	—
208	C₄×C₈	—	47	2	no	—
209	C₄×C₈	—	47	2	no	—
210	C₂⁴	—	4	1	no	—
211	C₂×Q₈	—	376	4	no	—
212	C₂×Q₈	—	188	4	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	47	1	no	—
219	C₂²×C₄	—	47	1	no	—
220	C₂²×C₄	—	376	4	no	—
221	C₂²×C₄	—	188	4	no	—
222	C₂²×C₄	—	188	1	no	—
223	C₂²×C₄	—	188	1	no	—
224	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
225	D₈	—	2	4	no	—
226	D₈	—	2	2	no	—
227	D₈	C₂×Dic₉₄	1	4	yes	complemented
228	C₂×C₈	—	47	2	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
231	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
232	C₂×C₈	#10625352	1	2	yes	—
233	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	94	2	no	—
237	C₄⋊C₄	—	94	2	no	—
238	C₄⋊C₄	—	188	4	no	—
239	C₄⋊C₄	—	188	1	no	—
240	C₄⋊C₄	—	188	1	no	—
241	C₂²⋊C₄	—	188	4	no	—
242	C₄²	—	47	2	no	—
243	C₄²	—	47	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	94	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	8	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	D₈×D₄₇	1	1	yes	characteristic
251	Q₈	—	376	8	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	47	4	no	—
258	C₂×C₄	—	47	2	no	—
259	C₂×C₄	—	47	2	no	—
260	C₂×C₄	—	94	2	no	—
261	C₂×C₄	—	376	8	no	—
262	C₂×C₄	—	376	2	no	—
263	C₂×C₄	—	376	2	no	—
264	C₂×C₄	—	188	2	no	—
265	C₂×C₄	—	188	2	no	—
266	C₂×C₄	#13094001	1	2	yes	—
267	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
268	C₂×C₄	#13094053	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
270	C₈	—	94	2	no	—
271	C₈	#13094001	1	2	yes	—
272	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
273	C₈	C₂×#10625352	1	1	yes	characteristic
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	8	no	—
277	C₂²	—	8	2	no	—
278	C₂²	#15689401	1	2	yes	—
279	C₂²	D₁₆×D₄₇	1	2	yes	—
280	C₂²	#15689562	1	2	yes	—
281	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
282	C₄	—	8	2	no	—
283	C₄	—	47	4	no	—
284	C₄	—	94	2	no	—
285	C₄	—	376	2	no	—
286	C₄	—	376	2	no	—
287	C₄	#15689401	1	2	yes	—
288	C₄	C₂×#13094001	1	1	yes	characteristic
289	C₄	C₂×#13094053	1	1	yes	characteristic
290	C₂	—	8	4	no	—
291	C₂	#20245523	1	4	yes	direct product factor
292	C₂	C₂×#15689401	1	1	yes	characteristic
293	C₂	C₂×D₁₆×D₄₇	1	1	yes	characteristic
294	C₂	C₂×#15689562	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#357	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
hypercenter	C₂×#357	D₄₇
hypocenter	C₄₇	C₂×#2224
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
1. C₂×#15689527	C₂	semidirect
3. #20245523	C₂	direct (N × H)
4. C₂×#15688949	C₂	semidirect
5. C₂×#15688948	C₂	semidirect
6. C₂×Dic₄₇⋊C₁₆	C₂	semidirect
13. #15689527	C₂²	semidirect
21. #15688949	C₂²	semidirect
22. #15688948	C₂²	semidirect
23. Dic₄₇⋊C₁₆	C₂²	semidirect
54. C₂×D₈×C₄₇	Q₈	semidirect
79. D₈×C₄₇	C₂×Q₈	semidirect
152. C₂×C₄₇	#2224	semidirect
176. C₄₇	C₂×#2224	semidirect
182. C₂×D₈	Dic₉₄	semidirect
227. D₈	C₂×Dic₉₄	semidirect
291. C₂	#20245523	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×#15689527	2
C₂×D₈×Dic₄₇	2
#20245523	2
C₂×#15688949	2
C₂×#15688948	2
C₂×Dic₄₇⋊C₁₆	2
C₂×#15688924	2
C₂×C₄₇×#357	2
C₂×#2224	47

Subgroups — C2×#20245523 #26370879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245523 #26370879