Subgroups — C₂×#20242777 #26198535

Order 12032.

790 conjugacy classes of subgroups.

The subgroup lattice has 314 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×#15689328	C₂	1	2	yes	maximal
3	#20242777	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688868	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625335	—	2	2	no	—
8	C₂×#13094024	C₂²	1	2	yes	—
9	#15689528	—	2	2	no	—
10	#15689528	C₂²	1	4	yes	complemented
11	#15689328	—	2	4	no	—
12	#15689328	C₂²	1	8	yes	—
13	C₂×#13093958	—	2	1	no	—
14	C₂×#13093958	—	2	1	no	—
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13093947	—	2	2	no	—
17	#15688876	C₂²	1	4	yes	complemented
18	#15688868	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₄₇	—	2	1	no	—
20	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₈×D₄×C₄₇	—	2	2	no	—
22	C₈×D₄×C₄₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×Dic₉₄	—	2	2	no	—
27	C₂×#10625335	—	2	8	no	—
28	C₂×#10625335	—	4	8	no	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	#13094024	—	2	2	no	—
31	#13094024	—	2	2	no	—
32	#13094024	—	2	2	no	—
33	#13094024	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	#13093958	—	2	4	no	—
37	#13093958	—	2	4	no	—
38	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
40	#13093947	—	2	8	no	—
41	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	—	2	1	no	—
43	C₂²×C₈×C₄₇	—	2	1	no	—
44	C₂²×C₈×C₄₇	—	2	4	no	—
45	C₂²×C₈×C₄₇	—	4	4	no	—
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₄×D₄×C₄₇	—	2	1	no	—
49	C₄×D₄×C₄₇	—	2	1	no	—
50	C₄×D₄×C₄₇	D₄	1	2	yes	—
51	C₄×D₄×C₄₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	—	2	1	no	—
56	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
57	C₂²⋊C₈×C₄₇	—	2	4	no	—
58	C₄×C₈×C₄₇	D₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂³	1	2	yes	—
60	C₂⁴×C₄₇	—	2	1	no	—
61	C₂×D₄×C₄₇	—	2	1	no	—
62	C₂×D₄×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	D₈	1	4	yes	—
64	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	—	2	1	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	—	4	2	no	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	4	2	no	—
74	C₂×Dic₉₄	—	2	4	no	—
75	C₂×Dic₉₄	—	4	4	no	—
76	C₂×Dic₉₄	—	4	4	no	—
77	C₂×Dic₉₄	—	4	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	4	2	no	—
81	C₂×C₈×C₄₇	—	4	8	no	—
82	C₂×C₈×C₄₇	SD₁₆	1	4	yes	complemented
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₄₇	—	2	1	no	—
86	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂²⋊C₄×C₄₇	—	2	2	no	—
88	C₂²⋊C₄×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₇	#41	1	1	yes	characteristic
92	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
93	#10625335	—	4	16	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₄⋊Dic₄₇	—	4	2	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	4	4	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	2	4	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	—	4	2	no	—
104	C₂³×C₄₇	#121	1	1	yes	characteristic
105	D₄×C₄₇	—	2	2	no	—
106	D₄×C₄₇	—	2	4	no	—
107	D₄×C₄₇	C₂×D₈	1	4	yes	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₄₇	#135	1	2	yes	—
120	C₂×C₄×C₄₇	#137	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	8	4	no	—
126	Dic₉₄	—	4	4	no	—
127	Dic₉₄	—	4	4	no	—
128	Dic₉₄	—	8	4	no	—
129	C₈×C₄₇	—	2	2	no	—
130	C₈×C₄₇	—	4	2	no	—
131	C₈×C₄₇	C₂×SD₁₆	1	4	yes	complemented
132	C₄₇⋊C₈	—	4	4	no	—
133	C₂×#1673	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	4	8	no	—
136	C₂²×C₄₇	—	4	2	no	—
137	C₂²×C₄₇	—	4	2	no	—
138	C₂²×C₄₇	#461	1	2	yes	—
139	C₂²×C₄₇	#466	1	2	yes	—
140	C₂²×C₄₇	#473	1	2	yes	—
141	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	—	4	2	no	—
145	C₄×C₄₇	#461	1	2	yes	—
146	C₄×C₄₇	#466	1	2	yes	—
147	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
149	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
150	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
151	Dic₄₇	—	8	4	no	—
152	C₂×#500	—	47	1	no	—
153	C₂×#461	—	47	2	no	—
154	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic
155	#1673	—	47	8	no	—
156	C₂×#335	—	47	1	no	—
157	C₂×#332	—	47	2	no	—
158	C₂×C₄₇	—	4	4	no	—
159	C₂×C₄₇	#1673	1	4	yes	complemented
160	C₂×C₄₇	C₂×#461	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
162	C₂×C₄₇	C₂×#473	1	1	yes	characteristic
163	C₂²×SD₁₆	—	94	2	no	—
164	C₂³×C₈	—	2	1	no	—
165	C₂×C₄⋊Q₈	—	47	2	no	—
166	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
167	#500	—	47	4	no	—
168	#500	—	94	2	no	—
169	#461	—	47	8	no	—
170	#461	—	94	4	no	—
171	C₈×D₄	—	2	2	no	—
172	C₈×D₄	C₂×D₄₇	1	4	yes	—
173	C₂×#107	—	94	1	no	—
174	C₂×#107	—	94	1	no	—
175	C₂×C₄⋊C₈	—	47	2	no	—
176	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
177	C₂×#103	—	94	2	no	—
178	C₂×C₂²⋊C₈	—	2	1	no	—
179	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
180	#335	—	47	4	no	—
181	#332	—	47	8	no	—
182	C₄₇	C₂×#1673	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	94	2	no	—
184	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
185	C₂³×C₄	—	2	1	no	—
186	C₂×SD₁₆	—	94	8	no	—
187	C₂×SD₁₆	—	188	8	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	4	4	no	—
191	C₂²×C₈	—	94	2	no	—
192	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₄⋊Q₈	—	47	4	no	—
195	C₄⋊Q₈	—	94	2	no	—
196	C₄⋊Q₈	—	94	2	no	—
197	C₄⋊Q₈	—	94	2	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	D₁₈₈	1	2	yes	—
201	C₄×D₄	C₂²×D₄₇	1	2	yes	—
202	C₂×C₄⋊C₄	—	94	2	no	—
203	C₂×C₄⋊C₄	—	94	2	no	—
204	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
207	#107	—	94	4	no	—
208	#107	—	94	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	47	4	no	—
211	C₄⋊C₈	—	94	2	no	—
212	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
213	#103	—	94	8	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
216	C₄×C₈	C₂²×D₄₇	1	2	yes	—
217	C₂⁴	—	2	1	no	—
218	C₂×Q₈	—	94	4	no	—
219	C₂×Q₈	—	188	4	no	—
220	C₂×Q₈	—	188	4	no	—
221	C₂×Q₈	—	188	2	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	D₃₇₆	1	4	yes	—
225	C₂×D₄	C₂×D₁₈₈	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	188	2	no	—
232	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
233	C₂²×C₄	#10625350	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	SD₁₆	—	188	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	8	no	—
240	C₂×C₈	—	94	4	no	—
241	C₂×C₈	—	188	4	no	—
242	C₂×C₈	#10625335	1	4	yes	complemented
243	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
244	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	94	4	no	—
247	C₄⋊C₄	—	94	4	no	—
248	C₄⋊C₄	—	188	2	no	—
249	C₄⋊C₄	—	188	2	no	—
250	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
254	C₄²	#10625350	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
256	C₄²	C₂³×D₄₇	1	1	yes	characteristic
257	C₂³	—	2	4	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	#13093998	1	1	yes	characteristic
262	Q₈	—	376	4	no	—
263	Q₈	—	188	4	no	—
264	Q₈	—	188	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂×D₃₇₆	1	4	yes	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	376	4	no	—
275	C₂×C₄	—	188	4	no	—
276	C₂×C₄	#13093998	1	1	yes	characteristic
277	C₂×C₄	#13093998	1	1	yes	characteristic
278	C₂×C₄	#13093998	1	1	yes	characteristic
279	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
280	C₂×C₄	C₂×#10625335	1	2	yes	—
281	C₂×C₄	#13094046	1	2	yes	—
282	C₂×C₄	#13094054	1	2	yes	—
283	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
285	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	4	2	no	—
288	C₈	—	188	4	no	—
289	C₈	C₂×#10625335	1	4	yes	complemented
290	C₂²	—	2	4	no	—
291	C₂²	—	4	8	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	#15689328	1	2	yes	—
295	C₂²	#15689359	1	2	yes	—
296	C₂²	#15689398	1	2	yes	—
297	C₂²	C₂×#13093998	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	376	4	no	—
302	C₄	#15689328	1	2	yes	—
303	C₄	#15689359	1	2	yes	—
304	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
305	C₄	C₂²×#10625335	1	1	yes	characteristic
306	C₄	C₂×#13094046	1	1	yes	characteristic
307	C₄	C₂×#13094054	1	1	yes	characteristic
308	C₂	—	4	4	no	—
309	C₂	#20242777	1	4	yes	direct product factor
310	C₂	C₂×#15689328	1	1	yes	characteristic
311	C₂	C₂×#15689359	1	1	yes	characteristic
312	C₂	C₂×#15689398	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈×D₄×C₄₇	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₄×C₄₇	D₄
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	C₂×#1673
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15689528	C₂	semidirect
3. #20242777	C₂	direct (N × H)
4. C₂×#15688876	C₂	semidirect
5. C₂×#15688868	C₂	semidirect
10. #15689528	C₂²	semidirect
17. #15688876	C₂²	semidirect
18. #15688868	C₂²	semidirect
82. C₂×C₈×C₄₇	SD₁₆	semidirect
131. C₈×C₄₇	C₂×SD₁₆	semidirect
159. C₂×C₄₇	#1673	semidirect
182. C₄₇	C₂×#1673	semidirect
242. C₂×C₈	#10625335	semidirect
289. C₈	C₂×#10625335	semidirect
309. C₂	#20242777	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×#15689328	2
#20242777	2
C₂×#15688876	2
C₂×#15688868	2
C₂×C₈×D₄×C₄₇	2
C₂×#1673	47

Subgroups — C2×#20242777 #26198535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242777 #26198535