Subgroups — C₂×#20253911 #27085893

Order 12032.

1902 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689676	C₂	1	2	yes	maximal
2	#20253911	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689531	C₂	1	4	yes	maximal, complemented
4	C₂×#15689530	C₂	1	4	yes	maximal, complemented
5	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
6	C₂×#15689159	C₂	1	2	yes	maximal
7	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15689676	—	2	8	no	—
11	#15689676	C₂²	1	16	yes	—
12	C₂×#13094023	C₂²	1	4	yes	—
13	C₂×#13094015	C₂²	1	4	yes	—
14	C₂×#13094001	—	2	4	no	—
15	C₂×C₄×Dic₉₄	—	2	4	no	—
16	C₂×#13093983	C₂²	1	2	yes	—
17	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
18	#15689531	C₂²	1	32	yes	complemented
19	#15689530	C₂²	1	32	yes	complemented
20	#15689161	C₂²	1	8	yes	—
21	#15689159	C₂²	1	16	yes	—
22	C₂×#13093958	C₂²	1	2	yes	—
23	C₂×#13093957	C₂²	1	4	yes	—
24	C₂×#13093956	C₂²	1	2	yes	—
25	C₂×#13093955	C₂²	1	4	yes	—
26	C₂×#13093934	C₂²	1	2	yes	—
27	C₂×#13093934	C₂²	1	2	yes	—
28	C₂×#13093934	C₂²	1	8	yes	—
29	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#117×C₄₇	C₂²	1	2	yes	—
31	C₄₇×#427	C₂²	1	8	yes	—
32	C₂×#107×C₄₇	C₂²	1	2	yes	—
33	C₂×#106×C₄₇	C₂²	1	2	yes	—
34	C₂²×Dic₉₄	—	2	4	no	—
35	C₂³×Dic₄₇	D₄	1	2	yes	—
36	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
39	#13094023	—	2	8	no	—
40	#13094023	C₂³	1	16	yes	—
41	#13094015	—	2	8	no	—
42	#13094015	C₂³	1	16	yes	—
43	#13094001	—	2	32	no	—
44	C₄×Dic₉₄	—	2	32	no	—
45	#13093983	—	2	4	no	—
46	#13093983	C₂³	1	8	yes	—
47	C₂×C₄⋊Dic₄₇	—	2	4	no	—
48	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
49	C₂×Dic₄₇⋊C₄	—	2	4	no	—
50	C₂×Dic₄₇⋊C₄	—	2	4	no	—
51	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
52	C₂×C₄×Dic₄₇	—	2	2	no	—
53	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
54	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
55	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
56	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
57	#13093958	C₂³	1	8	yes	—
58	#13093957	C₂³	1	16	yes	—
59	#13093956	C₂³	1	8	yes	—
60	#13093955	C₂³	1	16	yes	—
61	#13093934	C₂³	1	8	yes	—
62	#13093934	C₂³	1	8	yes	—
63	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	Q₈	1	8	yes	complemented
65	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
68	#117×C₄₇	—	2	4	no	—
69	#117×C₄₇	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
71	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
72	C₂×C₄²×C₄₇	—	2	2	no	—
73	#107×C₄₇	C₂³	1	8	yes	—
74	#106×C₄₇	C₂³	1	8	yes	—
75	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂²×Dic₄₇	—	2	4	no	—
84	C₂²×Dic₄₇	—	4	4	no	—
85	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
86	C₂²×Dic₄₇	C₂×D₄	1	8	yes	—
87	C₂×Dic₉₄	—	2	16	no	—
88	C₂×Dic₉₄	—	4	16	no	—
89	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
90	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
91	C₂×C₈×C₄₇	C₂×Q₈	1	16	yes	complemented
92	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
93	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
94	C₄⋊C₄×C₄₇	—	2	4	no	—
95	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
96	C₂²⋊C₄×C₄₇	—	2	8	no	—
97	C₄²×C₄₇	—	2	8	no	—
98	C₂²⋊Dic₄₇	—	2	8	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	2	16	no	—
101	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
102	Dic₄₇⋊C₄	—	2	16	no	—
103	Dic₄₇⋊C₄	—	2	16	no	—
104	C₄×Dic₄₇	—	2	8	no	—
105	C₄×Dic₄₇	C₂×D₄	1	8	yes	—
106	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
107	C₄×Dic₄₇	C₂⁴	1	4	yes	—
108	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
109	C₂×C₄₇⋊C₈	C₂×Q₈	1	16	yes	complemented
110	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
111	C₂³×C₄₇	—	2	4	no	—
112	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
113	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
114	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	4	8	no	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
119	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
123	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
125	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	2	8	no	—
128	C₂×Dic₄₇	—	4	8	no	—
129	C₂×Dic₄₇	—	4	8	no	—
130	C₂×Dic₄₇	C₂²×D₄	1	8	yes	—
131	Dic₉₄	—	4	32	no	—
132	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
133	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
134	C₄₇⋊C₈	C₂²×Q₈	1	8	yes	—
135	C₂×#3162	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	8	no	—
137	C₂²×C₄₇	—	2	4	no	—
138	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
139	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
140	C₂²×C₄₇	#575	1	4	yes	—
141	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
142	C₄×C₄₇	—	4	8	no	—
143	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
144	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
145	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
146	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
147	Dic₄₇	—	2	8	no	—
148	Dic₄₇	—	4	8	no	—
149	C₂×#533	—	47	2	no	—
150	#3162	—	47	16	no	—
151	C₂×#501	—	47	4	no	—
152	C₂×#501	—	47	4	no	—
153	C₂×#427	—	47	1	no	—
154	C₂×#427	—	47	2	no	—
155	C₂×#427	D₄₇	1	1	yes	characteristic
156	C₂²×#97	—	47	1	no	—
157	C₂×C₄₇	—	2	4	no	—
158	C₂×C₄₇	#3162	1	4	yes	complemented
159	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#575	1	2	yes	—
161	C₂³×C₈	—	47	1	no	—
162	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
163	#533	—	47	16	no	—
164	#533	—	94	8	no	—
165	C₂×C₄⋊Q₈	—	47	4	no	—
166	C₂×#125	—	47	4	no	—
167	C₂×C₂²⋊Q₈	—	94	4	no	—
168	C₂×C₄×Q₈	—	94	4	no	—
169	C₂×#117	—	47	2	no	—
170	C₂×#117	C₂×D₄₇	1	2	yes	—
171	C₂²×C₄²	—	47	1	no	—
172	#501	—	47	32	no	—
173	#501	—	47	32	no	—
174	#427	—	47	8	no	—
175	#427	—	47	16	no	—
176	#427	C₂×D₄₇	1	8	yes	—
177	C₂×#107	—	47	2	no	—
178	C₂×#107	—	47	4	no	—
179	C₂×#107	C₂×D₄₇	1	2	yes	—
180	C₂×#106	—	47	2	no	—
181	C₂×#106	—	47	4	no	—
182	C₂×#106	C₂×D₄₇	1	2	yes	—
183	C₂×#97	—	47	2	no	—
184	C₂×#97	—	47	2	no	—
185	C₂×#97	—	47	8	no	—
186	C₄₇	C₂×#3162	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₈	—	94	4	no	—
188	C₂³×C₄	—	47	2	no	—
189	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
190	C₂²×C₈	—	47	8	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	47	4	no	—
193	C₂²×C₈	Dic₉₄	1	8	yes	complemented
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
197	C₄⋊Q₈	—	47	16	no	—
198	C₄⋊Q₈	—	94	8	no	—
199	#125	—	47	16	no	—
200	#125	—	94	8	no	—
201	C₂²⋊Q₈	—	94	32	no	—
202	C₄×Q₈	—	94	32	no	—
203	#117	—	2	4	no	—
204	#117	—	47	8	no	—
205	#117	—	94	4	no	—
206	#117	C₂²×D₄₇	1	8	yes	—
207	C₂×C₄⋊C₄	—	47	4	no	—
208	C₂×C₄⋊C₄	—	94	4	no	—
209	C₂×C₄⋊C₄	—	94	4	no	—
210	C₂×C₄⋊C₄	—	94	4	no	—
211	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	94	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	47	4	no	—
216	C₂×C₄²	—	47	2	no	—
217	C₂×C₄²	—	47	2	no	—
218	C₂×C₄²	—	47	4	no	—
219	C₂×C₄²	—	94	2	no	—
220	#107	—	47	8	no	—
221	#107	—	47	16	no	—
222	#107	C₂²×D₄₇	1	8	yes	—
223	#106	—	47	8	no	—
224	#106	—	47	16	no	—
225	#106	C₂²×D₄₇	1	8	yes	—
226	#97	—	47	8	no	—
227	#97	—	47	8	no	—
228	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
229	C₂×Q₈	—	94	16	no	—
230	C₂×Q₈	—	188	16	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	47	4	no	—
233	C₂²×C₄	—	47	8	no	—
234	C₂²×C₄	—	94	4	no	—
235	C₂²×C₄	—	188	4	no	—
236	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
237	C₂²×C₄	D₄×D₄₇	1	2	yes	—
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
240	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
242	C₂×C₈	—	47	8	no	—
243	C₂×C₈	—	47	16	no	—
244	C₂×C₈	—	47	4	no	—
245	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
246	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
247	C₂×C₈	C₂×Dic₉₄	1	16	yes	complemented
248	C₂×C₈	C₂³×D₄₇	1	2	yes	—
249	C₂×C₈	C₂³×D₄₇	1	2	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	47	8	no	—
252	C₄⋊C₄	—	94	16	no	—
253	C₄⋊C₄	—	94	16	no	—
254	C₄⋊C₄	—	94	16	no	—
255	C₄⋊C₄	—	94	4	no	—
256	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
257	C₂²⋊C₄	—	2	8	no	—
258	C₂²⋊C₄	—	94	8	no	—
259	C₄²	—	2	8	no	—
260	C₄²	—	47	8	no	—
261	C₄²	—	47	4	no	—
262	C₄²	—	47	4	no	—
263	C₄²	—	94	8	no	—
264	C₂³	—	2	4	no	—
265	C₂³	#13094023	1	4	yes	—
266	C₂³	C₂×D₄×D₄₇	1	2	yes	—
267	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
268	Q₈	—	188	32	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	47	8	no	—
272	C₂×C₄	—	94	8	no	—
273	C₂×C₄	—	94	8	no	—
274	C₂×C₄	—	188	8	no	—
275	C₂×C₄	—	188	8	no	—
276	C₂×C₄	#13094023	1	8	yes	—
277	C₂×C₄	#13094023	1	4	yes	—
278	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
279	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
280	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
281	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
282	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
283	C₂×C₄	C₂²×Dic₉₄	1	4	yes	—
284	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
285	C₈	—	47	8	no	—
286	C₈	C₂²×Dic₉₄	1	4	yes	—
287	C₈	C₂²×Dic₉₄	1	4	yes	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	8	no	—
290	C₂²	C₂×#13094023	1	4	yes	—
291	C₂²	C₂×#13094023	1	2	yes	—
292	C₂²	#15689920	1	4	yes	—
293	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
294	C₄	—	4	8	no	—
295	C₄	—	94	8	no	—
296	C₄	—	188	8	no	—
297	C₄	C₂×#13094023	1	2	yes	—
298	C₄	C₂×#13094023	1	4	yes	—
299	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
300	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
301	C₂	—	2	4	no	—
302	C₂	#20253911	1	4	yes	direct product factor
303	C₂	C₂²×#13094023	1	1	yes	characteristic
304	C₂	C₂×#15689920	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	C₂×#3162
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #20253911	C₂	direct (N × H)
3. C₂×#15689531	C₂	semidirect
4. C₂×#15689530	C₂	semidirect
18. #15689531	C₂²	semidirect
19. #15689530	C₂²	semidirect
36. C₂²×C₄₇⋊C₈	Q₈	semidirect
64. C₂²×C₈×C₄₇	Q₈	semidirect
91. C₂×C₈×C₄₇	C₂×Q₈	semidirect
109. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
158. C₂×C₄₇	#3162	semidirect
186. C₄₇	C₂×#3162	semidirect
193. C₂²×C₈	Dic₉₄	semidirect
247. C₂×C₈	C₂×Dic₉₄	semidirect
302. C₂	#20253911	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×#15689676	2
#20253911	2
C₂×#15689531	2
C₂×#15689530	2
C₂×#15689161	2
C₂×#15689159	2
C₂²×#13093934	2
C₂×C₄₇×#427	2
C₂×#3162	47

Subgroups — C2×#20253911 #27085893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253911 #27085893