Subgroups — C₂×#20253941 #27085935

Order 12032.

1902 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094016	C₂	1	1	yes	maximal, characteristic
2	#20253941	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689529	C₂	1	4	yes	maximal, complemented
4	C₂×#15689522	C₂	1	4	yes	maximal, complemented
5	C₂×#15689166	C₂	1	4	yes	maximal
6	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13094016	—	2	2	no	—
10	C₂×#13094016	—	2	4	no	—
11	C₂×#13094016	C₂²	1	2	yes	—
12	C₂×#13094016	C₂²	1	2	yes	—
13	C₂×#13094016	C₂²	1	8	yes	—
14	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂²×Dic₄₇⋊C₄	—	2	2	no	—
16	#15689529	C₂²	1	32	yes	complemented
17	#15689522	C₂²	1	32	yes	complemented
18	#15689166	C₂²	1	32	yes	—
19	C₂×#13093957	C₂²	1	8	yes	—
20	C₂×#13093955	C₂²	1	8	yes	—
21	C₄×#10625329	C₂²	1	8	yes	—
22	C₂×#13093935	C₂²	1	2	yes	—
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
24	C₄₇×#533	—	2	4	no	—
25	C₄₇×#533	C₂²	1	8	yes	—
26	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
27	C₂×#125×C₄₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
29	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
30	C₂×#117×C₄₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
32	C₂³×Dic₄₇	—	2	2	no	—
33	C₂×#10625329	Q₈	1	16	yes	complemented
34	C₂×#10625329	C₂³	1	8	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
36	#13094016	—	2	16	no	—
37	#13094016	—	2	4	no	—
38	#13094016	—	2	4	no	—
39	#13094016	C₂³	1	8	yes	—
40	#13094016	C₂³	1	8	yes	—
41	C₂×C₄⋊Dic₄₇	—	2	2	no	—
42	C₂×C₄⋊Dic₄₇	—	2	4	no	—
43	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
44	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
45	C₂×Dic₄₇⋊C₄	—	2	4	no	—
46	C₂×Dic₄₇⋊C₄	—	2	4	no	—
47	C₂×Dic₄₇⋊C₄	—	2	16	no	—
48	#13093957	C₂³	1	32	yes	—
49	#13093955	C₂³	1	32	yes	—
50	#13093935	C₂³	1	8	yes	—
51	C₄×C₄₇⋊C₈	C₂³	1	8	yes	—
52	C₂²×Q₈×C₄₇	—	2	2	no	—
53	C₂³×C₄×C₄₇	D₄	1	2	yes	—
54	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₄₇	—	2	4	no	—
56	C₄⋊Q₈×C₄₇	C₂³	1	8	yes	—
57	#125×C₄₇	—	2	4	no	—
58	#125×C₄₇	C₂³	1	8	yes	—
59	C₂²⋊Q₈×C₄₇	—	2	16	no	—
60	C₄×Q₈×C₄₇	—	2	16	no	—
61	#117×C₄₇	—	2	4	no	—
62	#117×C₄₇	C₂³	1	8	yes	—
63	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
66	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
67	C₂×C₄²×C₄₇	—	2	2	no	—
68	C₂×C₄²×C₄₇	D₄	1	2	yes	—
69	C₂×C₄²×C₄₇	D₄	1	4	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
73	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×Q₈×C₄₇	—	2	8	no	—
75	C₂×Q₈×C₄₇	—	4	8	no	—
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	4	4	no	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
86	C₂²×Dic₄₇	—	2	8	no	—
87	C₂²×Dic₄₇	—	2	4	no	—
88	C₂²×Dic₄₇	—	4	8	no	—
89	C₄⋊C₄×C₄₇	—	2	16	no	—
90	C₄⋊C₄×C₄₇	—	2	8	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
93	C₂²⋊C₄×C₄₇	—	2	8	no	—
94	C₄²×C₄₇	—	2	8	no	—
95	C₄²×C₄₇	C₂×D₄	1	8	yes	—
96	C₄²×C₄₇	C₂×D₄	1	2	yes	—
97	C₄²×C₄₇	C₂×D₄	1	2	yes	—
98	C₄²×C₄₇	C₂⁴	1	2	yes	—
99	C₄²×C₄₇	C₂⁴	1	2	yes	—
100	C₄⋊Dic₄₇	—	2	8	no	—
101	C₄⋊Dic₄₇	—	2	4	no	—
102	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
103	Dic₄₇⋊C₄	—	2	16	no	—
104	Dic₄₇⋊C₄	—	2	16	no	—
105	#10625329	C₂×Q₈	1	32	yes	complemented
106	C₂×C₄₇⋊C₈	C₂×Q₈	1	16	yes	—
107	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
108	C₂³×C₄₇	—	2	4	no	—
109	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
110	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
111	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₄₇	—	4	16	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	4	8	no	—
117	C₂×C₄×C₄₇	—	4	8	no	—
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
119	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
122	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
125	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
126	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
127	C₂×Dic₄₇	—	2	8	no	—
128	C₂×Dic₄₇	—	4	16	no	—
129	C₂×Dic₄₇	—	4	8	no	—
130	C₄₇⋊C₈	C₂²×Q₈	1	16	yes	—
131	C₂×#3166	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	8	no	—
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
135	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
136	C₂²×C₄₇	#575	1	4	yes	—
137	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	2	4	no	—
140	C₄×C₄₇	—	4	8	no	—
141	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
142	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
143	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
144	Dic₄₇	—	4	8	no	—
145	C₂×#533	D₄₇	1	1	yes	characteristic
146	C₂²×#125	—	47	1	no	—
147	#3166	—	47	16	no	—
148	C₂×#501	—	47	4	no	—
149	C₂×#499	—	47	4	no	—
150	C₂×#428	—	47	4	no	—
151	C₂×C₄×M₁₆	—	47	1	no	—
152	C₂×C₄₇	—	2	4	no	—
153	C₂×C₄₇	#3166	1	4	yes	complemented
154	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#575	1	2	yes	—
156	C₂²×M₁₆	—	47	2	no	—
157	#533	—	2	4	no	—
158	#533	C₂×D₄₇	1	8	yes	—
159	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
160	C₂×#125	—	47	2	no	—
161	C₂×#125	—	47	2	no	—
162	C₂×#125	—	47	8	no	—
163	C₂×#125	—	94	2	no	—
164	C₂×#125	—	94	4	no	—
165	C₂×#125	C₂×D₄₇	1	2	yes	—
166	C₂×C₂²⋊Q₈	—	2	2	no	—
167	C₂×C₄×Q₈	—	2	2	no	—
168	C₂×#117	C₂×D₄₇	1	2	yes	—
169	C₂²×C₄⋊C₄	—	47	2	no	—
170	C₂²×C₄⋊C₄	—	94	2	no	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#501	—	47	32	no	—
173	#499	—	47	32	no	—
174	#428	—	47	32	no	—
175	C₂×#107	—	47	8	no	—
176	C₂×#106	—	47	8	no	—
177	C₄×M₁₆	—	47	8	no	—
178	C₂×#97	—	47	2	no	—
179	C₂×C₄×C₈	—	47	2	no	—
180	C₄₇	C₂×#3166	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	2	2	no	—
182	C₂³×C₄	—	94	2	no	—
183	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
184	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂×M₁₆	—	47	16	no	—
186	C₂×M₁₆	—	47	8	no	—
187	C₂²×C₈	—	47	4	no	—
188	C₄⋊Q₈	—	2	4	no	—
189	C₄⋊Q₈	C₂²×D₄₇	1	8	yes	—
190	#125	—	2	4	no	—
191	#125	—	47	8	no	—
192	#125	—	47	8	no	—
193	#125	—	94	16	no	—
194	#125	—	94	4	no	—
195	#125	—	94	4	no	—
196	#125	C₂²×D₄₇	1	8	yes	—
197	C₂²⋊Q₈	—	2	16	no	—
198	C₄×Q₈	—	2	16	no	—
199	#117	—	2	4	no	—
200	#117	C₂²×D₄₇	1	8	yes	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	47	4	no	—
204	C₂×C₄⋊C₄	—	47	8	no	—
205	C₂×C₄⋊C₄	—	94	4	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	16	no	—
208	C₂×C₄⋊C₄	—	94	4	no	—
209	C₂×C₄⋊C₄	—	94	4	no	—
210	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
214	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
218	#107	—	47	32	no	—
219	#106	—	47	32	no	—
220	#97	—	47	8	no	—
221	C₄×C₈	—	47	8	no	—
222	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	94	8	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	—	188	8	no	—
230	C₂²×C₄	Q₈×D₄₇	1	8	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
237	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
238	M₁₆	—	47	32	no	—
239	C₂×C₈	—	47	16	no	—
240	C₂×C₈	—	47	8	no	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	2	8	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	47	8	no	—
245	C₄⋊C₄	—	94	16	no	—
246	C₄⋊C₄	—	94	16	no	—
247	C₄⋊C₄	—	94	8	no	—
248	C₄⋊C₄	—	94	4	no	—
249	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
250	C₂²⋊C₄	—	2	8	no	—
251	C₄²	—	2	8	no	—
252	C₄²	C₂×C₂²⋊D₄₇	1	8	yes	—
253	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
254	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
255	C₄²	C₂³×D₄₇	1	2	yes	—
256	C₄²	C₂³×D₄₇	1	2	yes	—
257	C₂³	—	2	4	no	—
258	C₂³	#13094022	1	4	yes	—
259	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
260	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
261	Q₈	—	4	16	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	94	8	no	—
268	C₂×C₄	—	188	16	no	—
269	C₂×C₄	—	188	8	no	—
270	C₂×C₄	#13094022	1	8	yes	—
271	C₂×C₄	#13094022	1	4	yes	—
272	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
273	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
274	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
275	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
277	C₂×C₄	C₂×Q₈×D₄₇	1	8	yes	—
278	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
279	C₈	—	47	16	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	8	no	—
282	C₂²	C₂×#13094022	1	4	yes	—
283	C₂²	C₂×#13094022	1	2	yes	—
284	C₂²	#15689921	1	4	yes	—
285	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	8	no	—
289	C₄	—	188	8	no	—
290	C₄	C₂×#13094022	1	4	yes	—
291	C₄	C₂×#13094022	1	2	yes	—
292	C₄	C₂²×Q₈×D₄₇	1	2	yes	—
293	C₂	—	2	4	no	—
294	C₂	#20253941	1	4	yes	direct product factor
295	C₂	C₂²×#13094022	1	1	yes	characteristic
296	C₂	C₂×#15689921	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#3166
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #20253941	C₂	direct (N × H)
3. C₂×#15689529	C₂	semidirect
4. C₂×#15689522	C₂	semidirect
16. #15689529	C₂²	semidirect
17. #15689522	C₂²	semidirect
33. C₂×#10625329	Q₈	semidirect
105. #10625329	C₂×Q₈	semidirect
153. C₂×C₄₇	#3166	semidirect
180. C₄₇	C₂×#3166	semidirect
294. C₂	#20253941	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×#13094016	2
#20253941	2
C₂×#15689529	2
C₂×#15689522	2
C₂×#15689166	2
C₂×C₄×#10625329	2
C₂×C₄₇×#533	2
C₂×#3166	47

Subgroups — C2×#20253941 #27085935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253941 #27085935