Subgroups — C₂×#20246130 #26375110

Order 12032.

2398 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246130	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689022	C₂	1	2	yes	maximal
3	C₂×#15689022	C₂	1	4	yes	maximal
4	C₂×#15689009	C₂	1	4	yes	maximal
5	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×Dic₄₇	C₂	1	2	yes	maximal
7	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
8	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
12	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
14	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
15	#15689022	C₂²	1	16	yes	—
16	#15689022	C₂²	1	32	yes	—
17	#15689009	C₂²	1	32	yes	—
18	#15688993	C₂²	1	8	yes	—
19	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
20	#15688969	C₂²	1	8	yes	—
21	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
22	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
23	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
24	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
25	C₂×#125×C₄₇	C₄	1	16	yes	complemented
26	C₂×#125×C₄₇	C₂²	1	8	yes	—
27	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
28	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
29	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
30	C₂³×Dic₄₇	C₂³	1	1	yes	characteristic
31	C₂³×Dic₄₇	C₂³	1	1	yes	characteristic
32	C₂³×Dic₄₇	C₂³	1	4	yes	—
33	C₂³×Dic₄₇	C₂³	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	—	4	8	no	—
36	C₂×C₄⋊Dic₄₇	—	4	8	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
39	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
40	C₂×C₄×Dic₄₇	—	2	4	no	—
41	C₂×C₄×Dic₄₇	—	4	8	no	—
42	C₂×C₄×Dic₄₇	Q₈	1	8	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
44	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
46	C₄₇⋊#95	C₂³	1	8	yes	—
47	C₄₇⋊#95	C₂³	1	8	yes	—
48	C₄₇⋊#95	C₂³	1	16	yes	—
49	C₄₇⋊#95	C₂³	1	16	yes	—
50	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
52	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	#125×C₄₇	C₂×C₄	1	64	yes	complemented
54	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
57	C₂×C₄⋊C₄×C₄₇	C₂³	1	16	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
60	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	8	no	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	—	2	8	no	—
78	C₂²×Dic₄₇	—	2	8	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	2	8	no	—
81	C₂²×Dic₄₇	—	2	4	no	—
82	C₂²×Dic₄₇	C₂×Q₈	1	8	yes	—
83	C₂²×Dic₄₇	C₂⁴	1	2	yes	—
84	C₂²×Dic₄₇	C₂⁴	1	2	yes	—
85	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
86	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
87	C₄⋊C₄×C₄₇	C₂²×C₄	1	64	yes	—
88	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	16	yes	—
90	C₄⋊Dic₄₇	—	2	16	no	—
91	C₄⋊Dic₄₇	—	4	16	no	—
92	C₄⋊Dic₄₇	—	4	16	no	—
93	C₄×Dic₄₇	—	2	8	no	—
94	C₄×Dic₄₇	—	4	16	no	—
95	C₄×Dic₄₇	C₂×Q₈	1	16	yes	—
96	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
97	C₂³×C₄₇	C₂²×Q₈	1	2	yes	—
98	C₂³×C₄₇	C₂×#41	1	2	yes	—
99	C₂³×C₄₇	H(5,𝔽₂)	1	4	yes	—
100	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
101	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	32	no	—
103	C₂×C₄×C₄₇	—	2	16	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	8	no	—
108	C₂×C₄×C₄₇	C₄×Q₈	1	16	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
110	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
112	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
114	C₂×Dic₄₇	—	2	16	no	—
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	—	2	2	no	—
118	C₂×Dic₄₇	—	2	2	no	—
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	2	8	no	—
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	16	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	C₂²×Q₈	1	8	yes	—
128	C₂×#2330	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	C₂×C₄×Q₈	1	4	yes	—
130	C₂²×C₄₇	#520	1	8	yes	—
131	C₂²×C₄₇	#540	1	4	yes	—
132	C₂²×C₄₇	#541	1	4	yes	—
133	C₂²×C₄₇	#544	1	8	yes	—
134	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
135	C₂²×C₄₇	C₂³×Q₈	1	1	yes	characteristic
136	C₂²×C₄₇	C₂²×#41	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
138	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
139	C₄×C₄₇	—	2	16	no	—
140	C₄×C₄₇	—	2	8	no	—
141	C₄×C₄₇	C₂×C₄×Q₈	1	8	yes	—
142	Dic₄₇	—	2	8	no	—
143	Dic₄₇	—	4	2	no	—
144	Dic₄₇	—	4	8	no	—
145	Dic₄₇	—	4	2	no	—
146	C₂²×#125	D₄₇	1	1	yes	characteristic
147	#2330	—	47	16	no	—
148	C₂×#389	—	47	2	no	—
149	C₂×#389	—	47	4	no	—
150	C₂×#387	—	47	4	no	—
151	C₂×#384	—	47	1	no	—
152	C₂×C₄×C₄⋊C₄	—	47	2	no	—
153	C₂×#376	—	47	1	no	—
154	C₂×C₄₇	#2330	1	8	yes	complemented
155	C₂×C₄₇	C₂²×C₄×Q₈	1	1	yes	characteristic
156	C₂×C₄₇	C₂×#520	1	2	yes	—
157	C₂×C₄₇	C₂×#540	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#541	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#544	1	2	yes	—
160	C₂×#125	Dic₄₇	1	16	yes	complemented
161	C₂×#125	C₂×D₄₇	1	8	yes	—
162	C₂²×C₄⋊C₄	—	47	2	no	—
163	C₂²×C₄⋊C₄	—	47	4	no	—
164	C₂²×C₄⋊C₄	—	47	4	no	—
165	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
166	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
167	C₂²×C₄²	—	47	1	no	—
168	C₂²×C₄²	—	47	1	no	—
169	C₂²×C₄²	—	47	4	no	—
170	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
171	#389	—	47	32	no	—
172	#389	—	47	16	no	—
173	#387	—	47	32	no	—
174	#384	—	47	8	no	—
175	C₄×C₄⋊C₄	—	47	16	no	—
176	#376	—	47	8	no	—
177	C₂×#95	—	47	2	no	—
178	C₂×#95	—	47	2	no	—
179	C₂×#95	—	47	4	no	—
180	C₂×#95	—	47	4	no	—
181	C₄₇	C₂×#2330	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	—	47	2	no	—
183	C₂³×C₄	—	47	1	no	—
184	C₂³×C₄	—	47	4	no	—
185	C₂³×C₄	—	47	1	no	—
186	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
187	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
188	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	#125	C₂×Dic₄₇	1	64	yes	complemented
190	C₂×C₄⋊C₄	—	47	8	no	—
191	C₂×C₄⋊C₄	—	47	16	no	—
192	C₂×C₄⋊C₄	—	47	16	no	—
193	C₂×C₄⋊C₄	—	94	8	no	—
194	C₂×C₄⋊C₄	—	188	8	no	—
195	C₂×C₄⋊C₄	—	188	8	no	—
196	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
197	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
198	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
199	C₂×C₄⋊C₄	C₂²×D₄₇	1	16	yes	—
200	C₂×C₄²	—	47	8	no	—
201	C₂×C₄²	—	47	4	no	—
202	C₂×C₄²	—	47	4	no	—
203	C₂×C₄²	—	47	16	no	—
204	C₂×C₄²	—	94	4	no	—
205	C₂×C₄²	—	188	8	no	—
206	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
207	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
208	#95	—	47	8	no	—
209	#95	—	47	8	no	—
210	#95	—	47	16	no	—
211	#95	—	47	16	no	—
212	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	47	8	no	—
218	C₂²×C₄	—	47	4	no	—
219	C₂²×C₄	—	47	2	no	—
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	47	8	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	—	94	4	no	—
224	C₂²×C₄	—	94	2	no	—
225	C₂²×C₄	—	94	2	no	—
226	C₂²×C₄	—	94	8	no	—
227	C₂²×C₄	—	94	8	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	—	94	8	no	—
230	C₂²×C₄	—	94	4	no	—
231	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
232	C₂²×C₄	#10625354	1	4	yes	—
233	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
234	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
235	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
236	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
237	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
238	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
239	C₄⋊C₄	—	94	16	no	—
240	C₄⋊C₄	—	188	16	no	—
241	C₄⋊C₄	—	188	16	no	—
242	C₄⋊C₄	C₂²×Dic₄₇	1	64	yes	—
243	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
244	C₄²	—	47	16	no	—
245	C₄²	—	94	8	no	—
246	C₄²	—	188	16	no	—
247	C₄²	C₂²×Dic₄₇	1	16	yes	—
248	C₂³	C₂³×Dic₄₇	1	2	yes	—
249	C₂³	C₂×Q₈×D₄₇	1	2	yes	—
250	C₂³	C₂×#10625354	1	2	yes	—
251	C₂³	#13094070	1	4	yes	—
252	C₂³	#13094071	1	4	yes	—
253	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
254	C₂×C₄	—	2	32	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	47	8	no	—
261	C₂×C₄	—	94	16	no	—
262	C₂×C₄	—	94	4	no	—
263	C₂×C₄	—	94	4	no	—
264	C₂×C₄	—	94	2	no	—
265	C₂×C₄	—	94	2	no	—
266	C₂×C₄	—	94	8	no	—
267	C₂×C₄	—	94	8	no	—
268	C₂×C₄	—	94	4	no	—
269	C₂×C₄	—	94	8	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	188	4	no	—
272	C₂×C₄	—	188	16	no	—
273	C₂×C₄	—	188	4	no	—
274	C₂×C₄	Q₈×Dic₄₇	1	16	yes	—
275	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
276	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
277	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
278	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
279	C₂×C₄	C₂×#10625354	1	4	yes	—
280	C₂²	C₂×Q₈×Dic₄₇	1	4	yes	—
281	C₂²	#15689606	1	8	yes	—
282	C₂²	#15689729	1	4	yes	—
283	C₂²	#15689736	1	4	yes	—
284	C₂²	#15689753	1	8	yes	—
285	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
286	C₂²	C₂²×Q₈×D₄₇	1	1	yes	characteristic
287	C₂²	C₂²×#10625354	1	1	yes	characteristic
288	C₂²	C₂×#13094070	1	2	yes	—
289	C₂²	C₂×#13094071	1	2	yes	—
290	C₄	—	2	16	no	—
291	C₄	—	2	8	no	—
292	C₄	—	94	8	no	—
293	C₄	—	188	2	no	—
294	C₄	—	188	8	no	—
295	C₄	—	188	2	no	—
296	C₄	C₂×Q₈×Dic₄₇	1	8	yes	—
297	C₂	#20246130	1	8	yes	direct product factor
298	C₂	C₂²×Q₈×Dic₄₇	1	1	yes	characteristic
299	C₂	C₂×#15689606	1	2	yes	—
300	C₂	C₂×#15689729	1	1	yes	characteristic
301	C₂	C₂×#15689736	1	1	yes	characteristic
302	C₂	C₂×#15689753	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	C₂×#2330
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
1. #20246130	C₂	direct (N × H)
25. C₂×#125×C₄₇	C₄	semidirect
53. #125×C₄₇	C₂×C₄	semidirect
154. C₂×C₄₇	#2330	semidirect
160. C₂×#125	Dic₄₇	semidirect
181. C₄₇	C₂×#2330	semidirect
189. #125	C₂×Dic₄₇	semidirect
297. C₂	#20246130	direct (N × H)

Maximal subgroups

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

subgroup	index
#20246130	2
C₂×#15689022	2
C₂×#15689009	2
C₂×#15688993	2
C₂×C₄⋊C₄×Dic₄₇	2
C₂×#15688969	2
C₂²×#125×C₄₇	2
C₂×#2330	47

Subgroups — C2×#20246130 #26375110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246130 #26375110