Subgroups — #26368891

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689529	C₂	1	2	yes	maximal
2	C₂×#15689525	C₂	1	1	yes	maximal, characteristic
3	#20249508	C₂	1	2	yes	maximal
4	#20243555	C₂	1	4	yes	maximal
5	#20243551	C₂	1	2	yes	maximal
6	#20243546	C₂	1	2	yes	maximal
7	#20243074	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#500	C₂	1	1	yes	maximal, characteristic
9	C₂×#13094016	C₂²	1	2	yes	—
10	#15689529	—	2	8	no	—
11	#15689525	—	2	4	no	—
12	C₂×#13093958	C₂²	1	1	yes	characteristic
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093957	C₂²	1	4	yes	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	C₂×#13093955	C₂²	1	4	yes	—
17	C₂×#13093935	C₂²	1	1	yes	characteristic
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
20	#15689054	—	2	4	no	—
21	#15689042	—	2	4	no	—
22	#15688992	C₂²	1	4	yes	—
23	#15688988	C₂²	1	4	yes	—
24	#15688969	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#500	—	2	4	no	—
27	C₂×#107×C₄₇	C₂²	1	2	yes	—
28	C₂×#107×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	#13094016	—	2	4	no	—
32	#13094016	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	2	no	—
36	C₂×Dic₄₇⋊C₄	—	2	2	no	—
37	C₂×Dic₄₇⋊C₄	—	2	8	no	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093958	—	2	2	no	—
40	#13093958	—	2	2	no	—
41	#13093957	—	2	8	no	—
42	#13093956	—	2	4	no	—
43	#13093955	—	2	8	no	—
44	#13093935	D₄	1	4	yes	—
45	#13093934	Q₈	1	8	yes	complemented
46	C₈×Dic₄₇	Q₈	1	8	yes	complemented
47	C₄₇⋊#95	—	2	8	no	—
48	C₄₇⋊#95	D₄	1	2	yes	—
49	C₄₇⋊#95	D₄	1	2	yes	—
50	C₂²×Q₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₄₇	—	2	4	no	—
54	C₄⋊Q₈×C₄₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#107×C₄₇	—	2	4	no	—
59	#107×C₄₇	—	2	4	no	—
60	C₄×C₈×C₄₇	D₄	1	4	yes	—
61	C₂×Q₈×C₄₇	—	4	4	no	—
62	C₂×Q₈×C₄₇	—	8	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	2	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	4	no	—
77	C₄²×C₄₇	C₂×D₄	1	2	yes	—
78	C₄²×C₄₇	C₂×D₄	1	2	yes	—
79	C₄⋊Dic₄₇	—	2	2	no	—
80	C₄⋊Dic₄₇	—	2	2	no	—
81	C₄⋊Dic₄₇	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	2	no	—
83	Dic₄₇⋊C₄	—	4	4	no	—
84	Dic₄₇⋊C₄	—	4	4	no	—
85	Dic₄₇⋊C₄	—	8	8	no	—
86	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
87	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
88	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
89	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
90	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
91	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	Q₈×C₄₇	—	8	8	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	8	no	—
97	C₂×C₄×C₄₇	#127	1	2	yes	—
98	C₂×C₄×C₄₇	#127	1	2	yes	—
99	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
102	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	4	2	no	—
106	C₂×Dic₄₇	—	4	2	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	4	no	—
109	C₂×Dic₄₇	C₄⋊Q₈	1	8	yes	—
110	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
111	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
112	C₄₇⋊C₈	C₄⋊Q₈	1	8	yes	—
113	#33319	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
116	C₂²×C₄₇	#576	1	2	yes	—
117	C₂²×C₄₇	#578	1	2	yes	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	8	4	no	—
121	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
122	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	Dic₄₇	—	2	8	no	—
125	Dic₄₇	—	8	2	no	—
126	Dic₄₇	—	8	2	no	—
127	C₂×#501	—	47	2	no	—
128	C₂×#500	D₄₇	1	1	yes	characteristic
129	C₂×#499	—	47	1	no	—
130	#2660	—	47	2	no	—
131	#1847	—	47	4	no	—
132	#1846	—	47	2	no	—
133	#1845	—	47	2	no	—
134	#1726	—	47	1	no	—
135	C₂×C₄₇	#2868	1	1	yes	characteristic
136	C₂×C₄₇	#3155	1	1	yes	characteristic
137	C₂×C₄₇	#3156	1	1	yes	characteristic
138	C₂×C₄₇	#3164	1	4	yes	—
139	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
140	C₂×#125	—	47	2	no	—
141	#501	—	94	8	no	—
142	#500	—	2	4	no	—
143	#499	—	94	4	no	—
144	C₂×#107	—	47	1	no	—
145	C₂×#107	—	47	1	no	—
146	C₂×#107	—	47	4	no	—
147	C₂×#107	C₂×D₄₇	1	2	yes	—
148	C₂×#107	C₂×D₄₇	1	2	yes	—
149	C₂×#106	—	47	2	no	—
150	C₂×#106	—	47	4	no	—
151	C₂×#97	—	47	2	no	—
152	C₂×#97	—	47	1	no	—
153	C₂×C₄×C₈	—	47	2	no	—
154	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
155	#398	—	94	4	no	—
156	#395	—	94	4	no	—
157	#384	—	47	4	no	—
158	#382	—	47	4	no	—
159	#376	—	47	1	no	—
160	C₄₇	#33319	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	2	2	no	—
162	C₂²×C₈	—	47	2	no	—
163	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	4	no	—
166	C₄⋊Q₈	—	2	4	no	—
167	#125	—	94	4	no	—
168	#125	—	94	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	47	2	no	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₄⋊C₄	—	94	8	no	—
173	C₂×C₄⋊C₄	—	94	2	no	—
174	C₂×C₄⋊C₄	—	94	2	no	—
175	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
178	#107	—	2	4	no	—
179	#107	—	2	4	no	—
180	#107	—	94	8	no	—
181	#107	—	94	2	no	—
182	#107	—	94	2	no	—
183	#106	—	94	8	no	—
184	#106	—	94	4	no	—
185	#97	—	47	8	no	—
186	#97	—	47	4	no	—
187	C₄×C₈	—	47	8	no	—
188	C₄×C₈	C₂²⋊D₄₇	1	4	yes	—
189	#95	—	47	2	no	—
190	#95	—	47	2	no	—
191	#95	—	94	8	no	—
192	C₂×Q₈	—	4	4	no	—
193	C₂×Q₈	—	8	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	47	4	no	—
196	C₂²×C₄	—	94	2	no	—
197	C₂²×C₄	—	94	2	no	—
198	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
201	C₂×C₈	—	47	4	no	—
202	C₂×C₈	—	47	4	no	—
203	C₂×C₈	—	47	4	no	—
204	C₂×C₈	Q₈×D₄₇	1	4	yes	—
205	C₂×C₈	Q₈×D₄₇	1	4	yes	—
206	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	94	2	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₄⋊C₄	—	94	2	no	—
214	C₄⋊C₄	—	94	2	no	—
215	C₄⋊C₄	—	376	8	no	—
216	C₄⋊C₄	—	188	4	no	—
217	C₄⋊C₄	—	188	4	no	—
218	C₄²	—	47	4	no	—
219	C₄²	—	47	4	no	—
220	C₄²	D₄×D₄₇	1	2	yes	—
221	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
222	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
223	Q₈	—	8	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	47	8	no	—
229	C₂×C₄	—	94	4	no	—
230	C₂×C₄	—	94	4	no	—
231	C₂×C₄	—	188	2	no	—
232	C₂×C₄	—	188	2	no	—
233	C₂×C₄	—	188	4	no	—
234	C₂×C₄	—	188	4	no	—
235	C₂×C₄	#13094020	1	2	yes	—
236	C₂×C₄	#13094020	1	2	yes	—
237	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
240	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
241	C₈	—	47	8	no	—
242	C₈	#13094022	1	4	yes	—
243	C₈	#13094022	1	4	yes	—
244	C₂²	C₂×#13094020	1	1	yes	characteristic
245	C₂²	C₄⋊Q₈×D₄₇	1	2	yes	—
246	C₂²	#15689924	1	2	yes	—
247	C₂²	#15689936	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	8	4	no	—
251	C₄	—	94	8	no	—
252	C₄	—	376	2	no	—
253	C₄	—	376	2	no	—
254	C₄	C₄⋊Q₈×D₄₇	1	2	yes	—
255	C₄	C₂×#13094022	1	1	yes	characteristic
256	C₄	C₂×#13094022	1	1	yes	characteristic
257	C₂	#20251784	1	1	yes	characteristic
258	C₂	#20253869	1	1	yes	characteristic
259	C₂	#20253877	1	1	yes	characteristic
260	C₂	#20253921	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#500	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₂×#500	D₄₇
hypocenter	C₄₇	#33319
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
45. #13093934	Q₈	semidirect
46. C₈×Dic₄₇	Q₈	semidirect
160. C₄₇	#33319	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689529	2
C₂×#15689525	2
#20249508	2
#20243555	2
#20243551	2
#20243546	2
#20243074	2
C₂×C₄₇×#500	2
#33319	47