Subgroups — #26109347

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243516	C₂	1	1	yes	maximal, characteristic
2	#20243393	C₂	1	2	yes	maximal
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
4	#20241272	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13093948	—	2	2	no	—
8	#15688993	C₂²	1	2	yes	—
9	#15688886	—	2	1	no	—
10	#15688886	—	2	1	no	—
11	#15688885	—	2	2	no	—
12	#15688874	—	2	2	no	—
13	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₄₇	—	2	2	no	—
15	C₈×Q₈×C₄₇	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	—	2	1	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	#13093948	—	4	4	no	—
27	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	—	2	1	no	—
29	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₄₇	—	2	1	no	—
32	C₄×Q₈×C₄₇	—	2	1	no	—
33	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₄₇	Q₈	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	—	2	1	no	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₄₇	—	2	4	no	—
40	C₄⋊C₈×C₄₇	—	2	1	no	—
41	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₄₇	—	2	4	no	—
43	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₂×Q₈×C₄₇	—	2	1	no	—
46	C₂×Q₈×C₄₇	—	2	2	no	—
47	C₂×Q₈×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂×Q₈×C₄₇	SD₁₆	1	4	yes	—
49	C₂²×C₄×C₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	4	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	4	2	no	—
59	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₄⋊C₄×C₄₇	—	2	1	no	—
63	C₄⋊C₄×C₄₇	—	2	2	no	—
64	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₄²×C₄₇	—	2	2	no	—
66	C₄²×C₄₇	—	2	2	no	—
67	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₄₇	#41	1	1	yes	characteristic
71	C₄⋊Dic₄₇	—	4	2	no	—
72	C₄⋊Dic₄₇	—	4	2	no	—
73	C₄⋊Dic₄₇	—	4	2	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₄⋊Dic₄₇	—	8	2	no	—
76	C₂×C₄₇⋊C₈	—	4	2	no	—
77	C₂×C₄₇⋊C₈	—	4	2	no	—
78	C₂×C₄₇⋊C₈	—	4	2	no	—
79	C₂³×C₄₇	#121	1	1	yes	characteristic
80	Q₈×C₄₇	—	2	2	no	—
81	Q₈×C₄₇	—	2	4	no	—
82	Q₈×C₄₇	#106	1	4	yes	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	4	1	no	—
89	C₂×C₄×C₄₇	—	4	1	no	—
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₄₇	#135	1	2	yes	—
98	C₂×C₄×C₄₇	#135	1	2	yes	—
99	C₂×C₄×C₄₇	#137	1	2	yes	—
100	C₂×Dic₄₇	—	8	2	no	—
101	C₂×Dic₄₇	—	8	2	no	—
102	C₂×Dic₄₇	—	8	2	no	—
103	C₈×C₄₇	—	2	2	no	—
104	C₈×C₄₇	—	2	2	no	—
105	C₈×C₄₇	—	4	2	no	—
106	C₄₇⋊C₈	—	8	2	no	—
107	#8647	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	#380	1	1	yes	characteristic
109	C₂²×C₄₇	#464	1	2	yes	—
110	C₂²×C₄₇	#465	1	2	yes	—
111	C₂²×C₄₇	#472	1	1	yes	characteristic
112	C₂²×C₄₇	#473	1	1	yes	characteristic
113	C₄×C₄₇	—	2	4	no	—
114	C₄×C₄₇	—	2	1	no	—
115	C₄×C₄₇	—	2	1	no	—
116	C₄×C₄₇	—	4	2	no	—
117	C₄×C₄₇	#416	1	1	yes	characteristic
118	C₄×C₄₇	#419	1	1	yes	characteristic
119	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
120	C₄×C₄₇	#427	1	1	yes	characteristic
121	C₄×C₄₇	#465	1	2	yes	—
122	C₄×C₄₇	#479	1	2	yes	—
123	Dic₄₇	—	16	2	no	—
124	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
125	#1834	—	47	1	no	—
126	#1803	—	47	2	no	—
127	C₂×#334	—	47	1	no	—
128	#1296	—	47	2	no	—
129	C₂×C₄₇	#1678	1	2	yes	—
130	C₂×C₄₇	#1680	1	2	yes	—
131	C₂×C₄₇	#1805	1	1	yes	characteristic
132	C₂×C₄₇	#1807	1	1	yes	characteristic
133	C₂×C₄₇	#1816	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
135	C₈×Q₈	—	2	2	no	—
136	C₈×Q₈	Dic₄₇	1	4	yes	complemented
137	C₂×C₄⋊C₈	—	2	1	no	—
138	C₂×C₄⋊C₈	—	47	2	no	—
139	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
140	C₂×#103	—	94	2	no	—
141	C₂×C₄×C₈	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
143	#384	—	47	2	no	—
144	#340	—	94	1	no	—
145	#340	—	94	1	no	—
146	#340	—	94	2	no	—
147	#334	—	94	2	no	—
148	C₄₇	#8647	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	94	2	no	—
152	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	Dic₉₄	1	2	yes	—
157	C₄×Q₈	C₂×Dic₄₇	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	94	2	no	—
160	C₂×C₄⋊C₄	—	94	2	no	—
161	C₂×C₄⋊C₄	—	94	2	no	—
162	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	1	no	—
167	C₄⋊C₈	—	94	2	no	—
168	C₄⋊C₈	—	94	2	no	—
169	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
170	#103	—	188	4	no	—
171	C₄×C₈	—	2	4	no	—
172	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
173	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	#10625325	1	4	yes	—
177	C₂×Q₈	C₄⋊Dic₄₇	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	188	2	no	—
181	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
182	C₂²×C₄	#10625350	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	188	2	no	—
189	C₂×C₈	—	188	2	no	—
190	C₂×C₈	—	188	2	no	—
191	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	376	2	no	—
197	C₄⋊C₄	—	188	2	no	—
198	C₄⋊C₄	—	188	2	no	—
199	C₄⋊C₄	—	188	2	no	—
200	C₄⋊C₄	—	188	2	no	—
201	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
205	C₄²	#10625350	1	1	yes	characteristic
206	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂³	#13093998	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	2	2	no	—
211	Q₈	#13093956	1	4	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	376	2	no	—
220	C₂×C₄	—	376	2	no	—
221	C₂×C₄	—	376	2	no	—
222	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
224	C₂×C₄	#13093983	1	1	yes	characteristic
225	C₂×C₄	#13093998	1	1	yes	characteristic
226	C₂×C₄	#13094003	1	1	yes	characteristic
227	C₂×C₄	#13094003	1	1	yes	characteristic
228	C₂×C₄	C₂×#10625325	1	2	yes	—
229	C₂×C₄	#13094044	1	2	yes	—
230	C₂×C₄	#13094046	1	2	yes	—
231	C₂×C₄	#13094055	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	376	2	no	—
236	C₂²	#15688983	1	1	yes	characteristic
237	C₂²	#15689345	1	2	yes	—
238	C₂²	#15689352	1	2	yes	—
239	C₂²	#15689393	1	1	yes	characteristic
240	C₂²	#15689398	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	752	2	no	—
246	C₄	#15689119	1	1	yes	characteristic
247	C₄	#15689132	1	1	yes	characteristic
248	C₄	C₂×#13093956	1	1	yes	characteristic
249	C₄	#15689161	1	1	yes	characteristic
250	C₄	#15689352	1	2	yes	—
251	C₄	#15689430	1	2	yes	—
252	C₂	#20242804	1	2	yes	—
253	C₂	#20242818	1	2	yes	—
254	C₂	#20243403	1	1	yes	characteristic
255	C₂	#20243413	1	1	yes	characteristic
256	C₂	#20243452	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	#8647
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₄₇	C₄	semidirect
136. C₈×Q₈	Dic₄₇	semidirect
148. C₄₇	#8647	semidirect

Maximal subgroups

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

subgroup	index
#20243516	2
#20243393	2
C₂×#15688874	2
#20241272	2
C₂×C₈×Q₈×C₄₇	2
#8647	47