Subgroups — #26115220

Order 12032.

434 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20242136	C₂	1	1	yes	maximal, characteristic
2	#20241783	C₂	1	2	yes	maximal
3	#20241712	C₂	1	2	yes	maximal
4	#20241680	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	#15689176	C₂²	1	1	yes	characteristic
8	#15689176	C₂²	1	1	yes	characteristic
9	#15688876	—	2	1	no	—
10	#15688874	—	2	1	no	—
11	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
12	#15688849	—	2	4	no	—
13	#15688845	—	2	1	no	—
14	#15688845	—	2	1	no	—
15	C₂²×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₄₇	—	2	1	no	—
18	C₄×Q₁₆×C₄₇	—	2	1	no	—
19	C₄×Q₁₆×C₄₇	—	2	1	no	—
20	C₄×Q₁₆×C₄₇	—	2	1	no	—
21	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
22	C₂×#103×C₄₇	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
27	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
30	C₄×C₄₇⋊C₈	—	2	1	no	—
31	C₄×C₄₇⋊C₈	—	2	1	no	—
32	C₄×C₄₇⋊C₈	—	2	2	no	—
33	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₄₇	—	2	4	no	—
35	C₂×Q₁₆×C₄₇	—	2	1	no	—
36	C₂×Q₁₆×C₄₇	—	2	1	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₄₇	—	2	2	no	—
40	C₄×Q₈×C₄₇	—	2	2	no	—
41	C₄×Q₈×C₄₇	—	2	2	no	—
42	C₄×Q₈×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	#107×C₄₇	—	2	1	no	—
48	#107×C₄₇	—	2	1	no	—
49	#103×C₄₇	—	2	2	no	—
50	#103×C₄₇	—	2	2	no	—
51	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₇	—	2	2	no	—
56	C₂×Q₈×C₄₇	—	2	2	no	—
57	C₂×Q₈×C₄₇	—	2	2	no	—
58	C₂×Q₈×C₄₇	—	2	2	no	—
59	C₂×Q₈×C₄₇	—	2	2	no	—
60	C₂×Q₈×C₄₇	M₁₆	1	4	yes	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	Q₁₆×C₄₇	—	2	4	no	—
67	Q₁₆×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	4	2	no	—
79	C₄²×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₄₇	#41	1	1	yes	characteristic
84	C₂×C₄₇⋊C₈	—	2	2	no	—
85	C₂×C₄₇⋊C₈	—	2	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
88	Q₈×C₄₇	—	2	4	no	—
89	Q₈×C₄₇	—	2	4	no	—
90	Q₈×C₄₇	—	4	2	no	—
91	Q₈×C₄₇	—	4	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	4	1	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	8	2	no	—
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₄₇	#131	1	2	yes	—
107	C₈×C₄₇	—	2	1	no	—
108	C₈×C₄₇	—	2	1	no	—
109	C₈×C₄₇	—	2	1	no	—
110	C₈×C₄₇	—	2	1	no	—
111	C₄₇⋊C₈	—	4	2	no	—
112	C₄₇⋊C₈	—	4	2	no	—
113	C₄₇⋊C₈	—	8	2	no	—
114	#9671	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	1	no	—
116	C₂²×C₄₇	—	2	1	no	—
117	C₂²×C₄₇	#390	1	1	yes	characteristic
118	C₂²×C₄₇	#435	1	2	yes	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	—	2	1	no	—
122	C₄×C₄₇	—	4	4	no	—
123	C₄×C₄₇	—	8	2	no	—
124	C₄×C₄₇	#407	1	1	yes	characteristic
125	C₄×C₄₇	#408	1	1	yes	characteristic
126	C₄×C₄₇	#435	1	2	yes	—
127	C₄×C₄₇	#473	1	2	yes	—
128	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
129	#1564	—	47	1	no	—
130	#1479	—	47	2	no	—
131	#1461	—	47	2	no	—
132	#1452	—	47	1	no	—
133	C₂×C₄₇	—	2	2	no	—
134	C₂×C₄₇	#1955	1	1	yes	characteristic
135	C₂×C₄₇	#1976	1	1	yes	characteristic
136	C₂×C₄₇	#1979	1	1	yes	characteristic
137	C₂²×Q₁₆	Dic₄₇	1	1	yes	characteristic
138	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
139	C₄×Q₁₆	—	2	1	no	—
140	C₄×Q₁₆	—	2	1	no	—
141	C₄×Q₁₆	—	2	1	no	—
142	C₄×Q₁₆	—	2	1	no	—
143	#431	—	47	1	no	—
144	#431	—	47	1	no	—
145	#431	—	47	2	no	—
146	C₂×#107	Dic₄₇	1	1	yes	characteristic
147	C₂×#103	Dic₄₇	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
149	#335	—	94	1	no	—
150	#334	—	94	1	no	—
151	Q₈⋊C₈	—	94	4	no	—
152	#324	—	94	4	no	—
153	#322	—	94	1	no	—
154	#322	—	94	1	no	—
155	C₄₇	#9671	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
157	C₂×Q₁₆	—	2	4	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
161	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	#107	—	2	1	no	—
171	#107	—	2	1	no	—
172	C₄⋊C₈	—	47	2	no	—
173	C₄⋊C₈	—	94	1	no	—
174	C₄⋊C₈	—	94	2	no	—
175	#103	—	2	2	no	—
176	#103	—	2	2	no	—
177	C₂²⋊C₈	—	47	2	no	—
178	C₂²⋊C₈	—	94	1	no	—
179	C₂²⋊C₈	—	94	2	no	—
180	C₄×C₈	—	94	2	no	—
181	C₄×C₈	—	94	1	no	—
182	C₄×C₈	—	94	1	no	—
183	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	#10625329	1	4	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	#10625352	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
198	Q₁₆	—	2	4	no	—
199	Q₁₆	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	94	2	no	—
203	C₂×C₈	—	94	2	no	—
204	C₂×C₈	—	188	2	no	—
205	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₄²	—	4	2	no	—
215	C₄²	D₄×D₄₇	1	1	yes	characteristic
216	C₄²	#10625352	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
220	Q₈	—	2	4	no	—
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
233	C₂×C₄	#13093995	1	1	yes	characteristic
234	C₂×C₄	#13093995	1	1	yes	characteristic
235	C₂×C₄	#13094011	1	1	yes	characteristic
236	C₂×C₄	#13094020	1	1	yes	characteristic
237	C₂×C₄	C₂×#10625329	1	2	yes	—
238	C₂×C₄	#13094031	1	2	yes	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	376	2	no	—
244	C₈	—	188	2	no	—
245	C₈	—	188	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#15689026	1	1	yes	characteristic
249	C₂²	#15689195	1	2	yes	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	#15689087	1	1	yes	characteristic
256	C₄	#15689090	1	1	yes	characteristic
257	C₄	#15689195	1	2	yes	—
258	C₄	#15689395	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	#20244187	1	1	yes	characteristic
261	C₂	#20244287	1	1	yes	characteristic
262	C₂	#20244304	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093995
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#9671
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
155. C₄₇	#9671	semidirect

Maximal subgroups

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

subgroup	index
#20242136	2
#20241783	2
#20241712	2
#20241680	2
C₂×C₄×Q₁₆×C₄₇	2
#9671	47