Subgroups — #26316150

Order 12032.

894 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251766	C₂	1	1	yes	maximal, characteristic, complemented
2	#20244524	C₂	1	4	yes	maximal, complemented
3	#20244213	C₂	1	2	yes	maximal
4	#20243523	C₂	1	2	yes	maximal, complemented
5	#20243398	C₂	1	4	yes	maximal, complemented
6	#20243241	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13094024	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13093945	C₂²	1	8	yes	—
11	C₂×#13093935	C₂²	1	2	yes	—
12	#15689029	—	2	2	no	—
13	#15689029	—	2	2	no	—
14	#15688989	C₂²	1	2	yes	—
15	#15688989	C₂²	1	2	yes	—
16	#15688885	C₂²	1	8	yes	—
17	#15688878	C₂²	1	2	yes	—
18	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₇	—	2	2	no	—
20	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#390	—	2	2	no	—
22	C₄₇×#390	C₂²	1	2	yes	—
23	C₄₇×#384	C₂²	1	1	yes	characteristic
24	C₄³×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉₄	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	#13094024	—	4	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
31	#13093945	—	4	8	no	—
32	#13093935	—	2	4	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂²×D₄×C₄₇	—	2	2	no	—
37	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₄₇	—	2	2	no	—
39	#127×C₄₇	—	2	2	no	—
40	#127×C₄₇	Q₈	1	4	yes	complemented
41	C₄×D₄×C₄₇	—	2	8	no	—
42	C₄×D₄×C₄₇	—	4	8	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂⁴×C₄₇	—	2	2	no	—
53	C₂×D₄×C₄₇	—	2	8	no	—
54	C₂×D₄×C₄₇	—	2	4	no	—
55	C₂×D₄×C₄₇	—	2	2	no	—
56	C₂×D₄×C₄₇	—	2	4	no	—
57	C₂×D₄×C₄₇	—	4	4	no	—
58	C₂×D₄×C₄₇	—	4	4	no	—
59	C₂×D₄×C₄₇	C₂×Q₈	1	4	yes	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	4	8	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂×Dic₉₄	—	4	4	no	—
70	C₂×Dic₉₄	—	4	4	no	—
71	C₂×Dic₉₄	—	4	4	no	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄⋊C₄×C₄₇	—	4	4	no	—
74	C₂²⋊C₄×C₄₇	—	4	4	no	—
75	C₂²⋊C₄×C₄₇	—	4	8	no	—
76	C₄²×C₄₇	—	2	4	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	4	2	no	—
79	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
80	C₄²×C₄₇	#41	1	2	yes	—
81	C₄²×C₄₇	#41	1	4	yes	—
82	C₄⋊Dic₄₇	—	2	8	no	—
83	C₄⋊Dic₄₇	—	4	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	—	4	2	no	—
89	C₂³×C₄₇	—	4	2	no	—
90	C₂³×C₄₇	—	4	4	no	—
91	C₂³×C₄₇	—	4	4	no	—
92	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
93	D₄×C₄₇	—	2	8	no	—
94	D₄×C₄₇	—	4	16	no	—
95	D₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	8	8	no	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
106	C₂×C₄×C₄₇	#124	1	2	yes	—
107	C₂×C₄×C₄₇	#124	1	2	yes	—
108	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
109	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	Dic₉₄	—	8	8	no	—
117	C₄₇⋊C₈	—	4	4	no	—
118	#29145	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	4	4	no	—
120	C₂²×C₄₇	—	4	8	no	—
121	C₂²×C₄₇	—	8	2	no	—
122	C₂²×C₄₇	—	8	2	no	—
123	C₂²×C₄₇	—	8	4	no	—
124	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
126	C₂²×C₄₇	#535	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
128	C₂²×C₄₇	#575	1	2	yes	—
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	4	no	—
133	C₄×C₄₇	—	8	2	no	—
134	C₄×C₄₇	#554	1	2	yes	—
135	C₄×C₄₇	#555	1	2	yes	—
136	Dic₄₇	—	8	4	no	—
137	#2862	—	47	1	no	—
138	C₄×#127	D₄₇	1	1	yes	characteristic
139	#2023	—	47	4	no	—
140	#1961	—	47	2	no	—
141	#1836	—	47	2	no	—
142	#1804	—	47	4	no	—
143	#1767	—	47	1	no	—
144	C₂×C₄₇	—	8	4	no	—
145	C₂×C₄₇	#2563	1	1	yes	characteristic
146	C₂×C₄₇	#3077	1	2	yes	—
147	C₂×C₄₇	#3135	1	1	yes	characteristic
148	C₂×C₄₇	#3138	1	1	yes	characteristic
149	C₂×C₄₇	#3213	1	2	yes	—
150	C₂×C₄⋊Q₈	—	47	2	no	—
151	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
152	C₂×C₄×D₄	—	2	2	no	—
153	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
154	C₂×C₄⋊C₈	—	47	2	no	—
155	C₂×#102	—	47	8	no	—
156	C₂×#97	—	47	2	no	—
157	#391	—	94	2	no	—
158	#391	—	94	2	no	—
159	#390	—	2	2	no	—
160	#390	C₂×D₄₇	1	2	yes	—
161	#384	C₂×D₄₇	1	1	yes	characteristic
162	#382	—	47	2	no	—
163	#382	—	47	2	no	—
164	C₄³	C₂×D₄₇	1	1	yes	characteristic
165	#340	—	47	8	no	—
166	#336	—	47	2	no	—
167	C₄₇	#29145	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₈	—	94	2	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
171	C₂³×C₄	—	2	2	no	—
172	C₂²×C₈	—	47	4	no	—
173	C₄⋊Q₈	—	188	4	no	—
174	#127	—	2	2	no	—
175	#127	Dic₉₄	1	4	yes	complemented
176	C₄×D₄	—	2	8	no	—
177	C₄×D₄	—	4	8	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	47	4	no	—
180	C₂×C₄⋊C₄	—	94	2	no	—
181	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
185	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
189	C₄⋊C₈	—	94	4	no	—
190	#102	—	188	8	no	—
191	#97	—	94	4	no	—
192	#95	—	94	4	no	—
193	#95	—	94	2	no	—
194	#95	—	94	2	no	—
195	C₂⁴	—	2	2	no	—
196	C₂×Q₈	—	188	4	no	—
197	C₂×Q₈	—	188	4	no	—
198	C₂×Q₈	—	188	4	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	C₂×Dic₉₄	1	4	yes	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
214	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
215	C₂×C₈	—	94	4	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	94	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	94	8	no	—
221	C₄⋊C₄	—	188	4	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₂²⋊C₄	—	4	8	no	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	2	no	—
227	C₄²	C₂×Dic₉₄	1	2	yes	—
228	C₄²	#10625350	1	2	yes	—
229	C₄²	#10625352	1	4	yes	—
230	C₂³	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
236	Q₈	—	376	8	no	—
237	D₄	—	2	8	no	—
238	D₄	—	4	16	no	—
239	D₄	—	4	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	8	no	—
249	C₂×C₄	—	188	4	no	—
250	C₂×C₄	—	188	4	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	#13094003	1	4	yes	—
253	C₂×C₄	#13094011	1	2	yes	—
254	C₂×C₄	#13094011	1	2	yes	—
255	C₂×C₄	#13094050	1	4	yes	—
256	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
257	C₂×C₄	C₂×#10625352	1	2	yes	—
258	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
259	C₂×C₄	#13094070	1	2	yes	—
260	C₈	—	188	4	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	4	no	—
266	C₂²	C₂×#13094003	1	1	yes	characteristic
267	C₂²	C₂×#13094011	1	1	yes	characteristic
268	C₂²	#15689687	1	1	yes	characteristic
269	C₂²	C₂×#13094050	1	2	yes	—
270	C₂²	#15689921	1	2	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	8	2	no	—
276	C₄	—	376	4	no	—
277	C₄	#15689830	1	2	yes	—
278	C₄	#15689838	1	2	yes	—
279	C₂	—	8	4	no	—
280	C₂	#20248561	1	1	yes	characteristic
281	C₂	#20253185	1	2	yes	—
282	C₂	#20253733	1	1	yes	characteristic
283	C₂	#20253762	1	1	yes	characteristic
284	C₂	#20254371	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂×#127×C₄₇	C₂²
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#29145
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #20251766	C₂	semidirect
2. #20244524	C₂	semidirect
4. #20243523	C₂	semidirect
5. #20243398	C₂	semidirect
6. #20243241	C₂	semidirect
40. #127×C₄₇	Q₈	semidirect
167. C₄₇	#29145	semidirect
175. #127	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20251766	2
#20244524	2
#20244213	2
#20243523	2
#20243398	2
#20243241	2
C₄×#127×C₄₇	2
#29145	47