Subgroups — #26316182

Order 12032.

878 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251751	C₂	1	1	yes	maximal, characteristic, complemented
2	#20244534	C₂	1	4	yes	maximal, complemented
3	#20244213	C₂	1	2	yes	maximal
4	#20243523	C₂	1	2	yes	maximal, complemented
5	#20243388	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×#13094016	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	#15689021	—	2	2	no	—
13	#15689021	—	2	2	no	—
14	#15688993	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×C₄₇⋊C₈	C₂³	1	4	yes	—
26	#13094016	—	4	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	C₂×Dic₄₇⋊C₄	—	2	4	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
32	#13093945	—	4	8	no	—
33	#13093935	—	2	4	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₂²×D₄×C₄₇	—	2	2	no	—
36	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₄₇	—	2	2	no	—
38	#127×C₄₇	—	2	2	no	—
39	#127×C₄₇	Q₈	1	4	yes	—
40	C₄×D₄×C₄₇	—	2	8	no	—
41	C₄×D₄×C₄₇	—	4	8	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂⁴×C₄₇	—	2	2	no	—
52	C₂×D₄×C₄₇	—	2	8	no	—
53	C₂×D₄×C₄₇	—	2	4	no	—
54	C₂×D₄×C₄₇	—	2	2	no	—
55	C₂×D₄×C₄₇	—	2	4	no	—
56	C₂×D₄×C₄₇	—	4	4	no	—
57	C₂×D₄×C₄₇	—	4	4	no	—
58	C₂×D₄×C₄₇	C₂×Q₈	1	4	yes	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	—	4	8	no	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
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	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₄⋊C₄×C₄₇	—	2	4	no	—
69	C₄⋊C₄×C₄₇	—	4	4	no	—
70	C₂²⋊C₄×C₄₇	—	4	8	no	—
71	C₂²⋊C₄×C₄₇	—	4	4	no	—
72	C₄²×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	4	2	no	—
75	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
76	C₄²×C₄₇	#41	1	2	yes	—
77	C₄²×C₄₇	#41	1	4	yes	—
78	C₄⋊Dic₄₇	—	2	8	no	—
79	C₄⋊Dic₄₇	—	4	4	no	—
80	C₄⋊Dic₄₇	—	4	4	no	—
81	Dic₄₇⋊C₄	—	4	8	no	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂³×C₄₇	—	2	4	no	—
86	C₂³×C₄₇	—	4	2	no	—
87	C₂³×C₄₇	—	4	2	no	—
88	C₂³×C₄₇	—	4	4	no	—
89	C₂³×C₄₇	—	4	4	no	—
90	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	D₄×C₄₇	—	2	8	no	—
92	D₄×C₄₇	—	4	16	no	—
93	D₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	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	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	8	8	no	—
103	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
104	C₂×C₄×C₄₇	#124	1	2	yes	—
105	C₂×C₄×C₄₇	#124	1	2	yes	—
106	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₄₇⋊C₈	—	4	4	no	—
115	#29149	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	—	4	4	no	—
117	C₂²×C₄₇	—	4	8	no	—
118	C₂²×C₄₇	—	8	2	no	—
119	C₂²×C₄₇	—	8	2	no	—
120	C₂²×C₄₇	—	8	4	no	—
121	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
123	C₂²×C₄₇	#535	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
125	C₂²×C₄₇	#575	1	2	yes	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	4	2	no	—
129	C₄×C₄₇	—	8	2	no	—
130	C₄×C₄₇	#490	1	4	yes	—
131	C₄×C₄₇	#554	1	2	yes	—
132	C₄×C₄₇	#555	1	2	yes	—
133	Dic₄₇	—	8	4	no	—
134	#2859	—	47	1	no	—
135	C₄×#127	D₄₇	1	1	yes	characteristic
136	#2025	—	47	4	no	—
137	#1961	—	47	2	no	—
138	#1836	—	47	2	no	—
139	#1802	—	47	4	no	—
140	#1767	—	47	1	no	—
141	C₂×C₄₇	—	8	4	no	—
142	C₂×C₄₇	#2563	1	1	yes	characteristic
143	C₂×C₄₇	#3077	1	2	yes	—
144	C₂×C₄₇	C₂×#490	1	1	yes	characteristic
145	C₂×C₄₇	#3142	1	1	yes	characteristic
146	C₂×C₄₇	#3214	1	2	yes	—
147	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
148	C₂×#125	—	47	2	no	—
149	C₂×C₄×D₄	—	2	2	no	—
150	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
151	C₂×C₄⋊C₈	—	47	2	no	—
152	C₂×#102	—	47	8	no	—
153	C₂×#97	—	47	2	no	—
154	#390	—	2	2	no	—
155	#390	C₂×D₄₇	1	2	yes	—
156	#389	—	94	2	no	—
157	#389	—	94	2	no	—
158	#384	—	47	2	no	—
159	#384	C₂×D₄₇	1	1	yes	characteristic
160	#382	—	47	2	no	—
161	C₄³	C₂×D₄₇	1	1	yes	characteristic
162	#340	—	47	8	no	—
163	#336	—	47	2	no	—
164	C₄₇	#29149	1	1	yes	characteristic, complemented, Sylow
165	C₂²×D₄	—	2	2	no	—
166	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂²×C₈	—	47	4	no	—
169	#127	—	2	2	no	—
170	#127	Dic₉₄	1	4	yes	—
171	#125	—	188	4	no	—
172	C₄×D₄	—	2	8	no	—
173	C₄×D₄	—	4	8	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	47	4	no	—
176	C₂×C₄⋊C₄	—	94	4	no	—
177	C₂×C₄⋊C₄	—	94	2	no	—
178	C₂×C₄⋊C₄	—	94	2	no	—
179	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
180	C₂×C₂²⋊C₄	—	2	4	no	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	C₄⋊C₈	—	94	4	no	—
188	#102	—	188	8	no	—
189	#97	—	94	4	no	—
190	#95	—	94	4	no	—
191	C₂⁴	—	2	2	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	4	4	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	C₂×Dic₉₄	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	8	no	—
202	C₂²×C₄	—	94	4	no	—
203	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
205	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	94	4	no	—
210	C₂×C₈	—	94	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	94	8	no	—
214	C₄⋊C₄	—	188	8	no	—
215	C₄⋊C₄	—	188	4	no	—
216	C₄⋊C₄	—	188	4	no	—
217	C₂²⋊C₄	—	4	8	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	C₂×Dic₉₄	1	2	yes	—
223	C₄²	#10625350	1	2	yes	—
224	C₄²	#10625352	1	4	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
231	D₄	—	2	8	no	—
232	D₄	—	4	16	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	8	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	—	188	4	no	—
245	C₂×C₄	—	188	4	no	—
246	C₂×C₄	#13094003	1	4	yes	—
247	C₂×C₄	#13094011	1	2	yes	—
248	C₂×C₄	#13094011	1	2	yes	—
249	C₂×C₄	#13094050	1	4	yes	—
250	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
251	C₂×C₄	C₂×#10625352	1	2	yes	—
252	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
253	C₂×C₄	#13094070	1	2	yes	—
254	C₈	—	188	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₂×#13094003	1	1	yes	characteristic
261	C₂²	C₂×#13094011	1	1	yes	characteristic
262	C₂²	#15689687	1	1	yes	characteristic
263	C₂²	C₂×#13094050	1	2	yes	—
264	C₂²	#15689921	1	2	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	376	4	no	—
270	C₄	#15689487	1	4	yes	—
271	C₄	#15689830	1	2	yes	—
272	C₄	#15689838	1	2	yes	—
273	C₂	—	8	4	no	—
274	C₂	#20248561	1	1	yes	characteristic
275	C₂	#20253185	1	2	yes	—
276	C₂	C₂×#15689487	1	1	yes	characteristic
277	C₂	#20253788	1	1	yes	characteristic
278	C₂	#20254379	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	#29149
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. #20251751	C₂	semidirect
2. #20244534	C₂	semidirect
4. #20243523	C₂	semidirect
5. #20243388	C₂	semidirect
6. #20243241	C₂	semidirect
164. C₄₇	#29149	semidirect

Maximal subgroups

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

subgroup	index
#20251751	2
#20244534	2
#20244213	2
#20243523	2
#20243388	2
#20243241	2
C₄×#127×C₄₇	2
#29149	47