Subgroups — #26114927

Order 12032.

382 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	#20244147	C₂	1	2	yes	maximal
2	#20243548	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20241272	C₂	1	2	yes	maximal
5	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	—	2	2	no	—
7	C₂×#13093958	—	2	1	no	—
8	C₂×#13093958	—	2	1	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₄₇	—	2	1	no	—
11	#15688993	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
13	#15688886	—	2	2	no	—
14	#15688877	—	2	1	no	—
15	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#335	C₄	1	4	yes	complemented
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₄₇	—	2	1	no	—
25	#13093958	—	4	2	no	—
26	#13093958	—	4	2	no	—
27	#13093958	—	4	4	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
29	C₄⋊(C₄₇⋊C₈)	Q₈	1	2	yes	—
30	C₈×Dic₄₇	—	4	2	no	—
31	C₄₇⋊#95	—	2	1	no	—
32	C₂²×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₇	—	2	2	no	—
37	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₄₇	Q₈	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	2	1	no	—
44	C₂²×Dic₄₇	—	2	1	no	—
45	C₂²×Dic₄₇	—	4	2	no	—
46	C₂×C₈×C₄₇	—	2	1	no	—
47	C₂×C₈×C₄₇	—	2	1	no	—
48	C₂×C₈×C₄₇	—	2	4	no	—
49	C₂×C₈×C₄₇	—	4	2	no	—
50	C₂×C₈×C₄₇	—	4	2	no	—
51	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₄₇	D₈	1	2	yes	—
54	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
55	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₄₇	#41	1	1	yes	characteristic
59	C₄⋊Dic₄₇	—	2	1	no	—
60	C₄⋊Dic₄₇	—	2	1	no	—
61	C₄⋊Dic₄₇	—	4	2	no	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	4	2	no	—
65	C₄⋊Dic₄₇	—	8	2	no	—
66	C₄⋊Dic₄₇	D₈	1	2	yes	—
67	C₄⋊Dic₄₇	Q₁₆	1	2	yes	—
68	C₄×Dic₄₇	—	4	1	no	—
69	C₄×Dic₄₇	—	4	1	no	—
70	C₂×C₄₇⋊C₈	—	2	2	no	—
71	C₂×C₄₇⋊C₈	—	4	1	no	—
72	C₂×C₄₇⋊C₈	—	4	1	no	—
73	C₂³×C₄₇	#121	1	1	yes	characteristic
74	C₂×C₄×C₄₇	—	2	1	no	—
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
88	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
90	C₂×Dic₄₇	—	2	2	no	—
91	C₂×Dic₄₇	—	2	2	no	—
92	C₂×Dic₄₇	—	4	1	no	—
93	C₂×Dic₄₇	—	4	1	no	—
94	C₂×Dic₄₇	—	4	1	no	—
95	C₂×Dic₄₇	—	4	1	no	—
96	C₂×Dic₄₇	—	8	2	no	—
97	C₂×Dic₄₇	—	8	2	no	—
98	C₂×Dic₄₇	—	8	2	no	—
99	C₈×C₄₇	—	2	2	no	—
100	C₈×C₄₇	—	4	4	no	—
101	C₈×C₄₇	#107	1	4	yes	complemented
102	C₄₇⋊C₈	—	4	2	no	—
103	#9614	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	#389	1	1	yes	characteristic
105	C₂²×C₄₇	#459	1	1	yes	characteristic
106	C₂²×C₄₇	#462	1	1	yes	characteristic
107	C₂²×C₄₇	#466	1	1	yes	characteristic
108	C₂²×C₄₇	#467	1	1	yes	characteristic
109	C₂²×C₄₇	#472	1	1	yes	characteristic
110	C₂²×C₄₇	#473	1	1	yes	characteristic
111	C₄×C₄₇	—	2	1	no	—
112	C₄×C₄₇	—	2	1	no	—
113	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
114	C₄×C₄₇	#428	1	1	yes	characteristic
115	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
116	C₄×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
117	C₄×C₄₇	#474	1	1	yes	characteristic
118	C₄×C₄₇	#477	1	1	yes	characteristic
119	C₄×C₄₇	#499	1	1	yes	characteristic
120	C₄×C₄₇	#500	1	1	yes	characteristic
121	Dic₄₇	—	4	2	no	—
122	Dic₄₇	—	4	2	no	—
123	Dic₄₇	—	16	2	no	—
124	#1947	—	47	2	no	—
125	#1845	—	47	1	no	—
126	C₈×C₄⋊C₄	—	47	1	no	—
127	C₂×#335	D₄₇	1	1	yes	characteristic
128	#1296	—	47	2	no	—
129	C₂×C₄₇	#1667	1	1	yes	characteristic
130	C₂×C₄₇	#1669	1	1	yes	characteristic
131	C₂×C₄₇	#1674	1	1	yes	characteristic
132	C₂×C₄₇	#1676	1	1	yes	characteristic
133	C₂×C₄₇	#1942	1	1	yes	characteristic
134	C₂×C₄₇	#1947	1	1	yes	characteristic
135	C₂×C₄₇	#1952	1	1	yes	characteristic
136	C₂×#107	—	94	2	no	—
137	C₂×#107	—	94	1	no	—
138	C₂×#107	—	94	1	no	—
139	C₂×C₄⋊C₈	—	47	1	no	—
140	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
141	C₂×C₄×C₈	—	94	1	no	—
142	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
143	#384	—	47	2	no	—
144	C₄×C₄⋊C₄	—	47	1	no	—
145	#340	—	94	2	no	—
146	#336	—	94	1	no	—
147	#335	Dic₄₇	1	4	yes	complemented
148	C₄₇	#9614	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	94	1	no	—
151	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
152	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	47	1	no	—
154	C₂×C₄⋊C₄	—	47	1	no	—
155	C₂×C₄⋊C₄	—	94	2	no	—
156	C₂×C₄⋊C₄	—	94	2	no	—
157	C₂×C₄⋊C₄	—	94	2	no	—
158	C₂×C₄²	—	94	1	no	—
159	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
160	#107	—	188	4	no	—
161	#107	—	188	2	no	—
162	#107	—	188	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	47	2	no	—
165	C₄⋊C₈	—	94	1	no	—
166	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
167	C₄×C₈	—	188	2	no	—
168	C₄×C₈	Dic₉₄	1	2	yes	—
169	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
170	#95	—	94	1	no	—
171	C₂²×C₄	—	94	1	no	—
172	C₂²×C₄	—	94	1	no	—
173	C₂²×C₄	—	188	2	no	—
174	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
175	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
176	C₂²×C₄	#10625354	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	94	2	no	—
183	C₂×C₈	—	188	1	no	—
184	C₂×C₈	—	188	1	no	—
185	C₂×C₈	D₃₇₆	1	2	yes	—
186	C₂×C₈	Dic₁₈₈	1	2	yes	—
187	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
188	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
189	C₄⋊C₄	—	47	2	no	—
190	C₄⋊C₄	—	47	2	no	—
191	C₄⋊C₄	—	94	1	no	—
192	C₄⋊C₄	—	94	1	no	—
193	C₄⋊C₄	—	376	2	no	—
194	C₄⋊C₄	—	188	2	no	—
195	C₄⋊C₄	—	188	2	no	—
196	C₄⋊C₄	—	188	2	no	—
197	C₄⋊C₄	—	188	2	no	—
198	C₄²	—	188	1	no	—
199	C₄²	—	188	1	no	—
200	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
201	C₄²	#10625352	1	1	yes	characteristic
202	C₄²	Q₈×D₄₇	1	1	yes	characteristic
203	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
204	C₂³	#13093999	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	94	2	no	—
208	C₂×C₄	—	94	2	no	—
209	C₂×C₄	—	376	2	no	—
210	C₂×C₄	—	376	2	no	—
211	C₂×C₄	—	376	2	no	—
212	C₂×C₄	—	188	1	no	—
213	C₂×C₄	—	188	1	no	—
214	C₂×C₄	—	188	1	no	—
215	C₂×C₄	—	188	1	no	—
216	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
217	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
218	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
219	C₂×C₄	#13094004	1	1	yes	characteristic
220	C₂×C₄	#13094015	1	1	yes	characteristic
221	C₂×C₄	#13094023	1	1	yes	characteristic
222	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
224	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
225	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
226	C₂×C₄	#13094043	1	1	yes	characteristic
227	C₂×C₄	#13094047	1	1	yes	characteristic
228	C₂×C₄	#13094052	1	1	yes	characteristic
229	C₂×C₄	#13094057	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	—	188	2	no	—
233	C₈	#13093958	1	4	yes	complemented
234	C₂²	#15689022	1	1	yes	characteristic
235	C₂²	#15689319	1	1	yes	characteristic
236	C₂²	#15689334	1	1	yes	characteristic
237	C₂²	#15689360	1	1	yes	characteristic
238	C₂²	#15689365	1	1	yes	characteristic
239	C₂²	#15689394	1	1	yes	characteristic
240	C₂²	#15689399	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	188	2	no	—
244	C₄	—	188	2	no	—
245	C₄	—	752	2	no	—
246	C₄	C₂×#13093958	1	1	yes	characteristic
247	C₄	#15689165	1	1	yes	characteristic
248	C₄	D₈×Dic₄₇	1	1	yes	characteristic
249	C₄	Q₁₆×Dic₄₇	1	1	yes	characteristic
250	C₄	#15689404	1	1	yes	characteristic
251	C₄	#15689419	1	1	yes	characteristic
252	C₄	#15689524	1	1	yes	characteristic
253	C₄	#15689527	1	1	yes	characteristic
254	C₂	#20242744	1	1	yes	characteristic
255	C₂	#20242754	1	1	yes	characteristic
256	C₂	#20242783	1	1	yes	characteristic
257	C₂	#20242793	1	1	yes	characteristic
258	C₂	#20244122	1	1	yes	characteristic
259	C₂	#20244147	1	1	yes	characteristic
260	C₂	#20244174	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093999
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	#9614
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. C₄₇×#335	C₄	semidirect
101. C₈×C₄₇	#107	semidirect
147. #335	Dic₄₇	semidirect
148. C₄₇	#9614	semidirect
233. C₈	#13093958	semidirect

Maximal subgroups

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

subgroup	index
#20244147	2
#20243548	2
C₈×C₄⋊Dic₄₇	2
#20241272	2
C₂×C₄₇×#335	2
#9614	47