Subgroups — #26109216

Order 12032.

378 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689253	C₂	1	1	yes	maximal, characteristic
2	#20243381	C₂	1	2	yes	maximal
3	#20242147	C₂	1	1	yes	maximal, characteristic
4	#20241262	C₂	1	2	yes	maximal
5	C₄₇×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	#15689253	—	2	1	no	—
8	#15689253	—	2	1	no	—
9	#15689253	—	2	1	no	—
10	#15689253	—	2	1	no	—
11	#15689147	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₇⋊C₈	C₄	1	2	yes	—
14	C₂×#13093942	—	2	2	no	—
15	C₂×#13093934	C₄	1	2	yes	—
16	#15688891	—	2	2	no	—
17	#15688876	—	2	1	no	—
18	#15688876	—	2	1	no	—
19	C₄₇×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#334	—	2	1	no	—
22	C₄₇×#334	—	2	1	no	—
23	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
24	C₂×#10625329	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×Dic₉₄	—	2	1	no	—
27	C₄×Dic₉₄	—	2	1	no	—
28	C₄×Dic₉₄	—	2	1	no	—
29	C₄×Dic₉₄	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
31	C₂×Dic₄₇⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
33	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	Dic₄₇⋊C₈	—	2	2	no	—
38	Dic₄₇⋊C₈	—	2	2	no	—
39	#13093942	—	4	4	no	—
40	#13093934	—	2	2	no	—
41	#13093934	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₄₇	—	2	2	no	—
47	C₄⋊C₈×C₄₇	—	2	2	no	—
48	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
56	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₉₄	—	2	1	no	—
58	C₂×Dic₉₄	—	2	1	no	—
59	C₂×Dic₉₄	—	2	2	no	—
60	C₂×Dic₉₄	—	2	2	no	—
61	M₁₆×C₄₇	—	4	2	no	—
62	M₁₆×C₄₇	—	4	2	no	—
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	4	1	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₇	#41	1	1	yes	characteristic
73	C₄²×C₄₇	#41	1	1	yes	characteristic
74	C₄⋊Dic₄₇	—	2	1	no	—
75	C₄⋊Dic₄₇	—	2	1	no	—
76	Dic₄₇⋊C₄	—	2	2	no	—
77	Dic₄₇⋊C₄	—	2	2	no	—
78	C₄×Dic₄₇	—	2	2	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	#10625329	—	4	2	no	—
81	#10625329	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	2	2	no	—
83	C₂×C₄₇⋊C₈	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	4	2	no	—
85	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	—	4	1	no	—
89	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₄₇	#137	1	2	yes	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	—	2	4	no	—
100	C₂×Dic₄₇	—	4	2	no	—
101	Dic₉₄	—	4	2	no	—
102	Dic₉₄	—	4	1	no	—
103	Dic₉₄	—	4	1	no	—
104	C₈×C₄₇	—	2	1	no	—
105	C₈×C₄₇	—	2	1	no	—
106	C₈×C₄₇	—	2	1	no	—
107	C₈×C₄₇	—	8	2	no	—
108	C₈×C₄₇	#101	1	2	yes	complemented
109	C₄₇⋊C₈	—	4	2	no	—
110	C₄₇⋊C₈	—	8	2	no	—
111	#8619	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	—	2	1	no	—
113	C₂²×C₄₇	—	2	1	no	—
114	C₂²×C₄₇	#380	1	1	yes	characteristic
115	C₂²×C₄₇	#428	1	2	yes	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	C₂×#101	1	1	yes	characteristic
120	C₄×C₄₇	#413	1	1	yes	characteristic
121	C₄×C₄₇	#468	1	2	yes	—
122	C₄×C₄₇	#469	1	1	yes	characteristic
123	C₄×C₄₇	#470	1	1	yes	characteristic
124	Dic₄₇	—	4	2	no	—
125	Dic₄₇	—	4	2	no	—
126	C₂×#446	—	47	1	no	—
127	#1801	—	47	2	no	—
128	#1567	—	47	1	no	—
129	#1566	D₄₇	1	1	yes	characteristic
130	#1292	—	47	2	no	—
131	C₂×C₄₇	—	2	2	no	—
132	C₂×C₄₇	#1801	1	1	yes	characteristic
133	C₂×C₄₇	#1809	1	1	yes	characteristic
134	C₂×C₄₇	#1813	1	1	yes	characteristic
135	C₂×C₄×Q₈	—	47	1	no	—
136	#446	—	94	1	no	—
137	#446	—	94	1	no	—
138	#446	—	94	1	no	—
139	#446	—	94	1	no	—
140	#423	—	47	2	no	—
141	#423	C₂×D₄₇	1	2	yes	—
142	C₂×C₄⋊C₈	—	47	2	no	—
143	C₂×C₄⋊C₈	—	47	1	no	—
144	C₂×#101	—	94	2	no	—
145	C₂×#97	—	47	2	no	—
146	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
147	#343	—	94	2	no	—
148	#335	—	94	1	no	—
149	#335	—	94	1	no	—
150	#334	—	2	1	no	—
151	#334	—	2	1	no	—
152	C₄₇	#8619	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	47	1	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	94	2	no	—
156	C₂²×C₈	—	47	2	no	—
157	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
159	C₄×Q₈	—	94	1	no	—
160	C₄×Q₈	—	94	1	no	—
161	C₄×Q₈	—	94	1	no	—
162	C₄×Q₈	—	94	1	no	—
163	C₂×C₄⋊C₄	—	47	1	no	—
164	C₂×C₄⋊C₄	—	47	2	no	—
165	C₂×C₄²	—	47	2	no	—
166	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	94	2	no	—
170	C₄⋊C₈	—	94	2	no	—
171	C₄⋊C₈	—	94	1	no	—
172	C₄⋊C₈	—	94	2	no	—
173	C₄⋊C₈	—	94	2	no	—
174	C₄⋊C₈	—	94	1	no	—
175	#101	—	188	4	no	—
176	#97	—	94	2	no	—
177	#97	—	94	2	no	—
178	C₄×C₈	D₁₈₈	1	1	yes	characteristic
179	C₄×C₈	D₁₈₈	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂×Q₈	—	94	1	no	—
183	C₂×Q₈	—	94	1	no	—
184	C₂×Q₈	—	94	2	no	—
185	C₂×Q₈	—	94	2	no	—
186	C₂²×C₄	—	47	2	no	—
187	C₂²×C₄	—	47	2	no	—
188	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
190	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
191	M₁₆	—	4	2	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	188	2	no	—
194	M₁₆	—	188	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	94	2	no	—
199	C₂×C₈	—	188	2	no	—
200	C₂×C₈	—	188	2	no	—
201	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
202	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
205	C₄⋊C₄	—	94	1	no	—
206	C₄⋊C₄	—	94	1	no	—
207	C₄⋊C₄	—	94	2	no	—
208	C₄⋊C₄	—	94	2	no	—
209	C₄²	—	94	2	no	—
210	C₄²	—	94	2	no	—
211	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
212	C₄²	#10625352	1	1	yes	characteristic
213	C₄²	#10625354	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
216	Q₈	—	188	2	no	—
217	Q₈	—	188	1	no	—
218	Q₈	—	188	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	94	2	no	—
223	C₂×C₄	—	94	2	no	—
224	C₂×C₄	—	94	4	no	—
225	C₂×C₄	—	188	2	no	—
226	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	#13093982	1	1	yes	characteristic
228	C₂×C₄	#13093995	1	1	yes	characteristic
229	C₂×C₄	#13093999	1	1	yes	characteristic
230	C₂×C₄	#13094000	1	1	yes	characteristic
231	C₂×C₄	#13094004	1	1	yes	characteristic
232	C₂×C₄	#13094051	1	2	yes	—
233	C₂×C₄	#13094053	1	2	yes	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	8	2	no	—
238	C₈	—	376	2	no	—
239	C₈	—	188	2	no	—
240	C₈	#13093942	1	2	yes	complemented
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#15688980	1	1	yes	characteristic
244	C₂²	#15689162	1	2	yes	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	188	2	no	—
249	C₄	—	188	2	no	—
250	C₄	C₂×#13093942	1	1	yes	characteristic
251	C₄	#15689105	1	1	yes	characteristic
252	C₄	#15689366	1	2	yes	—
253	C₄	#15689377	1	1	yes	characteristic
254	C₄	#15689382	1	1	yes	characteristic
255	C₂	—	2	2	no	—
256	C₂	#20243381	1	1	yes	characteristic
257	C₂	#20243418	1	1	yes	characteristic
258	C₂	#20243435	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688980
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1566	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1566	D₄₇
hypocenter	C₄₇	#8619
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
108. C₈×C₄₇	#101	semidirect
152. C₄₇	#8619	semidirect
240. C₈	#13093942	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689253	2
#20243381	2
#20242147	2
#20241262	2
C₄₇×#1566	2
#8619	47