Subgroups — #26104744

Order 12032.

432 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	#20251751	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	#20241210	C₂	1	4	yes	maximal
4	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	—	2	1	no	—
6	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
7	#15689021	—	2	1	no	—
8	#15689021	—	2	1	no	—
9	#15689021	C₄	1	2	yes	—
10	#15689021	C₂²	1	2	yes	—
11	#15688993	C₄	1	1	yes	characteristic
12	#15688993	C₄	1	1	yes	characteristic
13	#15688989	—	2	1	no	—
14	#15688878	—	2	1	no	—
15	#15688878	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
17	C₂×#97×C₄₇	—	2	1	no	—
18	C₄³×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#336	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	#13094016	—	4	4	no	—
22	C₂×C₄⋊Dic₄₇	—	2	1	no	—
23	C₂×C₄⋊Dic₄₇	—	2	1	no	—
24	C₂×C₄⋊Dic₄₇	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×Dic₄₇⋊C₄	—	2	2	no	—
29	C₂×Dic₄₇⋊C₄	—	2	2	no	—
30	C₄×C₄₇⋊C₈	—	2	4	no	—
31	C₄₇⋊#95	—	2	2	no	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₂²×C₈×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₇	—	2	4	no	—
42	#97×C₄₇	—	4	2	no	—
43	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₄₇	—	2	2	no	—
51	C₂²×Dic₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	—	2	4	no	—
53	C₂×C₈×C₄₇	—	4	2	no	—
54	C₂×C₈×C₄₇	—	4	2	no	—
55	C₄²×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	—	2	2	no	—
57	C₄²×C₄₇	—	2	2	no	—
58	C₄²×C₄₇	—	2	2	no	—
59	C₄²×C₄₇	—	2	2	no	—
60	C₄²×C₄₇	—	4	1	no	—
61	C₄²×C₄₇	SD₁₆	1	2	yes	—
62	C₄²×C₄₇	Q₁₆	1	2	yes	—
63	C₄⋊Dic₄₇	—	2	4	no	—
64	C₄⋊Dic₄₇	—	4	2	no	—
65	C₄⋊Dic₄₇	—	4	2	no	—
66	C₄⋊Dic₄₇	—	4	2	no	—
67	C₄⋊Dic₄₇	—	4	2	no	—
68	C₄⋊Dic₄₇	—	4	2	no	—
69	Dic₄₇⋊C₄	—	4	4	no	—
70	Dic₄₇⋊C₄	—	4	4	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	4	2	no	—
73	C₂×C₄₇⋊C₈	—	4	2	no	—
74	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	1	no	—
76	C₂×C₄×C₄₇	—	2	1	no	—
77	C₂×C₄×C₄₇	—	2	1	no	—
78	C₂×C₄×C₄₇	—	2	1	no	—
79	C₂×C₄×C₄₇	—	2	1	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	#101	1	2	yes	—
89	C₂×C₄×C₄₇	#103	1	2	yes	—
90	C₂×C₄×C₄₇	#103	1	2	yes	—
91	C₂×C₄×C₄₇	#104	1	2	yes	—
92	C₂×C₄×C₄₇	#104	1	2	yes	—
93	C₂×C₄×C₄₇	#131	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×Dic₄₇	—	4	2	no	—
98	C₂×Dic₄₇	—	4	2	no	—
99	C₂×Dic₄₇	—	4	2	no	—
100	C₂×Dic₄₇	—	4	2	no	—
101	C₂×Dic₄₇	—	4	2	no	—
102	C₂×Dic₄₇	—	4	2	no	—
103	C₈×C₄₇	—	4	4	no	—
104	C₄₇⋊C₈	—	4	4	no	—
105	#7683	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	#408	1	1	yes	characteristic
107	C₂²×C₄₇	#410	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
110	C₂²×C₄₇	#418	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
113	C₄×C₄₇	—	2	2	no	—
114	C₄×C₄₇	—	2	2	no	—
115	C₄×C₄₇	—	2	2	no	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	4	1	no	—
120	C₄×C₄₇	#330	1	4	yes	—
121	Dic₄₇	—	8	2	no	—
122	Dic₄₇	—	8	2	no	—
123	#2859	—	47	1	no	—
124	#1767	D₄₇	1	1	yes	characteristic
125	#1744	—	47	1	no	—
126	#1271	—	47	4	no	—
127	C₂×C₄₇	#1470	1	1	yes	characteristic
128	C₂×C₄₇	#1484	1	1	yes	characteristic
129	C₂×C₄₇	#1493	1	1	yes	characteristic
130	C₂×C₄₇	#1503	1	1	yes	characteristic
131	C₂×C₄₇	#1505	1	1	yes	characteristic
132	C₂×C₄₇	C₂×#330	1	1	yes	characteristic
133	C₂×C₄₇	#1549	1	1	yes	characteristic
134	C₂×#125	—	94	1	no	—
135	C₂×C₄⋊C₈	—	2	1	no	—
136	C₂×#97	—	2	1	no	—
137	C₂×C₄×C₈	—	94	1	no	—
138	#389	—	47	2	no	—
139	#389	—	47	2	no	—
140	#389	—	94	1	no	—
141	#389	—	94	1	no	—
142	#384	—	47	1	no	—
143	#384	—	47	1	no	—
144	#382	—	94	1	no	—
145	C₄³	C₂×D₄₇	1	1	yes	characteristic
146	#336	—	47	2	no	—
147	#336	—	94	1	no	—
148	#336	C₂×D₄₇	1	2	yes	—
149	C₄₇	#7683	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	94	2	no	—
152	#125	—	188	4	no	—
153	C₂×C₄⋊C₄	—	47	2	no	—
154	C₂×C₄⋊C₄	—	47	2	no	—
155	C₂×C₄⋊C₄	—	94	2	no	—
156	C₂×C₄⋊C₄	—	94	1	no	—
157	C₂×C₄⋊C₄	—	94	1	no	—
158	C₂×C₄⋊C₄	—	94	1	no	—
159	C₂×C₄⋊C₄	—	94	1	no	—
160	C₂×C₄⋊C₄	—	94	2	no	—
161	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
162	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
163	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
164	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
168	C₄⋊C₈	—	2	4	no	—
169	#97	—	4	2	no	—
170	C₄×C₈	—	94	4	no	—
171	#95	—	94	2	no	—
172	#95	—	94	2	no	—
173	C₂²×C₄	—	94	2	no	—
174	C₂²×C₄	—	94	2	no	—
175	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	94	4	no	—
186	C₂×C₈	—	188	2	no	—
187	C₂×C₈	—	188	2	no	—
188	C₄⋊C₄	—	94	4	no	—
189	C₄⋊C₄	—	188	2	no	—
190	C₄⋊C₄	—	188	4	no	—
191	C₄⋊C₄	—	188	2	no	—
192	C₄⋊C₄	—	188	2	no	—
193	C₄⋊C₄	—	188	2	no	—
194	C₄⋊C₄	—	188	2	no	—
195	C₄⋊C₄	—	188	4	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	1	no	—
202	C₄²	#10625335	1	2	yes	—
203	C₄²	#10625337	1	2	yes	—
204	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	188	2	no	—
219	C₂×C₄	—	188	2	no	—
220	C₂×C₄	—	188	2	no	—
221	C₂×C₄	—	188	2	no	—
222	C₂×C₄	—	188	2	no	—
223	C₂×C₄	—	188	2	no	—
224	C₂×C₄	#13093942	1	2	yes	—
225	C₂×C₄	#13093947	1	2	yes	—
226	C₂×C₄	#13093947	1	2	yes	—
227	C₂×C₄	#13093950	1	2	yes	—
228	C₂×C₄	#13093952	1	2	yes	—
229	C₂×C₄	#13094032	1	2	yes	—
230	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
231	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
232	C₂×C₄	#13094052	1	2	yes	—
233	C₈	—	4	4	no	—
234	C₈	—	188	4	no	—
235	C₂²	#15689091	1	1	yes	characteristic
236	C₂²	#15689096	1	1	yes	characteristic
237	C₂²	C₂×#13093942	1	1	yes	characteristic
238	C₂²	C₂×#13093947	1	1	yes	characteristic
239	C₂²	#15689129	1	1	yes	characteristic
240	C₂²	C₂×#13093950	1	1	yes	characteristic
241	C₂²	C₂×#13093952	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	1	no	—
249	C₄	—	376	2	no	—
250	C₄	—	376	2	no	—
251	C₄	#15688864	1	4	yes	—
252	C₂	#20241745	1	1	yes	characteristic
253	C₂	#20241806	1	1	yes	characteristic
254	C₂	#20241842	1	1	yes	characteristic
255	C₂	#20241886	1	1	yes	characteristic
256	C₂	#20241897	1	1	yes	characteristic
257	C₂	C₂×#15688864	1	1	yes	characteristic
258	C₂	#20242074	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1767	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1767	D₄₇
hypocenter	C₄₇	#7683
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
149. C₄₇	#7683	semidirect

Maximal subgroups

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

subgroup	index
#20251751	2
#20243134	2
#20241210	2
C₄₇×#1767	2
#7683	47