Subgroups — #26379169

Order 12032.

1910 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246574	C₂	1	4	yes	maximal, complemented
2	#20246494	C₂	1	4	yes	maximal, complemented
3	#20246265	C₂	1	8	yes	maximal
4	#20246194	C₂	1	8	yes	maximal, complemented
5	#20245991	C₂	1	4	yes	maximal
6	#20245786	C₂	1	2	yes	maximal
7	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#15689022	C₂²	1	16	yes	—
9	#15689009	C₂²	1	16	yes	—
10	#15689009	C₂²	1	16	yes	—
11	#15689009	C₂²	1	16	yes	—
12	#15689009	C₂²	1	16	yes	—
13	#15688991	C₂²	1	4	yes	—
14	#15688989	C₂²	1	4	yes	—
15	#15688986	C₂²	1	4	yes	—
16	#15688983	C₂²	1	4	yes	—
17	#15688983	C₂²	1	8	yes	—
18	C₄⋊C₄×Dic₄₇	C₂²	1	16	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
20	C₄×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
21	C₄×C₂²⋊Dic₄₇	C₂²	1	8	yes	—
22	#15688969	C₂²	1	4	yes	—
23	C₂×#125×C₄₇	C₄	1	16	yes	complemented
24	C₂×#125×C₄₇	C₂²	1	2	yes	—
25	C₂×#125×C₄₇	C₂²	1	4	yes	—
26	C₂×#125×C₄₇	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
29	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
32	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
33	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
35	C₄₇⋊#95	C₂³	1	8	yes	—
36	C₄₇⋊#95	C₂³	1	8	yes	—
37	C₄₇⋊#95	C₂³	1	8	yes	—
38	C₄₇⋊#95	C₂³	1	16	yes	—
39	C₄₇⋊#95	C₂³	1	8	yes	—
40	C₄₇⋊#95	C₂³	1	16	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
44	#125×C₄₇	C₂×C₄	1	16	yes	—
45	#125×C₄₇	C₂×C₄	1	32	yes	—
46	#125×C₄₇	C₂×C₄	1	16	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	32	yes	—
49	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	8	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
69	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
70	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
71	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
72	C₄⋊C₄×C₄₇	C₂²×C₄	1	16	yes	—
73	C₄⋊C₄×C₄₇	C₂²×C₄	1	32	yes	—
74	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²⋊Dic₄₇	—	2	16	no	—
78	C₄⋊Dic₄₇	—	4	16	no	—
79	C₄⋊Dic₄₇	—	4	8	no	—
80	C₄×Dic₄₇	—	4	8	no	—
81	C₄×Dic₄₇	—	4	16	no	—
82	C₂³×C₄₇	—	2	4	no	—
83	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
84	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
85	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	—	2	16	no	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	4	8	no	—
93	C₂×C₄×C₄₇	—	4	16	no	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	16	yes	—
99	C₂×Dic₄₇	—	2	16	no	—
100	C₂×Dic₄₇	—	2	16	no	—
101	C₂×Dic₄₇	—	2	16	no	—
102	#34372	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	—	2	4	no	—
104	C₂²×C₄₇	—	2	8	no	—
105	C₂²×C₄₇	#519	1	4	yes	—
106	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
108	C₂²×C₄₇	#585	1	4	yes	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	—	2	4	no	—
112	C₄×C₄₇	—	2	16	no	—
113	C₄×C₄₇	—	2	4	no	—
114	Dic₄₇	—	4	16	no	—
115	C₂²×#125	D₄₇	1	1	yes	characteristic
116	#2382	—	47	4	no	—
117	#2371	—	47	4	no	—
118	#2345	—	47	8	no	—
119	#2336	—	47	8	no	—
120	#2313	—	47	4	no	—
121	#2283	—	47	2	no	—
122	C₂×C₄₇	—	2	4	no	—
123	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
124	C₂×C₄₇	#3428	1	2	yes	—
125	C₂×C₄₇	#3553	1	4	yes	—
126	C₂×#125	Dic₄₇	1	16	yes	complemented
127	C₂×#125	C₂×D₄₇	1	4	yes	—
128	C₂×#125	C₂×D₄₇	1	2	yes	—
129	C₂×#125	C₂×D₄₇	1	2	yes	—
130	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
131	C₂²×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
132	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
133	#389	—	47	16	no	—
134	#387	—	47	16	no	—
135	#387	—	47	16	no	—
136	#387	—	47	16	no	—
137	#387	—	47	16	no	—
138	#383	—	47	4	no	—
139	#382	—	47	4	no	—
140	#381	—	47	4	no	—
141	#380	—	47	8	no	—
142	#380	—	47	4	no	—
143	C₄×C₄⋊C₄	—	47	16	no	—
144	C₄×C₄⋊C₄	—	47	4	no	—
145	C₄×C₂²⋊C₄	—	47	8	no	—
146	C₄×C₂²⋊C₄	—	47	4	no	—
147	#376	—	47	4	no	—
148	C₄₇	#34372	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
151	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
152	#125	C₂×Dic₄₇	1	32	yes	—
153	#125	C₂×Dic₄₇	1	16	yes	—
154	#125	C₂×Dic₄₇	1	16	yes	—
155	C₂×C₄⋊C₄	—	47	8	no	—
156	C₂×C₄⋊C₄	—	47	16	no	—
157	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₄₇	1	32	yes	—
159	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
160	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
161	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
162	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
163	C₂×C₂²⋊C₄	—	47	8	no	—
164	C₂×C₄²	—	47	8	no	—
165	C₂×C₄²	—	47	16	no	—
166	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
167	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	#95	—	47	8	no	—
171	#95	—	47	8	no	—
172	#95	—	47	8	no	—
173	#95	—	47	16	no	—
174	#95	—	47	16	no	—
175	#95	—	47	8	no	—
176	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	—	47	16	no	—
181	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
182	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
183	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
184	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
186	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
187	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
188	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
189	C₄⋊C₄	—	188	16	no	—
190	C₄⋊C₄	—	188	8	no	—
191	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
192	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
193	C₄⋊C₄	C₂²×Dic₄₇	1	16	yes	—
194	C₄⋊C₄	C₂²×Dic₄₇	1	32	yes	—
195	C₂²⋊C₄	—	94	16	no	—
196	C₄²	—	188	8	no	—
197	C₄²	—	188	16	no	—
198	C₄²	C₂²×Dic₄₇	1	4	yes	—
199	C₄²	C₂²×Dic₄₇	1	8	yes	—
200	C₄²	C₂²×Dic₄₇	1	4	yes	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂³×Dic₄₇	1	2	yes	—
203	C₂³	#13094072	1	4	yes	—
204	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	4	16	no	—
213	C₂×C₄	—	94	16	no	—
214	C₂×C₄	—	94	16	no	—
215	C₂×C₄	—	94	16	no	—
216	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
217	C₂×C₄	C₂³×Dic₄₇	1	2	yes	—
218	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
219	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
220	C₂×C₄	C₂³×Dic₄₇	1	16	yes	—
221	C₂²	—	2	8	no	—
222	C₂²	—	2	4	no	—
223	C₂²	#15689601	1	4	yes	—
224	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
225	C₂²	C₂×#13094072	1	2	yes	—
226	C₂²	#15689959	1	4	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	16	no	—
231	C₄	—	2	4	no	—
232	C₄	—	188	16	no	—
233	C₂	—	2	4	no	—
234	C₂	C₂×#15689601	1	1	yes	characteristic
235	C₂	#20256591	1	2	yes	—
236	C₂	#20257844	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Fitting	C₂²×#125×C₄₇	C₂
socle, cosocle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	#34372
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
1. #20246574	C₂	semidirect
2. #20246494	C₂	semidirect
4. #20246194	C₂	semidirect
23. C₂×#125×C₄₇	C₄	semidirect
126. C₂×#125	Dic₄₇	semidirect
148. C₄₇	#34372	semidirect

Maximal subgroups

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

subgroup	index
#20246574	2
#20246494	2
#20246265	2
#20246194	2
#20245991	2
#20245786	2
C₂²×#125×C₄₇	2
#34372	47