Subgroups — #26216478

Order 12032.

910 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
2	#20245791	C₂	1	2	yes	maximal
3	#20243520	C₂	1	8	yes	maximal, complemented
4	#20243105	C₂	1	1	yes	maximal, characteristic
5	#20243104	C₂	1	2	yes	maximal
6	C₄₇×#1735	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13093934	C₄	1	8	yes	complemented
11	C₂×#13093934	C₂²	1	4	yes	—
12	#15688992	C₂²	1	4	yes	—
13	#15688988	C₂²	1	4	yes	—
14	#15688981	—	2	4	no	—
15	C₄×Dic₄₇⋊C₄	—	2	4	no	—
16	#15688886	C₂²	1	4	yes	—
17	#15688885	C₂²	1	8	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#340	C₂²	1	4	yes	—
21	C₂³×Dic₄₇	D₄	1	2	yes	—
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
25	#13093983	—	2	4	no	—
26	#13093983	—	2	4	no	—
27	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
28	C₂×Dic₄₇⋊C₄	—	2	8	no	—
29	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
33	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
35	#13093934	C₂×C₄	1	16	yes	—
36	C₄₇⋊#95	—	2	8	no	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	—	2	4	no	—
39	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
41	#117×C₄₇	—	2	4	no	—
42	#117×C₄₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₇	#41	1	4	yes	—
52	C₂²×C₄×C₄₇	#41	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	4	no	—
55	C₂²×Dic₄₇	—	4	4	no	—
56	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
57	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₄₇	—	2	8	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
61	C₄⋊C₄×C₄₇	—	2	4	no	—
62	C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₂²⋊C₄×C₄₇	—	4	4	no	—
64	C₄²×C₄₇	—	4	4	no	—
65	C₂²⋊Dic₄₇	—	4	4	no	—
66	C₄⋊Dic₄₇	—	2	4	no	—
67	C₄⋊Dic₄₇	—	2	4	no	—
68	Dic₄₇⋊C₄	—	4	16	no	—
69	C₄×Dic₄₇	—	4	4	no	—
70	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
71	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
72	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
73	C₂×C₄₇⋊C₈	—	2	8	no	—
74	C₂×C₄₇⋊C₈	—	2	4	no	—
75	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
76	C₂³×C₄₇	—	2	4	no	—
77	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₄₇	#124	1	4	yes	—
79	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	—	4	4	no	—
81	C₂×C₄×C₄₇	—	4	8	no	—
82	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₄₇	#117	1	4	yes	—
84	C₂×C₄×C₄₇	#117	1	4	yes	—
85	C₂×C₄×C₄₇	#124	1	4	yes	—
86	C₂×C₄×C₄₇	#125	1	8	yes	—
87	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
91	C₂×Dic₄₇	—	2	8	no	—
92	C₂×Dic₄₇	—	4	8	no	—
93	C₂×Dic₄₇	—	4	4	no	—
94	C₂×Dic₄₇	—	4	8	no	—
95	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
96	C₈×C₄₇	—	2	4	no	—
97	C₄₇⋊C₈	—	2	4	no	—
98	#20762	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	8	no	—
100	C₂²×C₄₇	—	2	4	no	—
101	C₂²×C₄₇	#382	1	4	yes	—
102	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×#124	1	2	yes	—
104	C₂²×C₄₇	#575	1	4	yes	—
105	C₄×C₄₇	—	8	4	no	—
106	C₄×C₄₇	#382	1	4	yes	—
107	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
108	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
109	C₄×C₄₇	C₂×#125	1	2	yes	—
110	Dic₄₇	—	2	8	no	—
111	Dic₄₇	—	8	4	no	—
112	C₂²×#97	—	47	1	no	—
113	#2285	—	47	2	no	—
114	#1836	—	47	8	no	—
115	#1735	—	47	1	no	—
116	#1735	—	47	2	no	—
117	#1735	D₄₇	1	1	yes	characteristic
118	C₂×C₄₇	—	2	4	no	—
119	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
120	C₂×C₄₇	#2911	1	2	yes	—
121	C₂×C₄₇	#3135	1	4	yes	—
122	C₂³×C₈	—	47	1	no	—
123	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
124	C₂×#117	—	47	2	no	—
125	C₂×#117	C₂×D₄₇	1	2	yes	—
126	C₂²×C₄²	—	47	1	no	—
127	C₂×#97	—	47	8	no	—
128	C₂×#97	—	47	4	no	—
129	#384	—	47	4	no	—
130	#382	—	47	4	no	—
131	#380	—	94	4	no	—
132	C₄×C₄⋊C₄	—	94	4	no	—
133	#340	—	47	4	no	—
134	#340	—	47	8	no	—
135	#340	C₂×D₄₇	1	4	yes	—
136	C₄₇	#20762	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	47	2	no	—
138	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	—	47	4	no	—
141	C₂²×C₈	—	47	2	no	—
142	C₂²×C₈	—	94	4	no	—
143	C₂²×C₈	C₄×D₄₇	1	4	yes	—
144	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
145	#117	—	2	4	no	—
146	#117	—	2	4	no	—
147	#117	—	94	4	no	—
148	#117	—	94	4	no	—
149	C₂×C₄⋊C₄	—	47	4	no	—
150	C₂×C₄⋊C₄	—	94	8	no	—
151	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
152	C₂×C₂²⋊C₄	—	2	2	no	—
153	C₂×C₂²⋊C₄	—	94	2	no	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	—	47	4	no	—
156	C₂×C₄²	—	47	2	no	—
157	C₂×C₄²	—	47	4	no	—
158	C₂×C₄²	—	47	2	no	—
159	C₂×C₄²	—	94	2	no	—
160	#97	—	47	16	no	—
161	#95	—	94	8	no	—
162	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	47	4	no	—
165	C₂²×C₄	—	47	8	no	—
166	C₂²×C₄	—	94	4	no	—
167	C₂²×C₄	—	188	4	no	—
168	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
169	C₂²×C₄	#10625350	1	4	yes	—
170	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₄	#10625352	1	4	yes	—
172	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
173	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	47	4	no	—
177	C₂×C₈	—	94	8	no	—
178	C₂×C₈	—	94	4	no	—
179	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	94	4	no	—
183	C₄⋊C₄	—	94	4	no	—
184	C₄⋊C₄	—	188	16	no	—
185	C₂²⋊C₄	—	4	4	no	—
186	C₂²⋊C₄	—	188	4	no	—
187	C₄²	—	4	4	no	—
188	C₄²	—	47	8	no	—
189	C₄²	—	47	4	no	—
190	C₄²	—	47	4	no	—
191	C₄²	—	188	4	no	—
192	C₂³	—	2	4	no	—
193	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
194	C₂³	#13094010	1	4	yes	—
195	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	8	no	—
198	C₂×C₄	—	47	8	no	—
199	C₂×C₄	—	94	8	no	—
200	C₂×C₄	—	188	8	no	—
201	C₂×C₄	—	188	4	no	—
202	C₂×C₄	—	188	8	no	—
203	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
204	C₂×C₄	#13093980	1	4	yes	—
205	C₂×C₄	#13093981	1	4	yes	—
206	C₂×C₄	#13094010	1	4	yes	—
207	C₂×C₄	#13094014	1	8	yes	—
208	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
209	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
210	C₂×C₄	C₂×#10625352	1	2	yes	—
211	C₂×C₄	C₂×#10625350	1	2	yes	—
212	C₈	—	2	4	no	—
213	C₈	—	94	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	#15688987	1	4	yes	—
217	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
218	C₂²	C₂×#13094010	1	2	yes	—
219	C₂²	#15689920	1	4	yes	—
220	C₄	—	8	4	no	—
221	C₄	—	94	8	no	—
222	C₄	—	376	4	no	—
223	C₄	#15688987	1	4	yes	—
224	C₄	C₂×#13093980	1	1	yes	characteristic
225	C₄	C₂×#13093981	1	1	yes	characteristic
226	C₄	C₂×#13094014	1	2	yes	—
227	C₂	—	2	4	no	—
228	C₂	C₂×#15688987	1	1	yes	characteristic
229	C₂	#20251976	1	2	yes	—
230	C₂	#20253731	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1735	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1735	D₄₇
hypocenter	C₄₇	#20762
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1735

Complements

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

N	H ≅ G/N	type
3. #20243520	C₂	semidirect
10. C₂×#13093934	C₄	semidirect
136. C₄₇	#20762	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093934	2
#20245791	2
#20243520	2
#20243105	2
#20243104	2
C₄₇×#1735	2
#20762	47