Subgroups — #26369699

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256524	C₂	1	1	yes	maximal, characteristic
2	#20252306	C₂	1	2	yes	maximal, complemented
3	#20245275	C₂	1	4	yes	maximal, complemented
4	#20245270	C₂	1	4	yes	maximal, complemented
5	#20245183	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688931	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	C₂²	1	2	yes	—
9	#15689889	—	2	2	no	—
10	#15689566	C₄	1	4	yes	complemented
11	#15689561	C₄	1	4	yes	complemented
12	#15689555	C₂²	1	4	yes	—
13	#15689246	—	2	4	no	—
14	#15689241	C₂²	1	8	yes	complemented
15	C₂×#13093960	C₂²	1	2	yes	—
16	#15689141	—	2	2	no	—
17	#15688953	C₂²	1	4	yes	complemented
18	#15688931	C₂²	1	4	yes	complemented
19	#15688931	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#405	C₂²	1	1	yes	characteristic
23	C₄₇×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₄₇	C₂²	1	2	yes	complemented
25	C₂×#10625350	—	2	2	no	—
26	#13094046	—	2	2	no	—
27	#13094046	—	2	2	no	—
28	#13094046	C₂×C₄	1	4	yes	—
29	#13094046	C₂³	1	4	yes	—
30	C₂×Dic₁₈₈	C₂×C₄	1	2	yes	—
31	C₂×#10625325	—	2	2	no	—
32	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
33	#13094032	—	4	4	no	—
34	#13094030	—	2	8	no	—
35	C₂×#10625329	—	2	2	no	—
36	M₁₆×D₄₇	—	4	4	no	—
37	Dic₃₇₆	C₂×C₄	1	4	yes	—
38	#13093969	C₂×C₄	1	8	yes	—
39	D₇₅₂	C₂×C₄	1	4	yes	—
40	#13093960	—	2	2	no	—
41	#13093960	C₂³	1	4	yes	—
42	#13093952	—	2	8	no	—
43	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	#117×C₄₇	D₄	1	1	yes	characteristic
46	M₃₂×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₄₇	—	2	2	no	—
48	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
51	#97×C₄₇	D₄	1	2	yes	—
52	C₄×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₇	—	4	2	no	—
55	#10625350	—	2	4	no	—
56	#10625350	—	2	4	no	—
57	#10625350	—	4	2	no	—
58	#10625350	—	4	2	no	—
59	C₂×D₁₈₈	—	2	2	no	—
60	C₂×C₄×D₄₇	—	4	2	no	—
61	C₂×Dic₉₄	—	2	2	no	—
62	M₁₆×C₄₇	D₈	1	4	yes	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₄₇	#41	1	2	yes	—
67	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	1	no	—
69	C₂²⋊C₄×C₄₇	—	2	1	no	—
70	C₄²×C₄₇	C₂×D₄	1	2	yes	—
71	#10625329	—	2	4	no	—
72	#10625329	—	4	2	no	—
73	C₂×C₄₇⋊C₈	—	4	2	no	—
74	Dic₁₈₈	—	2	2	no	—
75	Dic₁₈₈	C₂²×C₄	1	4	yes	—
76	D₃₇₆	—	2	2	no	—
77	D₃₇₆	C₂²×C₄	1	4	yes	—
78	#10625325	—	2	4	no	—
79	#10625325	—	2	4	no	—
80	#10625324	—	4	8	no	—
81	C₈×D₄₇	—	4	8	no	—
82	C₁₆×C₄₇	—	2	1	no	—
83	C₁₆×C₄₇	—	2	1	no	—
84	C₁₆×C₄₇	C₂²×C₄	1	4	yes	—
85	C₂³×C₄₇	#135	1	1	yes	characteristic
86	C₂²×D₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	C₂²⋊D₄₇	—	4	4	no	—
93	C₂²⋊D₄₇	—	4	4	no	—
94	C₂×Dic₄₇	—	4	2	no	—
95	D₁₈₈	—	2	4	no	—
96	D₁₈₈	—	4	2	no	—
97	C₄×D₄₇	—	4	4	no	—
98	C₄×D₄₇	—	4	4	no	—
99	Dic₉₄	—	2	4	no	—
100	Dic₉₄	—	4	2	no	—
101	C₈×C₄₇	C₄×D₄	1	2	yes	—
102	C₈×C₄₇	C₂×D₈	1	4	yes	—
103	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
105	C₄₇⋊C₈	—	4	4	no	—
106	#33464	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	—	4	1	no	—
108	C₂²×C₄₇	C₄×D₈	1	2	yes	—
109	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
110	C₂×D₄₇	—	4	4	no	—
111	C₂×D₄₇	—	16	2	no	—
112	C₄×C₄₇	—	4	2	no	—
113	C₄×C₄₇	C₄×D₈	1	2	yes	—
114	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
116	Dic₄₇	—	4	4	no	—
117	#3414	—	47	1	no	—
118	#2957	—	47	2	no	—
119	#2179	—	47	4	no	—
120	#2178	—	47	4	no	—
121	#2161	—	47	1	no	—
122	C₂×#359	—	47	2	no	—
123	#2108	D₄₇	1	1	yes	characteristic, complemented
124	C₂×C₄₇	—	2	1	no	—
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
127	D₄₇	—	8	4	no	—
128	C₂×#135	—	47	2	no	—
129	#568	—	94	2	no	—
130	#510	—	47	4	no	—
131	#509	—	47	4	no	—
132	#508	—	47	4	no	—
133	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
134	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
135	#444	—	94	4	no	—
136	#443	—	47	8	no	—
137	C₂×#108	—	47	2	no	—
138	#421	—	94	2	no	—
139	#405	C₂×D₄₇	1	1	yes	characteristic
140	#368	—	47	4	no	—
141	#359	—	47	4	no	—
142	#359	—	47	4	no	—
143	#346	C₂×D₄₇	1	2	yes	complemented
144	C₄×C₁₆	C₂×D₄₇	1	2	yes	complemented
145	C₄₇	#33464	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	94	2	no	—
147	#135	—	47	4	no	—
148	#135	—	47	4	no	—
149	#135	—	94	2	no	—
150	#135	—	94	2	no	—
151	C₂×Q₁₆	—	47	2	no	—
152	C₂×SD₁₆	—	94	2	no	—
153	C₂×D₈	—	47	2	no	—
154	#131	—	94	8	no	—
155	#131	—	188	4	no	—
156	C₂×M₁₆	—	94	2	no	—
157	C₂×M₁₆	—	188	4	no	—
158	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
160	#117	D₁₈₈	1	1	yes	characteristic
161	Q₃₂	—	47	4	no	—
162	SD₃₂	—	47	8	no	—
163	D₁₆	—	47	4	no	—
164	M₃₂	C₄×D₄₇	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
167	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
169	#108	—	47	4	no	—
170	#108	—	94	2	no	—
171	#104	—	94	8	no	—
172	#97	D₁₈₈	1	2	yes	—
173	C₄×C₈	C₂²×D₄₇	1	2	yes	—
174	#41	—	94	4	no	—
175	#41	—	94	4	no	—
176	#41	—	188	2	no	—
177	#41	—	188	2	no	—
178	C₂×Q₈	—	94	2	no	—
179	C₂×D₄	—	94	2	no	—
180	C₂×D₄	—	188	2	no	—
181	C₂²×C₄	—	188	2	no	—
182	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
183	Q₁₆	—	47	4	no	—
184	Q₁₆	—	94	2	no	—
185	SD₁₆	—	94	4	no	—
186	SD₁₆	—	94	4	no	—
187	D₈	—	47	4	no	—
188	D₈	—	94	2	no	—
189	M₁₆	—	94	4	no	—
190	M₁₆	—	188	2	no	—
191	M₁₆	—	188	8	no	—
192	M₁₆	D₃₇₆	1	4	yes	—
193	C₂×C₈	—	188	2	no	—
194	C₂×C₈	—	188	8	no	—
195	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
196	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
197	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
198	C₂×C₈	#10625350	1	2	yes	—
199	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₁₈₈	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂×C₄×D₄₇	1	4	yes	—
206	C₂³	—	188	2	no	—
207	C₂³	#13094046	1	1	yes	characteristic
208	Q₈	—	94	4	no	—
209	Q₈	—	188	2	no	—
210	D₄	—	94	4	no	—
211	D₄	—	188	4	no	—
212	D₄	—	188	4	no	—
213	D₄	—	188	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	188	2	no	—
216	C₂×C₄	—	188	4	no	—
217	C₂×C₄	—	188	4	no	—
218	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
219	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
220	C₂×C₄	#13094046	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
222	C₈	—	188	4	no	—
223	C₈	C₄×D₁₈₈	1	2	yes	—
224	C₈	C₂×D₃₇₆	1	4	yes	—
225	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
226	C₈	C₂×#10625350	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	188	4	no	—
229	C₂²	—	752	2	no	—
230	C₂²	C₄×D₃₇₆	1	2	yes	—
231	C₂²	C₂×#13094046	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	188	4	no	—
234	C₄	C₄×D₃₇₆	1	2	yes	—
235	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
236	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	376	4	no	—
240	C₂	C₂×C₄×D₃₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₃₇₆
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2108	C₂
socle	C₂×C₄₇	C₂×C₄×D₈
prime radical	#20256524	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2108	D₄₇
hypocenter	C₄₇	#33464
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #20252306	C₂	semidirect
3. #20245275	C₂	semidirect
4. #20245270	C₂	semidirect
5. #20245183	C₂	semidirect
6. C₂×#15688931	C₂	semidirect
7. C₄₇×#2108	C₂	semidirect
10. #15689566	C₄	semidirect
11. #15689561	C₄	semidirect
14. #15689241	C₂²	semidirect
17. #15688953	C₂²	semidirect
18. #15688931	C₂²	semidirect
19. #15688931	C₂²	semidirect
23. C₄₇×#346	C₂²	semidirect
24. C₄×C₁₆×C₄₇	C₂²	semidirect
123. #2108	D₄₇	semidirect
143. #346	C₂×D₄₇	semidirect
144. C₄×C₁₆	C₂×D₄₇	semidirect
145. C₄₇	#33464	semidirect

Maximal subgroups

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

subgroup	index
#20256524	2
#20252306	2
#20245275	2
#20245270	2
#20245183	2
C₂×#15688931	2
C₄₇×#2108	2
#33464	47