Subgroups — #26369698

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	#20252305	C₂	1	2	yes	maximal, complemented
2	#20245274	C₂	1	4	yes	maximal, complemented
3	#20245269	C₂	1	4	yes	maximal, complemented
4	#20245183	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688930	C₂	1	2	yes	maximal, complemented
6	#20244987	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#3414	C₂	1	1	yes	maximal, characteristic
8	#15689890	—	2	2	no	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	#15689245	—	2	4	no	—
11	#15689240	C₂²	1	8	yes	complemented
12	C₂×#13093960	C₂²	1	2	yes	—
13	#15689140	—	2	2	no	—
14	#15689078	C₂²	1	1	yes	characteristic
15	#15688953	C₂²	1	4	yes	complemented
16	#15688930	C₂²	1	4	yes	complemented
17	#15688930	C₂²	1	4	yes	complemented
18	#15688898	C₂²	1	2	yes	complemented
19	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
20	C₂×#135×C₄₇	C₂²	1	2	yes	—
21	C₄₇×#510	C₄	1	4	yes	complemented
22	C₄₇×#509	C₄	1	4	yes	complemented
23	C₄₇×#508	C₂²	1	4	yes	—
24	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
25	#13094031	—	2	8	no	—
26	#13094031	—	4	4	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	4	4	no	—
29	C₂×#10625329	D₄	1	1	yes	characteristic
30	#13093983	D₄	1	1	yes	characteristic
31	C₂×C₄₇⋊C₁₆	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
33	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
34	#13093960	—	2	2	no	—
35	#13093960	C₂³	1	4	yes	—
36	#13093951	—	2	8	no	—
37	#13093934	D₄	1	2	yes	—
38	C₈×Dic₄₇	C₂³	1	2	yes	—
39	C₂×#41×C₄₇	—	2	2	no	—
40	#135×C₄₇	—	2	2	no	—
41	#135×C₄₇	—	2	2	no	—
42	#135×C₄₇	C₂×C₄	1	4	yes	—
43	#135×C₄₇	C₂³	1	4	yes	—
44	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×SD₁₆×C₄₇	—	2	2	no	—
46	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	Q₃₂×C₄₇	C₂×C₄	1	4	yes	—
49	SD₃₂×C₄₇	C₂×C₄	1	8	yes	—
50	D₁₆×C₄₇	C₂×C₄	1	4	yes	—
51	M₃₂×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
53	#41×C₄₇	—	2	4	no	—
54	#41×C₄₇	—	2	4	no	—
55	#41×C₄₇	—	4	2	no	—
56	#41×C₄₇	—	4	2	no	—
57	C₂×Q₈×C₄₇	—	2	2	no	—
58	C₂×D₄×C₄₇	—	2	2	no	—
59	C₂×D₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₄₇	—	2	2	no	—
63	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	—
64	SD₁₆×C₄₇	—	2	4	no	—
65	SD₁₆×C₄₇	—	2	4	no	—
66	D₈×C₄₇	—	2	2	no	—
67	D₈×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	#41	1	2	yes	—
71	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₂²⋊Dic₄₇	—	2	1	no	—
73	C₄⋊Dic₄₇	—	2	1	no	—
74	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
75	#10625329	—	2	4	no	—
76	#10625329	—	4	2	no	—
77	#10625329	—	4	8	no	—
78	#10625329	D₈	1	4	yes	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	4	8	no	—
81	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
82	C₁₆×C₄₇	C₂²×C₄	1	4	yes	—
83	C₄₇⋊C₁₆	—	2	1	no	—
84	C₄₇⋊C₁₆	—	2	1	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	#135	1	1	yes	characteristic
87	Q₈×C₄₇	—	2	4	no	—
88	Q₈×C₄₇	—	4	2	no	—
89	D₄×C₄₇	—	2	4	no	—
90	D₄×C₄₇	—	4	4	no	—
91	D₄×C₄₇	—	4	4	no	—
92	D₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₄₇	—	2	2	no	—
101	C₈×C₄₇	C₄×D₄	1	2	yes	—
102	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
104	C₄₇⋊C₈	—	4	4	no	—
105	C₄₇⋊C₈	C₂×D₈	1	4	yes	—
106	#33464	—	47	1	no	maximal, Sylow
107	C₂²×C₄₇	—	4	1	no	—
108	C₂²×C₄₇	—	4	4	no	—
109	C₂²×C₄₇	—	16	2	no	—
110	C₂²×C₄₇	C₄×D₈	1	2	yes	—
111	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
112	C₄×C₄₇	—	4	4	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	2	no	—
117	#3414	D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
124	C₂×C₄₇	—	2	1	no	—
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	—	8	4	no	—
127	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
128	C₂×#135	C₂×D₄₇	1	2	yes	—
129	#568	—	94	2	no	—
130	#510	Dic₄₇	1	4	yes	complemented
131	#509	Dic₄₇	1	4	yes	complemented
132	#508	C₂×D₄₇	1	4	yes	—
133	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
134	C₂²×C₁₆	—	47	1	no	—
135	#444	—	94	4	no	—
136	#443	—	47	8	no	—
137	C₂×#108	—	47	2	no	—
138	#421	—	94	2	no	—
139	#405	—	47	1	no	—
140	#368	—	47	4	no	—
141	#359	—	47	4	no	—
142	#359	—	47	4	no	—
143	#346	—	47	2	no	—
144	C₄×C₁₆	—	47	2	no	—
145	C₄₇	#33464	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	2	2	no	—
147	#135	—	2	2	no	—
148	#135	—	2	2	no	—
149	#135	C₂×Dic₄₇	1	4	yes	—
150	#135	C₂²×D₄₇	1	4	yes	—
151	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×D₈	C₂×Dic₄₇	1	2	yes	—
154	#131	—	94	8	no	—
155	#131	—	188	4	no	—
156	C₂×M₁₆	—	47	1	no	—
157	C₂×M₁₆	—	94	2	no	—
158	C₂×M₁₆	—	188	4	no	—
159	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
160	#117	—	47	1	no	—
161	Q₃₂	C₂×Dic₄₇	1	4	yes	—
162	SD₃₂	C₂×Dic₄₇	1	8	yes	—
163	D₁₆	C₂×Dic₄₇	1	4	yes	—
164	M₃₂	C₂×Dic₄₇	1	4	yes	—
165	C₂×C₁₆	—	47	1	no	—
166	C₂×C₁₆	—	47	1	no	—
167	C₂×C₁₆	—	94	2	no	—
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	—	47	2	no	—
173	C₄×C₈	—	47	2	no	—
174	#41	—	2	4	no	—
175	#41	—	2	4	no	—
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
183	Q₁₆	—	2	2	no	—
184	Q₁₆	C₂²×Dic₄₇	1	4	yes	—
185	SD₁₆	—	2	4	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	2	no	—
188	D₈	C₂²×Dic₄₇	1	4	yes	—
189	M₁₆	—	47	4	no	—
190	M₁₆	—	94	4	no	—
191	M₁₆	—	188	8	no	—
192	M₁₆	—	188	2	no	—
193	C₂×C₈	—	47	2	no	—
194	C₂×C₈	—	188	8	no	—
195	C₂×C₈	—	188	2	no	—
196	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
197	C₂×C₈	#10625352	1	2	yes	—
198	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
199	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
200	C₄⋊C₄	—	94	1	no	—
201	C₂²⋊C₄	—	94	1	no	—
202	C₄²	—	47	2	no	—
203	C₁₆	—	94	1	no	—
204	C₁₆	—	94	1	no	—
205	C₁₆	C₂²×Dic₄₇	1	4	yes	—
206	C₂³	—	4	2	no	—
207	C₂³	#13094043	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	94	2	no	—
218	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
219	C₂×C₄	D₈×D₄₇	1	2	yes	—
220	C₂×C₄	#13094043	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
222	C₈	—	47	4	no	—
223	C₈	—	188	4	no	—
224	C₈	D₄×Dic₄₇	1	2	yes	—
225	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
226	C₈	C₂×#10625352	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	16	2	no	—
230	C₂²	D₈×Dic₄₇	1	2	yes	—
231	C₂²	C₂×#13094043	1	1	yes	characteristic
232	C₄	—	4	4	no	—
233	C₄	—	188	2	no	—
234	C₄	D₈×Dic₄₇	1	2	yes	—
235	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
236	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	8	4	no	—
240	C₂	C₂×D₈×Dic₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3414	C₂
socle	C₂×C₄₇	C₂×C₄×D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#3414	D₄₇
hypocenter	C₄₇	#33464
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #3414

Complements

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

N	H ≅ G/N	type
1. #20252305	C₂	semidirect
2. #20245274	C₂	semidirect
3. #20245269	C₂	semidirect
4. #20245183	C₂	semidirect
5. C₂×#15688930	C₂	semidirect
6. #20244987	C₂	semidirect
11. #15689240	C₂²	semidirect
15. #15688953	C₂²	semidirect
16. #15688930	C₂²	semidirect
17. #15688930	C₂²	semidirect
18. #15688898	C₂²	semidirect
19. C₁₆×Dic₄₇	C₂²	semidirect
21. C₄₇×#510	C₄	semidirect
22. C₄₇×#509	C₄	semidirect
130. #510	Dic₄₇	semidirect
131. #509	Dic₄₇	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
#20252305	2
#20245274	2
#20245269	2
#20245183	2
C₂×#15688930	2
#20244987	2
C₄₇×#3414	2
#33464	47