Subgroups — #26203936

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253943	C₂	1	4	yes	maximal, complemented
2	#20253902	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242987	C₂	1	2	yes	maximal, complemented
4	#20242982	C₂	1	2	yes	maximal, complemented
5	#20242977	C₂	1	2	yes	maximal, complemented
6	#20242974	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic
9	#15689680	C₂²	1	2	yes	—
10	#15689528	C₂²	1	2	yes	—
11	#15689525	—	2	2	no	—
12	#15689521	C₂²	1	2	yes	—
13	#15689519	C₂²	1	4	yes	—
14	#15689495	C₂²	1	4	yes	—
15	#15689475	C₂²	1	4	yes	—
16	#15689469	C₂²	1	4	yes	—
17	#15689350	—	2	8	no	—
18	C₄×#10625335	—	2	8	no	—
19	#15689161	—	2	2	no	—
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
21	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
23	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×#135×C₄₇	—	2	2	no	—
25	C₄₇×#532	C₂²	1	2	yes	—
26	C₄₇×#495	—	2	2	no	—
27	C₄₇×#494	C₂²	1	1	yes	characteristic
28	C₄₇×#493	C₂²	1	1	yes	characteristic
29	C₄₇×#492	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#10625335	—	2	8	no	—
32	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
33	#13094024	C₂³	1	4	yes	—
34	#13094016	—	2	2	no	—
35	#13094003	—	2	4	no	—
36	C₄×Dic₉₄	—	2	4	no	—
37	#13093983	—	2	2	no	—
38	#13093958	—	2	4	no	—
39	#13093956	—	2	4	no	—
40	#13093955	—	2	8	no	—
41	#13093947	—	2	8	no	—
42	#13093945	—	2	8	no	—
43	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
44	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
45	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
46	C₂×#41×C₄₇	—	2	2	no	—
47	#135×C₄₇	—	4	8	no	—
48	C₂×Q₁₆×C₄₇	—	2	2	no	—
49	C₂×SD₁₆×C₄₇	—	2	4	no	—
50	C₂×D₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	D₄	1	2	yes	—
52	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
53	#127×C₄₇	C₂³	1	2	yes	—
54	#124×C₄₇	—	2	2	no	—
55	#121×C₄₇	—	2	4	no	—
56	C₄×D₄×C₄₇	—	2	4	no	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₄₇	D₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	#41×C₄₇	—	4	8	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×Dic₉₄	—	2	4	no	—
70	Q₁₆×C₄₇	—	4	4	no	—
71	SD₁₆×C₄₇	—	4	8	no	—
72	D₈×C₄₇	—	4	4	no	—
73	C₂×C₈×C₄₇	SD₁₆	1	8	yes	complemented
74	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₂²⋊C₄×C₄₇	—	4	4	no	—
78	C₄²×C₄₇	C₂×D₄	1	2	yes	—
79	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²⋊Dic₄₇	—	4	2	no	—
82	#10625335	—	4	16	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	Dic₄₇⋊C₄	—	4	4	no	—
86	C₄×Dic₄₇	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	2	8	no	—
88	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
89	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
90	C₂³×C₄₇	—	4	2	no	—
91	C₂³×C₄₇	#127	1	1	yes	characteristic
92	Q₈×C₄₇	—	4	4	no	—
93	D₄×C₄₇	—	4	4	no	—
94	D₄×C₄₇	—	4	8	no	—
95	D₄×C₄₇	—	8	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	8	4	no	—
99	C₂×C₄×C₄₇	#127	1	2	yes	—
100	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
102	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×Dic₄₇	—	4	4	no	—
105	Dic₉₄	—	4	8	no	—
106	C₈×C₄₇	C₂×SD₁₆	1	4	yes	—
107	C₈×C₄₇	C₂×SD₁₆	1	4	yes	—
108	C₄₇⋊C₈	—	2	4	no	—
109	C₄₇⋊C₈	#135	1	8	yes	complemented
110	#19596	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	8	2	no	—
113	C₂²×C₄₇	—	8	4	no	—
114	C₂²×C₄₇	#492	1	2	yes	—
115	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
116	C₄×C₄₇	—	2	2	no	—
117	C₄×C₄₇	—	8	2	no	—
118	C₄×C₄₇	#492	1	2	yes	—
119	C₄×C₄₇	C₂²×SD₁₆	1	2	yes	—
120	C₄×C₄₇	C₂×#135	1	4	yes	—
121	Dic₄₇	—	8	4	no	—
122	#3167	—	47	4	no	—
123	#3160	—	47	1	no	—
124	#3147	D₄₇	1	1	yes	characteristic
125	#1708	—	47	2	no	—
126	#1707	—	47	2	no	—
127	#1706	—	47	2	no	—
128	#1705	—	47	2	no	—
129	C₂×C₈²	—	47	1	no	—
130	C₂×C₄₇	—	2	2	no	—
131	C₂×C₄₇	—	16	2	no	—
132	C₂×C₄₇	C₂×#492	1	1	yes	characteristic
133	C₂×C₄₇	#3147	1	2	yes	—
134	C₂×#135	—	2	2	no	—
135	#533	—	47	2	no	—
136	#532	C₂×D₄₇	1	2	yes	—
137	#500	—	47	2	no	—
138	#499	—	94	2	no	—
139	#498	—	47	4	no	—
140	#498	—	47	2	no	—
141	#495	—	2	2	no	—
142	#494	C₂×D₄₇	1	1	yes	characteristic
143	#493	C₂×D₄₇	1	1	yes	characteristic
144	#492	—	47	4	no	—
145	#492	C₂×D₄₇	1	2	yes	—
146	#487	—	47	4	no	—
147	#486	—	47	4	no	—
148	#465	—	94	8	no	—
149	C₄×SD₁₆	—	94	8	no	—
150	#427	—	94	2	no	—
151	C₂×C₄×C₈	—	47	2	no	—
152	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₈²	—	47	1	no	—
154	C₈²	—	47	2	no	—
155	C₈²	—	47	1	no	—
156	C₄₇	#19596	1	1	yes	characteristic, complemented, Sylow
157	C₂×#41	—	2	2	no	—
158	#135	—	4	8	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×SD₁₆	—	94	8	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂²×C₈	—	47	4	no	—
164	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
165	C₄⋊Q₈	—	47	4	no	—
166	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
167	#127	C₂²×D₄₇	1	2	yes	—
168	#125	—	94	2	no	—
169	#124	—	2	2	no	—
170	C₂²⋊Q₈	—	94	4	no	—
171	#121	—	2	4	no	—
172	C₄×Q₈	—	94	4	no	—
173	C₄×D₄	—	2	4	no	—
174	#117	—	94	2	no	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	#107	—	94	4	no	—
177	#106	—	94	8	no	—
178	#106	—	94	4	no	—
179	#103	—	94	8	no	—
180	#102	—	94	8	no	—
181	C₄×C₈	—	47	4	no	—
182	C₄×C₈	—	47	2	no	—
183	C₄×C₈	—	47	2	no	—
184	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
185	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
187	#41	—	4	8	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	94	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	D₄×D₄₇	1	2	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
196	Q₁₆	—	4	4	no	—
197	SD₁₆	—	4	8	no	—
198	SD₁₆	—	188	16	no	—
199	D₈	—	4	4	no	—
200	C₂×C₈	—	47	4	no	—
201	C₂×C₈	—	47	4	no	—
202	C₂×C₈	—	94	8	no	—
203	C₂×C₈	#10625335	1	8	yes	complemented
204	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
205	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	94	4	no	—
208	C₄⋊C₄	—	94	4	no	—
209	C₄⋊C₄	—	188	4	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₂²⋊C₄	—	188	2	no	—
212	C₄²	—	188	2	no	—
213	C₄²	D₄×D₄₇	1	2	yes	—
214	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
215	C₄²	C₂³×D₄₇	1	1	yes	characteristic
216	C₂³	—	4	2	no	—
217	C₂³	#13094020	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	188	8	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	—	188	4	no	—
227	C₂×C₄	#13094020	1	2	yes	—
228	C₂×C₄	#13094020	1	1	yes	characteristic
229	C₂×C₄	C₂×#10625335	1	4	yes	—
230	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
231	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₈	—	47	8	no	—
233	C₈	—	94	4	no	—
234	C₈	C₂×#10625335	1	4	yes	—
235	C₈	C₂×#10625335	1	4	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	4	no	—
239	C₂²	#15689495	1	2	yes	—
240	C₂²	C₂×#13094020	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	376	4	no	—
244	C₄	#15689495	1	2	yes	—
245	C₄	C₂²×#10625335	1	2	yes	—
246	C₄	#135×D₄₇	1	4	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	16	2	no	—
249	C₂	C₂×#15689495	1	1	yes	characteristic
250	C₂	#20253812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3147	C₂
socle	C₂²×C₄₇	C₂×#127
hypercenter	#3147	D₄₇
hypocenter	C₄₇	#19596
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #20253943	C₂	semidirect
2. #20253902	C₂	semidirect
3. #20242987	C₂	semidirect
4. #20242982	C₂	semidirect
5. #20242977	C₂	semidirect
6. #20242974	C₂	semidirect
7. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
73. C₂×C₈×C₄₇	SD₁₆	semidirect
109. C₄₇⋊C₈	#135	semidirect
156. C₄₇	#19596	semidirect
203. C₂×C₈	#10625335	semidirect

Maximal subgroups

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

subgroup	index
#20253943	2
#20253902	2
#20242987	2
#20242982	2
#20242977	2
#20242974	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#3147	2
#19596	47