Subgroups — #26115657

Order 12032.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243552	C₂	1	1	yes	maximal, characteristic, complemented
2	#20241948	C₂	1	4	yes	maximal, complemented
3	#20241671	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic
5	#15689176	C₂²	1	2	yes	—
6	C₂×#13093958	—	2	1	no	—
7	C₂×#13093956	—	2	1	no	—
8	#15688989	C₂²	1	2	yes	—
9	#15688857	—	2	8	no	—
10	#15688844	—	2	1	no	—
11	C₈×C₄₇⋊C₈	—	2	1	no	—
12	C₂×#135×C₄₇	—	2	1	no	—
13	C₂×#135×C₄₇	—	2	1	no	—
14	C₄₇×#532	C₂²	1	2	yes	—
15	C₄₇×#495	—	2	1	no	—
16	C₄₇×#495	—	2	1	no	—
17	C₄₇×#494	C₄	1	1	yes	characteristic, complemented
18	C₄₇×#493	C₄	1	1	yes	characteristic, complemented
19	C₄₇×#492	C₄	1	2	yes	complemented
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	#13093958	—	4	2	no	—
23	#13093956	—	4	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄₇⋊#95	—	2	4	no	—
28	C₂×#41×C₄₇	—	2	2	no	—
29	#135×C₄₇	—	4	2	no	—
30	#135×C₄₇	—	4	2	no	—
31	#135×C₄₇	—	4	2	no	—
32	#135×C₄₇	—	4	2	no	—
33	C₂×Q₁₆×C₄₇	—	2	1	no	—
34	C₂×Q₁₆×C₄₇	—	2	1	no	—
35	C₂×SD₁₆×C₄₇	—	2	2	no	—
36	C₂×SD₁₆×C₄₇	—	2	2	no	—
37	C₂×D₈×C₄₇	—	2	1	no	—
38	C₂×D₈×C₄₇	—	2	1	no	—
39	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
42	#127×C₄₇	C₂×C₄	1	2	yes	—
43	#124×C₄₇	—	2	2	no	—
44	#121×C₄₇	—	2	4	no	—
45	C₄×D₄×C₄₇	—	2	4	no	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
51	#41×C₄₇	—	4	2	no	—
52	#41×C₄₇	—	4	2	no	—
53	#41×C₄₇	—	4	2	no	—
54	#41×C₄₇	—	4	2	no	—
55	C₂×Q₈×C₄₇	—	2	2	no	—
56	C₂×D₄×C₄₇	—	2	2	no	—
57	C₂×D₄×C₄₇	—	2	4	no	—
58	C₂×D₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	4	2	no	—
63	Q₁₆×C₄₇	—	4	2	no	—
64	Q₁₆×C₄₇	—	4	2	no	—
65	SD₁₆×C₄₇	—	4	4	no	—
66	SD₁₆×C₄₇	—	4	4	no	—
67	D₈×C₄₇	—	4	2	no	—
68	D₈×C₄₇	—	4	2	no	—
69	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	D₈	1	2	yes	—
74	C₂×C₈×C₄₇	D₈	1	2	yes	—
75	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
76	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₂²⋊C₄×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₇	#41	1	2	yes	—
82	C₄⋊Dic₄₇	—	4	2	no	—
83	C₄⋊Dic₄₇	—	4	2	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	#127	1	1	yes	characteristic
87	Q₈×C₄₇	—	4	2	no	—
88	Q₈×C₄₇	—	4	2	no	—
89	D₄×C₄₇	—	4	2	no	—
90	D₄×C₄₇	—	4	2	no	—
91	D₄×C₄₇	—	8	4	no	—
92	D₄×C₄₇	—	8	2	no	—
93	D₄×C₄₇	—	8	2	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	8	2	no	—
97	C₂×C₄×C₄₇	—	8	2	no	—
98	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₄₇	#121	1	2	yes	—
101	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×Dic₄₇	—	8	4	no	—
107	C₂×Dic₄₇	—	8	2	no	—
108	C₈×C₄₇	#102	1	2	yes	—
109	C₈×C₄₇	#102	1	2	yes	—
110	C₈×C₄₇	#102	1	2	yes	—
111	C₈×C₄₇	#102	1	2	yes	—
112	C₄₇⋊C₈	—	4	4	no	—
113	#9756	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	8	2	no	—
116	C₂²×C₄₇	—	8	4	no	—
117	C₂²×C₄₇	#390	1	1	yes	characteristic
118	C₂²×C₄₇	#492	1	1	yes	characteristic
119	C₂²×C₄₇	#493	1	1	yes	characteristic
120	C₄×C₄₇	—	2	2	no	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	8	2	no	—
123	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
124	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
125	C₄×C₄₇	#486	1	1	yes	characteristic
126	C₄×C₄₇	#486	1	1	yes	characteristic
127	Dic₄₇	—	16	2	no	—
128	#3147	D₄₇	1	1	yes	characteristic
129	#1846	—	47	1	no	—
130	#1517	—	47	4	no	—
131	#1450	—	47	1	no	—
132	C₂×C₄₇	—	2	1	no	—
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	16	2	no	—
135	C₂×C₄₇	#1957	1	1	yes	characteristic
136	C₂×C₄₇	#1976	1	2	yes	—
137	C₂×#135	—	2	1	no	—
138	C₂×#135	—	2	1	no	—
139	#532	C₂×D₄₇	1	2	yes	—
140	#495	—	2	1	no	—
141	#495	—	2	1	no	—
142	#494	Dic₄₇	1	1	yes	characteristic, complemented
143	#493	Dic₄₇	1	1	yes	characteristic, complemented
144	#492	Dic₄₇	1	2	yes	complemented
145	#431	—	47	2	no	—
146	C₂×#107	—	94	1	no	—
147	C₂×#106	—	94	1	no	—
148	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
149	#382	—	47	2	no	—
150	#327	—	94	8	no	—
151	#322	—	94	1	no	—
152	C₈²	—	94	1	no	—
153	C₄₇	#9756	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	—	2	2	no	—
155	#135	—	4	2	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	#135	—	4	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
167	C₄⋊Q₈	C₂×Dic₄₇	1	2	yes	—
168	#127	C₂×Dic₄₇	1	2	yes	—
169	#124	—	2	2	no	—
170	#121	—	2	4	no	—
171	C₄×D₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	94	2	no	—
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	#107	—	188	2	no	—
175	#106	—	188	2	no	—
176	C₄⋊C₈	—	94	2	no	—
177	C₂²⋊C₈	—	47	4	no	—
178	C₄×C₈	—	94	2	no	—
179	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
180	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
183	#95	—	94	4	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	188	2	no	—
194	C₂²×C₄	D₄×D₄₇	1	2	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
196	Q₁₆	—	4	2	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	2	no	—
202	C₂×C₈	—	94	4	no	—
203	C₂×C₈	#10625334	1	2	yes	—
204	C₂×C₈	#10625334	1	2	yes	—
205	C₂×C₈	#10625335	1	2	yes	—
206	C₂×C₈	#10625335	1	2	yes	—
207	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
209	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	188	2	no	—
213	C₄⋊C₄	—	188	2	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	#10625352	1	2	yes	—
216	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂³	—	4	2	no	—
219	C₂³	#13094020	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	4	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	376	4	no	—
232	C₂×C₄	—	376	2	no	—
233	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
235	C₂×C₄	#13093995	1	2	yes	—
236	C₂×C₄	#13094011	1	1	yes	characteristic
237	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
238	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
239	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
240	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
241	C₈	—	188	4	no	—
242	C₈	#13093945	1	2	yes	—
243	C₈	#13093945	1	2	yes	—
244	C₈	#13093945	1	2	yes	—
245	C₈	#13093945	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	#15689026	1	1	yes	characteristic
250	C₂²	#15689495	1	1	yes	characteristic
251	C₂²	#15689499	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	752	2	no	—
256	C₄	C₂×#13093945	1	1	yes	characteristic
257	C₄	C₂×#13093945	1	1	yes	characteristic
258	C₄	#15689469	1	1	yes	characteristic
259	C₄	#15689469	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	#20244195	1	1	yes	characteristic
264	C₂	#20244287	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689026
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting, prime radical	C₄₇×#3147	C₂
socle	C₂²×C₄₇	#390
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3147	D₄₇
hypocenter	C₄₇	#9756
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #20243552	C₂	semidirect
2. #20241948	C₂	semidirect
3. #20241671	C₂	semidirect
17. C₄₇×#494	C₄	semidirect
18. C₄₇×#493	C₄	semidirect
19. C₄₇×#492	C₄	semidirect
142. #494	Dic₄₇	semidirect
143. #493	Dic₄₇	semidirect
144. #492	Dic₄₇	semidirect
153. C₄₇	#9756	semidirect

Maximal subgroups

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

subgroup	index
#20243552	2
#20241948	2
#20241671	2
C₄₇×#3147	2
#9756	47