Subgroups — #26181999

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252355	C₂	1	2	yes	maximal
2	#20241691	C₂	1	2	yes	maximal, complemented
3	#20241685	C₂	1	2	yes	maximal, complemented
4	#20241675	C₂	1	2	yes	maximal, complemented
5	#20241671	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#10625329	C₂	1	4	yes	maximal, complemented
7	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689254	C₄	1	4	yes	—
10	#15689201	C₄	1	4	yes	—
11	#15689195	C₄	1	8	yes	—
12	#15689176	C₂²	1	4	yes	—
13	#15689176	C₂²	1	2	yes	—
14	C₄×#10625329	C₂²	1	4	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#15688845	C₂²	1	4	yes	—
17	#15688845	C₂²	1	4	yes	—
18	#15688844	C₂²	1	2	yes	—
19	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
21	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×#131×C₄₇	C₄	1	2	yes	—
24	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#405	C₄	1	4	yes	—
26	C₄₇×#405	C₄	1	2	yes	—
27	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#10625329	C₂×C₄	1	8	yes	—
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
33	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
35	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
36	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
37	#13093935	C₂×C₄	1	8	yes	—
38	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
39	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
40	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
42	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
43	#131×C₄₇	—	2	8	no	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
50	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
51	#117×C₄₇	C₂×C₄	1	2	yes	—
52	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	#97×C₄₇	C₂×C₄	1	2	yes	—
56	#97×C₄₇	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₄₇	C₂³	1	2	yes	—
61	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	#41×C₄₇	—	2	4	no	—
63	C₂×Q₈×C₄₇	C₄²	1	1	yes	characteristic
64	C₂×D₄×C₄₇	C₄²	1	2	yes	—
65	C₂×D₄×C₄₇	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₄₇	—	2	4	no	—
71	M₁₆×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	—	2	8	no	—
73	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
74	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
78	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
79	C₂²⋊C₄×C₄₇	C₄²	1	4	yes	—
80	C₂²⋊C₄×C₄₇	C₄²	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
86	#10625329	—	2	16	no	—
87	C₂×C₄₇⋊C₈	—	2	8	no	—
88	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
89	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
90	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
91	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
92	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
93	Q₈×C₄₇	—	2	2	no	—
94	D₄×C₄₇	—	2	4	no	—
95	D₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₄₇	—	2	4	no	—
107	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
108	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
109	C₄₇⋊C₈	—	2	4	no	—
110	C₄₇⋊C₈	—	2	8	no	—
111	C₄₇⋊C₈	#131	1	8	yes	complemented
112	#17335	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
117	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	2	4	no	—
121	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
122	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
123	C₄×C₄₇	C₂×#131	1	4	yes	—
124	#2965	—	47	2	no	—
125	C₄×#131	D₄₇	1	1	yes	characteristic
126	#1455	—	47	2	no	—
127	#1453	—	47	2	no	—
128	#1451	—	47	2	no	—
129	#1450	—	47	1	no	—
130	C₈×M₁₆	—	47	4	no	—
131	C₂×C₈²	—	47	1	no	—
132	C₂×C₄₇	—	2	2	no	—
133	C₂×C₄₇	—	4	2	no	—
134	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
135	C₂×C₄₇	C₄×#131	1	2	yes	—
136	C₂×#131	Dic₄₇	1	2	yes	—
137	C₄×#41	C₂×D₄₇	1	1	yes	characteristic
138	#446	—	47	4	no	—
139	#436	—	47	4	no	—
140	#435	—	47	8	no	—
141	#431	—	47	4	no	—
142	#431	—	47	2	no	—
143	#405	Dic₄₇	1	4	yes	—
144	#405	Dic₄₇	1	2	yes	—
145	C₄×M₁₆	—	47	4	no	—
146	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
147	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
148	C₂×C₄×C₈	—	47	2	no	—
149	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
151	#322	—	47	2	no	—
152	#322	—	47	4	no	—
153	#322	—	47	4	no	—
154	C₈²	—	47	1	no	—
155	C₈²	—	47	1	no	—
156	C₈²	—	47	2	no	—
157	C₈²	—	47	2	no	—
158	C₄₇	#17335	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
160	#131	—	2	8	no	—
161	C₂×M₁₆	—	47	8	no	—
162	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
163	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
164	C₂²×C₈	—	47	4	no	—
165	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
166	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
167	C₄×Q₈	C₄×D₄₇	1	2	yes	—
168	C₄×D₄	C₄×D₄₇	1	4	yes	—
169	C₄×D₄	C₄×D₄₇	1	2	yes	—
170	#117	C₂×Dic₄₇	1	2	yes	—
171	#117	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
174	C₄⋊C₈	—	47	8	no	—
175	C₄⋊C₈	—	47	4	no	—
176	C₂²⋊C₈	—	47	4	no	—
177	C₂²⋊C₈	—	47	8	no	—
178	#97	—	47	8	no	—
179	#97	C₂×Dic₄₇	1	2	yes	—
180	#97	C₂×Dic₄₇	1	4	yes	—
181	C₄×C₈	—	47	4	no	—
182	C₄×C₈	—	47	2	no	—
183	C₄×C₈	—	47	2	no	—
184	C₄×C₈	—	47	4	no	—
185	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
186	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
187	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₄₇	1	2	yes	—
189	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	#41	—	2	4	no	—
191	C₂×Q₈	C₄×Dic₄₇	1	1	yes	characteristic
192	C₂×D₄	C₄×Dic₄₇	1	2	yes	—
193	C₂×D₄	C₄×Dic₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
196	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	8	no	—
200	M₁₆	—	94	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	47	4	no	—
203	C₂×C₈	—	47	4	no	—
204	C₂×C₈	—	47	8	no	—
205	C₂×C₈	—	94	8	no	—
206	C₂×C₈	#10625329	1	8	yes	complemented
207	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
208	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
209	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
210	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
211	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
212	C₂²⋊C₄	C₄×Dic₄₇	1	4	yes	—
213	C₂²⋊C₄	C₄×Dic₄₇	1	2	yes	—
214	C₄²	C₂×C₄×D₄₇	1	2	yes	—
215	C₄²	C₂×C₄×D₄₇	1	4	yes	—
216	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₄²	C₂²×Dic₄₇	1	2	yes	—
218	C₄²	C₂³×D₄₇	1	1	yes	characteristic
219	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
220	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
230	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
231	C₂×C₄	C₂×#10625329	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
233	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
234	C₈	—	2	4	no	—
235	C₈	—	47	8	no	—
236	C₈	—	94	4	no	—
237	C₈	—	94	8	no	—
238	C₈	C₂×#10625329	1	4	yes	—
239	C₈	C₂×#10625329	1	4	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₄×#10625329	1	2	yes	—
244	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	C₄×#10625329	1	2	yes	—
249	C₄	C₂²×#10625329	1	2	yes	—
250	C₄	#131×D₄₇	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₂×C₄×#10625329	1	1	yes	characteristic
254	C₂	#131×Dic₄₇	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄×#131×C₄₇	C₂
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₄×#131	D₄₇
hypocenter	C₄₇	#17335
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #20241691	C₂	semidirect
3. #20241685	C₂	semidirect
4. #20241675	C₂	semidirect
5. #20241671	C₂	semidirect
6. C₈×#10625329	C₂	semidirect
7. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
73. C₂×C₈×C₄₇	M₁₆	semidirect
111. C₄₇⋊C₈	#131	semidirect
158. C₄₇	#17335	semidirect
206. C₂×C₈	#10625329	semidirect

Maximal subgroups

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

subgroup	index
#20252355	2
#20241691	2
#20241685	2
#20241675	2
#20241671	2
C₈×#10625329	2
C₂×C₈×C₄₇⋊C₈	2
C₄×#131×C₄₇	2
#17335	47