Subgroups — #26369525

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256490	C₂	1	2	yes	maximal, complemented
2	#20252292	C₂	1	1	yes	maximal, characteristic
3	#20245229	C₂	1	4	yes	maximal, complemented
4	#20245224	C₂	1	4	yes	maximal, complemented
5	C₂×#15688908	C₂	1	2	yes	maximal, complemented
6	#20244985	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689923	C₄	1	1	yes	characteristic
9	C₂×#13094030	C₂²	1	2	yes	—
10	#15689541	C₂²	1	8	yes	complemented
11	#15689539	—	2	4	no	—
12	C₂×#13093964	C₂²	1	1	yes	characteristic
13	M₃₂×D₄₇	—	2	4	no	—
14	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	#15689246	C₄	1	4	yes	—
16	#15689241	C₂²	1	4	yes	—
17	#15689172	C₄	1	1	yes	characteristic
18	#15689141	C₄	1	2	yes	—
19	#15688908	C₂²	1	4	yes	complemented
20	#15688908	C₂²	1	4	yes	complemented
21	#15688899	C₂²	1	2	yes	complemented
22	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
24	C₄₇×#405	C₂²	1	1	yes	characteristic
25	C₄₇×#364	C₂²	1	4	yes	complemented
26	C₂×#10625350	C₂×C₄	1	2	yes	—
27	#13094057	C₈	1	4	yes	—
28	#13094052	C₈	1	4	yes	—
29	#13094046	C₂×C₄	1	4	yes	—
30	#13094030	—	2	2	no	—
31	#13094030	—	2	2	no	—
32	#13094030	C₂×C₄	1	4	yes	—
33	#13094030	C₂³	1	4	yes	—
34	C₂×#10625329	C₂×C₄	1	2	yes	—
35	C₂×#10625324	—	2	2	no	—
36	C₂×C₈×D₄₇	—	2	2	no	—
37	#13093964	D₄	1	4	yes	—
38	#13093963	—	2	8	no	—
39	C₂×C₄₇⋊C₁₆	—	2	2	no	—
40	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
43	C₁₆×D₄₇	—	2	8	no	—
44	#13093960	C₂×C₄	1	4	yes	—
45	#13093952	C₂×C₄	1	8	yes	—
46	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
49	M₃₂×C₄₇	—	2	2	no	—
50	M₃₂×C₄₇	C₂³	1	4	yes	—
51	C₂×C₁₆×C₄₇	—	2	2	no	—
52	#97×C₄₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₄₇	—	2	2	no	—
56	#10625350	—	2	2	no	—
57	#10625350	—	2	2	no	—
58	#10625350	C₂×C₈	1	4	yes	—
59	#10625350	C₂²×C₄	1	4	yes	—
60	C₂×D₁₈₈	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₄₇	—	2	2	no	—
62	C₂×Dic₉₄	C₂×C₈	1	2	yes	—
63	M₁₆×C₄₇	C₂×C₈	1	4	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₄₇	#41	1	2	yes	—
68	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₄₇	—	2	1	no	—
70	C₂²⋊C₄×C₄₇	—	2	1	no	—
71	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
72	#10625329	—	2	2	no	—
73	#10625329	C₂²×C₄	1	4	yes	—
74	C₂×C₄₇⋊C₈	—	2	2	no	—
75	Dic₁₈₈	C₂×C₈	1	4	yes	—
76	D₃₇₆	C₂×C₈	1	4	yes	—
77	#10625325	C₂×C₈	1	8	yes	—
78	#10625324	—	2	4	no	—
79	#10625324	—	2	4	no	—
80	C₈×D₄₇	—	2	4	no	—
81	C₈×D₄₇	—	2	4	no	—
82	C₁₆×C₄₇	—	2	4	no	—
83	C₄₇⋊C₁₆	—	2	1	no	—
84	C₄₇⋊C₁₆	—	2	1	no	—
85	C₄₇⋊C₁₆	C₂×D₄	1	4	yes	—
86	C₂³×C₄₇	#131	1	1	yes	characteristic
87	C₂²×D₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂²⋊D₄₇	—	2	4	no	—
94	C₂²⋊D₄₇	—	2	4	no	—
95	C₂×Dic₄₇	—	2	2	no	—
96	D₁₈₈	—	2	2	no	—
97	D₁₈₈	C₂²×C₈	1	4	yes	—
98	C₄×D₄₇	—	2	4	no	—
99	C₄×D₄₇	—	2	4	no	—
100	Dic₉₄	—	2	2	no	—
101	Dic₉₄	C₂²×C₈	1	4	yes	—
102	C₈×C₄₇	C₄×D₄	1	2	yes	—
103	C₈×C₄₇	C₂²×C₈	1	4	yes	—
104	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₄₇⋊C₈	—	2	4	no	—
107	#33442	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	4	1	no	—
109	C₂²×C₄₇	C₈×D₄	1	2	yes	—
110	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
111	C₂×D₄₇	—	2	4	no	—
112	C₂×D₄₇	—	8	2	no	—
113	C₄×C₄₇	—	4	2	no	—
114	C₄×C₄₇	C₈×D₄	1	2	yes	—
115	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
116	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
117	Dic₄₇	—	2	4	no	—
118	#3407	—	47	2	no	—
119	#2955	—	47	1	no	—
120	#2171	—	47	4	no	—
121	#2170	—	47	4	no	—
122	#2153	D₄₇	1	1	yes	characteristic, complemented
123	C₂×#350	—	47	2	no	—
124	#2108	—	47	1	no	—
125	C₂×C₄₇	—	2	1	no	—
126	C₂×C₄₇	—	2	2	no	—
127	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
128	D₄₇	—	4	4	no	—
129	#575	—	47	1	no	—
130	C₂×#131	—	47	2	no	—
131	#504	—	47	8	no	—
132	#504	—	94	4	no	—
133	C₂×M₃₂	—	47	1	no	—
134	C₂×M₃₂	—	94	4	no	—
135	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
136	C₂²×C₁₆	—	47	1	no	—
137	#444	—	47	4	no	—
138	#443	—	47	4	no	—
139	#430	—	47	1	no	—
140	#421	—	47	2	no	—
141	#405	C₂×D₄₇	1	1	yes	characteristic
142	#364	C₂×D₄₇	1	4	yes	complemented
143	#350	—	47	4	no	—
144	#350	—	47	4	no	—
145	#346	—	47	2	no	—
146	C₄×C₁₆	—	47	2	no	—
147	C₄₇	#33442	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	47	2	no	—
149	#137	—	47	4	no	—
150	AGL(1,ℤ/8)	—	47	4	no	—
151	#135	—	47	4	no	—
152	#131	—	47	4	no	—
153	#131	—	47	4	no	—
154	#131	—	94	2	no	—
155	#131	—	94	2	no	—
156	C₂×M₁₆	—	47	2	no	—
157	C₂×M₁₆	—	94	2	no	—
158	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
159	C₂²×C₈	—	94	2	no	—
160	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
161	#117	C₄×D₄₇	1	1	yes	characteristic
162	M₃₂	—	2	2	no	—
163	M₃₂	—	47	4	no	—
164	M₃₂	—	94	8	no	—
165	M₃₂	C₂²×D₄₇	1	4	yes	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	47	2	no	—
168	C₂×C₁₆	—	47	1	no	—
169	C₂×C₁₆	—	47	1	no	—
170	C₂×C₁₆	—	94	8	no	—
171	C₂×C₁₆	—	94	2	no	—
172	#108	—	47	4	no	—
173	#104	—	47	8	no	—
174	#97	C₄×D₄₇	1	2	yes	—
175	C₄×C₈	C₂²×D₄₇	1	2	yes	—
176	#41	—	47	4	no	—
177	#41	—	47	4	no	—
178	#41	—	94	2	no	—
179	#41	—	94	2	no	—
180	C₂×Q₈	—	47	2	no	—
181	C₂×D₄	—	47	2	no	—
182	C₂×D₄	—	94	2	no	—
183	C₂²×C₄	—	94	2	no	—
184	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
185	Q₁₆	—	47	4	no	—
186	SD₁₆	—	47	8	no	—
187	D₈	—	47	4	no	—
188	M₁₆	—	47	4	no	—
189	M₁₆	—	94	2	no	—
190	M₁₆	—	94	4	no	—
191	M₁₆	—	94	4	no	—
192	M₁₆	C₈×D₄₇	1	4	yes	—
193	C₂×C₈	—	94	2	no	—
194	C₂×C₈	—	94	4	no	—
195	C₂×C₈	—	94	4	no	—
196	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
198	C₂×C₈	D₄×D₄₇	1	2	yes	—
199	C₂×C₈	#10625354	1	2	yes	—
200	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₄²	C₂×C₄×D₄₇	1	2	yes	—
204	C₁₆	—	2	4	no	—
205	C₁₆	—	47	4	no	—
206	C₁₆	—	94	1	no	—
207	C₁₆	—	94	1	no	—
208	C₂³	—	94	2	no	—
209	C₂³	#13094032	1	1	yes	characteristic
210	Q₈	—	47	4	no	—
211	Q₈	—	94	2	no	—
212	D₄	—	47	4	no	—
213	D₄	—	94	4	no	—
214	D₄	—	94	4	no	—
215	D₄	—	94	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	94	2	no	—
218	C₂×C₄	—	94	4	no	—
219	C₂×C₄	—	94	4	no	—
220	C₂×C₄	#13093986	1	2	yes	—
221	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
222	C₂×C₄	#13094032	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
224	C₈	—	94	4	no	—
225	C₈	#13093986	1	2	yes	—
226	C₈	C₂×C₈×D₄₇	1	4	yes	—
227	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
228	C₈	C₂×#10625354	1	1	yes	characteristic
229	C₂²	—	4	1	no	—
230	C₂²	—	94	4	no	—
231	C₂²	—	376	2	no	—
232	C₂²	#15689188	1	2	yes	—
233	C₂²	C₂×#13094032	1	1	yes	characteristic
234	C₄	—	4	2	no	—
235	C₄	—	94	4	no	—
236	C₄	#15689188	1	2	yes	—
237	C₄	C₂×#13093986	1	1	yes	characteristic
238	C₄	C₂²×C₈×D₄₇	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	188	4	no	—
242	C₂	C₂×#15689188	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2153	C₂
socle	C₂×C₄₇	C₂×C₈×D₄
prime radical	#15689923	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2153	D₄₇
hypocenter	C₄₇	#33442
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
1. #20256490	C₂	semidirect
3. #20245229	C₂	semidirect
4. #20245224	C₂	semidirect
5. C₂×#15688908	C₂	semidirect
6. #20244985	C₂	semidirect
7. C₄₇×#2153	C₂	semidirect
10. #15689541	C₂²	semidirect
19. #15688908	C₂²	semidirect
20. #15688908	C₂²	semidirect
21. #15688899	C₂²	semidirect
22. C₄×C₄₇⋊C₁₆	C₂²	semidirect
25. C₄₇×#364	C₂²	semidirect
122. #2153	D₄₇	semidirect
142. #364	C₂×D₄₇	semidirect
147. C₄₇	#33442	semidirect

Maximal subgroups

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

subgroup	index
#20256490	2
#20252292	2
#20245229	2
#20245224	2
C₂×#15688908	2
#20244985	2
C₄₇×#2153	2
#33442	47