Subgroups — #26369365

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245195	C₂	1	8	yes	maximal, complemented
2	#20245144	C₂	1	2	yes	maximal, complemented
3	#20244999	C₂	1	2	yes	maximal
4	C₄×#13093964	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	4	yes	—
7	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	8	yes	complemented
8	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	8	yes	—
9	#15688899	C₂²	1	4	yes	complemented
10	C₄×C₄₇⋊C₁₆	C₂²	1	4	yes	complemented
11	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
14	C₄₇×#405	C₄	1	4	yes	—
15	C₄₇×#405	C₂²	1	4	yes	—
16	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
17	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	#13093964	—	2	8	no	—
21	C₂×C₄₇⋊C₁₆	C₂³	1	8	yes	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₄₇	—	2	4	no	—
24	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₄₇	—	2	4	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	#117×C₄₇	C₈	1	4	yes	—
33	#117×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₄₇	C₈	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	#97×C₄₇	C₂×C₄	1	4	yes	—
39	#97×C₄₇	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₄₇	C₂³	1	2	yes	—
43	C₄×C₈×C₄₇	C₂³	1	2	yes	—
44	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₄₇	—	2	8	no	—
54	C₂×C₈×C₄₇	—	2	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	2	4	no	—
57	C₂×C₈×C₄₇	—	2	4	no	—
58	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
59	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₄₇	#41	1	4	yes	—
64	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
66	C₂²⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
67	C₄²×C₄₇	C₂×C₈	1	2	yes	—
68	C₄²×C₄₇	C₂×C₈	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
71	C₄₇⋊C₁₆	—	2	8	no	—
72	C₂³×C₄₇	—	2	2	no	—
73	C₂³×C₄₇	—	4	2	no	—
74	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
75	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	2	no	—
82	C₂×C₄×C₄₇	#117	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₄₇	—	2	2	no	—
90	C₈×C₄₇	—	2	2	no	—
91	C₈×C₄₇	#117	1	2	yes	—
92	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
93	C₈×C₄₇	C₂×#41	1	2	yes	—
94	#33419	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	—	2	2	no	—
96	C₂²×C₄₇	—	2	4	no	—
97	C₂²×C₄₇	—	2	2	no	—
98	C₂²×C₄₇	—	4	4	no	—
99	C₂²×C₄₇	#431	1	2	yes	—
100	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
101	C₄×C₄₇	—	2	4	no	—
102	C₄×C₄₇	—	2	4	no	—
103	C₄×C₄₇	—	2	2	no	—
104	C₄×C₄₇	#431	1	2	yes	—
105	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
106	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
107	C₂×#405	D₄₇	1	1	yes	characteristic
108	#2164	—	47	8	no	—
109	#2150	—	47	2	no	—
110	#2112	—	47	2	no	—
111	C₄×M₃₂	—	47	2	no	—
112	C₂×C₄₇	—	2	2	no	—
113	C₂×C₄₇	—	4	2	no	—
114	C₂×C₄₇	C₂×#431	1	1	yes	characteristic
115	C₂×C₄₇	#3407	1	2	yes	—
116	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
117	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
118	C₂×#117	Dic₄₇	1	1	yes	characteristic
119	C₂×M₃₂	—	47	4	no	—
120	#405	Dic₄₇	1	4	yes	—
121	#405	C₂×D₄₇	1	4	yes	—
122	C₂×#97	Dic₄₇	1	1	yes	characteristic
123	C₂×#97	Dic₄₇	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
126	C₄⋊C₁₆	—	47	8	no	—
127	C₂²⋊C₁₆	—	47	8	no	—
128	#346	—	47	4	no	—
129	C₄×C₁₆	—	47	4	no	—
130	C₄₇	#33419	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
136	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
137	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
138	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
140	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
141	#117	C₄₇⋊C₈	1	4	yes	—
142	#117	C₂×Dic₄₇	1	4	yes	—
143	C₂×C₄⋊C₄	C₄₇⋊C₈	1	2	yes	—
144	C₂×C₂²⋊C₄	C₄₇⋊C₈	1	2	yes	—
145	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
147	M₃₂	—	94	8	no	—
148	C₂×C₁₆	—	47	8	no	—
149	#97	C₂×Dic₄₇	1	4	yes	—
150	#97	C₂×Dic₄₇	1	4	yes	—
151	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
152	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
153	C₄×C₈	C₂²×D₄₇	1	2	yes	—
154	C₄×C₈	C₂²×D₄₇	1	2	yes	—
155	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
160	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
161	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
162	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
163	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	#10625329	1	4	yes	—
170	C₂×C₈	#10625350	1	4	yes	—
171	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
172	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
173	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
175	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₄₇⋊C₈	1	8	yes	—
177	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	8	yes	—
178	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
179	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
180	C₄²	C₂²×Dic₄₇	1	2	yes	—
181	C₄²	C₂²×Dic₄₇	1	2	yes	—
182	C₁₆	—	94	8	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
186	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	#13093983	1	2	yes	—
194	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
195	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
196	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
197	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#10625329	1	2	yes	—
199	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	#13093983	1	2	yes	—
203	C₈	C₂×#10625329	1	4	yes	—
204	C₈	C₂×#10625350	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#15689176	1	2	yes	—
210	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	4	no	—
214	C₄	#15689176	1	2	yes	—
215	C₄	C₂×#13093983	1	1	yes	characteristic
216	C₄	C₂²×#10625329	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×#15689176	1	1	yes	characteristic
220	C₂	#20256491	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₂×C₄₇×#405	C₂
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#33419
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #20245195	C₂	semidirect
2. #20245144	C₂	semidirect
4. C₄×#13093964	C₂	semidirect
7. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
9. #15688899	C₂²	semidirect
10. C₄×C₄₇⋊C₁₆	C₂²	semidirect
130. C₄₇	#33419	semidirect

Maximal subgroups

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

subgroup	index
#20245195	2
#20245144	2
#20244999	2
C₄×#13093964	2
C₂×C₄₇×#405	2
#33419	47