Subgroups — #26131374

Order 12032.

410 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245009	C₂	1	1	yes	maximal, characteristic
2	#20244984	C₂	1	2	yes	maximal, complemented
3	#20243188	C₂	1	1	yes	maximal, characteristic
4	#20241473	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	#15689107	C₄	1	1	yes	characteristic
8	#15689097	C₄	1	1	yes	characteristic
9	#15689078	C₂²	1	2	yes	—
10	#15688905	C₄	1	4	yes	complemented
11	#15688898	—	2	2	no	—
12	C₁₆×Dic₄₇	—	2	2	no	—
13	#15688894	C₄	1	2	yes	—
14	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#504	—	2	2	no	—
16	C₄₇×#504	—	2	2	no	—
17	C₂×M₃₂×C₄₇	C₄	1	2	yes	complemented
18	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₄₇	C₄	1	2	yes	complemented
20	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#10625329	C₂×C₄	1	2	yes	—
22	#13093983	C₂×C₄	1	2	yes	—
23	#13093964	—	2	2	no	—
24	#13093964	C₂×C₄	1	4	yes	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	#13093943	C₈	1	2	yes	—
27	#13093941	C₈	1	2	yes	—
28	#13093939	C₈	1	4	yes	—
29	#13093934	—	2	2	no	—
30	C₈×Dic₄₇	—	2	2	no	—
31	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
32	#131×C₄₇	—	2	2	no	—
33	#131×C₄₇	—	2	2	no	—
34	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
38	M₃₂×C₄₇	—	2	2	no	—
39	M₃₂×C₄₇	—	2	4	no	—
40	C₂×C₁₆×C₄₇	—	2	2	no	—
41	C₂×C₁₆×C₄₇	—	2	2	no	—
42	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
46	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
47	#41×C₄₇	—	2	2	no	—
48	#41×C₄₇	—	2	2	no	—
49	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
50	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₄₇	—	2	2	no	—
55	M₁₆×C₄₇	—	2	4	no	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	C₄²	1	2	yes	—
59	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₄₇	#41	1	2	yes	—
63	C₂²⋊Dic₄₇	C₂×C₈	1	4	yes	—
64	C₄⋊Dic₄₇	—	2	2	no	—
65	C₄×Dic₄₇	—	2	2	no	—
66	#10625329	—	2	2	no	—
67	#10625329	C₂×C₈	1	4	yes	—
68	C₂×C₄₇⋊C₈	—	2	2	no	—
69	C₁₆×C₄₇	—	2	2	no	—
70	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₁₆×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄₇⋊C₁₆	—	2	4	no	—
74	C₂³×C₄₇	C₄×C₈	1	2	yes	—
75	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
76	Q₈×C₄₇	—	2	2	no	—
77	D₄×C₄₇	—	2	4	no	—
78	D₄×C₄₇	—	2	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₄₇	#131	1	2	yes	—
86	C₂×Dic₄₇	—	2	4	no	—
87	C₈×C₄₇	—	2	2	no	—
88	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₄₇	#117	1	1	yes	characteristic
90	C₈×C₄₇	C₄×D₄	1	2	yes	—
91	C₄₇⋊C₈	—	2	4	no	—
92	#12613	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	4	1	no	—
95	C₂²×C₄₇	—	4	2	no	—
96	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂²×C₄₇	C₈×D₄	1	2	yes	—
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₄₇	#405	1	1	yes	characteristic
101	C₄×C₄₇	C₈×D₄	1	2	yes	—
102	Dic₄₇	—	4	4	no	—
103	C₂×#504	D₄₇	1	1	yes	characteristic
104	#2115	—	47	1	no	—
105	#2108	—	47	2	no	—
106	#1757	—	47	1	no	—
107	#1378	—	47	2	no	—
108	C₂×C₄₇	—	2	2	no	—
109	C₂×C₄₇	—	2	1	no	—
110	C₂×C₄₇	—	4	2	no	—
111	C₂×C₄₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
113	#504	—	2	2	no	—
114	#504	—	2	2	no	—
115	C₂×M₃₂	—	47	2	no	—
116	C₂×M₃₂	Dic₄₇	1	2	yes	complemented
117	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
118	C₂²×C₁₆	Dic₄₇	1	2	yes	complemented
119	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
120	#413	—	47	1	no	—
121	#410	—	47	1	no	—
122	#405	—	47	2	no	—
123	#349	—	47	4	no	—
124	#346	—	94	2	no	—
125	C₄×C₁₆	—	94	2	no	—
126	#344	—	47	2	no	—
127	C₄₇	#12613	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
129	#131	—	2	2	no	—
130	#131	—	2	2	no	—
131	C₂×M₁₆	—	47	2	no	—
132	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
133	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
134	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
135	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
136	#117	—	47	2	no	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	4	no	—
139	M₃₂	—	47	4	no	—
140	M₃₂	—	94	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	94	2	no	—
144	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
145	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
146	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
148	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
149	#101	—	47	2	no	—
150	#100	—	47	2	no	—
151	#99	—	47	4	no	—
152	#97	—	94	2	no	—
153	C₄×C₈	—	94	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₈×D₄₇	1	1	yes	characteristic
157	C₂×D₄	C₈×D₄₇	1	2	yes	—
158	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
159	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
160	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	47	4	no	—
164	M₁₆	—	94	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	94	2	no	—
168	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
169	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
170	C₂×C₈	#10625350	1	2	yes	—
171	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
173	C₄⋊C₄	—	94	2	no	—
174	C₂²⋊C₄	—	47	4	no	—
175	C₄²	—	94	2	no	—
176	C₁₆	—	2	2	no	—
177	C₁₆	—	94	4	no	—
178	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
179	C₁₆	C₂²⋊Dic₄₇	1	2	yes	—
180	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
181	C₂³	C₈×Dic₄₇	1	2	yes	—
182	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
183	Q₈	—	2	2	no	—
184	D₄	—	2	4	no	—
185	D₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	94	4	no	—
189	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
191	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
192	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
193	C₂×C₄	#13094030	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	94	4	no	—
196	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
197	C₈	#13093983	1	1	yes	characteristic
198	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
203	C₂²	C₈×C₂²⋊D₄₇	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	188	4	no	—
206	C₄	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
207	C₄	#15689078	1	1	yes	characteristic
208	C₄	C₈×C₂²⋊D₄₇	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	C₈×C₂²⋊Dic₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#504	C₂
socle	C₂×C₄₇	C₈×C₂²⋊C₄
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#504	D₄₇
hypocenter	C₄₇	#12613
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #20244984	C₂	semidirect
4. #20241473	C₂	semidirect
10. #15688905	C₄	semidirect
17. C₂×M₃₂×C₄₇	C₄	semidirect
19. C₂²×C₁₆×C₄₇	C₄	semidirect
116. C₂×M₃₂	Dic₄₇	semidirect
118. C₂²×C₁₆	Dic₄₇	semidirect
127. C₄₇	#12613	semidirect

Maximal subgroups

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

subgroup	index
#20245009	2
#20244984	2
#20243188	2
#20241473	2
C₂×C₄₇×#504	2
#12613	47