Subgroups — #26131655

Order 12032.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244992	C₂	1	2	yes	maximal
2	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
3	#20243267	C₂	1	1	yes	maximal, characteristic
4	#20241475	C₂	1	2	yes	maximal
5	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	C₂²	1	1	yes	characteristic
7	C₂×#13093964	C₂²	1	1	yes	characteristic
8	C₂×#13093960	C₄	1	1	yes	characteristic
9	#15689161	C₄	1	1	yes	characteristic
10	#15689078	C₂²	1	2	yes	—
11	#15688900	—	2	4	no	—
12	#15688899	C₄	1	2	yes	complemented
13	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	complemented
14	#15688883	C₄	1	2	yes	—
15	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#364	C₄	1	2	yes	complemented
18	C₄₇×#364	C₄	1	2	yes	complemented
19	C₂×#10625329	C₂×C₄	1	2	yes	—
20	#13093983	C₂×C₄	1	2	yes	—
21	#13093964	—	2	1	no	—
22	#13093964	—	2	1	no	—
23	#13093964	D₄	1	2	yes	—
24	#13093964	Q₈	1	2	yes	—
25	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
27	#13093960	—	2	1	no	—
28	#13093960	—	2	1	no	—
29	#13093958	—	2	1	no	—
30	#13093956	—	2	1	no	—
31	#13093934	—	2	2	no	—
32	C₈×Dic₄₇	—	2	2	no	—
33	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₄₇	—	2	2	no	—
37	M₃₂×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₄₇	—	2	2	no	—
39	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	—	2	1	no	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
48	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
56	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
57	C₄²×C₄₇	—	2	1	no	—
58	C₄²×C₄₇	C₄²	1	1	yes	characteristic
59	C₄²×C₄₇	C₄²	1	1	yes	characteristic
60	C₂²⋊Dic₄₇	—	2	2	no	—
61	C₄⋊Dic₄₇	—	2	2	no	—
62	C₄×Dic₄₇	—	2	2	no	—
63	#10625329	—	2	2	no	—
64	#10625329	—	2	2	no	—
65	C₂×C₄₇⋊C₈	—	2	2	no	—
66	C₁₆×C₄₇	—	2	4	no	—
67	C₄₇⋊C₁₆	—	2	2	no	—
68	C₄₇⋊C₁₆	C₄⋊C₄	1	4	yes	—
69	C₂³×C₄₇	#131	1	1	yes	characteristic
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	—	4	2	no	—
73	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
79	C₂×Dic₄₇	—	4	2	no	—
80	C₈×C₄₇	#97	1	1	yes	characteristic
81	C₈×C₄₇	#97	1	1	yes	characteristic
82	C₈×C₄₇	#97	1	1	yes	characteristic
83	C₈×C₄₇	#97	1	1	yes	characteristic
84	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₄₇	#117	1	1	yes	characteristic
86	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
87	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
88	C₄₇⋊C₈	—	4	2	no	—
89	#12667	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	4	1	no	—
91	C₂²×C₄₇	#405	1	1	yes	characteristic
92	C₂²×C₄₇	#436	1	1	yes	characteristic
93	C₂²×C₄₇	#446	1	1	yes	characteristic
94	C₄×C₄₇	—	2	2	no	—
95	C₄×C₄₇	—	4	1	no	—
96	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
98	C₄×C₄₇	#436	1	1	yes	characteristic
99	C₄×C₄₇	#446	1	1	yes	characteristic
100	Dic₄₇	—	8	2	no	—
101	C₂×#364	D₄₇	1	1	yes	characteristic
102	#2110	—	47	2	no	—
103	C₄×M₃₂	—	47	1	no	—
104	#1772	—	47	1	no	—
105	#1378	—	47	2	no	—
106	C₂×C₄₇	—	2	1	no	—
107	C₂×C₄₇	—	2	1	no	—
108	C₂×C₄₇	—	2	1	no	—
109	C₂×C₄₇	#1770	1	1	yes	characteristic
110	C₂×M₃₂	—	47	1	no	—
111	C₂×M₃₂	—	47	1	no	—
112	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
113	C₂×#108	—	47	1	no	—
114	#427	—	47	1	no	—
115	#405	—	47	2	no	—
116	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
117	#364	Dic₄₇	1	2	yes	complemented
118	#364	Dic₄₇	1	2	yes	complemented
119	#347	—	94	4	no	—
120	#346	—	47	2	no	—
121	C₄×C₁₆	—	47	2	no	—
122	#339	—	47	2	no	—
123	C₄₇	#12667	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	—	47	2	no	—
125	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	#117	—	47	2	no	—
128	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	47	2	no	—
131	M₃₂	—	47	2	no	—
132	M₃₂	—	94	1	no	—
133	M₃₂	—	94	1	no	—
134	M₃₂	C₂×Dic₄₇	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	47	2	no	—
137	C₂×C₁₆	—	47	2	no	—
138	#108	—	94	1	no	—
139	#108	—	94	1	no	—
140	#107	—	94	1	no	—
141	#106	—	94	1	no	—
142	#97	—	94	2	no	—
143	C₄×C₈	—	94	2	no	—
144	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
145	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
150	M₁₆	—	94	2	no	—
151	M₁₆	—	94	2	no	—
152	C₂×C₈	—	94	2	no	—
153	C₂×C₈	#10625324	1	1	yes	characteristic
154	C₂×C₈	#10625324	1	1	yes	characteristic
155	C₂×C₈	#10625324	1	1	yes	characteristic
156	C₂×C₈	#10625324	1	1	yes	characteristic
157	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
158	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
160	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
161	C₂×C₈	#10625352	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
163	C₂×C₈	#10625354	1	1	yes	characteristic
164	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
165	C₄⋊C₄	—	94	2	no	—
166	C₂²⋊C₄	—	94	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	94	2	no	—
169	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
170	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
171	C₁₆	—	2	4	no	—
172	C₁₆	—	47	4	no	—
173	C₁₆	—	94	2	no	—
174	C₂³	#13094032	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	188	2	no	—
179	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
180	C₂×C₄	#13093986	1	1	yes	characteristic
181	C₂×C₄	#13093989	1	1	yes	characteristic
182	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
183	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
184	C₂×C₄	#13094032	1	1	yes	characteristic
185	C₈	—	188	2	no	—
186	C₈	#13093934	1	1	yes	characteristic
187	C₈	#13093934	1	1	yes	characteristic
188	C₈	#13093934	1	1	yes	characteristic
189	C₈	#13093934	1	1	yes	characteristic
190	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
191	C₈	#13093982	1	1	yes	characteristic
192	C₈	D₄×Dic₄₇	1	1	yes	characteristic
193	C₈	Q₈×Dic₄₇	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#15689081	1	1	yes	characteristic
196	C₂²	#15689200	1	1	yes	characteristic
197	C₂²	#15689250	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	376	2	no	—
201	C₄	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
202	C₄	C₂×#13093934	1	1	yes	characteristic
203	C₄	#15689200	1	1	yes	characteristic
204	C₄	#15689250	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#20243258	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13093982
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂×C₄₇	#1770
prime radical	C₂³×C₄₇	#131
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	#12667
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
12. #15688899	C₄	semidirect
13. C₄×C₄₇⋊C₁₆	C₄	semidirect
17. C₄₇×#364	C₄	semidirect
18. C₄₇×#364	C₄	semidirect
117. #364	Dic₄₇	semidirect
118. #364	Dic₄₇	semidirect
123. C₄₇	#12667	semidirect

Maximal subgroups

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

subgroup	index
#20244992	2
C₄×#13093964	2
#20243267	2
#20241475	2
C₂×C₄₇×#364	2
#12667	47