Subgroups — #26369656

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252256	C₂	1	2	yes	maximal
2	#20245259	C₂	1	4	yes	maximal, complemented
3	Q₃₂×Dic₄₇	C₂	1	4	yes	maximal, complemented
4	#20245172	C₂	1	1	yes	maximal, characteristic
5	#20245108	C₂	1	2	yes	maximal
6	#20244984	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#15689601	—	2	2	no	—
9	C₂×#13093964	C₂²	1	1	yes	characteristic
10	#15689230	—	2	4	no	—
11	Q₁₆×Dic₄₇	C₂²	1	8	yes	—
12	#15689161	C₂²	1	2	yes	—
13	#15689132	—	2	2	no	—
14	#15689078	C₂²	1	1	yes	characteristic
15	#15688951	C₂²	1	2	yes	—
16	#15688949	C₂²	1	2	yes	—
17	#15688927	C₂²	1	8	yes	—
18	#15688898	C₂²	1	2	yes	—
19	C₁₆×Dic₄₇	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂×Q₃₂×C₄₇	C₄	1	8	yes	complemented
22	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
23	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#10625329	D₄	1	1	yes	characteristic
26	Q₈×Dic₄₇	—	2	8	no	—
27	Q₈×Dic₄₇	—	4	4	no	—
28	#13093983	—	2	2	no	—
29	#13093983	—	4	4	no	—
30	#13093983	D₄	1	1	yes	characteristic
31	#13093964	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
33	#13093958	C₂³	1	4	yes	—
34	#13093956	—	2	2	no	—
35	#13093948	—	2	8	no	—
36	#13093934	D₄	1	2	yes	—
37	C₈×Dic₄₇	C₂³	1	2	yes	—
38	C₂²×Q₈×C₄₇	—	2	2	no	—
39	C₂×Q₁₆×C₄₇	—	2	2	no	—
40	C₂×Q₁₆×C₄₇	—	2	4	no	—
41	C₂×Q₁₆×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	Q₃₂×C₄₇	C₂×C₄	1	8	yes	—
45	Q₃₂×C₄₇	C₂×C₄	1	8	yes	—
46	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₄₇	—	2	8	no	—
50	C₂×Q₈×C₄₇	—	2	4	no	—
51	C₂×Q₈×C₄₇	—	4	2	no	—
52	C₂×Q₈×C₄₇	—	4	4	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₄₇	—	2	8	no	—
56	Q₁₆×C₄₇	—	2	4	no	—
57	Q₁₆×C₄₇	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	#41	1	2	yes	—
61	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²⋊Dic₄₇	—	4	2	no	—
63	C₂²⋊Dic₄₇	D₈	1	2	yes	—
64	C₄⋊Dic₄₇	—	2	4	no	—
65	C₄⋊Dic₄₇	—	4	8	no	—
66	C₄⋊Dic₄₇	D₈	1	2	yes	—
67	C₄×Dic₄₇	—	4	2	no	—
68	C₄×Dic₄₇	—	4	8	no	—
69	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
70	#10625329	—	2	2	no	—
71	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
72	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
73	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄₇⋊C₁₆	—	2	2	no	—
75	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₄₇	—	2	8	no	—
77	Q₈×C₄₇	—	4	4	no	—
78	Q₈×C₄₇	—	4	8	no	—
79	C₂×C₄×C₄₇	—	4	4	no	—
80	C₂×C₄×C₄₇	—	4	8	no	—
81	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	#135	1	2	yes	—
84	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₄₇	—	4	4	no	—
86	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
87	C₈×C₄₇	C₄×D₄	1	2	yes	—
88	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
90	C₄₇⋊C₈	—	2	2	no	—
91	#33458	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	2	no	—
93	C₂²×C₄₇	C₄×D₈	1	2	yes	—
94	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₄₇	—	4	8	no	—
96	C₄×C₄₇	C₄×D₈	1	2	yes	—
97	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
99	Dic₄₇	—	2	4	no	—
100	Dic₄₇	—	8	4	no	—
101	C₂²×Q₃₂	D₄₇	1	1	yes	characteristic
102	#2950	—	47	2	no	—
103	#2176	—	47	4	no	—
104	C₄×Q₃₂	—	47	4	no	—
105	#2158	—	47	1	no	—
106	#2141	—	47	2	no	—
107	#2108	—	47	1	no	—
108	C₂×C₄₇	—	2	2	no	—
109	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₄₇	#3417	1	2	yes	—
111	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
112	#519	—	94	2	no	—
113	C₂×Q₃₂	Dic₄₇	1	8	yes	complemented
114	C₂×Q₃₂	C₂×D₄₇	1	2	yes	—
115	C₂×Q₃₂	C₂×D₄₇	1	2	yes	—
116	C₂×M₃₂	—	47	1	no	—
117	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
118	#441	—	94	4	no	—
119	C₄×Q₁₆	—	47	8	no	—
120	#427	—	47	2	no	—
121	#419	—	94	2	no	—
122	#405	—	47	1	no	—
123	#367	—	47	2	no	—
124	#366	—	47	2	no	—
125	#358	—	47	8	no	—
126	#346	—	47	2	no	—
127	C₄×C₁₆	—	47	2	no	—
128	C₄₇	#33458	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	C₂×Dic₄₇	1	8	yes	—
133	C₂×Q₁₆	C₂²×D₄₇	1	4	yes	—
134	C₂×M₁₆	—	47	1	no	—
135	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
136	C₄×Q₈	—	94	8	no	—
137	C₄×Q₈	—	188	4	no	—
138	#117	—	47	1	no	—
139	#117	—	94	2	no	—
140	#117	—	188	4	no	—
141	Q₃₂	C₂×Dic₄₇	1	8	yes	—
142	Q₃₂	C₂×Dic₄₇	1	8	yes	—
143	M₃₂	—	94	2	no	—
144	C₂×C₁₆	—	47	2	no	—
145	C₂×C₁₆	C₂×Dic₄₇	1	4	yes	—
146	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
148	#107	—	47	4	no	—
149	#106	—	94	2	no	—
150	#103	—	94	8	no	—
151	#97	—	47	2	no	—
152	C₄×C₈	—	47	2	no	—
153	C₂×Q₈	—	2	8	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×Dic₄₇	1	8	yes	—
162	M₁₆	—	94	2	no	—
163	C₂×C₈	—	47	2	no	—
164	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
165	C₂×C₈	#10625352	1	2	yes	—
166	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
167	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
168	C₄⋊C₄	—	47	2	no	—
169	C₄⋊C₄	—	94	4	no	—
170	C₄⋊C₄	—	188	8	no	—
171	C₂²⋊C₄	—	47	2	no	—
172	C₂²⋊C₄	—	188	2	no	—
173	C₄²	—	47	2	no	—
174	C₄²	—	188	8	no	—
175	C₄²	—	188	2	no	—
176	C₁₆	—	94	2	no	—
177	C₁₆	C₂²×Dic₄₇	1	2	yes	—
178	C₁₆	C₂²×Dic₄₇	1	2	yes	—
179	C₂³	D₈×D₄₇	1	1	yes	characteristic
180	Q₈	—	2	8	no	—
181	Q₈	—	4	4	no	—
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	47	4	no	—
186	C₂×C₄	—	188	4	no	—
187	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
188	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
189	C₂×C₄	#13094043	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
191	C₈	—	94	2	no	—
192	C₈	D₄×Dic₄₇	1	2	yes	—
193	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
194	C₈	C₂×#10625352	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	D₈×Dic₄₇	1	2	yes	—
197	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	94	4	no	—
200	C₄	—	376	4	no	—
201	C₄	D₈×Dic₄₇	1	2	yes	—
202	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
203	C₄	C₂×#13094043	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×D₈×Dic₄₇	1	1	yes	characteristic
206	C₂	#20256537	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×Q₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×Q₃₂	D₄₇
hypocenter	C₄₇	#33458
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #20245259	C₂	semidirect
3. Q₃₂×Dic₄₇	C₂	semidirect
21. C₂×Q₃₂×C₄₇	C₄	semidirect
113. C₂×Q₃₂	Dic₄₇	semidirect
128. C₄₇	#33458	semidirect

Maximal subgroups

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

subgroup	index
#20252256	2
#20245259	2
Q₃₂×Dic₄₇	2
#20245172	2
#20245108	2
#20244984	2
C₂²×Q₃₂×C₄₇	2
#33458	47