Subgroups — #26371796

Order 12032.

474 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	#20253928	C₂	1	2	yes	maximal
2	#20245672	C₂	1	4	yes	maximal, complemented
3	#20245669	C₂	1	2	yes	maximal, complemented
4	#20245666	C₂	1	2	yes	maximal, complemented
5	#20245177	C₂	1	2	yes	maximal
6	C₂×#15688951	C₂	1	1	yes	maximal, characteristic
7	#20244984	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#2158	C₂	1	1	yes	maximal, characteristic
9	#15689751	—	2	2	no	—
10	C₂×#13093964	C₂²	1	1	yes	characteristic
11	#15689530	—	2	4	no	—
12	#15689527	C₂²	1	4	yes	—
13	#15689524	C₂²	1	4	yes	—
14	#15689166	—	2	2	no	—
15	C₂×#13093958	C₂²	1	2	yes	—
16	#15689078	C₂²	1	1	yes	characteristic
17	#15688951	C₂²	1	2	yes	—
18	#15688951	C₂²	1	2	yes	—
19	#15688950	C₂²	1	4	yes	—
20	#15688948	C₂²	1	4	yes	—
21	#15688898	C₂²	1	2	yes	—
22	C₁₆×Dic₄₇	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#427	C₂²	1	2	yes	—
25	C₄₇×#367	C₂²	1	1	yes	characteristic
26	C₄₇×#367	C₂²	1	1	yes	characteristic
27	C₄₇×#366	C₂²	1	2	yes	—
28	C₂×#10625329	D₄	1	1	yes	characteristic
29	#13094024	—	4	2	no	—
30	#13094023	—	2	4	no	—
31	#13094016	—	4	2	no	—
32	#13094015	—	2	4	no	—
33	#13094001	—	4	4	no	—
34	#13093983	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₄₇	—	2	2	no	—
36	#13093964	Q₈	1	4	yes	complemented
37	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
38	#13093958	—	2	2	no	—
39	#13093958	C₂³	1	4	yes	—
40	#13093957	—	2	4	no	—
41	#13093955	—	2	4	no	—
42	#13093934	D₄	1	2	yes	—
43	C₈×Dic₄₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	#117×C₄₇	—	2	2	no	—
46	C₂×C₁₆×C₄₇	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
49	#107×C₄₇	C₂³	1	2	yes	—
50	#107×C₄₇	C₂³	1	2	yes	—
51	#106×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₇	—	4	2	no	—
54	C₂×Dic₉₄	—	4	4	no	—
55	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₄⋊C₄×C₄₇	—	2	2	no	—
61	C₂²⋊C₄×C₄₇	—	4	2	no	—
62	C₄²×C₄₇	—	4	2	no	—
63	C₂²⋊Dic₄₇	D₈	1	2	yes	—
64	C₄⋊Dic₄₇	—	2	4	no	—
65	C₄⋊Dic₄₇	—	4	2	no	—
66	C₄⋊Dic₄₇	—	4	4	no	—
67	C₄⋊Dic₄₇	D₈	1	2	yes	—
68	Dic₄₇⋊C₄	—	4	4	no	—
69	Dic₄₇⋊C₄	—	4	4	no	—
70	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
71	#10625329	Q₁₆	1	4	yes	—
72	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
73	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
74	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
75	C₄₇⋊C₁₆	C₂×Q₈	1	4	yes	—
76	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	—	4	2	no	—
78	C₂×C₄×C₄₇	—	4	2	no	—
79	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₄₇	—	4	4	no	—
84	C₂×Dic₄₇	—	16	2	no	—
85	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
86	Dic₉₄	—	16	4	no	—
87	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
88	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
89	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
90	C₄₇⋊C₈	C₂×Q₁₆	1	4	yes	—
91	#33730	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	—	2	2	no	—
93	C₂²×C₄₇	#500	1	2	yes	—
94	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₄₇	—	8	2	no	—
96	C₄×C₄₇	—	8	2	no	—
97	C₄×C₄₇	#500	1	2	yes	—
98	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
100	Dic₄₇	—	2	4	no	—
101	Dic₄₇	—	8	4	no	—
102	#3164	—	47	2	no	—
103	#2255	—	47	4	no	—
104	#2254	—	47	2	no	—
105	#2253	—	47	2	no	—
106	#2159	—	47	2	no	—
107	#2158	D₄₇	1	1	yes	characteristic
108	C₂×#367	—	47	1	no	—
109	#2108	—	47	1	no	—
110	C₂×C₄₇	—	2	2	no	—
111	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
112	C₂×C₄₇	#3416	1	2	yes	—
113	#544	—	94	2	no	—
114	C₂×M₃₂	—	47	1	no	—
115	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
116	#501	—	94	4	no	—
117	#500	—	47	4	no	—
118	#499	—	47	4	no	—
119	#428	—	94	2	no	—
120	#427	C₂×D₄₇	1	2	yes	—
121	C₂×#107	—	47	2	no	—
122	#405	—	47	1	no	—
123	#367	—	47	2	no	—
124	#367	—	47	2	no	—
125	#367	—	47	4	no	—
126	#367	C₂×D₄₇	1	1	yes	characteristic
127	#367	C₂×D₄₇	1	1	yes	characteristic
128	#366	—	47	4	no	—
129	#366	C₂×D₄₇	1	2	yes	—
130	#346	—	47	2	no	—
131	C₄×C₁₆	—	47	2	no	—
132	C₄₇	#33730	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	47	1	no	—
134	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	94	4	no	—
136	C₄⋊Q₈	—	188	2	no	—
137	#125	—	94	4	no	—
138	#125	—	188	2	no	—
139	C₂²⋊Q₈	—	188	4	no	—
140	#117	—	2	2	no	—
141	#117	—	47	1	no	—
142	C₂×C₄⋊C₄	—	94	2	no	—
143	M₃₂	—	47	4	no	—
144	C₂×C₁₆	—	47	2	no	—
145	C₂×C₁₆	Dic₉₄	1	4	yes	complemented
146	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
148	#107	—	47	4	no	—
149	#107	—	94	4	no	—
150	#107	—	94	2	no	—
151	#107	C₂²×D₄₇	1	2	yes	—
152	#107	C₂²×D₄₇	1	2	yes	—
153	#106	—	2	2	no	—
154	#106	—	94	4	no	—
155	#97	—	47	2	no	—
156	C₄×C₈	—	47	2	no	—
157	C₂×Q₈	—	188	4	no	—
158	C₂²×C₄	—	188	2	no	—
159	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
160	M₁₆	—	47	4	no	—
161	C₂×C₈	—	47	2	no	—
162	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
163	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
164	C₂×C₈	Q₈×D₄₇	1	2	yes	—
165	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	47	2	no	—
169	C₄⋊C₄	—	94	4	no	—
170	C₄⋊C₄	—	188	4	no	—
171	C₄⋊C₄	—	188	4	no	—
172	C₄⋊C₄	—	188	4	no	—
173	C₄⋊C₄	—	188	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	—	47	2	no	—
176	C₄²	—	4	2	no	—
177	C₄²	—	47	2	no	—
178	C₁₆	—	47	4	no	—
179	C₁₆	C₂×Dic₉₄	1	2	yes	—
180	C₁₆	C₂×Dic₉₄	1	2	yes	—
181	C₂³	D₈×D₄₇	1	1	yes	characteristic
182	Q₈	—	752	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	47	4	no	—
186	C₂×C₄	—	188	4	no	—
187	C₂×C₄	—	752	2	no	—
188	C₂×C₄	#13094023	1	2	yes	—
189	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
191	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
192	C₈	—	47	4	no	—
193	C₈	#13094023	1	2	yes	—
194	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
195	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#15689527	1	2	yes	—
198	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
199	C₄	—	8	2	no	—
200	C₄	—	8	2	no	—
201	C₄	—	94	4	no	—
202	C₄	—	376	4	no	—
203	C₄	#15689527	1	2	yes	—
204	C₄	C₂×#13094023	1	1	yes	characteristic
205	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#15689527	1	1	yes	characteristic
208	C₂	#20256531	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2158	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂³×C₄₇	C₂×D₈
hypercenter	#2158	D₄₇
hypocenter	C₄₇	#33730
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #20245672	C₂	semidirect
3. #20245669	C₂	semidirect
4. #20245666	C₂	semidirect
36. #13093964	Q₈	semidirect
46. C₂×C₁₆×C₄₇	Q₈	semidirect
132. C₄₇	#33730	semidirect
145. C₂×C₁₆	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253928	2
#20245672	2
#20245669	2
#20245666	2
#20245177	2
C₂×#15688951	2
#20244984	2
C₄₇×#2158	2
#33730	47