Subgroups — #26371769

Order 12032.

490 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253901	C₂	1	2	yes	maximal
2	#20245669	C₂	1	2	yes	maximal, complemented
3	#20245667	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245664	C₂	1	2	yes	maximal, complemented
5	#20245662	C₂	1	2	yes	maximal, complemented
6	#20245171	C₂	1	2	yes	maximal
7	#20245169	C₂	1	2	yes	maximal
8	C₄×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689679	—	2	2	no	—
11	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
13	#15689527	C₂²	1	4	yes	—
14	#15689525	C₂²	1	2	yes	—
15	#15689522	—	2	2	no	—
16	#15689520	—	2	2	no	—
17	#15689160	C₂²	1	4	yes	—
18	#15689158	—	2	2	no	—
19	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
20	#15688951	C₂²	1	2	yes	—
21	#15688950	C₂²	1	4	yes	—
22	#15688949	C₂²	1	2	yes	—
23	#15688948	C₂²	1	4	yes	—
24	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#499	C₂²	1	2	yes	—
28	C₄₇×#367	C₂²	1	2	yes	—
29	C₄₇×#366	C₂²	1	2	yes	—
30	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
32	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
33	#13094023	—	2	4	no	—
34	#13094016	—	2	2	no	—
35	#13094002	—	4	4	no	—
36	#13093989	—	4	4	no	—
37	#13093982	—	2	4	no	—
38	C₄²×D₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
41	C₁₆×D₄₇	Q₈	1	4	yes	complemented
42	C₁₆×D₄₇	Q₈	1	4	yes	complemented
43	#13093958	C₂³	1	4	yes	—
44	#13093957	—	2	4	no	—
45	#13093956	—	2	2	no	—
46	#13093955	—	2	4	no	—
47	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
48	C₈×Dic₄₇	D₄	1	1	yes	characteristic
49	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
50	#125×C₄₇	—	2	2	no	—
51	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
53	#107×C₄₇	C₂³	1	4	yes	—
54	#106×C₄₇	—	2	2	no	—
55	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₄₇	D₈	1	2	yes	—
57	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₉₄	—	4	4	no	—
59	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	—	2	4	no	—
62	C₄⋊C₄×C₄₇	—	4	4	no	—
63	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
64	(C₂×D₄₇)⋊C₄	—	4	4	no	—
65	C₄⋊Dic₄₇	—	2	4	no	—
66	Dic₄₇⋊C₄	—	4	4	no	—
67	Dic₄₇⋊C₄	—	4	4	no	—
68	C₄×Dic₄₇	—	4	4	no	—
69	C₄×Dic₄₇	D₈	1	2	yes	—
70	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
73	C₈×D₄₇	Q₁₆	1	4	yes	—
74	C₈×D₄₇	C₂×Q₈	1	2	yes	—
75	C₈×D₄₇	C₂×Q₈	1	2	yes	—
76	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
77	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
78	C₄₇⋊C₁₆	C₂×Q₈	1	2	yes	—
79	C₄₇⋊C₁₆	C₂×Q₈	1	2	yes	—
80	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₄₇	—	4	4	no	—
85	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
87	C₄×D₄₇	—	2	4	no	—
88	C₄×D₄₇	C₄⋊Q₈	1	2	yes	—
89	C₄×D₄₇	C₂×Q₁₆	1	2	yes	—
90	Dic₉₄	—	8	8	no	—
91	C₈×C₄₇	C₂×Q₁₆	1	2	yes	—
92	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
94	C₄₇⋊C₈	C₄⋊Q₈	1	2	yes	—
95	C₄₇⋊C₈	C₂×Q₁₆	1	2	yes	—
96	#33726	—	47	1	no	maximal, Sylow
97	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
98	C₂×D₄₇	—	2	2	no	—
99	C₂×D₄₇	#500	1	2	yes	—
100	C₄×C₄₇	—	2	2	no	—
101	C₄×C₄₇	—	8	4	no	—
102	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₄₇	—	8	4	no	—
105	Dic₄₇	#500	1	2	yes	—
106	Dic₄₇	#508	1	4	yes	—
107	#3160	—	47	2	no	—
108	#2254	—	47	2	no	—
109	#2253	—	47	2	no	—
110	#2253	—	47	1	no	—
111	#2253	D₄₇	1	1	yes	characteristic, complemented
112	#2252	—	47	2	no	—
113	#2158	—	47	2	no	—
114	#2158	—	47	2	no	—
115	C₂×C₄×C₁₆	—	47	1	no	—
116	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
117	C₂×C₄₇	C₂×#508	1	2	yes	—
118	D₄₇	—	2	2	no	—
119	#533	—	94	2	no	—
120	C₂²×C₁₆	—	47	1	no	—
121	C₂²×C₁₆	—	47	1	no	—
122	#500	—	47	4	no	—
123	#499	—	47	2	no	—
124	#499	—	94	2	no	—
125	#499	C₂×D₄₇	1	2	yes	—
126	#498	—	94	2	no	—
127	#427	—	47	4	no	—
128	#427	—	94	2	no	—
129	C₂×C₄×C₈	—	47	1	no	—
130	#367	—	47	2	no	—
131	#367	—	47	4	no	—
132	#367	C₂×D₄₇	1	2	yes	—
133	#366	—	47	2	no	—
134	#366	—	47	4	no	—
135	#366	C₂×D₄₇	1	2	yes	—
136	C₄×C₁₆	—	47	1	no	—
137	C₄×C₁₆	—	47	1	no	—
138	C₄×C₁₆	—	47	1	no	—
139	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
140	C₄₇	#33726	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₈	—	47	1	no	—
142	C₂²×C₈	—	47	1	no	—
143	C₄⋊Q₈	—	94	4	no	—
144	#125	—	2	2	no	—
145	#125	—	94	2	no	—
146	C₂²⋊Q₈	—	188	4	no	—
147	C₄×Q₈	—	188	4	no	—
148	#117	—	94	4	no	—
149	C₂×C₄²	—	47	1	no	—
150	C₂×C₁₆	—	47	1	no	—
151	C₂×C₁₆	—	47	1	no	—
152	C₂×C₁₆	—	47	4	no	—
153	C₂×C₁₆	—	47	4	no	—
154	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
156	#107	—	47	4	no	—
157	#107	—	94	4	no	—
158	#107	C₂²×D₄₇	1	4	yes	—
159	#106	—	2	2	no	—
160	#106	—	94	4	no	—
161	#106	—	94	2	no	—
162	C₄×C₈	—	47	1	no	—
163	C₄×C₈	—	47	1	no	—
164	C₄×C₈	—	47	1	no	—
165	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×Q₈	—	188	4	no	—
167	C₂²×C₄	—	47	1	no	—
168	C₂²×C₄	—	47	2	no	—
169	C₂×C₈	—	47	1	no	—
170	C₂×C₈	—	47	1	no	—
171	C₂×C₈	—	47	4	no	—
172	C₂×C₈	—	47	2	no	—
173	C₂×C₈	—	47	2	no	—
174	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	94	4	no	—
179	C₄⋊C₄	—	188	4	no	—
180	C₄⋊C₄	—	188	4	no	—
181	C₂²⋊C₄	—	188	4	no	—
182	C₄²	—	47	2	no	—
183	C₄²	—	47	1	no	—
184	C₄²	—	188	4	no	—
185	C₄²	D₄×D₄₇	1	1	yes	characteristic
186	C₁₆	—	47	2	no	—
187	C₁₆	—	47	2	no	—
188	C₁₆	Q₈×D₄₇	1	2	yes	—
189	C₁₆	Q₈×D₄₇	1	2	yes	—
190	C₂³	—	47	1	no	—
191	Q₈	—	376	8	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	47	1	no	—
194	C₂×C₄	—	47	2	no	—
195	C₂×C₄	—	47	2	no	—
196	C₂×C₄	—	47	2	no	—
197	C₂×C₄	—	94	4	no	—
198	C₂×C₄	—	188	4	no	—
199	C₂×C₄	D₈×D₄₇	1	2	yes	—
200	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
201	C₈	—	47	2	no	—
202	C₈	—	47	2	no	—
203	C₈	Q₁₆×D₄₇	1	2	yes	—
204	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
205	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
206	C₂²	—	47	2	no	—
207	C₂²	—	94	2	no	—
208	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	8	4	no	—
211	C₄	—	47	2	no	—
212	C₄	—	47	4	no	—
213	C₄	—	376	4	no	—
214	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
215	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
216	C₂	—	94	2	no	—
217	C₂	#500×D₄₇	1	1	yes	characteristic
218	C₂	#508×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2253	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂²×D₄₇	C₂×D₈
hypercenter	#2253	D₄₇
hypocenter	C₄₇	#33726
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #20245669	C₂	semidirect
3. #20245667	C₂	semidirect
4. #20245664	C₂	semidirect
5. #20245662	C₂	semidirect
9. C₄₇×#2253	C₂	semidirect
41. C₁₆×D₄₇	Q₈	semidirect
42. C₁₆×D₄₇	Q₈	semidirect
111. #2253	D₄₇	semidirect
140. C₄₇	#33726	semidirect

Maximal subgroups

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

subgroup	index
#20253901	2
#20245669	2
#20245667	2
#20245664	2
#20245662	2
#20245171	2
#20245169	2
C₄×C₁₆×D₄₇	2
C₄₇×#2253	2
#33726	47