Subgroups — #26371768

Order 12032.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253900	C₂	1	2	yes	maximal
2	#20245669	C₂	1	2	yes	maximal, complemented
3	#20245666	C₂	1	2	yes	maximal, complemented
4	#20245665	C₂	1	2	yes	maximal, complemented
5	#20245662	C₂	1	2	yes	maximal, complemented
6	#20245172	C₂	1	1	yes	maximal, characteristic
7	#20245170	C₂	1	2	yes	maximal
8	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2158	C₂	1	1	yes	maximal, characteristic
10	#15689676	—	2	2	no	—
11	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#15689527	C₂²	1	4	yes	—
13	#15689524	C₂²	1	4	yes	—
14	#15689523	—	2	2	no	—
15	#15689520	—	2	2	no	—
16	#15689161	C₂²	1	2	yes	—
17	#15689159	—	2	2	no	—
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	#15688951	C₂²	1	2	yes	—
20	#15688950	C₂²	1	4	yes	—
21	#15688949	C₂²	1	2	yes	—
22	#15688948	C₂²	1	4	yes	—
23	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#427	C₂²	1	2	yes	—
29	C₄₇×#367	C₂²	1	2	yes	—
30	C₄₇×#366	C₂²	1	2	yes	—
31	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
32	#13094023	—	2	4	no	—
33	#13094015	—	2	4	no	—
34	#13094001	—	4	4	no	—
35	C₄×Dic₉₄	—	4	4	no	—
36	#13093983	—	2	2	no	—
37	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₁₆	Q₈	1	4	yes	complemented
39	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
41	#13093958	C₂³	1	4	yes	—
42	#13093957	—	2	4	no	—
43	#13093956	—	2	2	no	—
44	#13093955	—	2	4	no	—
45	C₈×Dic₄₇	D₄	1	1	yes	characteristic
46	C₈×Dic₄₇	D₄	1	1	yes	characteristic
47	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	#117×C₄₇	—	2	2	no	—
51	C₂×C₁₆×C₄₇	Q₈	1	4	yes	complemented
52	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
54	#107×C₄₇	C₂³	1	4	yes	—
55	#106×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	D₈	1	2	yes	—
58	C₂×Dic₉₄	—	4	4	no	—
59	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₂²⋊C₄×C₄₇	—	4	2	no	—
65	C₄²×C₄₇	—	4	2	no	—
66	C₂²⋊Dic₄₇	—	4	2	no	—
67	C₄⋊Dic₄₇	—	2	4	no	—
68	C₄⋊Dic₄₇	—	4	4	no	—
69	Dic₄₇⋊C₄	—	4	4	no	—
70	Dic₄₇⋊C₄	—	4	4	no	—
71	C₄×Dic₄₇	—	4	2	no	—
72	C₄×Dic₄₇	D₈	1	2	yes	—
73	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄₇⋊C₈	Q₁₆	1	4	yes	—
76	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
79	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
80	C₄₇⋊C₁₆	C₂×Q₈	1	2	yes	—
81	C₄₇⋊C₁₆	C₂×Q₈	1	2	yes	—
82	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₄₇	—	4	4	no	—
84	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
85	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₄₇	—	2	4	no	—
89	C₂×Dic₄₇	—	4	4	no	—
90	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
91	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
92	Dic₉₄	—	8	8	no	—
93	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
94	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
95	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
96	C₄₇⋊C₈	C₂×Q₁₆	1	2	yes	—
97	C₄₇⋊C₈	C₂×Q₁₆	1	2	yes	—
98	#33726	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	#500	1	2	yes	—
101	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₄₇	—	8	4	no	—
103	C₄×C₄₇	#500	1	2	yes	—
104	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₄×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
106	Dic₄₇	—	2	2	no	—
107	Dic₄₇	—	8	4	no	—
108	Dic₄₇	#508	1	4	yes	—
109	#3160	—	47	2	no	—
110	#2254	—	47	2	no	—
111	#2253	—	47	2	no	—
112	#2253	—	47	2	no	—
113	#2252	—	47	2	no	—
114	#2158	—	47	1	no	—
115	#2158	—	47	2	no	—
116	#2158	D₄₇	1	1	yes	characteristic
117	C₂×C₄×C₁₆	—	47	1	no	—
118	C₂×C₄₇	—	2	2	no	—
119	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#508	1	2	yes	—
121	#533	—	94	2	no	—
122	C₂²×C₁₆	—	47	1	no	—
123	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
124	#500	—	47	4	no	—
125	#499	—	47	4	no	—
126	#499	—	94	2	no	—
127	#498	—	94	2	no	—
128	#427	—	47	2	no	—
129	#427	—	94	2	no	—
130	#427	C₂×D₄₇	1	2	yes	—
131	C₂×C₄×C₈	—	47	1	no	—
132	#367	—	47	2	no	—
133	#367	—	47	4	no	—
134	#367	C₂×D₄₇	1	2	yes	—
135	#366	—	47	2	no	—
136	#366	—	47	4	no	—
137	#366	C₂×D₄₇	1	2	yes	—
138	C₄×C₁₆	—	47	1	no	—
139	C₄×C₁₆	—	47	1	no	—
140	C₄×C₁₆	—	47	1	no	—
141	C₄×C₁₆	—	47	1	no	—
142	C₄₇	#33726	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₈	—	47	1	no	—
144	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
145	C₄⋊Q₈	—	94	4	no	—
146	#125	—	94	4	no	—
147	C₂²⋊Q₈	—	188	4	no	—
148	C₄×Q₈	—	188	4	no	—
149	#117	—	2	2	no	—
150	#117	—	94	2	no	—
151	C₂×C₄²	—	47	1	no	—
152	C₂×C₁₆	—	47	4	no	—
153	C₂×C₁₆	—	47	1	no	—
154	C₂×C₁₆	—	47	1	no	—
155	C₂×C₁₆	Dic₉₄	1	4	yes	complemented
156	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
157	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
158	#107	—	47	4	no	—
159	#107	—	94	4	no	—
160	#107	C₂²×D₄₇	1	4	yes	—
161	#106	—	2	2	no	—
162	#106	—	94	4	no	—
163	#106	—	94	2	no	—
164	C₄×C₈	—	47	1	no	—
165	C₄×C₈	—	47	1	no	—
166	C₄×C₈	—	47	1	no	—
167	C₄×C₈	—	47	1	no	—
168	C₂×Q₈	—	188	4	no	—
169	C₂²×C₄	—	47	2	no	—
170	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
171	C₂×C₈	—	47	4	no	—
172	C₂×C₈	—	47	1	no	—
173	C₂×C₈	—	47	1	no	—
174	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
175	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₄₇	1	2	yes	—
177	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	94	4	no	—
180	C₄⋊C₄	—	188	4	no	—
181	C₄⋊C₄	—	188	4	no	—
182	C₄⋊C₄	—	188	4	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	188	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	—	47	2	no	—
187	C₄²	—	47	1	no	—
188	C₄²	—	47	1	no	—
189	C₄²	—	188	2	no	—
190	C₁₆	—	47	2	no	—
191	C₁₆	—	47	2	no	—
192	C₁₆	C₂×Dic₉₄	1	2	yes	—
193	C₁₆	C₂×Dic₉₄	1	2	yes	—
194	C₂³	D₈×D₄₇	1	1	yes	characteristic
195	Q₈	—	376	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	47	2	no	—
198	C₂×C₄	—	47	2	no	—
199	C₂×C₄	—	94	4	no	—
200	C₂×C₄	—	188	4	no	—
201	C₂×C₄	#13094023	1	2	yes	—
202	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
204	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
205	C₈	—	47	2	no	—
206	C₈	—	47	2	no	—
207	C₈	#13094023	1	2	yes	—
208	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
209	C₈	C₂²×Dic₉₄	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	#15689527	1	2	yes	—
212	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
213	C₄	—	8	4	no	—
214	C₄	—	47	4	no	—
215	C₄	—	94	2	no	—
216	C₄	—	376	4	no	—
217	C₄	#15689527	1	2	yes	—
218	C₄	C₂×#13094023	1	1	yes	characteristic
219	C₄	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#15689527	1	1	yes	characteristic
222	C₂	#508×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	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	#33726
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #20245669	C₂	semidirect
3. #20245666	C₂	semidirect
4. #20245665	C₂	semidirect
5. #20245662	C₂	semidirect
38. C₂×C₄₇⋊C₁₆	Q₈	semidirect
51. C₂×C₁₆×C₄₇	Q₈	semidirect
142. C₄₇	#33726	semidirect
155. 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
#20253900	2
#20245669	2
#20245666	2
#20245665	2
#20245662	2
#20245172	2
#20245170	2
C₂×C₁₆×Dic₄₇	2
C₄₇×#2158	2
#33726	47