Subgroups — #26115312

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251786	C₂	1	1	yes	maximal, characteristic
2	#20243540	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688868	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	—	2	2	no	—
6	C₂×#13094024	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15688993	—	2	1	no	—
10	#15688993	—	2	1	no	—
11	#15688993	C₄	1	4	yes	—
12	#15688868	—	2	8	no	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	—	2	1	no	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	1	no	—
17	C₂²×Dic₉₄	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	#13094024	—	2	4	no	—
20	#13094024	—	2	4	no	—
21	#13094024	—	4	4	no	—
22	#13094024	—	4	4	no	—
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₂×C₄⋊Dic₄₇	—	2	4	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
27	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
28	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₄₇	—	2	2	no	—
36	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
37	C₄×C₈×C₄₇	—	4	2	no	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	2	4	no	—
44	C₂×Dic₉₄	—	4	2	no	—
45	C₂×Dic₉₄	—	4	2	no	—
46	C₂×Dic₉₄	—	4	4	no	—
47	C₂×Dic₉₄	—	4	4	no	—
48	C₂×C₈×C₄₇	—	2	4	no	—
49	C₂×C₈×C₄₇	—	4	2	no	—
50	C₂×C₈×C₄₇	—	4	2	no	—
51	C₄²×C₄₇	—	2	4	no	—
52	C₄²×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	D₈	1	4	yes	—
54	C₄²×C₄₇	SD₁₆	1	4	yes	—
55	C₄²×C₄₇	SD₁₆	1	2	yes	—
56	C₄²×C₄₇	Q₁₆	1	2	yes	—
57	C₄²×C₄₇	C₂×D₄	1	2	yes	—
58	C₄²×C₄₇	#41	1	2	yes	—
59	C₄⋊Dic₄₇	—	2	8	no	—
60	C₄⋊Dic₄₇	—	4	8	no	—
61	C₄⋊Dic₄₇	—	4	8	no	—
62	C₄⋊Dic₄₇	—	4	4	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂×C₄₇⋊C₈	—	4	4	no	—
65	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	#101	1	4	yes	—
70	C₂×C₄×C₄₇	#102	1	4	yes	—
71	C₂×C₄×C₄₇	#102	1	4	yes	—
72	C₂×C₄×C₄₇	#103	1	4	yes	—
73	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₄₇	#121	1	2	yes	—
75	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#137	1	2	yes	—
82	C₂×Dic₄₇	—	4	4	no	—
83	C₂×Dic₄₇	—	4	8	no	—
84	Dic₉₄	—	8	4	no	—
85	Dic₉₄	—	8	2	no	—
86	Dic₉₄	—	8	2	no	—
87	C₈×C₄₇	—	4	4	no	—
88	C₄₇⋊C₈	—	8	2	no	—
89	#9692	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	#390	1	1	yes	characteristic
91	C₂²×C₄₇	C₂×#101	1	2	yes	—
92	C₂²×C₄₇	C₂×#102	1	2	yes	—
93	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
94	C₂²×C₄₇	#419	1	1	yes	characteristic
95	C₄×C₄₇	—	2	4	no	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	#332	1	8	yes	complemented
98	C₄×C₄₇	#461	1	2	yes	—
99	C₄×C₄₇	#462	1	2	yes	—
100	C₄×C₄₇	#486	1	2	yes	—
101	C₄×C₄₇	#492	1	1	yes	characteristic
102	C₄×C₄₇	#493	1	1	yes	characteristic
103	Dic₄₇	—	8	4	no	—
104	#2868	—	47	1	no	—
105	#1842	—	47	1	no	—
106	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
107	C₂×#332	—	47	4	no	—
108	C₂×C₄₇	C₂×#332	1	2	yes	—
109	C₂×C₄₇	#1546	1	2	yes	—
110	C₂×C₄₇	#1957	1	1	yes	characteristic
111	C₂×C₄₇	#1966	1	1	yes	characteristic
112	C₂×C₄₇	#1981	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	47	2	no	—
114	C₂×C₄⋊Q₈	—	94	2	no	—
115	C₂×C₄⋊C₈	—	47	2	no	—
116	C₂×C₄⋊C₈	—	94	2	no	—
117	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
118	C₂×C₄×C₈	—	2	1	no	—
119	#384	—	47	4	no	—
120	#384	—	94	1	no	—
121	#384	—	94	1	no	—
122	C₄³	C₂×D₄₇	1	1	yes	characteristic
123	#336	—	2	1	no	—
124	#332	—	94	8	no	—
125	C₄₇	#9692	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	94	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	94	2	no	—
129	C₄⋊Q₈	—	94	4	no	—
130	C₄⋊Q₈	—	94	4	no	—
131	C₄⋊Q₈	—	188	4	no	—
132	C₄⋊Q₈	—	188	4	no	—
133	C₂×C₄⋊C₄	—	47	4	no	—
134	C₂×C₄⋊C₄	—	94	4	no	—
135	C₂×C₄⋊C₄	—	94	4	no	—
136	C₂×C₄⋊C₄	—	94	2	no	—
137	C₂×C₄²	C₄×D₄₇	1	2	yes	—
138	C₂×C₄²	D₁₈₈	1	2	yes	—
139	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
141	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	94	4	no	—
144	C₄⋊C₈	—	188	4	no	—
145	C₄⋊C₈	D₁₈₈	1	4	yes	—
146	C₄×C₈	—	4	2	no	—
147	C₂×Q₈	—	188	2	no	—
148	C₂×Q₈	—	188	2	no	—
149	C₂×Q₈	—	188	4	no	—
150	C₂×Q₈	—	188	4	no	—
151	C₂²×C₄	—	94	4	no	—
152	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
153	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	188	2	no	—
161	C₂×C₈	—	188	4	no	—
162	C₄⋊C₄	—	94	8	no	—
163	C₄⋊C₄	—	188	8	no	—
164	C₄⋊C₄	—	188	8	no	—
165	C₄⋊C₄	—	188	4	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	#10625325	1	4	yes	—
169	C₄²	D₃₇₆	1	4	yes	—
170	C₄²	#10625335	1	2	yes	—
171	C₄²	#10625337	1	2	yes	—
172	C₄²	C₂×D₁₈₈	1	2	yes	—
173	C₄²	#10625350	1	2	yes	—
174	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
175	Q₈	—	376	2	no	—
176	Q₈	—	376	2	no	—
177	Q₈	—	376	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	188	4	no	—
182	C₂×C₄	—	188	8	no	—
183	C₂×C₄	#13093942	1	4	yes	—
184	C₂×C₄	#13093946	1	4	yes	—
185	C₂×C₄	#13093946	1	4	yes	—
186	C₂×C₄	#13093947	1	4	yes	—
187	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
188	C₂×C₄	#13093998	1	2	yes	—
189	C₂×C₄	#13094012	1	1	yes	characteristic
190	C₂×C₄	#13094021	1	1	yes	characteristic
191	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
192	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
193	C₂×C₄	C₂×#10625325	1	2	yes	—
194	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
195	C₂×C₄	#13094054	1	2	yes	—
196	C₈	—	4	4	no	—
197	C₈	—	376	2	no	—
198	C₂²	#15689025	1	1	yes	characteristic
199	C₂²	C₂×#13093942	1	2	yes	—
200	C₂²	C₂×#13093946	1	2	yes	—
201	C₂²	C₂×#13093947	1	1	yes	characteristic
202	C₂²	#15689131	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	376	4	no	—
206	C₄	#15688868	1	8	yes	complemented
207	C₄	#15689328	1	2	yes	—
208	C₄	#15689333	1	2	yes	—
209	C₄	#15689470	1	2	yes	—
210	C₄	#15689496	1	1	yes	characteristic
211	C₄	#15689500	1	1	yes	characteristic
212	C₂	C₂×#15688868	1	2	yes	—
213	C₂	#20242059	1	2	yes	—
214	C₂	#20244194	1	1	yes	characteristic
215	C₂	#20244232	1	1	yes	characteristic
216	C₂	#20244311	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#9692
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
97. C₄×C₄₇	#332	semidirect
125. C₄₇	#9692	semidirect
206. C₄	#15688868	semidirect

Maximal subgroups

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

subgroup	index
#20251786	2
#20243540	2
C₂×#15688868	2
C₄×C₄⋊C₈×C₄₇	2
#9692	47