Subgroups — #26110027

Order 12032.

418 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	#20243540	C₂	1	1	yes	maximal, characteristic
2	#20243505	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688876	C₂	1	2	yes	maximal
4	C₂×#15688874	C₂	1	2	yes	maximal
5	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	#15688878	—	2	2	no	—
11	#15688876	—	2	4	no	—
12	#15688874	—	2	4	no	—
13	C₂×#97×C₄₇	C₄	1	2	yes	—
14	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
15	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
17	C₄³×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	C₂²×C₄₇⋊C₈	—	2	1	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#97×C₄₇	—	2	4	no	—
34	#97×C₄₇	—	2	2	no	—
35	#97×C₄₇	C₈	1	4	yes	complemented
36	C₄×C₈×C₄₇	D₄	1	4	yes	—
37	C₄×C₈×C₄₇	Q₈	1	4	yes	—
38	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₄⋊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×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
46	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
49	C₄²×C₄₇	—	2	2	no	—
50	C₄²×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	—	4	2	no	—
53	C₄²×C₄₇	C₂×C₈	1	2	yes	—
54	C₄²×C₄₇	M₁₆	1	2	yes	—
55	C₄²×C₄₇	C₂×D₄	1	2	yes	—
56	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄₇⋊C₈	—	4	2	no	—
58	C₂×C₄₇⋊C₈	—	4	1	no	—
59	C₂×C₄₇⋊C₈	—	4	1	no	—
60	C₂×C₄₇⋊C₈	—	4	4	no	—
61	C₂×C₄₇⋊C₈	—	4	2	no	—
62	C₂×C₄₇⋊C₈	—	4	2	no	—
63	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
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	—	2	2	no	—
70	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₄₇	#137	1	2	yes	—
80	C₈×C₄₇	—	2	4	no	—
81	C₈×C₄₇	C₄⋊C₈	1	8	yes	complemented
82	C₄₇⋊C₈	—	8	2	no	—
83	C₄₇⋊C₈	—	8	1	no	—
84	C₄₇⋊C₈	—	8	1	no	—
85	#8808	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	#384	1	1	yes	characteristic
87	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₄₇	#423	1	1	yes	characteristic
89	C₂²×C₄₇	#428	1	2	yes	—
90	C₂²×C₄₇	#430	1	2	yes	—
91	C₄×C₄₇	—	2	4	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	4	2	no	—
94	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
95	C₄×C₄₇	#423	1	2	yes	—
96	C₄×C₄₇	#496	1	1	yes	characteristic
97	C₄×C₄₇	#497	1	1	yes	characteristic
98	C₄×C₄₇	#501	1	2	yes	—
99	#1842	—	47	1	no	—
100	#1831	—	47	1	no	—
101	C₄×#97	D₄₇	1	1	yes	characteristic
102	C₂×#335	—	47	2	no	—
103	C₂×#334	—	47	2	no	—
104	C₂×C₄₇	#1565	1	2	yes	—
105	C₂×C₄₇	#1569	1	2	yes	—
106	C₂×C₄₇	#1842	1	1	yes	characteristic
107	C₂×C₄₇	#1847	1	1	yes	characteristic
108	C₂×C₄₇	#1849	1	1	yes	characteristic
109	C₂×C₄⋊C₈	—	47	1	no	—
110	C₂×C₄⋊C₈	—	47	2	no	—
111	C₂×C₄⋊C₈	—	47	1	no	—
112	C₂×C₄⋊C₈	—	94	2	no	—
113	C₂×#97	Dic₄₇	1	2	yes	—
114	C₂×#97	Dic₄₇	1	1	yes	characteristic
115	C₂×#97	Dic₄₇	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
117	C₄³	C₂×D₄₇	1	1	yes	characteristic
118	#336	—	94	2	no	—
119	#335	—	94	4	no	—
120	#334	—	94	4	no	—
121	C₄₇	#8808	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	—	94	1	no	—
123	C₂²×C₈	—	94	1	no	—
124	C₂²×C₈	—	94	2	no	—
125	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
126	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
127	C₂×C₄²	Dic₉₄	1	2	yes	—
128	C₂×C₄²	D₁₈₈	1	2	yes	—
129	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
131	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
132	C₄⋊C₈	—	94	2	no	—
133	C₄⋊C₈	—	94	2	no	—
134	C₄⋊C₈	—	94	2	no	—
135	C₄⋊C₈	—	94	2	no	—
136	C₄⋊C₈	—	188	4	no	—
137	#97	—	2	4	no	—
138	#97	—	2	2	no	—
139	#97	C₄₇⋊C₈	1	4	yes	complemented
140	C₄×C₈	Dic₉₄	1	4	yes	—
141	C₄×C₈	D₁₈₈	1	4	yes	—
142	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
143	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
144	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	188	2	no	—
150	C₂×C₈	—	188	1	no	—
151	C₂×C₈	—	188	1	no	—
152	C₂×C₈	—	188	4	no	—
153	C₂×C₈	—	188	2	no	—
154	C₂×C₈	—	188	2	no	—
155	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
156	C₂×C₈	#10625329	1	4	yes	—
157	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
158	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
164	C₄²	#10625329	1	2	yes	—
165	C₄²	C₂×Dic₉₄	1	2	yes	—
166	C₄²	C₂×D₁₈₈	1	2	yes	—
167	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	4	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
176	C₂×C₄	#13094021	1	1	yes	characteristic
177	C₂×C₄	#13094024	1	2	yes	—
178	C₂×C₄	#13094024	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×#10625329	1	2	yes	—
181	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
182	C₂×C₄	#13094052	1	2	yes	—
183	C₂×C₄	#13094057	1	2	yes	—
184	C₈	—	2	4	no	—
185	C₈	—	376	2	no	—
186	C₈	—	376	1	no	—
187	C₈	—	376	1	no	—
188	C₈	C₄⋊(C₄₇⋊C₈)	1	8	yes	complemented
189	C₂²	#15688993	1	1	yes	characteristic
190	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
191	C₂²	#15689147	1	1	yes	characteristic
192	C₂²	#15689165	1	2	yes	—
193	C₂²	#15689172	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
198	C₄	#15689147	1	2	yes	—
199	C₄	#15689512	1	1	yes	characteristic
200	C₄	#15689517	1	1	yes	characteristic
201	C₄	#15689532	1	2	yes	—
202	C₂	#20242140	1	2	yes	—
203	C₂	#20242155	1	2	yes	—
204	C₂	#20243540	1	1	yes	characteristic
205	C₂	#20243556	1	1	yes	characteristic
206	C₂	#20243563	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	#8808
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
35. #97×C₄₇	C₈	semidirect
81. C₈×C₄₇	C₄⋊C₈	semidirect
121. C₄₇	#8808	semidirect
139. #97	C₄₇⋊C₈	semidirect
188. C₈	C₄⋊(C₄₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
#20243505	2
C₂×#15688876	2
C₂×#15688874	2
C₄×#97×C₄₇	2
#8808	47