Subgroups — #26114977

Order 12032.

298 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244163	C₂	1	2	yes	maximal
2	#20242144	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
4	#20241284	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093959	—	2	2	no	—
7	#15689147	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15688887	—	2	2	no	—
10	#15688874	—	2	1	no	—
11	#15688874	—	2	1	no	—
12	#15688874	C₄	1	2	yes	—
13	#15688874	C₄	1	2	yes	—
14	C₄₇×#431	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#322	—	2	1	no	—
18	C₈²×C₄₇	—	2	1	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	#13093959	—	4	4	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)	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
27	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	1	no	—
31	C₂²⋊C₈×C₄₇	Q₈	1	2	yes	—
32	#97×C₄₇	—	2	1	no	—
33	C₄×C₈×C₄₇	—	2	1	no	—
34	C₄×C₈×C₄₇	—	2	1	no	—
35	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
38	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₄₇	—	2	2	no	—
43	M₁₆×C₄₇	SD₁₆	1	4	yes	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	2	2	no	—
47	C₂×C₈×C₄₇	—	4	1	no	—
48	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₄₇	#41	1	1	yes	characteristic
55	C₄²×C₄₇	#41	1	1	yes	characteristic
56	#10625329	—	4	2	no	—
57	#10625329	—	4	2	no	—
58	C₂×C₄₇⋊C₈	—	2	4	no	—
59	C₂×C₄₇⋊C₈	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	4	2	no	—
61	C₂×C₄₇⋊C₈	—	4	2	no	—
62	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	4	1	no	—
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₄₇	#135	1	2	yes	—
74	C₈×C₄₇	—	2	1	no	—
75	C₈×C₄₇	—	2	1	no	—
76	C₈×C₄₇	—	2	1	no	—
77	C₈×C₄₇	—	2	4	no	—
78	C₈×C₄₇	—	4	2	no	—
79	C₈×C₄₇	#108	1	2	yes	complemented
80	C₄₇⋊C₈	—	4	4	no	—
81	C₄₇⋊C₈	—	8	2	no	—
82	#9623	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	—	2	1	no	—
84	C₂²×C₄₇	—	2	1	no	—
85	C₂²×C₄₇	#389	1	1	yes	characteristic
86	C₂²×C₄₇	#498	1	1	yes	characteristic
87	C₂²×C₄₇	#499	1	1	yes	characteristic
88	C₄×C₄₇	—	2	1	no	—
89	C₄×C₄₇	—	2	1	no	—
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	C₂×#108	1	1	yes	characteristic
92	C₄×C₄₇	#430	1	1	yes	characteristic
93	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
94	C₄×C₄₇	#465	1	2	yes	—
95	#1950	—	47	2	no	—
96	#1566	—	47	1	no	—
97	C₂×#334	—	47	1	no	—
98	C₈×M₁₆	D₄₇	1	1	yes	characteristic
99	#1301	—	47	2	no	—
100	C₂×C₄₇	—	2	1	no	—
101	C₂×C₄₇	—	2	1	no	—
102	C₂×C₄₇	#1941	1	1	yes	characteristic
103	C₂×C₄₇	#1950	1	1	yes	characteristic
104	C₂×C₄₇	#1953	1	1	yes	characteristic
105	#431	C₂×D₄₇	1	1	yes	characteristic
106	C₂×#108	—	94	2	no	—
107	#423	—	47	2	no	—
108	C₂×C₄⋊C₈	—	47	2	no	—
109	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
111	#341	—	94	2	no	—
112	#334	—	47	2	no	—
113	#334	—	47	2	no	—
114	#334	—	94	1	no	—
115	#334	—	94	1	no	—
116	#322	—	2	1	no	—
117	C₈²	—	2	1	no	—
118	C₄₇	#9623	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	94	2	no	—
120	C₂×M₁₆	D₁₈₈	1	2	yes	—
121	C₂²×C₈	—	94	2	no	—
122	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
123	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
125	#108	—	188	4	no	—
126	C₄⋊C₈	—	2	1	no	—
127	C₄⋊C₈	—	47	4	no	—
128	C₄⋊C₈	—	94	2	no	—
129	C₄⋊C₈	—	94	2	no	—
130	C₄⋊C₈	—	94	2	no	—
131	C₂²⋊C₈	Dic₉₄	1	2	yes	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
136	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
139	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	188	2	no	—
144	M₁₆	—	188	2	no	—
145	M₁₆	#10625325	1	4	yes	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	94	4	no	—
151	C₂×C₈	—	188	2	no	—
152	C₂×C₈	—	188	2	no	—
153	C₂×C₈	—	188	2	no	—
154	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
157	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
158	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
159	C₄²	#10625350	1	1	yes	characteristic
160	C₄²	#10625350	1	1	yes	characteristic
161	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
162	C₂³	#13094024	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
167	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
169	C₂×C₄	#13093998	1	1	yes	characteristic
170	C₂×C₄	#13094003	1	1	yes	characteristic
171	C₂×C₄	#13094016	1	1	yes	characteristic
172	C₂×C₄	C₂×#10625325	1	2	yes	—
173	C₂×C₄	#13094046	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	2	no	—
179	C₈	—	376	2	no	—
180	C₈	—	188	4	no	—
181	C₈	#13093959	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#15689021	1	1	yes	characteristic
185	C₂²	#15689521	1	1	yes	characteristic
186	C₂²	#15689525	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#13093959	1	1	yes	characteristic
191	C₄	#15689171	1	1	yes	characteristic
192	C₄	C₄×#10625325	1	2	yes	—
193	C₄	#15689352	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#20244116	1	1	yes	characteristic
197	C₂	#20244163	1	1	yes	characteristic
198	C₂	#20244178	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₈×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	#389
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₈×M₁₆	D₄₇
hypocenter	C₄₇	#9623
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
79. C₈×C₄₇	#108	semidirect
118. C₄₇	#9623	semidirect
181. C₈	#13093959	semidirect

Maximal subgroups

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

subgroup	index
#20244163	2
#20242144	2
C₂×#15688874	2
#20241284	2
C₈×M₁₆×C₄₇	2
#9623	47