Subgroups — #26226614

Order 12032.

826 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	C₂×#15689176	C₂	1	2	yes	maximal
2	#20244107	C₂	1	8	yes	maximal, complemented
3	#20243372	C₂	1	4	yes	maximal
4	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
5	#15689176	—	2	8	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
10	#15688878	C₂²	1	8	yes	—
11	C₂×#117×C₄₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#384	C₄	1	2	yes	—
14	C₄₇×#382	C₄	1	2	yes	—
15	C₄₇×#380	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
17	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
18	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
19	C₄⋊(C₄₇⋊C₈)	—	4	8	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
21	C₄×C₄₇⋊C₈	—	2	8	no	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	#117×C₄₇	—	4	4	no	—
25	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
26	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	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	4	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
34	#95×C₄₇	C₂×C₄	1	8	yes	—
35	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	8	no	—
37	C₂²×C₄×C₄₇	—	2	2	no	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₄₇	—	2	16	no	—
48	C₄⋊C₄×C₄₇	—	4	4	no	—
49	C₂²⋊C₄×C₄₇	—	4	4	no	—
50	C₄²×C₄₇	—	2	8	no	—
51	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
53	C₄²×C₄₇	C₂×D₄	1	4	yes	—
54	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂×C₄₇⋊C₈	—	2	8	no	—
56	C₂×C₄₇⋊C₈	—	2	8	no	—
57	C₂×C₄₇⋊C₈	—	2	8	no	—
58	C₂³×C₄₇	—	2	1	no	—
59	C₂³×C₄₇	—	2	1	no	—
60	C₂³×C₄₇	—	2	4	no	—
61	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
62	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	4	no	—
66	C₂×C₄×C₄₇	—	2	4	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	2	16	no	—
69	C₂×C₄×C₄₇	—	2	8	no	—
70	C₂×C₄×C₄₇	—	2	16	no	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₄×C₄₇	#131	1	8	yes	—
75	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
78	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
79	C₄₇⋊C₈	—	4	8	no	—
80	#21952	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	—	2	2	no	—
82	C₂²×C₄₇	—	2	4	no	—
83	C₂²×C₄₇	—	4	2	no	—
84	C₂²×C₄₇	—	4	2	no	—
85	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄₇	#518	1	1	yes	characteristic
87	C₂²×C₄₇	#519	1	1	yes	characteristic
88	C₂²×C₄₇	C₂×#131	1	4	yes	—
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	—	4	8	no	—
92	C₄×C₄₇	#424	1	8	yes	—
93	C₄×C₄₇	#537	1	2	yes	—
94	C₄×C₄₇	#544	1	2	yes	—
95	C₂×#431	—	47	2	no	—
96	#2285	D₄₇	1	1	yes	characteristic
97	#1939	—	47	8	no	—
98	#1798	—	47	4	no	—
99	C₂×C₄₇	—	4	2	no	—
100	C₂×C₄₇	#2316	1	1	yes	characteristic
101	C₂×C₄₇	C₂×#424	1	2	yes	—
102	C₂×C₄₇	#2979	1	2	yes	—
103	C₂×C₄₇	#2986	1	2	yes	—
104	C₂×#117	C₂×D₄₇	1	2	yes	—
105	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
106	#431	—	94	8	no	—
107	C₂×C₄⋊C₈	—	47	4	no	—
108	C₂×C₄⋊C₈	—	47	8	no	—
109	C₂×C₂²⋊C₈	—	47	4	no	—
110	C₂×C₄×C₈	—	47	4	no	—
111	#384	Dic₄₇	1	2	yes	—
112	#382	Dic₄₇	1	2	yes	—
113	#380	Dic₄₇	1	4	yes	—
114	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
115	#336	—	47	8	no	—
116	C₄₇	#21952	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
118	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
119	C₂²×C₈	—	47	8	no	—
120	#117	—	4	4	no	—
121	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
122	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
123	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
124	C₂×C₄²	Dic₉₄	1	4	yes	—
125	C₂×C₄²	D₁₈₈	1	4	yes	—
126	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
127	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
129	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
130	C₄⋊C₈	—	94	8	no	—
131	C₄⋊C₈	—	188	8	no	—
132	C₂²⋊C₈	—	94	8	no	—
133	C₄×C₈	—	94	8	no	—
134	#95	C₂×Dic₄₇	1	8	yes	—
135	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
140	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
141	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
143	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
146	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
147	C₂×C₈	—	94	8	no	—
148	C₂×C₈	—	94	8	no	—
149	C₂×C₈	—	94	8	no	—
150	C₄⋊C₄	—	2	16	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	8	no	—
154	C₄²	C₄⋊Dic₄₇	1	4	yes	—
155	C₄²	C₄⋊Dic₄₇	1	4	yes	—
156	C₄²	C₂×Dic₉₄	1	4	yes	—
157	C₄²	C₂×D₁₈₈	1	4	yes	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	4	no	—
161	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
162	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	16	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
172	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
174	C₂×C₄	#13094031	1	8	yes	—
175	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
176	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
177	C₂×C₄	#13094069	1	2	yes	—
178	C₂×C₄	#13094072	1	2	yes	—
179	C₈	—	188	8	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
185	C₂²	#15689598	1	1	yes	characteristic
186	C₂²	#15689601	1	1	yes	characteristic
187	C₂²	C₂×#13094031	1	4	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	8	no	—
191	C₄	#15689150	1	8	yes	—
192	C₄	#15689704	1	2	yes	—
193	C₄	#15689754	1	2	yes	—
194	C₂	—	4	2	no	—
195	C₂	#20246006	1	1	yes	characteristic
196	C₂	C₂×#15689150	1	2	yes	—
197	C₂	#20252455	1	2	yes	—
198	C₂	#20252490	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2285	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#21952
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
2. #20244107	C₂	semidirect
116. C₄₇	#21952	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
#20244107	2
#20243372	2
C₄₇×#2285	2
#21952	47