Subgroups — #26215042

Order 12032.

866 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	2	yes	maximal
2	#20243263	C₂	1	8	yes	maximal, complemented
3	#20243154	C₂	1	4	yes	maximal
4	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
5	#15689180	C₄	1	16	yes	complemented
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	C₂×#13093935	C₂²	1	4	yes	—
9	C₂×#13093935	C₂²	1	4	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
11	#15688878	C₂²	1	8	yes	—
12	C₂×#117×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#384	C₄	1	2	yes	—
15	C₄₇×#382	C₄	1	2	yes	—
16	C₄₇×#380	C₄	1	4	yes	—
17	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
18	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
19	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
20	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	—
21	#13093935	—	4	4	no	—
22	#13093935	C₂×C₄	1	16	yes	—
23	C₄×C₄₇⋊C₈	—	4	4	no	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	#117×C₄₇	—	4	2	no	—
27	#117×C₄₇	—	4	2	no	—
28	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
30	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
36	#95×C₄₇	C₂×C₄	1	8	yes	—
37	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	4	4	no	—
41	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₄⋊C₄×C₄₇	—	4	8	no	—
49	C₄⋊C₄×C₄₇	—	4	4	no	—
50	C₂²⋊C₄×C₄₇	—	4	4	no	—
51	C₄²×C₄₇	—	2	4	no	—
52	C₄²×C₄₇	—	2	4	no	—
53	C₄²×C₄₇	C₄²	1	4	yes	—
54	C₄²×C₄₇	C₄²	1	4	yes	—
55	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
56	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂×C₄₇⋊C₈	—	2	8	no	—
58	C₂×C₄₇⋊C₈	—	2	8	no	—
59	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	—
60	C₂³×C₄₇	—	2	1	no	—
61	C₂³×C₄₇	—	2	1	no	—
62	C₂³×C₄₇	—	2	4	no	—
63	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
64	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	1	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	2	8	no	—
70	C₂×C₄×C₄₇	—	2	8	no	—
71	C₂×C₄×C₄₇	—	2	16	no	—
72	C₂×C₄×C₄₇	—	4	8	no	—
73	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
75	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
77	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
78	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
79	C₄₇⋊C₈	—	2	16	no	—
80	#20539	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	—	2	4	no	—
82	C₂²×C₄₇	—	2	2	no	—
83	C₂²×C₄₇	—	4	2	no	—
84	C₂²×C₄₇	—	4	2	no	—
85	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
86	C₂²×C₄₇	#518	1	1	yes	characteristic
87	C₂²×C₄₇	#519	1	1	yes	characteristic
88	C₂²×C₄₇	#568	1	4	yes	—
89	C₄×C₄₇	—	2	4	no	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	—	4	8	no	—
93	C₄×C₄₇	#520	1	4	yes	—
94	C₂×#432	—	47	2	no	—
95	#2285	D₄₇	1	1	yes	characteristic
96	#1771	—	47	8	no	—
97	#1750	—	47	4	no	—
98	C₂×C₄₇	—	4	2	no	—
99	C₂×C₄₇	#2274	1	1	yes	characteristic
100	C₂×C₄₇	#2873	1	2	yes	—
101	C₂×C₄₇	#2980	1	2	yes	—
102	C₂×C₄₇	#2987	1	2	yes	—
103	C₂×#117	C₂×D₄₇	1	2	yes	—
104	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
105	#432	—	47	16	no	—
106	C₂×C₄⋊C₈	—	47	4	no	—
107	C₂×C₂²⋊C₈	—	47	4	no	—
108	C₂×#97	—	47	4	no	—
109	C₂×#97	—	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	#20539	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	2	no	—
121	#117	—	4	2	no	—
122	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
123	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
124	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
125	C₂×C₄²	C₄×D₄₇	1	8	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	—	47	16	no	—
131	C₂²⋊C₈	—	47	16	no	—
132	#97	—	47	16	no	—
133	#97	—	188	4	no	—
134	C₄×C₈	—	188	4	no	—
135	#95	C₂×Dic₄₇	1	8	yes	—
136	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	4	4	no	—
140	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
141	C₂²×C₄	C₂×C₄×D₄₇	1	4	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	—	47	16	no	—
148	C₂×C₈	—	94	8	no	—
149	C₂×C₈	—	94	8	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	4	no	—
155	C₄²	C₄×Dic₄₇	1	4	yes	—
156	C₄²	C₄×Dic₄₇	1	4	yes	—
157	C₄²	C₂×C₄×D₄₇	1	4	yes	—
158	C₄²	C₂×C₄×D₄₇	1	4	yes	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	1	no	—
161	C₂³	—	2	4	no	—
162	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
163	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
173	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
174	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
175	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
176	C₂×C₄	#13094069	1	2	yes	—
177	C₂×C₄	#13094072	1	2	yes	—
178	C₈	—	94	16	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
184	C₂²	#15689598	1	1	yes	characteristic
185	C₂²	#15689601	1	1	yes	characteristic
186	C₂²	#15689890	1	4	yes	—
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	8	no	—
191	C₄	#15689604	1	4	yes	—
192	C₂	—	4	2	no	—
193	C₂	#20245745	1	1	yes	characteristic
194	C₂	#20251807	1	2	yes	—
195	C₂	#20252459	1	2	yes	—
196	C₂	#20252494	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#20539
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
2. #20243263	C₂	semidirect
5. #15689180	C₄	semidirect
116. C₄₇	#20539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
#20243263	2
#20243154	2
C₄₇×#2285	2
#20539	47