Subgroups — #26225693

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 192 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	#20244325	C₂	1	8	yes	maximal, complemented
3	#20243459	C₂	1	4	yes	maximal
4	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	#15689176	—	2	8	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
9	#15688878	C₂²	1	16	yes	—
10	C₄₇×#533	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
12	C₂×#125×C₄₇	C₄	1	2	yes	—
13	C₂×C₂²⋊Q₈×C₄₇	C₄	1	4	yes	—
14	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
15	C₂×#117×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
18	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
20	C₄×C₄₇⋊C₈	—	2	8	no	—
21	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₄⋊Q₈×C₄₇	—	2	8	no	—
25	#125×C₄₇	—	2	8	no	—
26	C₂²⋊Q₈×C₄₇	—	2	16	no	—
27	C₄×Q₈×C₄₇	—	2	8	no	—
28	C₄×Q₈×C₄₇	—	2	8	no	—
29	#117×C₄₇	—	2	8	no	—
30	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₇	D₄	1	8	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
39	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Q₈×C₄₇	—	2	8	no	—
41	C₂×Q₈×C₄₇	—	2	16	no	—
42	C₂²×C₄×C₄₇	—	2	8	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₄₇	—	2	8	no	—
52	C₄⋊C₄×C₄₇	—	2	16	no	—
53	C₄⋊C₄×C₄₇	—	2	8	no	—
54	C₂²⋊C₄×C₄₇	—	2	8	no	—
55	C₄²×C₄₇	—	2	8	no	—
56	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
58	C₄²×C₄₇	C₂×D₄	1	4	yes	—
59	C₄²×C₄₇	C₂×D₄	1	4	yes	—
60	C₂×C₄₇⋊C₈	—	2	8	no	—
61	C₂×C₄₇⋊C₈	—	2	16	no	—
62	C₂³×C₄₇	—	2	2	no	—
63	C₂³×C₄₇	—	2	4	no	—
64	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
65	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
66	Q₈×C₄₇	—	4	16	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	2	8	no	—
70	C₂×C₄×C₄₇	—	2	16	no	—
71	C₂×C₄×C₄₇	—	2	8	no	—
72	C₂×C₄×C₄₇	—	2	16	no	—
73	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
75	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
76	C₂×C₄×C₄₇	#131	1	8	yes	—
77	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
78	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
79	C₄₇⋊C₈	—	4	8	no	—
80	#21819	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	—	2	2	no	—
82	C₂²×C₄₇	—	2	4	no	—
83	C₂²×C₄₇	—	4	4	no	—
84	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄₇	#519	1	2	yes	—
86	C₂²×C₄₇	C₂×#131	1	4	yes	—
87	C₄×C₄₇	—	2	2	no	—
88	C₄×C₄₇	—	2	4	no	—
89	C₄×C₄₇	—	4	8	no	—
90	C₄×C₄₇	#408	1	8	yes	—
91	C₄×C₄₇	#536	1	4	yes	—
92	C₂×#533	D₄₇	1	1	yes	characteristic
93	C₂×#431	—	47	2	no	—
94	#1984	—	47	8	no	—
95	#1818	—	47	4	no	—
96	C₂×C₄₇	—	4	2	no	—
97	C₂×C₄₇	#2310	1	1	yes	characteristic
98	C₂×C₄₇	C₂×#408	1	2	yes	—
99	C₂×C₄₇	#2986	1	4	yes	—
100	#533	—	2	8	no	—
101	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
102	C₂×#125	Dic₄₇	1	2	yes	—
103	C₂×C₂²⋊Q₈	Dic₄₇	1	4	yes	—
104	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
105	C₂×#117	C₂×D₄₇	1	2	yes	—
106	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
107	#431	—	94	8	no	—
108	C₂×C₄⋊C₈	—	47	4	no	—
109	C₂×C₂²⋊C₈	—	47	4	no	—
110	C₂×C₄×C₈	—	47	4	no	—
111	#336	—	47	16	no	—
112	C₄₇	#21819	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
114	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
115	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
116	C₂²×C₈	—	47	8	no	—
117	C₄⋊Q₈	—	2	8	no	—
118	#125	—	2	8	no	—
119	C₂²⋊Q₈	—	2	16	no	—
120	C₄×Q₈	—	2	8	no	—
121	C₄×Q₈	—	2	8	no	—
122	#117	—	2	8	no	—
123	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
124	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
125	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
126	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
127	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
128	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
130	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
131	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
132	C₄⋊C₈	—	94	8	no	—
133	C₂²⋊C₈	—	94	8	no	—
134	C₄×C₈	—	94	8	no	—
135	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
136	C₂×Q₈	—	2	8	no	—
137	C₂×Q₈	—	2	16	no	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
141	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
142	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
146	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
147	C₂×C₈	—	94	8	no	—
148	C₂×C₈	—	94	16	no	—
149	C₄⋊C₄	—	2	8	no	—
150	C₄⋊C₄	—	2	16	no	—
151	C₄⋊C₄	—	2	8	no	—
152	C₂²⋊C₄	—	2	8	no	—
153	C₄²	—	2	8	no	—
154	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
155	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
156	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
157	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	4	no	—
160	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
161	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
162	Q₈	—	4	16	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
171	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
172	C₂×C₄	#13094031	1	8	yes	—
173	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
174	C₂×C₄	#13094072	1	4	yes	—
175	C₈	—	188	8	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	4	4	no	—
179	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
180	C₂²	#15689601	1	2	yes	—
181	C₂²	C₂×#13094031	1	4	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	8	no	—
185	C₄	#15689090	1	8	yes	—
186	C₄	#15689697	1	4	yes	—
187	C₂	—	4	2	no	—
188	C₂	#20245974	1	1	yes	characteristic
189	C₂	C₂×#15689090	1	2	yes	—
190	C₂	#20252490	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#533	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#21819
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #20244325	C₂	semidirect
112. C₄₇	#21819	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
#20244325	2
#20243459	2
C₂×C₄₇×#533	2
#21819	47