Subgroups — #26216471

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	2	yes	maximal
2	#20243519	C₂	1	4	yes	maximal, complemented
3	#20243516	C₂	1	4	yes	maximal, complemented
4	#20243105	C₂	1	4	yes	maximal
5	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	C₂²	1	4	yes	—
7	#15688993	C₂²	1	4	yes	—
8	#15688989	C₂²	1	4	yes	—
9	#15688983	—	2	4	no	—
10	C₄×C₄⋊Dic₄₇	—	2	4	no	—
11	#15688886	C₂²	1	16	yes	—
12	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
17	#13093983	—	2	16	no	—
18	C₂×C₄⋊Dic₄₇	—	2	8	no	—
19	C₂×C₄⋊Dic₄₇	C₂³	1	8	yes	—
20	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
21	C₂×C₄×Dic₄₇	—	2	4	no	—
22	C₄₇⋊#95	—	2	8	no	—
23	C₂³×C₄×C₄₇	D₄	1	2	yes	—
24	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	—	2	8	no	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
35	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	—	2	8	no	—
37	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
43	C₂²×C₄×C₄₇	#41	1	8	yes	—
44	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
45	C₂²×Dic₄₇	—	2	8	no	—
46	C₂×C₈×C₄₇	—	2	16	no	—
47	C₂×C₈×C₄₇	—	2	8	no	—
48	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
49	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
50	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
51	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
52	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
53	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
54	C₂²⋊Dic₄₇	—	4	8	no	—
55	C₄⋊Dic₄₇	—	2	16	no	—
56	C₄⋊Dic₄₇	—	4	16	no	—
57	C₄×Dic₄₇	—	4	8	no	—
58	C₂³×C₄₇	—	2	4	no	—
59	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₄₇	#124	1	4	yes	—
61	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	—	2	16	no	—
64	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
67	C₂×C₄×C₄₇	#117	1	8	yes	—
68	C₂×C₄×C₄₇	#124	1	4	yes	—
69	C₂×C₄×C₄₇	#125	1	8	yes	—
70	C₂×C₄×C₄₇	#135	1	8	yes	—
71	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
74	C₂×Dic₄₇	—	4	8	no	—
75	C₂×Dic₄₇	—	4	16	no	—
76	C₈×C₄₇	—	2	8	no	—
77	#20760	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	—	2	8	no	—
79	C₂²×C₄₇	—	2	4	no	—
80	C₂²×C₄₇	#382	1	4	yes	—
81	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄₇	C₂×#124	1	2	yes	—
83	C₂²×C₄₇	C₂×#135	1	4	yes	—
84	C₄×C₄₇	—	2	4	no	—
85	C₄×C₄₇	#382	1	4	yes	—
86	C₄×C₄₇	#427	1	8	yes	—
87	C₄×C₄₇	C₂×#117	1	2	yes	—
88	C₄×C₄₇	C₂×#125	1	2	yes	—
89	Dic₄₇	—	8	8	no	—
90	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
91	#2285	—	47	2	no	—
92	#1835	—	47	4	no	—
93	#1834	—	47	4	no	—
94	#1735	—	47	4	no	—
95	C₂×C₄₇	—	2	4	no	—
96	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
97	C₂×C₄₇	C₂×#427	1	2	yes	—
98	C₂×C₄₇	#3131	1	4	yes	—
99	C₂³×C₈	C₂×D₄₇	1	2	yes	—
100	C₂×#117	—	47	4	no	—
101	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
102	C₂×C₄×C₈	Dic₄₇	1	8	yes	complemented
103	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
105	#384	—	47	4	no	—
106	#382	—	47	4	no	—
107	#380	—	94	4	no	—
108	C₄×C₄⋊C₄	—	94	4	no	—
109	#340	—	47	16	no	—
110	C₄₇	#20760	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₄	D₁₈₈	1	2	yes	—
112	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
113	C₂²×C₈	—	2	8	no	—
114	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
115	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
116	#117	—	94	16	no	—
117	C₂×C₄⋊C₄	—	47	8	no	—
118	C₂×C₄⋊C₄	—	94	8	no	—
119	C₂×C₂²⋊C₄	—	94	4	no	—
120	C₂×C₄²	—	94	4	no	—
121	C₂×C₄²	Dic₉₄	1	4	yes	—
122	C₂×C₄²	D₁₈₈	1	2	yes	—
123	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
124	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₄₇	1	8	yes	—
127	C₄×C₈	C₂×Dic₄₇	1	8	yes	—
128	#95	—	94	8	no	—
129	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	—	94	8	no	—
132	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
133	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
134	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
135	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
136	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
137	C₂²×C₄	#10625350	1	8	yes	—
138	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
139	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
140	C₂×C₈	—	2	16	no	—
141	C₂×C₈	—	2	8	no	—
142	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
143	C₄⋊C₄	—	94	16	no	—
144	C₄⋊C₄	—	188	16	no	—
145	C₂²⋊C₄	—	188	8	no	—
146	C₄²	—	188	8	no	—
147	C₄²	C₄⋊Dic₄₇	1	8	yes	—
148	C₄²	C₂×Dic₉₄	1	2	yes	—
149	C₄²	C₂×Dic₉₄	1	2	yes	—
150	C₄²	C₂²×Dic₄₇	1	2	yes	—
151	C₄²	C₂²×Dic₄₇	1	2	yes	—
152	C₂³	—	2	4	no	—
153	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
154	C₂³	#13094012	1	4	yes	—
155	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	16	no	—
158	C₂×C₄	—	188	8	no	—
159	C₂×C₄	—	188	16	no	—
160	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
163	C₂×C₄	#13093983	1	8	yes	—
164	C₂×C₄	#13094012	1	4	yes	—
165	C₂×C₄	#13094016	1	8	yes	—
166	C₂×C₄	#13094046	1	8	yes	—
167	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
168	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
169	C₂×C₄	C₂×#10625350	1	4	yes	—
170	C₈	—	2	8	no	—
171	C₂²	—	2	8	no	—
172	C₂²	—	2	4	no	—
173	C₂²	#15688989	1	4	yes	—
174	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
175	C₂²	C₂×#13094012	1	2	yes	—
176	C₂²	C₂×#13094046	1	4	yes	—
177	C₄	—	2	4	no	—
178	C₄	—	376	8	no	—
179	C₄	#15688989	1	4	yes	—
180	C₄	#15689161	1	8	yes	—
181	C₄	C₂×#13093983	1	2	yes	—
182	C₄	C₂×#13094016	1	2	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₂×#15688989	1	1	yes	characteristic
185	C₂	C₂×#15689161	1	2	yes	—
186	C₂	#20253711	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#20760
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20243519	C₂	semidirect
3. #20243516	C₂	semidirect
14. C₂×C₄×C₈×C₄₇	C₄	semidirect
102. C₂×C₄×C₈	Dic₄₇	semidirect
110. C₄₇	#20760	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
#20243519	2
#20243516	2
#20243105	2
C₂²×C₄×C₈×C₄₇	2
#20760	47