Subgroups — #26213754

Order 12032.

1286 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	C₂×C₄×#10625329	C₂	1	2	yes	maximal
2	#20243137	C₂	1	4	yes	maximal, complemented
3	#20243134	C₂	1	4	yes	maximal, complemented
4	#20243082	C₂	1	4	yes	maximal
5	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	4	yes	—
7	C₄×#10625329	C₄	1	16	yes	complemented
8	C₂×#13093935	C₂²	1	4	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
10	#15688878	C₂²	1	16	yes	—
11	C₂²×C₄²×C₄₇	C₄	1	4	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄³×C₄₇	C₄	1	4	yes	—
15	C₄³×C₄₇	C₂²	1	2	yes	—
16	C₄³×C₄₇	C₂²	1	2	yes	—
17	C₂×#10625329	—	2	16	no	—
18	C₂×#10625329	C₂×C₄	1	16	yes	—
19	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
20	#13093935	C₂×C₄	1	16	yes	complemented
21	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
22	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	—	2	8	no	—
26	C₂×C₄²×C₄₇	—	2	16	no	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	—	2	8	no	—
38	C₂²×C₄×C₄₇	—	2	16	no	—
39	C₂²×C₄×C₄₇	—	2	8	no	—
40	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	2	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	4	yes	—
46	C₂²×C₄×C₄₇	#41	1	8	yes	—
47	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₄²×C₄₇	—	2	8	no	—
49	C₄²×C₄₇	—	2	16	no	—
50	C₄²×C₄₇	—	2	8	no	—
51	C₄²×C₄₇	—	2	8	no	—
52	C₄²×C₄₇	C₄²	1	8	yes	—
53	C₄²×C₄₇	M₁₆	1	16	yes	complemented
54	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
55	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
56	#10625329	—	2	32	no	—
57	C₂×C₄₇⋊C₈	—	2	16	no	—
58	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	—
59	C₂³×C₄₇	—	2	4	no	—
60	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
61	C₂³×C₄₇	#117	1	4	yes	—
62	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	32	no	—
64	C₂×C₄×C₄₇	—	2	8	no	—
65	C₂×C₄×C₄₇	—	2	4	no	—
66	C₂×C₄×C₄₇	—	2	8	no	—
67	C₂×C₄×C₄₇	—	2	16	no	—
68	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
71	C₂×C₄×C₄₇	#117	1	16	yes	—
72	C₂×C₄×C₄₇	#117	1	4	yes	—
73	C₂×C₄×C₄₇	C₂×M₁₆	1	16	yes	—
74	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
75	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
77	C₄₇⋊C₈	—	2	16	no	—
78	#20364	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	4	no	—
80	C₂²×C₄₇	—	2	8	no	—
81	C₂²×C₄₇	#376	1	4	yes	—
82	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
83	C₂²×C₄₇	C₂×#117	1	2	yes	—
84	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
85	C₄×C₄₇	—	2	4	no	—
86	C₄×C₄₇	—	2	16	no	—
87	C₄×C₄₇	#376	1	4	yes	—
88	C₄×C₄₇	C₄×M₁₆	1	8	yes	—
89	C₄×C₄₇	C₂×#117	1	4	yes	—
90	C₂×C₄×M₁₆	—	47	2	no	—
91	C₂×C₄³	D₄₇	1	1	yes	characteristic
92	#1745	—	47	4	no	—
93	#1744	—	47	4	no	—
94	#1728	—	47	4	no	—
95	C₂×C₄₇	—	2	4	no	—
96	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
97	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
98	C₂×C₄₇	#2920	1	4	yes	—
99	C₂²×M₁₆	—	47	4	no	—
100	C₂²×C₄²	Dic₄₇	1	4	yes	—
101	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
102	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
103	C₄×M₁₆	—	47	16	no	—
104	C₂×#97	—	47	4	no	—
105	C₂×C₄×C₈	—	47	4	no	—
106	C₄³	Dic₄₇	1	4	yes	—
107	C₄³	C₂×D₄₇	1	2	yes	—
108	C₄³	C₂×D₄₇	1	2	yes	—
109	#336	—	47	16	no	—
110	C₄₇	#20364	1	1	yes	characteristic, complemented, Sylow
111	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
112	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
113	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₂×M₁₆	—	47	16	no	—
115	C₂×M₁₆	—	94	16	no	—
116	C₂²×C₈	—	47	8	no	—
117	C₂×C₄²	—	2	8	no	—
118	C₂×C₄²	—	2	16	no	—
119	C₂×C₄²	C₄×D₄₇	1	8	yes	—
120	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
121	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
122	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
123	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
124	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
125	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
126	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
128	#97	—	47	16	no	—
129	C₄×C₈	—	47	16	no	—
130	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	16	no	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
135	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
136	C₂²×C₄	#10625350	1	8	yes	—
137	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
139	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
141	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
142	M₁₆	—	94	32	no	—
143	C₂×C₈	—	47	16	no	—
144	C₂×C₈	—	94	16	no	—
145	C₄²	—	2	8	no	—
146	C₄²	—	2	16	no	—
147	C₄²	—	2	8	no	—
148	C₄²	—	2	8	no	—
149	C₄²	#10625329	1	16	yes	complemented
150	C₄²	C₄×Dic₄₇	1	8	yes	—
151	C₄²	C₂×C₄×D₄₇	1	4	yes	—
152	C₄²	C₂×C₄×D₄₇	1	4	yes	—
153	C₂³	—	2	4	no	—
154	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
155	C₂³	#13093983	1	4	yes	—
156	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
157	C₂×C₄	—	2	32	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	16	no	—
162	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
163	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
164	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
165	C₂×C₄	#13093981	1	16	yes	—
166	C₂×C₄	#13093983	1	4	yes	—
167	C₂×C₄	C₂×#10625329	1	16	yes	—
168	C₂×C₄	C₂×#10625329	1	8	yes	—
169	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
170	C₂×C₄	C₂×#10625350	1	4	yes	—
171	C₈	—	94	16	no	—
172	C₂²	—	2	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	#15688969	1	4	yes	—
175	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
176	C₂²	C₂×#13093983	1	2	yes	—
177	C₂²	C₂²×#10625329	1	4	yes	—
178	C₄	—	2	16	no	—
179	C₄	—	2	4	no	—
180	C₄	#15688969	1	4	yes	—
181	C₄	C₄×#10625329	1	8	yes	—
182	C₄	C₂×#13093981	1	4	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₂×#15688969	1	1	yes	characteristic
185	C₂	C₂×C₄×#10625329	1	2	yes	—
186	C₂	#20252021	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄³×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#20364
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20243137	C₂	semidirect
3. #20243134	C₂	semidirect
7. C₄×#10625329	C₄	semidirect
20. #13093935	C₂×C₄	semidirect
21. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
53. C₄²×C₄₇	M₁₆	semidirect
110. C₄₇	#20364	semidirect
149. C₄²	#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10625329	2
#20243137	2
#20243134	2
#20243082	2
C₂×C₄³×C₄₇	2
#20364	47