Subgroups — #26147478

Order 12032.

1422 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688989	C₂	1	2	yes	maximal
2	#20241618	C₂	1	4	yes	maximal
3	#20241615	C₂	1	8	yes	maximal
4	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
6	#15688989	—	2	8	no	—
7	C₂×C₄₇⋊#95	C₂²	1	8	yes	—
8	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
9	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄³×C₄₇	C₄	1	8	yes	complemented
13	C₂³×Dic₄₇	C₂³	1	8	yes	—
14	C₂×C₄⋊Dic₄₇	—	2	8	no	—
15	C₂×C₄⋊Dic₄₇	—	2	16	no	—
16	C₄₇⋊#95	—	2	16	no	—
17	C₄₇⋊#95	—	2	32	no	—
18	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
19	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
20	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₄₇	—	2	16	no	—
22	C₂×C₄²×C₄₇	—	2	8	no	—
23	C₂×C₄²×C₄₇	C₂×C₄	1	16	yes	—
24	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₄₇	D₄	1	4	yes	—
27	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
28	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	—	2	8	no	—
30	C₂²×C₄×C₄₇	—	2	8	no	—
31	C₂²×C₄×C₄₇	—	2	8	no	—
32	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
33	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
34	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
38	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
39	C₂²×C₄×C₄₇	#41	1	16	yes	—
40	C₂²×C₄×C₄₇	#41	1	8	yes	—
41	C₂²×Dic₄₇	—	2	8	no	—
42	C₂²×Dic₄₇	—	2	32	no	—
43	C₂²×Dic₄₇	—	2	16	no	—
44	C₄²×C₄₇	—	2	32	no	—
45	C₄²×C₄₇	—	2	16	no	—
46	C₄²×C₄₇	C₄⋊C₄	1	16	yes	—
47	C₄⋊Dic₄₇	—	4	16	no	—
48	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
49	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
50	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂³×C₄₇	C₂×#41	1	4	yes	—
52	C₂³×C₄₇	C₂×#41	1	8	yes	—
53	C₂×C₄×C₄₇	—	2	8	no	—
54	C₂×C₄×C₄₇	—	2	8	no	—
55	C₂×C₄×C₄₇	—	2	8	no	—
56	C₂×C₄×C₄₇	—	2	32	no	—
57	C₂×C₄×C₄₇	—	2	8	no	—
58	C₂×C₄×C₄₇	—	2	8	no	—
59	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
60	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
61	C₂×C₄×C₄₇	#117	1	16	yes	—
62	C₂×C₄×C₄₇	#117	1	8	yes	—
63	C₂×C₄×C₄₇	#124	1	8	yes	—
64	C₂×C₄×C₄₇	#125	1	8	yes	—
65	C₂×Dic₄₇	—	4	8	no	—
66	C₂×Dic₄₇	—	4	32	no	—
67	C₂×Dic₄₇	—	4	16	no	—
68	#14689	—	47	1	no	maximal, Sylow
69	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄₇	C₂×#117	1	2	yes	—
71	C₂²×C₄₇	C₂×#117	1	4	yes	—
72	C₂²×C₄₇	C₂×#124	1	2	yes	—
73	C₂²×C₄₇	C₂×#125	1	2	yes	—
74	C₂²×C₄₇	#529	1	16	yes	—
75	C₂²×C₄₇	#532	1	4	yes	—
76	C₂²×C₄₇	#533	1	4	yes	—
77	C₄×C₄₇	—	2	16	no	—
78	C₄×C₄₇	—	2	8	no	—
79	C₄×C₄₇	#382	1	8	yes	—
80	Dic₄₇	—	8	8	no	—
81	C₂×#382	—	47	2	no	—
82	C₂×C₄³	D₄₇	1	1	yes	characteristic
83	C₄⋊#95	—	47	4	no	—
84	#1434	—	47	8	no	—
85	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
86	C₂×C₄₇	#2283	1	4	yes	—
87	C₂×C₄₇	#2285	1	2	yes	—
88	C₂×C₄₇	#2861	1	4	yes	—
89	C₂×C₄₇	#2864	1	4	yes	—
90	C₂²×C₄⋊C₄	—	47	4	no	—
91	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
92	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
93	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
94	#382	—	94	8	no	—
95	C₂×#95	—	47	16	no	—
96	C₂×#95	—	47	8	no	—
97	C₄³	Dic₄₇	1	8	yes	complemented
98	C₄₇	#14689	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	—	47	8	no	—
100	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
101	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
102	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	94	8	no	—
104	C₂×C₄⋊C₄	—	94	16	no	—
105	C₂×C₄²	—	2	16	no	—
106	C₂×C₄²	—	2	8	no	—
107	C₂×C₄²	Dic₉₄	1	4	yes	—
108	C₂×C₄²	D₁₈₈	1	4	yes	—
109	C₂×C₄²	C₂×Dic₄₇	1	16	yes	—
110	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
111	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
112	#95	—	94	16	no	—
113	#95	—	94	32	no	—
114	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	2	8	no	—
118	C₂²×C₄	—	94	8	no	—
119	C₂²×C₄	—	94	32	no	—
120	C₂²×C₄	—	94	16	no	—
121	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
122	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
123	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
124	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
125	C₂²×C₄	#10625350	1	16	yes	—
126	C₂²×C₄	#10625350	1	8	yes	—
127	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
128	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
129	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
130	C₄⋊C₄	—	188	16	no	—
131	C₄²	—	2	32	no	—
132	C₄²	—	2	16	no	—
133	C₄²	C₄⋊Dic₄₇	1	16	yes	—
134	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
135	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
136	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
137	C₂³	C₂×#10625350	1	4	yes	—
138	C₂³	C₂×#10625350	1	8	yes	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	32	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	188	8	no	—
146	C₂×C₄	—	188	32	no	—
147	C₂×C₄	—	188	16	no	—
148	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
149	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
150	C₂×C₄	#13093983	1	16	yes	—
151	C₂×C₄	#13093983	1	8	yes	—
152	C₂×C₄	#13094012	1	8	yes	—
153	C₂×C₄	#13094016	1	8	yes	—
154	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
155	C₂²	C₂×#13093983	1	2	yes	—
156	C₂²	C₂×#13093983	1	4	yes	—
157	C₂²	C₂×#13094012	1	2	yes	—
158	C₂²	C₂×#13094016	1	2	yes	—
159	C₂²	#15689657	1	16	yes	—
160	C₂²	#15689673	1	4	yes	—
161	C₂²	#15689680	1	4	yes	—
162	C₄	—	2	16	no	—
163	C₄	—	2	8	no	—
164	C₄	—	376	8	no	—
165	C₄	#15688989	1	8	yes	—
166	C₂	C₂×#15688989	1	1	yes	characteristic
167	C₂	#20245786	1	4	yes	—
168	C₂	#20245795	1	2	yes	—
169	C₂	#20251760	1	4	yes	—
170	C₂	#20251775	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄³×C₄₇	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#14689
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
12. C₄³×C₄₇	C₄	semidirect
97. C₄³	Dic₄₇	semidirect
98. C₄₇	#14689	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688989	2
#20241618	2
#20241615	2
C₂×C₄³×C₄₇	2
#14689	47