Subgroups — #26213657

Order 12032.

1956 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×#15689890	C₂	1	2	yes	maximal
2	C₂×C₄₇⋊#338	C₂	1	12	yes	maximal, complemented
3	C₂×C₄₇×#568	C₂	1	1	yes	maximal, characteristic
4	#15689890	—	2	8	no	—
5	#15689890	—	2	8	no	—
6	C₂×#13094031	C₄	1	16	yes	—
7	C₂²×#10625329	C₂²	1	12	yes	—
8	C₄₇⋊#338	C₂²	1	16	yes	—
9	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#568	—	2	8	no	—
11	C₂×#131×C₄₇	C₄	1	8	yes	—
12	C₂²×M₁₆×C₄₇	C₂²	1	6	yes	—
13	#13094031	—	2	64	no	—
14	C₂×#10625329	—	2	24	no	—
15	C₂×#10625329	—	2	24	no	—
16	C₂×#10625329	C₂×C₄	1	24	yes	—
17	C₂×#10625329	C₂³	1	8	yes	—
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	24	yes	—
19	C₂×#41×C₄₇	C₂×C₄	1	8	yes	—
20	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
21	C₂×#41×C₄₇	D₄	1	4	yes	—
22	C₂×#41×C₄₇	D₄	1	8	yes	—
23	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
24	C₂²×D₄×C₄₇	Q₈	1	3	yes	—
25	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
26	#131×C₄₇	—	2	32	no	—
27	C₂×M₁₆×C₄₇	—	2	24	no	—
28	C₂×M₁₆×C₄₇	C₂×C₄	1	12	yes	—
29	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	12	yes	—
31	C₂⁴×C₄₇	C₂×Q₈	1	3	yes	—
32	#41×C₄₇	C₄²	1	16	yes	—
33	#41×C₄₇	C₂²⋊C₄	1	32	yes	—
34	#41×C₄₇	C₂²⋊C₄	1	16	yes	—
35	C₂×Q₈×C₄₇	C₄⋊C₄	1	8	yes	—
36	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
37	C₂×D₄×C₄₇	C₄⋊C₄	1	24	yes	—
38	C₂×D₄×C₄₇	C₄⋊C₄	1	12	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
43	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
44	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
45	M₁₆×C₄₇	—	2	24	no	—
46	M₁₆×C₄₇	—	2	8	no	—
47	C₂×C₈×C₄₇	—	2	24	no	—
48	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
49	#10625329	—	2	48	no	—
50	#10625329	—	2	8	no	—
51	#10625329	—	2	8	no	—
52	C₂×C₄₇⋊C₈	—	2	48	no	—
53	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	—
54	C₂³×C₄₇	—	8	3	no	—
55	C₂³×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
56	C₂³×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
57	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
58	Q₈×C₄₇	#95	1	16	yes	—
59	D₄×C₄₇	#95	1	48	yes	—
60	C₂×C₄×C₄₇	C₂×C₄²	1	12	yes	—
61	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
62	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
64	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
65	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
66	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
68	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
69	C₈×C₄₇	—	2	8	no	—
70	C₄₇⋊C₈	—	2	16	no	—
71	#20340	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	4	1	no	—
73	C₂²×C₄₇	—	4	12	no	—
74	C₂²×C₄₇	—	4	6	no	—
75	C₂²×C₄₇	C₂×#95	1	12	yes	—
76	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
77	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₄₇	C₂×#95	1	12	yes	—
79	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
80	C₄×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
81	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×#568	—	47	2	no	—
83	C₂×#568	D₄₇	1	1	yes	characteristic
84	C₂×#338	—	47	12	no	—
85	C₂×C₄₇	—	2	2	no	—
86	C₂×C₄₇	—	2	1	no	—
87	C₂×C₄₇	—	2	12	no	—
88	C₂×C₄₇	C₂²×#95	1	1	yes	characteristic
89	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
90	#568	—	2	8	no	—
91	#568	—	94	8	no	—
92	#568	—	94	8	no	—
93	C₂×#131	—	47	16	no	—
94	C₂×#131	Dic₄₇	1	8	yes	—
95	C₂²×M₁₆	—	47	12	no	—
96	C₂²×M₁₆	C₂×D₄₇	1	6	yes	—
97	#338	—	47	16	no	—
98	C₄₇	#20340	1	1	yes	characteristic, complemented, Sylow
99	C₂×#41	C₄×D₄₇	1	8	yes	—
100	C₂×#41	D₁₈₈	1	4	yes	—
101	C₂×#41	C₂×Dic₄₇	1	4	yes	—
102	C₂×#41	C₂²⋊D₄₇	1	8	yes	—
103	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
104	C₂²×D₄	Dic₉₄	1	3	yes	—
105	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
106	#131	—	2	32	no	—
107	#131	—	94	64	no	—
108	C₂×M₁₆	—	2	24	no	—
109	C₂×M₁₆	—	47	24	no	—
110	C₂×M₁₆	—	47	8	no	—
111	C₂×M₁₆	—	94	24	no	—
112	C₂×M₁₆	—	94	24	no	—
113	C₂×M₁₆	C₂×Dic₄₇	1	12	yes	—
114	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
115	C₂²×C₈	—	47	24	no	—
116	C₂²×C₈	C₂×Dic₄₇	1	12	yes	—
117	C₂⁴	C₂×Dic₉₄	1	3	yes	—
118	#41	C₄×Dic₄₇	1	16	yes	—
119	#41	(C₂×D₄₇)⋊C₄	1	32	yes	—
120	#41	C₂²⋊Dic₄₇	1	16	yes	—
121	C₂×Q₈	Dic₄₇⋊C₄	1	8	yes	—
122	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
123	C₂×D₄	Dic₄₇⋊C₄	1	24	yes	—
124	C₂×D₄	C₄⋊Dic₄₇	1	12	yes	—
125	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
126	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
127	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
128	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
130	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
131	M₁₆	—	2	24	no	—
132	M₁₆	—	2	8	no	—
133	M₁₆	—	94	48	no	—
134	M₁₆	—	94	8	no	—
135	M₁₆	—	94	8	no	—
136	C₂×C₈	—	2	24	no	—
137	C₂×C₈	—	47	16	no	—
138	C₂×C₈	—	94	48	no	—
139	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
140	C₂³	—	8	3	no	—
141	C₂³	C₂×Dic₄₇⋊C₄	1	12	yes	—
142	C₂³	C₂×C₄⋊Dic₄₇	1	6	yes	—
143	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
144	Q₈	C₄₇⋊#95	1	16	yes	—
145	D₄	C₄₇⋊#95	1	48	yes	—
146	C₂×C₄	C₂×C₄×Dic₄₇	1	12	yes	—
147	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	24	yes	—
148	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
149	C₂×C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
150	C₂×C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
151	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
152	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
154	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
155	C₈	—	2	8	no	—
156	C₈	—	94	16	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	12	no	—
159	C₂²	—	4	6	no	—
160	C₂²	C₂×C₄₇⋊#95	1	12	yes	—
161	C₂²	C₂²×Dic₄₇⋊C₄	1	2	yes	—
162	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
163	C₄	C₂×C₄₇⋊#95	1	12	yes	—
164	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
165	C₄	C₂²×(C₂×D₄₇)⋊C₄	1	2	yes	—
166	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	—	2	1	no	—
169	C₂	—	2	12	no	—
170	C₂	C₂²×C₄₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#568	C₂
socle	C₂×C₄₇	C₂²×#95
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#568	D₄₇
hypocenter	C₄₇	#20340
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇⋊#338	C₂	semidirect
98. C₄₇	#20340	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689890	2
C₂×C₄₇⋊#338	2
C₂×C₄₇×#568	2
#20340	47