Subgroups — #26213703

Order 12032.

1978 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×#15689890	C₂	1	2	yes	maximal
2	#20243115	C₂	1	6	yes	maximal, complemented
3	C₂×#15688888	C₂	1	6	yes	maximal, complemented
4	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689890	—	2	8	no	—
6	#15689890	C₄	1	16	yes	—
7	C₂×#13094031	—	2	8	no	—
8	C₂²×#10625329	C₂²	1	12	yes	—
9	#15688888	C₂²	1	4	yes	—
10	#15688888	C₂²	1	12	yes	—
11	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#131×C₄₇	—	2	8	no	—
13	C₂×#131×C₄₇	C₄	1	8	yes	—
14	C₂²×M₁₆×C₄₇	C₂²	1	3	yes	—
15	C₂³×C₈×C₄₇	C₂²	1	3	yes	—
16	#13094031	—	2	64	no	—
17	C₂×#10625329	—	2	12	no	—
18	C₂×#10625329	—	2	24	no	—
19	C₂×#10625329	C₂×C₄	1	48	yes	—
20	C₂×#10625329	C₂³	1	8	yes	—
21	C₂²×C₄₇⋊C₈	—	2	12	no	—
22	C₂×#41×C₄₇	C₂×C₄	1	8	yes	—
23	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
24	C₂×#41×C₄₇	D₄	1	4	yes	—
25	C₂×#41×C₄₇	D₄	1	8	yes	—
26	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
27	C₂²×D₄×C₄₇	Q₈	1	3	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
29	#131×C₄₇	—	2	32	no	—
30	C₂×M₁₆×C₄₇	—	2	12	no	—
31	C₂×M₁₆×C₄₇	C₂×C₄	1	12	yes	—
32	C₂²×C₈×C₄₇	—	2	12	no	—
33	C₂²×C₈×C₄₇	C₂×C₄	1	12	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
36	C₂⁴×C₄₇	C₂×Q₈	1	3	yes	—
37	#41×C₄₇	C₄²	1	16	yes	—
38	#41×C₄₇	C₂²⋊C₄	1	32	yes	—
39	#41×C₄₇	C₂²⋊C₄	1	16	yes	—
40	C₂×Q₈×C₄₇	C₄⋊C₄	1	8	yes	—
41	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
42	C₂×D₄×C₄₇	C₄⋊C₄	1	24	yes	—
43	C₂×D₄×C₄₇	C₄⋊C₄	1	12	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	6	yes	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
48	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
49	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	M₁₆×C₄₇	—	2	24	no	—
51	C₂×C₈×C₄₇	—	2	24	no	—
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	—	2	6	no	—
54	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
56	#10625329	—	2	48	no	—
57	#10625329	—	2	8	no	—
58	#10625329	C₂²×C₄	1	16	yes	—
59	C₂×C₄₇⋊C₈	—	2	48	no	—
60	C₂×C₄₇⋊C₈	—	2	8	no	—
61	C₂³×C₄₇	—	4	6	no	—
62	C₂³×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
63	C₂³×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
65	Q₈×C₄₇	#95	1	16	yes	—
66	D₄×C₄₇	#95	1	48	yes	—
67	C₂×C₄×C₄₇	C₂×C₄²	1	12	yes	—
68	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	24	yes	—
69	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	12	yes	—
70	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
73	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
75	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
76	C₈×C₄₇	—	2	2	no	—
77	C₈×C₄₇	—	2	6	no	—
78	C₄₇⋊C₈	—	2	16	no	—
79	#20351	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	—	2	2	no	—
81	C₂²×C₄₇	—	2	12	no	—
82	C₂²×C₄₇	—	4	12	no	—
83	C₂²×C₄₇	C₂×#95	1	12	yes	—
84	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
85	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₄₇	C₂×#95	1	12	yes	—
87	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
88	C₄×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
89	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×#568	—	47	2	no	—
91	C₂²×#131	D₄₇	1	1	yes	characteristic
92	#1738	—	47	6	no	—
93	C₂×#341	—	47	6	no	—
94	C₂×C₄₇	—	2	2	no	—
95	C₂×C₄₇	—	2	12	no	—
96	C₂×C₄₇	C₂²×#95	1	1	yes	characteristic
97	C₂×C₄₇	#2916	1	2	yes	—
98	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
99	#568	—	47	16	no	—
100	#568	—	94	8	no	—
101	C₂×#131	—	2	8	no	—
102	C₂×#131	—	94	8	no	—
103	C₂×#131	Dic₄₇	1	8	yes	—
104	C₂²×M₁₆	—	47	12	no	—
105	C₂²×M₁₆	C₂×D₄₇	1	3	yes	—
106	C₂³×C₈	C₂×D₄₇	1	3	yes	—
107	#341	—	47	4	no	—
108	#341	—	47	12	no	—
109	C₄₇	#20351	1	1	yes	characteristic, complemented, Sylow
110	C₂×#41	C₄×D₄₇	1	8	yes	—
111	C₂×#41	D₁₈₈	1	4	yes	—
112	C₂×#41	C₂×Dic₄₇	1	4	yes	—
113	C₂×#41	C₂²⋊D₄₇	1	8	yes	—
114	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
115	C₂²×D₄	Dic₉₄	1	3	yes	—
116	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
117	#131	—	2	32	no	—
118	#131	—	94	64	no	—
119	C₂×M₁₆	—	2	12	no	—
120	C₂×M₁₆	—	47	48	no	—
121	C₂×M₁₆	—	47	8	no	—
122	C₂×M₁₆	—	94	12	no	—
123	C₂×M₁₆	—	94	24	no	—
124	C₂×M₁₆	C₂×Dic₄₇	1	12	yes	—
125	C₂²×C₈	—	2	12	no	—
126	C₂²×C₈	—	94	12	no	—
127	C₂²×C₈	C₂×Dic₄₇	1	12	yes	—
128	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
130	C₂⁴	C₂×Dic₉₄	1	3	yes	—
131	#41	C₄×Dic₄₇	1	16	yes	—
132	#41	(C₂×D₄₇)⋊C₄	1	32	yes	—
133	#41	C₂²⋊Dic₄₇	1	16	yes	—
134	C₂×Q₈	Dic₄₇⋊C₄	1	8	yes	—
135	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
136	C₂×D₄	Dic₄₇⋊C₄	1	24	yes	—
137	C₂×D₄	C₄⋊Dic₄₇	1	12	yes	—
138	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
139	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
140	C₂²×C₄	C₂×D₁₈₈	1	6	yes	—
141	C₂²×C₄	C₂²×Dic₄₇	1	6	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
143	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
144	M₁₆	—	2	24	no	—
145	M₁₆	—	47	16	no	—
146	M₁₆	—	94	48	no	—
147	M₁₆	—	94	8	no	—
148	C₂×C₈	—	2	24	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	6	no	—
151	C₂×C₈	—	94	48	no	—
152	C₂×C₈	—	94	8	no	—
153	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
154	C₂×C₈	C₂²×Dic₄₇	1	6	yes	—
155	C₂³	—	4	6	no	—
156	C₂³	C₂×Dic₄₇⋊C₄	1	12	yes	—
157	C₂³	C₂×C₄⋊Dic₄₇	1	6	yes	—
158	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
159	Q₈	C₄₇⋊#95	1	16	yes	—
160	D₄	C₄₇⋊#95	1	48	yes	—
161	C₂×C₄	C₂×C₄×Dic₄₇	1	12	yes	—
162	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	24	yes	—
163	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	12	yes	—
164	C₂×C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
166	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
167	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	6	no	—
172	C₈	—	94	16	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	12	no	—
175	C₂²	—	4	12	no	—
176	C₂²	C₂×C₄₇⋊#95	1	12	yes	—
177	C₂²	C₂²×Dic₄₇⋊C₄	1	2	yes	—
178	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
179	C₄	C₂×C₄₇⋊#95	1	12	yes	—
180	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
181	C₄	C₂²×(C₂×D₄₇)⋊C₄	1	2	yes	—
182	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₂	—	2	2	no	—
184	C₂	—	2	12	no	—
185	C₂	C₂²×C₄₇⋊#95	1	1	yes	characteristic
186	C₂	#20251999	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#20351
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #20243115	C₂	semidirect
3. C₂×#15688888	C₂	semidirect
109. C₄₇	#20351	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
#20243115	2
C₂×#15688888	2
C₂²×#131×C₄₇	2
#20351	47