Subgroups — #26216429

Order 12032.

1158 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	2	yes	maximal
2	#20243505	C₂	1	8	yes	maximal, complemented
3	#20243082	C₂	1	4	yes	maximal
4	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	4	yes	—
6	#15689147	—	2	8	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
8	#15688878	C₂²	1	16	yes	—
9	C₂²×C₄²×C₄₇	C₄	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	2	yes	—
13	C₄³×C₄₇	C₄	1	2	yes	—
14	C₄³×C₄₇	C₂²	1	4	yes	—
15	C₂×#10625329	—	2	8	no	—
16	C₂×#10625329	—	2	16	no	—
17	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
18	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
19	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	—	2	8	no	—
23	C₂×C₄²×C₄₇	—	2	16	no	—
24	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	D₄	1	4	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
34	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	2	8	no	—
36	C₂²×C₄×C₄₇	—	2	16	no	—
37	C₂²×C₄×C₄₇	—	2	8	no	—
38	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₄₇	#41	1	8	yes	—
46	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₄²×C₄₇	—	2	8	no	—
48	C₄²×C₄₇	—	2	16	no	—
49	C₄²×C₄₇	—	2	8	no	—
50	C₄²×C₄₇	—	2	8	no	—
51	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
53	C₄²×C₄₇	M₁₆	1	16	yes	complemented
54	C₄²×C₄₇	C₂×D₄	1	4	yes	—
55	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
56	#10625329	—	4	16	no	—
57	C₂×C₄₇⋊C₈	—	2	8	no	—
58	C₂×C₄₇⋊C₈	—	2	16	no	—
59	C₂³×C₄₇	—	2	4	no	—
60	C₂³×C₄₇	C₂×C₄⋊C₄	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×C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
71	C₂×C₄×C₄₇	#117	1	4	yes	—
72	C₂×C₄×C₄₇	#124	1	8	yes	—
73	C₂×C₄×C₄₇	#125	1	8	yes	—
74	C₂×C₄×C₄₇	C₂×M₁₆	1	16	yes	—
75	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
76	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
79	C₄₇⋊C₈	—	4	8	no	—
80	#20752	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	—	2	4	no	—
82	C₂²×C₄₇	—	2	8	no	—
83	C₂²×C₄₇	#382	1	4	yes	—
84	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×#117	1	2	yes	—
86	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
87	C₄×C₄₇	—	2	4	no	—
88	C₄×C₄₇	—	2	16	no	—
89	C₄×C₄₇	#382	1	4	yes	—
90	C₄×C₄₇	#423	1	8	yes	—
91	C₄×C₄₇	C₂×#124	1	2	yes	—
92	C₄×C₄₇	C₂×#125	1	2	yes	—
93	C₂×#423	—	47	2	no	—
94	C₂×C₄³	D₄₇	1	1	yes	characteristic
95	#1831	—	47	8	no	—
96	#1728	—	47	4	no	—
97	C₂×C₄₇	—	2	4	no	—
98	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
99	C₂×C₄₇	C₂×#423	1	2	yes	—
100	C₂×C₄₇	#2920	1	4	yes	—
101	C₂²×M₁₆	—	47	4	no	—
102	C₂²×C₄²	Dic₄₇	1	4	yes	—
103	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
104	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
105	#423	—	94	8	no	—
106	C₂×C₄⋊C₈	—	47	8	no	—
107	C₄³	Dic₄₇	1	2	yes	—
108	C₄³	Dic₄₇	1	2	yes	—
109	C₄³	C₂×D₄₇	1	4	yes	—
110	#336	—	47	16	no	—
111	C₄₇	#20752	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
113	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
114	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
115	C₂×M₁₆	—	94	8	no	—
116	C₂×M₁₆	—	94	16	no	—
117	C₂²×C₈	—	47	8	no	—
118	C₂×C₄²	—	2	8	no	—
119	C₂×C₄²	—	2	16	no	—
120	C₂×C₄²	Dic₉₄	1	4	yes	—
121	C₂×C₄²	D₁₈₈	1	4	yes	—
122	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
123	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
124	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
125	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
126	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
127	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
129	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
130	C₄⋊C₈	—	94	16	no	—
131	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	—	2	16	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
136	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
137	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
138	C₂²×C₄	#10625350	1	8	yes	—
139	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
141	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
143	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
144	M₁₆	—	188	16	no	—
145	C₂×C₈	—	94	8	no	—
146	C₂×C₈	—	94	16	no	—
147	C₄²	—	2	8	no	—
148	C₄²	—	2	16	no	—
149	C₄²	—	2	8	no	—
150	C₄²	—	2	8	no	—
151	C₄²	#10625329	1	16	yes	complemented
152	C₄²	C₄⋊Dic₄₇	1	4	yes	—
153	C₄²	C₄⋊Dic₄₇	1	4	yes	—
154	C₄²	C₂×Dic₉₄	1	4	yes	—
155	C₄²	C₂×D₁₈₈	1	4	yes	—
156	C₂³	—	2	4	no	—
157	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
158	C₂³	#13093983	1	4	yes	—
159	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
160	C₂×C₄	—	2	32	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	16	no	—
165	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
167	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
168	C₂×C₄	#13093983	1	4	yes	—
169	C₂×C₄	#13094012	1	8	yes	—
170	C₂×C₄	#13094016	1	8	yes	—
171	C₂×C₄	C₂×#10625329	1	16	yes	—
172	C₂×C₄	C₂×#10625329	1	8	yes	—
173	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
174	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
175	C₂×C₄	C₂×#10625350	1	4	yes	—
176	C₈	—	188	8	no	—
177	C₂²	—	2	4	no	—
178	C₂²	—	2	8	no	—
179	C₂²	#15688989	1	4	yes	—
180	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
181	C₂²	C₂×#13093983	1	2	yes	—
182	C₂²	C₂²×#10625329	1	4	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	2	16	no	—
185	C₄	#15688989	1	4	yes	—
186	C₄	#15689147	1	8	yes	—
187	C₄	C₂×#13094012	1	2	yes	—
188	C₄	C₂×#13094016	1	2	yes	—
189	C₂	—	2	4	no	—
190	C₂	C₂×#15688989	1	1	yes	characteristic
191	C₂	C₂×#15689147	1	2	yes	—
192	C₂	#20252021	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄³×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#20752
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20243505	C₂	semidirect
53. C₄²×C₄₇	M₁₆	semidirect
111. C₄₇	#20752	semidirect
151. 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×#15689147	2
#20243505	2
#20243082	2
C₂×C₄³×C₄₇	2
#20752	47