Subgroups — #26106747

Order 12032.

678 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#15688844	C₂	1	2	yes	maximal, complemented
3	C₄₇⋊#1273	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	4	yes	—
6	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
7	#15688878	C₄	1	4	yes	—
8	#15688844	—	2	4	no	—
9	C₂³×C₈×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	—	2	4	no	—
12	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
16	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
17	C₄×C₄₇⋊C₈	—	2	4	no	—
18	C₄×C₄₇⋊C₈	—	2	4	no	—
19	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₄₇	—	2	4	no	—
22	C₂²×C₈×C₄₇	—	2	4	no	—
23	C₂²×C₈×C₄₇	C₈	1	8	yes	complemented
24	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₄₇	—	2	2	no	—
27	C₂×C₄²×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₇	—	2	4	no	—
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₄×C₈×C₄₇	D₄	1	4	yes	—
34	C₂⁴×C₄₇	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	—	2	1	no	—
36	C₂²×C₄×C₄₇	—	2	1	no	—
37	C₂²×C₄×C₄₇	—	2	4	no	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	4	no	—
40	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₄₇	—	2	2	no	—
47	C₂×C₈×C₄₇	—	2	2	no	—
48	C₂×C₈×C₄₇	—	2	16	no	—
49	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
50	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
52	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
53	C₄²×C₄₇	—	2	1	no	—
54	C₄²×C₄₇	—	2	1	no	—
55	C₄²×C₄₇	—	2	4	no	—
56	C₄²×C₄₇	C₂×D₄	1	2	yes	—
57	C₄²×C₄₇	#41	1	2	yes	—
58	C₂×C₄₇⋊C₈	—	2	8	no	—
59	C₂×C₄₇⋊C₈	—	2	8	no	—
60	C₂×C₄₇⋊C₈	—	2	8	no	—
61	C₂³×C₄₇	—	2	1	no	—
62	C₂³×C₄₇	—	2	1	no	—
63	C₂³×C₄₇	—	2	2	no	—
64	C₂³×C₄₇	C₄×C₈	1	2	yes	—
65	C₂³×C₄₇	#97	1	4	yes	—
66	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	1	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	16	no	—
74	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
75	C₂×C₄×C₄₇	#97	1	4	yes	—
76	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
79	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
81	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₄₇	#131	1	4	yes	—
83	C₈×C₄₇	—	2	4	no	—
84	C₈×C₄₇	C₂²⋊C₈	1	8	yes	complemented
85	C₄₇⋊C₈	—	4	8	no	—
86	#8144	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	2	no	—
88	C₂²×C₄₇	—	2	4	no	—
89	C₂²×C₄₇	—	2	2	no	—
90	C₂²×C₄₇	—	2	4	no	—
91	C₂²×C₄₇	#322	1	4	yes	complemented
92	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²×C₄₇	C₂×#97	1	2	yes	—
95	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
96	C₂²×C₄₇	#405	1	2	yes	—
97	C₄×C₄₇	—	2	2	no	—
98	C₄×C₄₇	—	2	4	no	—
99	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
100	C₄×C₄₇	#431	1	2	yes	—
101	C₄×C₄₇	#435	1	8	yes	—
102	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
103	C₄×C₂²⋊C₈	—	47	2	no	—
104	C₂×#322	—	47	2	no	—
105	#1273	—	47	2	no	—
106	C₂×C₄₇	—	2	4	no	—
107	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
108	C₂×C₄₇	#1450	1	1	yes	characteristic
109	C₂×C₄₇	#1451	1	2	yes	—
110	C₂×C₄₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄₇	#1752	1	2	yes	—
112	C₂³×C₈	Dic₄₇	1	2	yes	—
113	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
114	C₂×C₂²⋊C₈	—	47	4	no	—
115	C₂×C₄×C₈	—	2	4	no	—
116	C₂×C₄×C₈	—	47	4	no	—
117	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
118	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
120	#336	—	47	4	no	—
121	#322	—	94	4	no	—
122	C₄₇	#8144	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₄	C₄×D₄₇	1	2	yes	—
124	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	47	8	no	—
128	C₂²×C₈	C₄₇⋊C₈	1	8	yes	complemented
129	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
130	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₄×D₄₇	1	2	yes	—
134	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
136	C₂²⋊C₈	—	94	8	no	—
137	C₄×C₈	—	2	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	94	4	no	—
140	C₄×C₈	—	94	4	no	—
141	C₄×C₈	C₂²⋊D₄₇	1	4	yes	—
142	C₂⁴	C₄×Dic₄₇	1	1	yes	characteristic
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	#10625324	1	8	yes	—
149	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
150	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
151	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
153	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	16	no	—
157	C₂×C₈	—	94	8	no	—
158	C₂×C₈	—	94	8	no	—
159	C₂×C₈	—	94	8	no	—
160	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
161	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
162	C₂×C₈	#10625329	1	4	yes	—
163	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	4	no	—
167	C₄²	#10625350	1	2	yes	—
168	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	1	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₄×C₄₇⋊C₈	1	2	yes	—
173	C₂³	#13093934	1	4	yes	—
174	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
183	C₂×C₄	#13093934	1	4	yes	—
184	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
185	C₂×C₄	#13093983	1	1	yes	characteristic
186	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
187	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
188	C₂×C₄	C₂×#10625324	1	4	yes	—
189	C₂×C₄	C₂×#10625329	1	2	yes	—
190	C₂×C₄	#13094030	1	4	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	188	8	no	—
193	C₈	C₂²⋊(C₄₇⋊C₈)	1	8	yes	complemented
194	C₂²	—	2	2	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	4	no	—
198	C₂²	#15688844	1	4	yes	complemented
199	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
200	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
201	C₂²	C₂×#13093934	1	2	yes	—
202	C₂²	C₄×#10625329	1	1	yes	characteristic
203	C₂²	#15689078	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
207	C₄	#15689176	1	2	yes	—
208	C₄	#15689194	1	8	yes	—
209	C₂	—	2	4	no	—
210	C₂	C₂×#15688844	1	1	yes	characteristic
211	C₂	#20241671	1	1	yes	characteristic
212	C₂	#20241677	1	2	yes	—
213	C₂	C₄×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
214	C₂	#20243164	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄²
prime radical	C₂⁴×C₄₇	C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#8144
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#15688844	C₂	semidirect
3. C₄₇⋊#1273	C₂	semidirect
23. C₂²×C₈×C₄₇	C₈	semidirect
84. C₈×C₄₇	C₂²⋊C₈	semidirect
91. C₂²×C₄₇	#322	semidirect
122. C₄₇	#8144	semidirect
128. C₂²×C₈	C₄₇⋊C₈	semidirect
193. C₈	C₂²⋊(C₄₇⋊C₈)	semidirect
198. C₂²	#15688844	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₄₇⋊C₈)	2
C₂×#15688844	2
C₄₇⋊#1273	2
C₂²×C₄×C₈×C₄₇	2
#8144	47