Subgroups — #26134434

Order 12032.

534 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
8	C₂²⋊(C₄₇⋊C₁₆)	—	2	8	no	—
9	C₄×C₄₇⋊C₁₆	—	2	2	no	—
10	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₄₇	C₄	1	1	yes	characteristic
12	C₈×D₄×C₄₇	—	2	2	no	—
13	C₈×D₄×C₄₇	—	2	2	no	—
14	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₄₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	—	2	4	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
22	C₂²×D₄×C₄₇	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₄₇	—	2	4	no	—
25	C₂²×C₈×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	—	2	4	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×D₄×C₄₇	—	2	2	no	—
32	C₄×D₄×C₄₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₇	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₄₇	C₈	1	2	yes	—
35	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₄₇	—	2	2	no	—
37	C₂²⋊C₈×C₄₇	—	2	4	no	—
38	C₄×C₈×C₄₇	D₄	1	2	yes	—
39	C₄×C₈×C₄₇	D₄	1	2	yes	—
40	C₂⁴×C₄₇	C₂×C₈	1	2	yes	—
41	C₂×D₄×C₄₇	—	2	2	no	—
42	C₂×D₄×C₄₇	—	2	2	no	—
43	C₂×D₄×C₄₇	C₁₆	1	4	yes	complemented
44	C₂²×C₄×C₄₇	—	2	2	no	—
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	—	2	2	no	—
54	C₂×C₈×C₄₇	—	2	4	no	—
55	C₂×C₈×C₄₇	—	4	4	no	—
56	C₂×C₈×C₄₇	—	4	2	no	—
57	C₂×C₈×C₄₇	—	4	2	no	—
58	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₄₇	#41	1	2	yes	—
63	C₄⋊C₄×C₄₇	—	2	2	no	—
64	C₂²⋊C₄×C₄₇	—	2	4	no	—
65	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₄₇	M₁₆	1	2	yes	—
67	C₄₇⋊C₁₆	—	2	4	no	—
68	C₄₇⋊C₁₆	—	4	2	no	—
69	C₂³×C₄₇	—	2	2	no	—
70	C₂³×C₄₇	—	2	4	no	—
71	C₂³×C₄₇	—	2	2	no	—
72	C₂³×C₄₇	—	2	4	no	—
73	C₂³×C₄₇	C₂×C₁₆	1	4	yes	—
74	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
75	D₄×C₄₇	—	2	8	no	—
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	2	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	4	2	no	—
80	C₂×C₄×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
85	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
86	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#131	1	2	yes	—
90	C₈×C₄₇	—	2	2	no	—
91	C₈×C₄₇	—	4	2	no	—
92	C₈×C₄₇	#121	1	2	yes	—
93	C₈×C₄₇	#124	1	1	yes	characteristic
94	C₈×C₄₇	#127	1	1	yes	characteristic
95	#13189	—	47	1	no	maximal, Sylow
96	C₂²×C₄₇	—	2	4	no	—
97	C₂²×C₄₇	—	2	8	no	—
98	C₂²×C₄₇	—	4	2	no	—
99	C₂²×C₄₇	—	4	4	no	—
100	C₂²×C₄₇	—	4	2	no	—
101	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₄₇	C₈×D₄	1	2	yes	—
103	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
105	C₂²×C₄₇	#504	1	2	yes	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	4	2	no	—
108	C₄×C₄₇	C₂²⋊C₁₆	1	4	yes	complemented
109	C₄×C₄₇	#390	1	1	yes	characteristic
110	C₄×C₄₇	#408	1	1	yes	characteristic
111	C₄×C₄₇	#436	1	2	yes	—
112	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic
113	C₂×C₄⋊C₁₆	—	47	1	no	—
114	C₂×C₂²⋊C₁₆	—	47	4	no	—
115	C₂×C₄×C₁₆	—	47	1	no	—
116	C₂×C₄₇	—	4	4	no	—
117	C₂×C₄₇	#1954	1	1	yes	characteristic
118	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₄₇	#2113	1	1	yes	characteristic
120	C₂×C₄₇	D₄×C₁₆	1	2	yes	—
121	C₂×C₄₇	#2169	1	2	yes	—
122	C₂³×C₈	C₂×D₄₇	1	2	yes	—
123	C₂×C₄×D₄	Dic₄₇	1	1	yes	characteristic
124	C₂²×C₁₆	—	47	2	no	—
125	C₂²×C₁₆	—	47	2	no	—
126	C₈×D₄	—	2	2	no	—
127	C₈×D₄	—	2	2	no	—
128	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
129	C₂×C₂²⋊C₈	Dic₄₇	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
131	C₄⋊C₁₆	—	94	2	no	—
132	C₂²⋊C₁₆	—	94	8	no	—
133	C₄×C₁₆	—	94	2	no	—
134	C₄₇	#13189	1	1	yes	characteristic, complemented, Sylow
135	C₂²×D₄	C₄₇⋊C₈	1	1	yes	characteristic
136	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
137	C₂²×C₈	—	2	4	no	—
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
142	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
144	C₄×D₄	—	2	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₂×C₄⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	C₄₇⋊C₈	1	2	yes	—
148	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
149	C₂×C₁₆	—	47	4	no	—
150	C₂×C₁₆	—	94	2	no	—
151	C₂×C₁₆	—	94	2	no	—
152	C₂×C₁₆	—	94	4	no	—
153	C₂×C₁₆	—	94	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₂²⋊C₈	—	2	4	no	—
156	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
157	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
158	C₂⁴	C₂×C₄₇⋊C₈	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	C₄₇⋊C₁₆	1	4	yes	complemented
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
167	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
169	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
177	C₂×C₈	D₄×D₄₇	1	2	yes	—
178	C₂×C₈	#10625352	1	2	yes	—
179	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₄²	#10625329	1	2	yes	—
184	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
185	C₁₆	—	94	4	no	—
186	C₁₆	—	188	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	4	no	—
191	C₂³	C₂×C₄₇⋊C₁₆	1	4	yes	—
192	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
193	D₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
201	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
202	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
203	C₂×C₄	#13093964	1	2	yes	—
204	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
205	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
206	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
207	C₂×C₄	#13094031	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	#13093995	1	2	yes	—
211	C₈	#13094011	1	1	yes	characteristic
212	C₈	#13094020	1	1	yes	characteristic
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
219	C₂²	D₄×C₄₇⋊C₈	1	2	yes	—
220	C₂²	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
221	C₂²	C₂×#13093964	1	1	yes	characteristic
222	C₂²	#15689540	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	C₂²⋊(C₄₇⋊C₁₆)	1	4	yes	complemented
226	C₄	#15689026	1	1	yes	characteristic
227	C₄	#15689090	1	1	yes	characteristic
228	C₄	#15689201	1	2	yes	—
229	C₂	—	4	4	no	—
230	C₂	#20244183	1	1	yes	characteristic
231	C₂	C₂×C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
232	C₂	#20245002	1	1	yes	characteristic
233	C₂	D₄×C₄₇⋊C₁₆	1	2	yes	—
234	C₂	#20245220	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₁₆
Fitting	C₂×C₈×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂²×D₄×C₄₇	C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	#13189
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
2. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
3. C₂×C₄×C₄₇⋊C₁₆	C₂	semidirect
21. C₂×C₄₇⋊C₁₆	D₄	semidirect
43. C₂×D₄×C₄₇	C₁₆	semidirect
108. C₄×C₄₇	C₂²⋊C₁₆	semidirect
134. C₄₇	#13189	semidirect
161. C₂×D₄	C₄₇⋊C₁₆	semidirect
225. C₄	C₂²⋊(C₄₇⋊C₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂×C₄×C₄₇⋊C₁₆	2
C₂×C₈×D₄×C₄₇	2
#13189	47