Subgroups — #26215987

Order 12032.

2022 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689087	C₂	1	2	yes	maximal
2	#20243456	C₂	1	8	yes	maximal, complemented
3	#20243082	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	4	yes	—
6	#15689087	—	2	8	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
8	#15688878	C₂²	1	16	yes	complemented
9	C₂⁴×C₄×C₄₇	C₄	1	2	yes	—
10	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₄₇	—	2	8	no	—
12	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₄	1	4	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
16	C₂×#10625329	—	2	8	no	—
17	C₂×#10625329	—	2	16	no	—
18	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	16	no	—
20	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₇	—	2	8	no	—
22	C₂³×C₄×C₄₇	—	2	16	no	—
23	C₂³×C₄×C₄₇	—	2	8	no	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₄₇	D₄	1	8	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	—	2	8	no	—
33	C₂×C₄²×C₄₇	—	2	16	no	—
34	C₂×C₄²×C₄₇	—	2	16	no	—
35	C₂×C₄²×C₄₇	—	2	8	no	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
39	C₂⁴×C₄₇	—	2	4	no	—
40	C₂⁴×C₄₇	—	2	8	no	—
41	C₂⁴×C₄₇	C₂²⋊C₄	1	4	yes	—
42	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₄₇	—	2	64	no	—
45	C₂²×C₄×C₄₇	—	2	8	no	—
46	C₂²×C₄×C₄₇	—	2	8	no	—
47	C₂²×C₄×C₄₇	—	2	8	no	—
48	C₂²×C₄×C₄₇	—	2	4	no	—
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	8	no	—
51	C₂²×C₄×C₄₇	—	2	32	no	—
52	C₂²×C₄×C₄₇	—	2	16	no	—
53	C₂²×C₄×C₄₇	—	2	16	no	—
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
57	C₂²×C₄×C₄₇	M₁₆	1	16	yes	complemented
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₇	#41	1	8	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₄²×C₄₇	—	2	16	no	—
66	C₄²×C₄₇	—	2	8	no	—
67	C₄²×C₄₇	—	2	8	no	—
68	#10625329	—	4	16	no	—
69	C₂×C₄₇⋊C₈	—	2	8	no	—
70	C₂×C₄₇⋊C₈	—	2	16	no	—
71	C₂³×C₄₇	—	2	32	no	—
72	C₂³×C₄₇	—	2	4	no	—
73	C₂³×C₄₇	—	2	4	no	—
74	C₂³×C₄₇	—	2	4	no	—
75	C₂³×C₄₇	—	2	8	no	—
76	C₂³×C₄₇	—	2	16	no	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
79	C₂³×C₄₇	#117	1	4	yes	—
80	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
81	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	—	2	32	no	—
83	C₂×C₄×C₄₇	—	2	32	no	—
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	8	no	—
88	C₂×C₄×C₄₇	—	2	32	no	—
89	C₂×C₄×C₄₇	—	2	16	no	—
90	C₂×C₄×C₄₇	—	2	16	no	—
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
92	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₇	#117	1	4	yes	—
94	C₂×C₄×C₄₇	#123	1	16	yes	—
95	C₂×C₄×C₄₇	C₂×M₁₆	1	16	yes	—
96	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
98	C₄₇⋊C₈	—	4	8	no	—
99	#20682	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	32	no	—
101	C₂²×C₄₇	—	2	4	no	—
102	C₂²×C₄₇	—	2	4	no	—
103	C₂²×C₄₇	—	2	8	no	—
104	C₂²×C₄₇	—	2	16	no	—
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	#381	1	4	yes	—
107	C₂²×C₄₇	#407	1	8	yes	—
108	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#117	1	2	yes	—
110	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
111	C₄×C₄₇	—	2	16	no	—
112	C₄×C₄₇	—	2	4	no	—
113	C₄×C₄₇	#381	1	4	yes	—
114	C₄×C₄₇	C₂×#123	1	4	yes	—
115	C₂³×C₄²	D₄₇	1	1	yes	characteristic
116	C₂×#407	—	47	2	no	—
117	#1817	—	47	8	no	—
118	#1728	—	47	4	no	—
119	C₂×C₄₇	—	2	4	no	—
120	C₂×C₄₇	—	2	8	no	—
121	C₂×C₄₇	C₂×#381	1	1	yes	characteristic
122	C₂×C₄₇	C₂×#407	1	2	yes	—
123	C₂×C₄₇	#2920	1	4	yes	—
124	C₂⁴×C₄	Dic₄₇	1	2	yes	—
125	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
126	C₂²×M₁₆	—	47	4	no	—
127	C₂²×C₄²	—	2	8	no	—
128	C₂²×C₄²	Dic₄₇	1	2	yes	—
129	C₂²×C₄²	Dic₄₇	1	4	yes	—
130	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
131	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
132	#407	—	94	8	no	—
133	C₂×C₂²⋊C₈	—	47	8	no	—
134	#336	—	47	16	no	—
135	C₄₇	#20682	1	1	yes	characteristic, complemented, Sylow
136	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
137	C₂³×C₄	—	2	8	no	—
138	C₂³×C₄	—	2	16	no	—
139	C₂³×C₄	—	2	8	no	—
140	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
141	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
142	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
143	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
144	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
145	C₂³×C₄	C₂²⋊D₄₇	1	8	yes	—
146	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
147	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
148	C₂×M₁₆	—	94	16	no	—
149	C₂×M₁₆	—	94	8	no	—
150	C₂²×C₈	—	47	8	no	—
151	C₂×C₄²	—	2	8	no	—
152	C₂×C₄²	—	2	16	no	—
153	C₂×C₄²	—	2	16	no	—
154	C₂×C₄²	—	2	8	no	—
155	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
156	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
157	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
158	C₂²⋊C₈	—	94	16	no	—
159	C₂⁴	—	2	4	no	—
160	C₂⁴	—	2	8	no	—
161	C₂⁴	C₂²⋊Dic₄₇	1	4	yes	—
162	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
163	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
164	C₂²×C₄	—	2	64	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	2	32	no	—
172	C₂²×C₄	—	2	16	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	#10625329	1	16	yes	complemented
176	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
177	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
178	C₂²×C₄	#10625350	1	8	yes	—
179	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
180	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
181	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
184	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
185	M₁₆	—	188	16	no	—
186	C₂×C₈	—	94	8	no	—
187	C₂×C₈	—	94	16	no	—
188	C₄²	—	2	16	no	—
189	C₄²	—	2	8	no	—
190	C₄²	—	2	8	no	—
191	C₂³	—	2	32	no	—
192	C₂³	—	2	8	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	16	no	—
195	C₂³	—	2	4	no	—
196	C₂³	—	2	4	no	—
197	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
198	C₂³	C₂×C₂²⋊Dic₄₇	1	4	yes	—
199	C₂³	#13093983	1	4	yes	—
200	C₂³	C₂×#10625329	1	8	yes	—
201	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
202	C₂×C₄	—	2	32	no	—
203	C₂×C₄	—	2	32	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	32	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	16	no	—
211	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
212	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
213	C₂×C₄	#13093983	1	4	yes	—
214	C₂×C₄	#13094007	1	16	yes	—
215	C₂×C₄	C₂×#10625329	1	16	yes	—
216	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
217	C₂×C₄	C₂×#10625350	1	4	yes	—
218	C₈	—	188	8	no	—
219	C₂²	—	2	32	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	8	no	—
222	C₂²	—	2	16	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	4	no	—
225	C₂²	#15688986	1	4	yes	—
226	C₂²	#15689087	1	8	yes	—
227	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
228	C₂²	C₂×#13093983	1	2	yes	—
229	C₂²	C₂²×#10625329	1	4	yes	—
230	C₄	—	2	4	no	—
231	C₄	—	2	16	no	—
232	C₄	#15688986	1	4	yes	—
233	C₄	C₂×#13094007	1	4	yes	—
234	C₂	—	2	4	no	—
235	C₂	—	2	8	no	—
236	C₂	C₂×#15688986	1	1	yes	characteristic
237	C₂	C₂×#15689087	1	2	yes	—
238	C₂	#20252021	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂³×C₄²×C₄₇	C₂
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	#20682
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. #20243456	C₂	semidirect
3. #20243082	C₂	semidirect
8. #15688878	C₂²	semidirect
57. C₂²×C₄×C₄₇	M₁₆	semidirect
135. C₄₇	#20682	semidirect
175. C₂²×C₄	#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
#20243456	2
#20243082	2
C₂³×C₄²×C₄₇	2
#20682	47