Subgroups — #26226523

Order 12032.

1074 conjugacy classes of subgroups.

The subgroup lattice has 228 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	#20243573	C₂	1	8	yes	maximal, complemented
3	#20243372	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2280	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10625329	C₂²	1	4	yes	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	complemented
7	#15689087	—	2	8	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	#15688878	C₂²	1	8	yes	complemented
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#381	C₄	1	2	yes	—
14	C₄₇×#380	C₄	1	4	yes	—
15	C₄₇×#379	C₄	1	2	yes	—
16	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
17	C₂×#10625329	—	2	8	no	—
18	C₂×#10625329	—	4	8	no	—
19	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
20	C₄⋊(C₄₇⋊C₈)	—	4	8	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
23	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	—	4	4	no	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	D₄	1	4	yes	—
28	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
31	#117×C₄₇	—	4	4	no	—
32	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
36	#95×C₄₇	C₂×C₄	1	8	yes	—
37	C₂⁴×C₄₇	—	4	2	no	—
38	C₂⁴×C₄₇	—	4	4	no	—
39	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
40	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	8	no	—
44	C₂²×C₄×C₄₇	—	4	8	no	—
45	C₂²×C₄×C₄₇	—	4	8	no	—
46	C₂²×C₄×C₄₇	—	4	4	no	—
47	C₂²×C₄×C₄₇	—	4	2	no	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₄₇	—	4	4	no	—
59	C₂²⋊C₄×C₄₇	—	4	8	no	—
60	C₂²⋊C₄×C₄₇	—	4	4	no	—
61	C₄²×C₄₇	—	2	8	no	—
62	#10625329	—	4	16	no	—
63	C₂×C₄₇⋊C₈	—	2	8	no	—
64	C₂×C₄₇⋊C₈	—	2	16	no	—
65	C₂³×C₄₇	—	2	2	no	—
66	C₂³×C₄₇	—	2	4	no	—
67	C₂³×C₄₇	—	2	4	no	—
68	C₂³×C₄₇	—	2	4	no	—
69	C₂³×C₄₇	—	2	4	no	—
70	C₂³×C₄₇	—	4	4	no	—
71	C₂³×C₄₇	—	4	8	no	—
72	C₂³×C₄₇	—	4	16	no	—
73	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
75	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	—	2	2	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	8	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	8	no	—
82	C₂×C₄×C₄₇	—	2	8	no	—
83	C₂×C₄×C₄₇	—	2	16	no	—
84	C₂×C₄×C₄₇	—	4	8	no	—
85	C₂×C₄×C₄₇	—	4	4	no	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
91	C₄₇⋊C₈	—	4	8	no	—
92	#21935	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	4	no	—
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	—	2	4	no	—
96	C₂²×C₄₇	—	2	4	no	—
97	C₂²×C₄₇	—	2	4	no	—
98	C₂²×C₄₇	—	4	16	no	—
99	C₂²×C₄₇	—	4	4	no	—
100	C₂²×C₄₇	—	4	8	no	—
101	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄₇	#518	1	2	yes	—
103	C₂²×C₄₇	#568	1	4	yes	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	2	4	no	—
106	C₄×C₄₇	—	4	8	no	—
107	C₄×C₄₇	#534	1	2	yes	—
108	C₄×C₄₇	#543	1	2	yes	—
109	C₂×#407	—	47	2	no	—
110	#2280	D₄₇	1	1	yes	characteristic
111	#1851	—	47	8	no	—
112	#1798	—	47	4	no	—
113	C₂×C₄₇	—	4	4	no	—
114	C₂×C₄₇	—	4	2	no	—
115	C₂×C₄₇	#2315	1	1	yes	characteristic
116	C₂×C₄₇	#2905	1	2	yes	—
117	C₂×C₄₇	#2978	1	4	yes	—
118	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
119	C₂²×M₁₆	—	47	4	no	—
120	C₂×#117	C₂×D₄₇	1	2	yes	—
121	C₂×C₄⋊C₈	—	47	8	no	—
122	#407	—	94	8	no	—
123	C₂×C₂²⋊C₈	—	47	4	no	—
124	C₂×C₂²⋊C₈	—	47	4	no	—
125	#381	Dic₄₇	1	2	yes	—
126	#380	Dic₄₇	1	4	yes	—
127	#379	Dic₄₇	1	2	yes	—
128	C₄×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
129	#336	—	47	8	no	—
130	C₄₇	#21935	1	1	yes	characteristic, complemented, Sylow
131	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
132	C₂³×C₄	—	4	4	no	—
133	C₂³×C₄	Dic₉₄	1	4	yes	—
134	C₂³×C₄	D₁₈₈	1	4	yes	—
135	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
136	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
137	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
139	C₂×M₁₆	—	94	8	no	—
140	C₂×M₁₆	—	188	8	no	—
141	C₂²×C₈	—	47	8	no	—
142	#117	—	4	4	no	—
143	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
144	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
145	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
146	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
147	C₄⋊C₈	—	188	8	no	—
148	C₂²⋊C₈	—	94	8	no	—
149	C₂²⋊C₈	—	94	8	no	—
150	#95	C₂×Dic₄₇	1	8	yes	—
151	C₂⁴	—	4	2	no	—
152	C₂⁴	—	4	4	no	—
153	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
154	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
155	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	8	no	—
158	C₂²×C₄	—	4	8	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	—	4	4	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
163	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
164	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
165	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
166	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
167	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
168	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
169	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
171	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
172	M₁₆	—	188	16	no	—
173	C₂×C₈	—	94	8	no	—
174	C₂×C₈	—	94	16	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₂²⋊C₄	—	4	8	no	—
177	C₂²⋊C₄	—	4	4	no	—
178	C₄²	—	2	8	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	4	no	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	4	16	no	—
187	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
188	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
189	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	16	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
201	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
202	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
203	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
204	C₂×C₄	#13094069	1	4	yes	—
205	C₈	—	188	8	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	16	no	—
212	C₂²	—	4	4	no	—
213	C₂²	—	4	8	no	—
214	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
215	C₂²	#15689598	1	2	yes	—
216	C₂²	#15689890	1	4	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	—	4	8	no	—
220	C₄	#15689685	1	2	yes	—
221	C₄	#15689749	1	2	yes	—
222	C₂	—	4	4	no	—
223	C₂	—	4	2	no	—
224	C₂	#20246002	1	1	yes	characteristic
225	C₂	#20251952	1	2	yes	—
226	C₂	#20252450	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2280	C₂
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2280	D₄₇
hypocenter	C₄₇	#21935
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2280

Complements

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

N	H ≅ G/N	type
2. #20243573	C₂	semidirect
3. #20243372	C₂	semidirect
6. C₂×C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
10. #15688878	C₂²	semidirect
130. C₄₇	#21935	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
#20243573	2
#20243372	2
C₄₇×#2280	2
#21935	47