Subgroups — #26216530

Order 12032.

830 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×#15688969	C₂	1	1	yes	maximal, characteristic
2	#20243532	C₂	1	8	yes	maximal, complemented
3	#20243082	C₂	1	1	yes	maximal, characteristic
4	#20243081	C₂	1	2	yes	maximal
5	#20243081	C₂	1	2	yes	maximal
6	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15688969	C₄	1	4	yes	—
11	#15688969	C₂²	1	4	yes	—
12	C₂×C₄₇⋊#95	C₄	1	4	yes	—
13	#15688878	C₂²	1	4	yes	—
14	#15688877	C₂²	1	8	yes	—
15	#15688877	C₂²	1	8	yes	—
16	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	C₂²	1	4	yes	—
19	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
25	C₂×C₄×Dic₄₇	—	2	4	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
32	C₄₇⋊#95	C₂×C₄	1	16	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	—	2	4	no	—
36	C₂×M₁₆×C₄₇	—	2	4	no	—
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₄₇	—	2	4	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
42	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	4	4	no	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	#41	1	4	yes	—
48	C₂²×C₄×C₄₇	#41	1	4	yes	—
49	C₂²×C₄×C₄₇	#41	1	2	yes	—
50	C₂²×C₄×C₄₇	#41	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂²×Dic₄₇	—	2	4	no	—
53	C₂²×Dic₄₇	—	2	4	no	—
54	C₂²×Dic₄₇	—	4	4	no	—
55	C₂²×Dic₄₇	—	4	4	no	—
56	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
58	M₁₆×C₄₇	—	4	8	no	—
59	C₂×C₈×C₄₇	—	2	4	no	—
60	C₂×C₈×C₄₇	—	2	4	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₄²×C₄₇	—	2	4	no	—
63	C₄²×C₄₇	—	2	4	no	—
64	C₄×Dic₄₇	—	2	4	no	—
65	C₄×Dic₄₇	—	2	4	no	—
66	C₄×Dic₄₇	—	2	2	no	—
67	C₄×Dic₄₇	—	2	4	no	—
68	C₄×Dic₄₇	—	2	2	no	—
69	#10625329	—	4	8	no	—
70	C₂×C₄₇⋊C₈	—	2	4	no	—
71	C₂×C₄₇⋊C₈	—	2	4	no	—
72	C₂×C₄₇⋊C₈	—	2	4	no	—
73	C₂³×C₄₇	—	2	4	no	—
74	C₂³×C₄₇	#117	1	2	yes	—
75	C₂³×C₄₇	#117	1	2	yes	—
76	C₂³×C₄₇	#117	1	2	yes	—
77	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	8	no	—
80	C₂×C₄×C₄₇	—	4	8	no	—
81	C₂×C₄×C₄₇	#117	1	2	yes	—
82	C₂×C₄×C₄₇	#117	1	2	yes	—
83	C₂×C₄×C₄₇	#117	1	2	yes	—
84	C₂×C₄×C₄₇	#126	1	8	yes	—
85	C₂×C₄×C₄₇	#126	1	8	yes	—
86	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
87	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
90	C₂×Dic₄₇	—	2	4	no	—
91	C₂×Dic₄₇	—	2	4	no	—
92	C₂×Dic₄₇	—	2	8	no	—
93	C₂×Dic₄₇	—	2	8	no	—
94	C₂×Dic₄₇	—	4	8	no	—
95	C₂×Dic₄₇	—	4	8	no	—
96	C₈×C₄₇	—	4	4	no	—
97	C₄₇⋊C₈	—	4	4	no	—
98	#20773	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	4	no	—
100	C₂²×C₄₇	—	2	4	no	—
101	C₂²×C₄₇	—	2	4	no	—
102	C₂²×C₄₇	#383	1	4	yes	—
103	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
104	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
105	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
106	C₂²×C₄₇	#568	1	4	yes	—
107	C₄×C₄₇	—	4	4	no	—
108	C₄×C₄₇	#383	1	4	yes	—
109	C₄×C₄₇	C₂×#126	1	2	yes	—
110	C₄×C₄₇	C₂×#126	1	2	yes	—
111	Dic₄₇	—	4	4	no	—
112	Dic₄₇	—	4	4	no	—
113	C₂×#376	—	47	1	no	—
114	#1840	—	47	8	no	—
115	#1728	—	47	1	no	—
116	#1728	—	47	2	no	—
117	#1728	—	47	2	no	—
118	#1728	D₄₇	1	1	yes	characteristic
119	C₂×C₄₇	—	2	4	no	—
120	C₂×C₄₇	C₂×#383	1	1	yes	characteristic
121	C₂×C₄₇	#2921	1	2	yes	—
122	C₂×C₄₇	#2921	1	2	yes	—
123	C₂×C₄₇	#2921	1	2	yes	—
124	C₂²×M₁₆	—	47	2	no	—
125	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
126	C₂²×C₄²	—	47	1	no	—
127	C₂²×C₄²	—	47	1	no	—
128	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
129	#376	—	47	4	no	—
130	#376	—	47	4	no	—
131	C₂×#95	—	47	4	no	—
132	#336	—	47	8	no	—
133	#336	—	47	8	no	—
134	#336	—	47	4	no	—
135	#336	C₂×D₄₇	1	4	yes	—
136	C₄₇	#20773	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	47	2	no	—
138	C₂³×C₄	—	47	2	no	—
139	C₂³×C₄	C₄×D₄₇	1	2	yes	—
140	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	4	no	—
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	94	4	no	—
145	C₂×M₁₆	—	94	4	no	—
146	C₂×M₁₆	—	94	4	no	—
147	C₂²×C₈	—	47	4	no	—
148	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
149	C₂×C₄²	—	2	4	no	—
150	C₂×C₄²	—	47	2	no	—
151	C₂×C₄²	—	47	2	no	—
152	C₂×C₄²	—	47	2	no	—
153	C₂×C₄²	—	47	2	no	—
154	C₂×C₄²	—	94	4	no	—
155	C₂×C₄²	—	94	2	no	—
156	C₂×C₄²	—	94	2	no	—
157	C₂×C₄²	C₄×D₄₇	1	2	yes	—
158	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
159	#95	—	47	16	no	—
160	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	—	47	4	no	—
164	C₂²×C₄	—	47	4	no	—
165	C₂²×C₄	—	94	4	no	—
166	C₂²×C₄	—	94	4	no	—
167	C₂²×C₄	—	188	4	no	—
168	C₂²×C₄	—	188	4	no	—
169	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
170	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
171	C₂²×C₄	#10625350	1	4	yes	—
172	C₂²×C₄	#10625352	1	4	yes	—
173	C₂²×C₄	#10625352	1	2	yes	—
174	C₂²×C₄	#10625354	1	2	yes	—
175	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
176	M₁₆	—	4	8	no	—
177	M₁₆	—	188	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	94	4	no	—
182	C₂×C₈	—	94	4	no	—
183	C₂×C₈	—	94	4	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	94	4	no	—
187	C₄²	—	94	4	no	—
188	C₄²	—	94	2	no	—
189	C₄²	—	94	4	no	—
190	C₄²	—	94	2	no	—
191	C₂³	—	2	4	no	—
192	C₂³	#13093980	1	2	yes	—
193	C₂³	#13093981	1	2	yes	—
194	C₂³	#13093982	1	2	yes	—
195	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	—	94	4	no	—
200	C₂×C₄	—	94	4	no	—
201	C₂×C₄	—	94	8	no	—
202	C₂×C₄	—	94	8	no	—
203	C₂×C₄	—	188	8	no	—
204	C₂×C₄	—	188	8	no	—
205	C₂×C₄	#13093980	1	2	yes	—
206	C₂×C₄	#13093981	1	2	yes	—
207	C₂×C₄	#13093982	1	2	yes	—
208	C₂×C₄	#13094017	1	8	yes	—
209	C₂×C₄	#13094019	1	8	yes	—
210	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
211	C₂×C₄	C₂×#10625352	1	2	yes	—
212	C₂×C₄	C₂×#10625350	1	2	yes	—
213	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
214	C₈	—	4	4	no	—
215	C₈	—	188	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	#15688990	1	4	yes	—
220	C₂²	C₂×#13093980	1	1	yes	characteristic
221	C₂²	C₂×#13093981	1	1	yes	characteristic
222	C₂²	C₂×#13093982	1	1	yes	characteristic
223	C₂²	#15689889	1	4	yes	—
224	C₄	—	4	4	no	—
225	C₄	—	188	4	no	—
226	C₄	—	188	4	no	—
227	C₄	#15688990	1	4	yes	—
228	C₄	C₂×#13094017	1	2	yes	—
229	C₄	C₂×#13094019	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	C₂×#15688990	1	1	yes	characteristic
232	C₂	#20252023	1	2	yes	—
233	C₂	#20252024	1	2	yes	—
234	C₂	#20252025	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1728	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#20773
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
2. #20243532	C₂	semidirect
136. C₄₇	#20773	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688969	2
#20243532	2
#20243082	2
#20243081	2
C₄₇×#1728	2
#20773	47