Subgroups — #26102816

Order 12032.

430 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	#20243368	C₂	1	2	yes	maximal
2	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
4	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
5	#15688878	C₄	1	4	yes	—
6	C₂³×C₈×C₄₇	C₄	1	2	yes	—
7	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#336	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
11	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
13	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₄₇	—	2	2	no	—
17	C₂²×C₈×C₄₇	—	2	2	no	—
18	C₂²×C₈×C₄₇	—	4	2	no	—
19	C₂²×C₈×C₄₇	—	4	2	no	—
20	C₂²×C₈×C₄₇	C₈	1	4	yes	complemented
21	C₂×C₄²×C₄₇	—	4	2	no	—
22	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₄₇	D₄	1	2	yes	—
24	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	—	2	1	no	—
28	C₂²×C₄×C₄₇	—	2	2	no	—
29	C₂²×C₄×C₄₇	—	4	1	no	—
30	C₂²×C₄×C₄₇	—	4	1	no	—
31	C₂²×C₄×C₄₇	—	4	1	no	—
32	C₂²×C₄×C₄₇	—	4	1	no	—
33	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
39	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
41	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
42	C₂×C₈×C₄₇	—	2	4	no	—
43	C₂×C₈×C₄₇	—	4	4	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	—	4	2	no	—
46	C₂×C₈×C₄₇	—	4	4	no	—
47	C₄²×C₄₇	—	2	4	no	—
48	C₄²×C₄₇	—	4	2	no	—
49	C₂×C₄₇⋊C₈	—	4	4	no	—
50	C₂×C₄₇⋊C₈	—	4	4	no	—
51	C₂×C₄₇⋊C₈	—	4	4	no	—
52	C₂³×C₄₇	—	2	1	no	—
53	C₂³×C₄₇	—	2	1	no	—
54	C₂³×C₄₇	—	2	1	no	—
55	C₂³×C₄₇	—	2	1	no	—
56	C₂³×C₄₇	#95	1	1	yes	characteristic
57	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
58	C₂³×C₄₇	#106	1	1	yes	characteristic
59	C₂³×C₄₇	#107	1	1	yes	characteristic
60	C₂³×C₄₇	#108	1	2	yes	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	4	1	no	—
73	C₂×C₄×C₄₇	—	4	2	no	—
74	C₂×C₄×C₄₇	—	4	1	no	—
75	C₂×C₄×C₄₇	—	4	4	no	—
76	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₄₇	#99	1	2	yes	—
80	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#102	1	2	yes	—
83	C₂×C₄×C₄₇	#103	1	2	yes	—
84	C₈×C₄₇	—	4	4	no	—
85	C₄₇⋊C₈	—	8	4	no	—
86	#7020	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	2	no	—
88	C₂²×C₄₇	—	2	1	no	—
89	C₂²×C₄₇	—	2	1	no	—
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	—	2	1	no	—
92	C₂²×C₄₇	—	2	1	no	—
93	C₂²×C₄₇	—	2	2	no	—
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	#334	1	2	yes	—
96	C₂²×C₄₇	#335	1	2	yes	—
97	C₂²×C₄₇	#336	1	1	yes	characteristic
98	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
99	C₂²×C₄₇	#338	1	1	yes	characteristic
100	C₂²×C₄₇	#340	1	1	yes	characteristic
101	C₂²×C₄₇	#341	1	1	yes	characteristic
102	C₂²×C₄₇	#342	1	1	yes	characteristic
103	C₂²×C₄₇	#343	1	1	yes	characteristic
104	C₄×C₄₇	—	2	1	no	—
105	C₄×C₄₇	—	2	1	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	4	2	no	—
109	C₄×C₄₇	#323	1	2	yes	—
110	C₄×C₄₇	#324	1	2	yes	—
111	#1797	—	47	2	no	—
112	C₂×#336	D₄₇	1	1	yes	characteristic
113	C₂×C₄₇	—	2	2	no	—
114	C₂×C₄₇	—	2	2	no	—
115	C₂×C₄₇	#1275	1	1	yes	characteristic
116	C₂×C₄₇	#1277	1	1	yes	characteristic
117	C₂×C₄₇	#1280	1	1	yes	characteristic
118	C₂×C₄₇	#1289	1	1	yes	characteristic
119	C₂×C₄₇	#1290	1	1	yes	characteristic
120	C₂×C₄₇	#1292	1	1	yes	characteristic
121	C₂×C₄₇	#1293	1	1	yes	characteristic
122	C₂³×C₈	Dic₄₇	1	2	yes	—
123	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
124	C₂×C₄⋊C₈	—	94	2	no	—
125	C₂×C₂²⋊C₈	—	94	2	no	—
126	#336	—	2	2	no	—
127	#336	—	47	4	no	—
128	C₄₇	#7020	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
130	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
131	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	4	2	no	—
135	C₂²×C₈	—	4	2	no	—
136	C₂²×C₈	—	94	4	no	—
137	C₂²×C₈	C₄₇⋊C₈	1	4	yes	complemented
138	C₂×C₄²	—	4	2	no	—
139	C₂×C₄²	C₄×D₄₇	1	2	yes	—
140	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
141	C₄⋊C₈	—	188	4	no	—
142	C₂²⋊C₈	—	188	2	no	—
143	C₂²⋊C₈	—	188	2	no	—
144	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	4	1	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	—	4	1	no	—
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	#10625325	1	2	yes	—
154	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
155	C₂²×C₄	Dic₁₈₈	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
157	C₂²×C₄	#10625329	1	2	yes	—
158	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
159	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
160	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
161	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	4	4	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	4	no	—
167	C₂×C₈	—	188	4	no	—
168	C₂×C₈	—	188	4	no	—
169	C₂×C₈	—	188	4	no	—
170	C₄²	—	2	4	no	—
171	C₄²	—	4	2	no	—
172	C₂³	—	2	1	no	—
173	C₂³	—	2	1	no	—
174	C₂³	—	2	1	no	—
175	C₂³	—	2	1	no	—
176	C₂³	C₄₇⋊#95	1	1	yes	characteristic
177	C₂³	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
178	C₂³	#13093956	1	1	yes	characteristic
179	C₂³	#13093958	1	1	yes	characteristic
180	C₂³	#13093960	1	2	yes	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
197	C₂×C₄	#13093935	1	1	yes	characteristic
198	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
199	C₂×C₄	#13093939	1	2	yes	—
200	C₂×C₄	#13093941	1	1	yes	characteristic
201	C₂×C₄	#13093943	1	1	yes	characteristic
202	C₂×C₄	#13093946	1	2	yes	—
203	C₂×C₄	#13093949	1	2	yes	—
204	C₈	—	4	4	no	—
205	C₈	—	376	4	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	2	no	—
214	C₂²	#15688874	1	2	yes	—
215	C₂²	#15688876	1	2	yes	—
216	C₂²	#15688878	1	1	yes	characteristic
217	C₂²	#15688880	1	1	yes	characteristic
218	C₂²	C₄₇⋊#338	1	1	yes	characteristic
219	C₂²	#15688886	1	1	yes	characteristic
220	C₂²	#15688888	1	1	yes	characteristic
221	C₂²	#15688890	1	1	yes	characteristic
222	C₂²	#15688892	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	#15688847	1	2	yes	—
229	C₄	#15688849	1	2	yes	—
230	C₂	—	2	2	no	—
231	C₂	—	2	2	no	—
232	C₂	#20241219	1	1	yes	characteristic
233	C₂	#20241223	1	1	yes	characteristic
234	C₂	#20241231	1	1	yes	characteristic
235	C₂	#20241256	1	1	yes	characteristic
236	C₂	#20241258	1	1	yes	characteristic
237	C₂	#20241263	1	1	yes	characteristic
238	C₂	#20241265	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₂×C₄₇×#336	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#7020
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
20. C₂²×C₈×C₄₇	C₈	semidirect
128. C₄₇	#7020	semidirect
137. C₂²×C₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₂×C₄₇×#336	2
#7020	47