Subgroups — #26102811

Order 12032.

294 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243505	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
5	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
6	#15688878	—	2	1	no	—
7	#15688878	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
11	C₄³×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#336	—	2	1	no	—
13	C₄₇×#336	C₄	1	2	yes	—
14	C₂²×C₄₇⋊C₈	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
17	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
18	C₄×C₄₇⋊C₈	—	2	2	no	—
19	C₄×C₄₇⋊C₈	C₈	1	4	yes	complemented
20	C₂²×C₈×C₄₇	—	2	2	no	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	—	4	1	no	—
29	C₄⋊C₈×C₄₇	—	4	1	no	—
30	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₄²×C₄₇	—	2	2	no	—
41	C₄²×C₄₇	—	4	1	no	—
42	C₄²×C₄₇	—	4	1	no	—
43	C₄²×C₄₇	—	4	1	no	—
44	C₄²×C₄₇	—	4	1	no	—
45	C₄²×C₄₇	C₂×C₈	1	2	yes	—
46	C₄²×C₄₇	M₁₆	1	2	yes	—
47	C₄²×C₄₇	D₈	1	1	yes	characteristic
48	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
49	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
50	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
51	C₂×C₄₇⋊C₈	—	2	4	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	4	2	no	—
55	C₂×C₄₇⋊C₈	—	4	2	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂³×C₄₇	#95	1	1	yes	characteristic
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	1	no	—
67	C₂×C₄×C₄₇	—	2	1	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	1	no	—
72	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
82	C₂×C₄×C₄₇	#108	1	2	yes	—
83	C₈×C₄₇	—	8	2	no	—
84	C₄₇⋊C₈	—	4	4	no	—
85	C₄₇⋊C₈	—	8	2	no	—
86	#7018	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	#336	1	1	yes	characteristic
88	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
89	C₂²×C₄₇	H(3,ℤ/4)	1	1	yes	characteristic
90	C₂²×C₄₇	#340	1	1	yes	characteristic
91	C₂²×C₄₇	#341	1	1	yes	characteristic
92	C₂²×C₄₇	#344	1	1	yes	characteristic
93	C₂²×C₄₇	#344	1	1	yes	characteristic
94	C₄×C₄₇	—	2	1	no	—
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	—	2	1	no	—
99	C₄×C₄₇	—	2	1	no	—
100	C₄×C₄₇	—	2	2	no	—
101	C₄×C₄₇	—	4	1	no	—
102	C₄×C₄₇	—	4	1	no	—
103	C₄×C₄₇	#334	1	2	yes	—
104	C₄×C₄₇	#335	1	2	yes	—
105	#1831	—	47	1	no	—
106	#1831	D₄₇	1	1	yes	characteristic
107	#1744	—	47	1	no	—
108	C₂×C₄₇	#1274	1	1	yes	characteristic
109	C₂×C₄₇	#1277	1	1	yes	characteristic
110	C₂×C₄₇	#1281	1	1	yes	characteristic
111	C₂×C₄₇	#1286	1	1	yes	characteristic
112	C₂×C₄₇	#1286	1	1	yes	characteristic
113	C₂×C₄₇	#1287	1	1	yes	characteristic
114	C₂×C₄₇	#1287	1	1	yes	characteristic
115	C₂×C₄⋊C₈	—	2	1	no	—
116	C₂×C₄⋊C₈	—	94	1	no	—
117	C₂×C₄×C₈	—	47	2	no	—
118	C₄³	C₂×D₄₇	1	1	yes	characteristic
119	#336	—	2	1	no	—
120	#336	—	47	2	no	—
121	#336	—	94	1	no	—
122	#336	—	94	1	no	—
123	#336	—	94	1	no	—
124	#336	Dic₄₇	1	2	yes	—
125	C₄₇	#7018	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	94	2	no	—
128	C₂²×C₈	—	94	2	no	—
129	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
130	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
132	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
133	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
136	C₄⋊C₈	—	4	1	no	—
137	C₄⋊C₈	—	4	1	no	—
138	C₄⋊C₈	—	188	1	no	—
139	C₄⋊C₈	—	188	1	no	—
140	C₄×C₈	—	47	4	no	—
141	C₄×C₈	—	94	2	no	—
142	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
143	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
146	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	94	4	no	—
153	C₂×C₈	—	188	2	no	—
154	C₂×C₈	—	188	2	no	—
155	C₂×C₈	—	188	2	no	—
156	C₂×C₈	—	188	2	no	—
157	C₂×C₈	—	188	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	4	1	no	—
163	C₄²	C₈×D₄₇	1	2	yes	—
164	C₄²	#10625324	1	2	yes	—
165	C₄²	#10625334	1	1	yes	characteristic
166	C₄²	#10625335	1	1	yes	characteristic
167	C₄²	#10625336	1	1	yes	characteristic
168	C₄²	#10625337	1	1	yes	characteristic
169	C₂³	C₄₇⋊#95	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
185	C₂×C₄	#13093934	1	1	yes	characteristic
186	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
187	C₂×C₄	#13093944	1	1	yes	characteristic
188	C₂×C₄	#13093945	1	1	yes	characteristic
189	C₂×C₄	#13093947	1	1	yes	characteristic
190	C₂×C₄	#13093948	1	1	yes	characteristic
191	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
192	C₂×C₄	#13093955	1	1	yes	characteristic
193	C₂×C₄	#13093957	1	1	yes	characteristic
194	C₂×C₄	#13093959	1	2	yes	—
195	C₈	—	8	2	no	—
196	C₈	—	376	2	no	—
197	C₈	—	188	4	no	—
198	C₂²	#15688877	1	1	yes	characteristic
199	C₂²	#15688879	1	1	yes	characteristic
200	C₂²	#15688880	1	1	yes	characteristic
201	C₂²	#15688885	1	1	yes	characteristic
202	C₂²	#15688887	1	1	yes	characteristic
203	C₂²	#15688893	1	1	yes	characteristic
204	C₂²	#15688895	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	#15688873	1	2	yes	—
215	C₄	#15688875	1	2	yes	—
216	C₂	#20241216	1	1	yes	characteristic
217	C₂	#20241222	1	1	yes	characteristic
218	C₂	#20241232	1	1	yes	characteristic
219	C₂	#20241246	1	1	yes	characteristic
220	C₂	#20241248	1	1	yes	characteristic
221	C₂	#20241249	1	1	yes	characteristic
222	C₂	#20241251	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#7018
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
19. C₄×C₄₇⋊C₈	C₈	semidirect
125. C₄₇	#7018	semidirect

Maximal subgroups

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

subgroup	index
#20243505	2
#20243134	2
C₄₇×#1831	2
#7018	47