Subgroups — #26115020

Order 12032.

310 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243560	C₂	1	1	yes	maximal, characteristic
2	#20243255	C₂	1	1	yes	maximal, characteristic
3	#20242159	C₂	1	2	yes	maximal
4	#20241290	C₂	1	2	yes	maximal
5	C₄₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093960	—	2	1	no	—
7	C₂×#13093960	—	2	1	no	—
8	#15689147	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#13093934	—	2	1	no	—
11	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	#15688877	—	2	1	no	—
13	#15688874	—	2	4	no	—
14	C₄₇×#431	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#335	C₄	1	2	yes	complemented
17	C₄₇×#334	C₄	1	2	yes	complemented
18	C₂×#10625329	—	2	2	no	—
19	C₂²×C₄₇⋊C₈	—	2	1	no	—
20	C₂×C₄⋊Dic₄₇	—	2	1	no	—
21	C₂×C₄×Dic₄₇	—	2	1	no	—
22	#13093960	—	4	2	no	—
23	#13093960	—	4	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
28	#13093934	—	4	2	no	—
29	C₄₇⋊#95	Q₈	1	2	yes	—
30	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	—	2	2	no	—
36	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
39	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	1	no	—
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	—	2	4	no	—
47	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
52	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
53	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₄₇	#41	1	1	yes	characteristic
57	C₄⋊Dic₄₇	—	4	2	no	—
58	C₄×Dic₄₇	—	4	1	no	—
59	C₄×Dic₄₇	—	4	1	no	—
60	#10625329	—	4	2	no	—
61	#10625329	—	4	2	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	4	1	no	—
64	C₂×C₄₇⋊C₈	—	4	1	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	4	1	no	—
69	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
79	C₂×Dic₄₇	—	4	2	no	—
80	C₂×Dic₄₇	—	4	2	no	—
81	C₂×Dic₄₇	—	4	2	no	—
82	C₈×C₄₇	—	2	1	no	—
83	C₈×C₄₇	—	2	1	no	—
84	C₈×C₄₇	—	4	4	no	—
85	C₈×C₄₇	#106	1	2	yes	—
86	C₈×C₄₇	#106	1	2	yes	—
87	C₄₇⋊C₈	—	4	2	no	—
88	C₄₇⋊C₈	—	8	2	no	—
89	#9631	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	1	no	—
91	C₂²×C₄₇	—	2	1	no	—
92	C₂²×C₄₇	#389	1	1	yes	characteristic
93	C₂²×C₄₇	#475	1	1	yes	characteristic
94	C₂²×C₄₇	#476	1	1	yes	characteristic
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
99	C₄×C₄₇	#428	1	1	yes	characteristic
100	C₄×C₄₇	#444	1	2	yes	—
101	C₄×C₄₇	#460	1	1	yes	characteristic
102	C₄×C₄₇	#461	1	1	yes	characteristic
103	Dic₄₇	—	8	2	no	—
104	#1848	—	47	1	no	—
105	#1770	—	47	1	no	—
106	#1570	—	47	2	no	—
107	#1564	D₄₇	1	1	yes	characteristic
108	#1303	—	47	2	no	—
109	C₂×C₄₇	—	2	1	no	—
110	C₂×C₄₇	—	2	1	no	—
111	C₂×C₄₇	#1946	1	1	yes	characteristic
112	C₂×C₄₇	#1948	1	1	yes	characteristic
113	C₂×C₄₇	#1953	1	1	yes	characteristic
114	#431	C₂×D₄₇	1	2	yes	—
115	C₂×#108	—	94	1	no	—
116	C₂×#108	—	94	1	no	—
117	#423	—	47	2	no	—
118	C₂×C₄⋊C₈	—	47	1	no	—
119	C₂×#97	—	94	1	no	—
120	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
121	C₄×C₄⋊C₄	—	47	1	no	—
122	#336	—	94	1	no	—
123	#335	Dic₄₇	1	2	yes	complemented
124	#334	—	94	4	no	—
125	#334	Dic₄₇	1	2	yes	complemented
126	C₄₇	#9631	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	94	2	no	—
128	C₂²×C₈	—	94	1	no	—
129	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
130	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	94	1	no	—
132	C₂×C₄²	—	94	1	no	—
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	#108	—	188	2	no	—
135	#108	—	188	2	no	—
136	C₄⋊C₈	—	47	2	no	—
137	C₄⋊C₈	—	94	2	no	—
138	C₄⋊C₈	—	94	1	no	—
139	C₄⋊C₈	—	94	2	no	—
140	C₄⋊C₈	C₂×Dic₄₇	1	4	yes	—
141	C₂²⋊C₈	—	2	2	no	—
142	#97	—	188	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	Dic₉₄	1	1	yes	characteristic
145	C₄×C₈	Dic₉₄	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
148	#95	—	47	2	no	—
149	C₂²×C₄	—	94	2	no	—
150	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
151	C₂²×C₄	#10625352	1	1	yes	characteristic
152	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
153	M₁₆	—	188	2	no	—
154	M₁₆	—	188	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	94	2	no	—
159	C₂×C₈	—	188	2	no	—
160	C₂×C₈	—	188	1	no	—
161	C₂×C₈	—	188	1	no	—
162	C₂×C₈	#10625325	1	2	yes	—
163	C₂×C₈	#10625325	1	2	yes	—
164	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
165	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
166	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
168	C₄⋊C₄	—	188	2	no	—
169	C₄²	—	188	1	no	—
170	C₄²	—	188	1	no	—
171	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
172	C₄²	D₄×D₄₇	1	1	yes	characteristic
173	C₄²	#10625354	1	1	yes	characteristic
174	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
175	C₂³	#13094004	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	188	2	no	—
179	C₂×C₄	—	188	2	no	—
180	C₂×C₄	—	188	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
184	C₂×C₄	#13093999	1	1	yes	characteristic
185	C₂×C₄	#13094015	1	1	yes	characteristic
186	C₂×C₄	#13094023	1	1	yes	characteristic
187	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
188	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
189	C₂×C₄	#13094052	1	1	yes	characteristic
190	C₂×C₄	#13094057	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	376	2	no	—
195	C₈	—	188	2	no	—
196	C₈	#13093956	1	2	yes	—
197	C₈	#13093956	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#15689022	1	1	yes	characteristic
201	C₂²	#15689409	1	1	yes	characteristic
202	C₂²	#15689414	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	376	2	no	—
207	C₄	C₂×#13093956	1	1	yes	characteristic
208	C₄	#15689165	1	1	yes	characteristic
209	C₄	#15689245	1	2	yes	—
210	C₄	#15689324	1	1	yes	characteristic
211	C₄	#15689329	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#20244142	1	1	yes	characteristic
215	C₂	#20244152	1	1	yes	characteristic
216	C₂	#20244179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093999
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1564	C₂
socle	C₂²×C₄₇	#389
prime radical	C₂³×C₄₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1564	D₄₇
hypocenter	C₄₇	#9631
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. C₄₇×#335	C₄	semidirect
17. C₄₇×#334	C₄	semidirect
123. #335	Dic₄₇	semidirect
125. #334	Dic₄₇	semidirect
126. C₄₇	#9631	semidirect

Maximal subgroups

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

subgroup	index
#20243560	2
#20243255	2
#20242159	2
#20241290	2
C₄₇×#1564	2
#9631	47