Subgroups — #26109948

Order 12032.

362 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	#20243540	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	#20241222	C₂	1	4	yes	maximal
4	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	C₄	1	2	yes	—
10	#15688878	C₄	1	2	yes	—
11	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
12	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₄³×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
17	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₄×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₈×C₄₇	—	2	2	no	—
23	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂×C₄	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	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	4	no	—
31	C₄⋊C₈×C₄₇	—	4	2	no	—
32	C₄⋊C₈×C₄₇	—	4	2	no	—
33	C₂²×C₄×C₄₇	C₄⋊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	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₇	—	4	2	no	—
41	C₂×C₈×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	C₄²×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄²×C₄₇	—	4	1	no	—
46	C₄²×C₄₇	—	4	1	no	—
47	C₄²×C₄₇	D₈	1	2	yes	—
48	C₄²×C₄₇	SD₁₆	1	2	yes	—
49	C₄²×C₄₇	SD₁₆	1	2	yes	—
50	C₄²×C₄₇	Q₁₆	1	2	yes	—
51	C₄²×C₄₇	#41	1	2	yes	—
52	C₄²×C₄₇	#41	1	2	yes	—
53	C₂×C₄₇⋊C₈	—	2	4	no	—
54	C₂×C₄₇⋊C₈	—	2	4	no	—
55	C₂×C₄₇⋊C₈	—	2	4	no	—
56	C₂×C₄₇⋊C₈	—	4	2	no	—
57	C₂×C₄₇⋊C₈	—	4	2	no	—
58	C₂×C₄₇⋊C₈	—	4	2	no	—
59	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#106	1	2	yes	—
71	C₂×C₄×C₄₇	#106	1	2	yes	—
72	C₂×C₄×C₄₇	#107	1	2	yes	—
73	C₂×C₄×C₄₇	#107	1	2	yes	—
74	C₂×C₄×C₄₇	#117	1	2	yes	—
75	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#125	1	2	yes	—
78	C₂×C₄×C₄₇	#131	1	2	yes	—
79	C₂×C₄×C₄₇	#131	1	2	yes	—
80	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
84	C₈×C₄₇	—	8	2	no	—
85	C₄₇⋊C₈	—	4	4	no	—
86	C₄₇⋊C₈	—	8	2	no	—
87	#8787	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	#382	1	1	yes	characteristic
89	C₂²×C₄₇	#424	1	1	yes	characteristic
90	C₂²×C₄₇	#424	1	1	yes	characteristic
91	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
92	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
93	C₂²×C₄₇	#430	1	1	yes	characteristic
94	C₂²×C₄₇	#430	1	1	yes	characteristic
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	—	2	2	no	—
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	—	2	2	no	—
100	C₄×C₄₇	#433	1	4	yes	—
101	C₄×C₄₇	#486	1	1	yes	characteristic
102	C₄×C₄₇	#486	1	1	yes	characteristic
103	C₄×C₄₇	#487	1	1	yes	characteristic
104	C₄×C₄₇	#487	1	1	yes	characteristic
105	#1842	—	47	1	no	—
106	#1842	D₄₇	1	1	yes	characteristic
107	#1744	—	47	1	no	—
108	#1277	—	47	4	no	—
109	C₂×C₄₇	#1570	1	1	yes	characteristic
110	C₂×C₄₇	#1570	1	1	yes	characteristic
111	C₂×C₄₇	#1571	1	1	yes	characteristic
112	C₂×C₄₇	#1571	1	1	yes	characteristic
113	C₂×C₄₇	#1832	1	1	yes	characteristic
114	C₂×C₄₇	#1834	1	1	yes	characteristic
115	C₂×C₄₇	#1838	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	2	1	no	—
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	—	47	2	no	—
119	C₂×C₄⋊C₈	—	94	1	no	—
120	C₂×C₄⋊C₈	—	94	1	no	—
121	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
122	C₂×C₄×C₈	—	47	2	no	—
123	C₄³	C₂×D₄₇	1	1	yes	characteristic
124	#336	—	47	2	no	—
125	#336	—	47	2	no	—
126	C₄₇	#8787	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	47	4	no	—
129	C₂²×C₈	—	94	2	no	—
130	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
131	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
133	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	—	4	2	no	—
140	C₄⋊C₈	—	94	4	no	—
141	C₄⋊C₈	—	188	2	no	—
142	C₄⋊C₈	—	188	2	no	—
143	C₄×C₈	—	94	2	no	—
144	C₄×C₈	—	94	2	no	—
145	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	94	4	no	—
156	C₂×C₈	—	94	4	no	—
157	C₂×C₈	—	94	4	no	—
158	C₂×C₈	—	188	2	no	—
159	C₂×C₈	—	188	2	no	—
160	C₂×C₈	—	188	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	#10625334	1	2	yes	—
166	C₄²	#10625335	1	2	yes	—
167	C₄²	#10625336	1	2	yes	—
168	C₄²	#10625337	1	2	yes	—
169	C₄²	#10625352	1	2	yes	—
170	C₄²	#10625354	1	2	yes	—
171	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
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	#13093955	1	2	yes	—
183	C₂×C₄	#13093955	1	2	yes	—
184	C₂×C₄	#13093957	1	2	yes	—
185	C₂×C₄	#13093957	1	2	yes	—
186	C₂×C₄	#13093982	1	2	yes	—
187	C₂×C₄	#13094011	1	1	yes	characteristic
188	C₂×C₄	#13094013	1	1	yes	characteristic
189	C₂×C₄	#13094015	1	2	yes	—
190	C₂×C₄	#13094030	1	2	yes	—
191	C₂×C₄	#13094032	1	2	yes	—
192	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
193	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
194	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
195	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
196	C₈	—	8	2	no	—
197	C₈	—	376	2	no	—
198	C₈	—	188	4	no	—
199	C₂²	#15688988	1	1	yes	characteristic
200	C₂²	#15689149	1	1	yes	characteristic
201	C₂²	#15689151	1	1	yes	characteristic
202	C₂²	C₂×#13093955	1	1	yes	characteristic
203	C₂²	C₂×#13093957	1	1	yes	characteristic
204	C₂²	#15689171	1	1	yes	characteristic
205	C₂²	#15689171	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	#15689183	1	4	yes	—
212	C₄	#15689469	1	1	yes	characteristic
213	C₄	#15689471	1	1	yes	characteristic
214	C₄	#15689473	1	1	yes	characteristic
215	C₄	#15689475	1	1	yes	characteristic
216	C₂	#20242158	1	1	yes	characteristic
217	C₂	#20242160	1	1	yes	characteristic
218	C₂	#20242162	1	1	yes	characteristic
219	C₂	#20242164	1	1	yes	characteristic
220	C₂	#20243507	1	1	yes	characteristic
221	C₂	#20243515	1	1	yes	characteristic
222	C₂	#20243528	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#8787
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
126. C₄₇	#8787	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
#20243134	2
#20241222	2
C₄₇×#1842	2
#8787	47