Subgroups — #26224628

Order 12032.

618 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246693	C₂	1	1	yes	maximal, characteristic
2	#20244111	C₂	1	2	yes	maximal
3	#20243952	C₂	1	4	yes	maximal
4	#20243508	C₂	1	1	yes	maximal, characteristic
5	#20243262	C₂	1	4	yes	maximal
6	#20243255	C₂	1	2	yes	maximal
7	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×Dic₄₇⋊C₈	C₂²	1	4	yes	—
10	C₂×#13093934	C₂²	1	4	yes	—
11	#15689065	C₄	1	4	yes	—
12	#15689057	C₄	1	4	yes	—
13	C₄×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	#15688878	C₂²	1	4	yes	—
16	#15688877	C₂²	1	4	yes	—
17	#15688877	C₂²	1	4	yes	—
18	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#376	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
24	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
25	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	Dic₄₇⋊C₈	—	2	8	no	—
30	#13093934	—	2	8	no	—
31	C₄₇⋊#95	C₂×C₄	1	2	yes	—
32	C₄₇⋊#95	C₂×C₄	1	2	yes	—
33	C₄₇⋊#95	C₂×C₄	1	4	yes	—
34	C₄₇⋊#95	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	#97×C₄₇	—	2	4	no	—
41	#97×C₄₇	—	2	2	no	—
42	C₄×C₈×C₄₇	—	2	4	no	—
43	C₄×C₈×C₄₇	Q₈	1	4	yes	—
44	#95×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
49	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₄₇	#41	1	4	yes	—
56	C₂×C₈×C₄₇	#41	1	4	yes	—
57	C₄²×C₄₇	—	2	2	no	—
58	C₄²×C₄₇	—	2	2	no	—
59	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
60	C₄²×C₄₇	#41	1	2	yes	—
61	C₄⋊Dic₄₇	—	4	4	no	—
62	Dic₄₇⋊C₄	—	4	4	no	—
63	Dic₄₇⋊C₄	—	4	4	no	—
64	C₄×Dic₄₇	—	2	4	no	—
65	C₄×Dic₄₇	—	2	4	no	—
66	C₂×C₄₇⋊C₈	—	2	4	no	—
67	C₂×C₄₇⋊C₈	—	2	4	no	—
68	C₂×C₄₇⋊C₈	—	2	4	no	—
69	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	—	2	4	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	4	no	—
74	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
75	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
79	C₂×Dic₄₇	—	2	4	no	—
80	C₂×Dic₄₇	—	2	4	no	—
81	C₂×Dic₄₇	—	2	4	no	—
82	C₂×Dic₄₇	—	2	4	no	—
83	C₂×Dic₄₇	—	2	4	no	—
84	C₂×Dic₄₇	—	2	4	no	—
85	C₈×C₄₇	—	2	4	no	—
86	C₈×C₄₇	—	2	2	no	—
87	C₈×C₄₇	#125	1	4	yes	—
88	C₄₇⋊C₈	—	4	4	no	—
89	#21677	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂²×C₄₇	C₄×#41	1	2	yes	—
92	C₂²×C₄₇	#568	1	4	yes	—
93	C₄×C₄₇	—	2	4	no	—
94	C₄×C₄₇	—	4	4	no	—
95	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
96	C₄×C₄₇	#529	1	2	yes	—
97	C₄×C₄₇	#533	1	1	yes	characteristic
98	Dic₄₇	—	4	4	no	—
99	Dic₄₇	—	4	4	no	—
100	#2402	—	47	1	no	—
101	#1940	—	47	2	no	—
102	#1917	—	47	4	no	—
103	#1832	—	47	1	no	—
104	#1771	—	47	4	no	—
105	#1770	—	47	2	no	—
106	#1726	D₄₇	1	1	yes	characteristic
107	C₂×C₄₇	C₄×#125	1	1	yes	characteristic
108	C₂×C₄₇	#2961	1	2	yes	—
109	C₂×C₄₇	#2967	1	4	yes	—
110	C₂×C₄⋊C₈	—	47	4	no	—
111	C₂×C₄⋊C₈	—	47	2	no	—
112	C₂×#97	—	47	4	no	—
113	C₂×#97	C₂×D₄₇	1	2	yes	—
114	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
117	#400	—	47	4	no	—
118	#398	—	47	4	no	—
119	C₄×C₄⋊C₄	—	47	2	no	—
120	C₄×C₄⋊C₄	—	47	4	no	—
121	#376	C₂×D₄₇	1	1	yes	characteristic
122	#336	—	47	4	no	—
123	#336	—	47	4	no	—
124	#336	—	47	4	no	—
125	C₄₇	#21677	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	47	4	no	—
127	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
129	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	47	4	no	—
131	C₂×C₄⋊C₄	—	47	4	no	—
132	C₂×C₄⋊C₄	—	47	4	no	—
133	C₂×C₄²	—	47	4	no	—
134	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
135	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
136	C₄⋊C₈	—	94	8	no	—
137	C₄⋊C₈	—	94	4	no	—
138	#97	—	2	4	no	—
139	#97	—	2	2	no	—
140	#97	—	94	8	no	—
141	C₄×C₈	—	2	4	no	—
142	C₄×C₈	Dic₉₄	1	4	yes	—
143	#95	—	47	2	no	—
144	#95	—	47	4	no	—
145	#95	—	47	4	no	—
146	#95	—	47	2	no	—
147	#95	C₄×D₄₇	1	4	yes	—
148	C₂²×C₄	—	47	4	no	—
149	C₂²×C₄	—	47	4	no	—
150	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
152	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	94	4	no	—
157	C₂×C₈	—	94	4	no	—
158	C₂×C₈	—	94	4	no	—
159	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
160	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
161	C₂×C₈	#10625350	1	4	yes	—
162	C₂×C₈	#10625350	1	4	yes	—
163	C₄⋊C₄	—	188	4	no	—
164	C₄⋊C₄	—	188	4	no	—
165	C₄⋊C₄	—	188	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	94	4	no	—
169	C₄²	—	94	4	no	—
170	C₄²	C₂×Dic₉₄	1	2	yes	—
171	C₄²	#10625350	1	2	yes	—
172	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	94	4	no	—
178	C₂×C₄	—	94	4	no	—
179	C₂×C₄	—	94	4	no	—
180	C₂×C₄	—	94	4	no	—
181	C₂×C₄	—	94	4	no	—
182	C₂×C₄	—	94	4	no	—
183	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
184	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
185	C₂×C₄	C₂×#10625350	1	2	yes	—
186	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
187	C₂×C₄	C₂×#10625350	1	2	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	2	2	no	—
190	C₈	—	188	4	no	—
191	C₈	#13094016	1	4	yes	—
192	C₂²	C₂×C₄×Dic₉₄	1	1	yes	characteristic
193	C₂²	C₄×#10625350	1	2	yes	—
194	C₂²	#15689888	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	4	no	—
197	C₄	—	188	4	no	—
198	C₄	—	188	4	no	—
199	C₄	C₂×#13094016	1	1	yes	characteristic
200	C₄	#15689657	1	2	yes	—
201	C₄	#15689680	1	1	yes	characteristic
202	C₂	C₄×#13094016	1	1	yes	characteristic
203	C₂	#20252326	1	2	yes	—
204	C₂	#20252369	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1726	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21677
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1726

Complements

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

N	H ≅ G/N	type
125. C₄₇	#21677	semidirect

Maximal subgroups

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

subgroup	index
#20246693	2
#20244111	2
#20243952	2
#20243508	2
#20243262	2
#20243255	2
C₄₇×#1726	2
#21677	47