Subgroups — #26224629

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	#20244112	C₂	1	2	yes	maximal
2	#20243954	C₂	1	4	yes	maximal
3	#20243508	C₂	1	1	yes	maximal, characteristic
4	#20243263	C₂	1	4	yes	maximal
5	#20243257	C₂	1	2	yes	maximal
6	#20243075	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#2402	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×#13093935	C₂²	1	4	yes	—
11	C₂×#13093935	C₂²	1	1	yes	characteristic
12	C₂×#13093935	C₂²	1	2	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	#15688878	C₂²	1	4	yes	—
16	#15688878	C₂²	1	4	yes	—
17	#15688878	C₂²	1	4	yes	—
18	C₄₇×#400	C₄	1	4	yes	—
19	C₄₇×#398	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
22	C₄₇×#376	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
24	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
27	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
29	#13093935	—	2	2	no	—
30	#13093935	—	2	4	no	—
31	#13093935	—	2	8	no	—
32	C₄×C₄₇⋊C₈	—	2	4	no	—
33	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	#95×C₄₇	C₂×C₄	1	4	yes	—
41	#95×C₄₇	C₂×C₄	1	2	yes	—
42	#95×C₄₇	C₂×C₄	1	4	yes	—
43	#95×C₄₇	C₂×C₄	1	4	yes	—
44	#95×C₄₇	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₄⋊C₄×C₄₇	—	4	4	no	—
51	C₄⋊C₄×C₄₇	—	4	4	no	—
52	C₄⋊C₄×C₄₇	—	4	4	no	—
53	C₄²×C₄₇	—	2	2	no	—
54	C₄²×C₄₇	—	2	4	no	—
55	C₄²×C₄₇	—	2	2	no	—
56	C₄²×C₄₇	—	2	4	no	—
57	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
58	C₄²×C₄₇	#41	1	2	yes	—
59	C₂×C₄₇⋊C₈	—	2	2	no	—
60	C₂×C₄₇⋊C₈	—	2	2	no	—
61	C₂×C₄₇⋊C₈	—	2	2	no	—
62	C₂×C₄₇⋊C₈	—	2	4	no	—
63	C₂×C₄₇⋊C₈	—	2	4	no	—
64	C₂×C₄₇⋊C₈	—	2	4	no	—
65	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
66	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
67	C₂×C₄₇⋊C₈	#41	1	4	yes	—
68	C₂×C₄₇⋊C₈	#41	1	4	yes	—
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	—	2	4	no	—
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	—	2	4	no	—
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
81	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
83	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
85	C₄₇⋊C₈	—	2	4	no	—
86	C₄₇⋊C₈	—	2	2	no	—
87	C₄₇⋊C₈	—	4	4	no	—
88	C₄₇⋊C₈	#125	1	4	yes	—
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	—	4	4	no	—
96	C₄×C₄₇	—	4	4	no	—
97	C₄×C₄₇	C₂×#125	1	1	yes	characteristic
98	C₄×C₄₇	#529	1	2	yes	—
99	C₄×C₄₇	#533	1	1	yes	characteristic
100	#2402	D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
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	2	no	—
111	C₂×C₄⋊C₈	—	47	4	no	—
112	C₂×#97	—	47	2	no	—
113	C₂×#97	—	47	1	no	—
114	C₂×#97	—	47	4	no	—
115	C₂×C₄×C₈	—	47	1	no	—
116	C₂×C₄×C₈	—	47	2	no	—
117	#400	Dic₄₇	1	4	yes	—
118	#398	Dic₄₇	1	4	yes	—
119	C₄×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
120	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
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	1	no	—
127	C₂²×C₈	—	47	1	no	—
128	C₂²×C₈	—	47	2	no	—
129	C₂²×C₈	—	47	4	no	—
130	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
131	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
132	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
133	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
134	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
135	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
136	C₄⋊C₈	—	94	8	no	—
137	C₄⋊C₈	—	94	4	no	—
138	#97	—	94	8	no	—
139	#97	—	94	4	no	—
140	#97	—	94	2	no	—
141	C₄×C₈	—	47	4	no	—
142	C₄×C₈	—	94	4	no	—
143	#95	C₂×Dic₄₇	1	4	yes	—
144	#95	C₂×Dic₄₇	1	2	yes	—
145	#95	C₂×Dic₄₇	1	4	yes	—
146	#95	C₂×Dic₄₇	1	4	yes	—
147	#95	C₂×Dic₄₇	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
151	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
152	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
153	C₂×C₈	—	47	2	no	—
154	C₂×C₈	—	47	4	no	—
155	C₂×C₈	—	47	2	no	—
156	C₂×C₈	—	47	4	no	—
157	C₂×C₈	—	94	2	no	—
158	C₂×C₈	—	94	2	no	—
159	C₂×C₈	—	94	2	no	—
160	C₂×C₈	—	94	4	no	—
161	C₂×C₈	—	94	4	no	—
162	C₂×C₈	—	94	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	Q₈×D₄₇	1	2	yes	—
171	C₄²	#10625354	1	2	yes	—
172	C₂³	C₂³×Dic₄₇	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	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
184	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
185	C₂×C₄	#41×D₄₇	1	2	yes	—
186	C₂×C₄	#41×D₄₇	1	2	yes	—
187	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
188	C₈	—	47	4	no	—
189	C₈	—	94	4	no	—
190	C₈	—	94	2	no	—
191	C₈	—	188	4	no	—
192	C₂²	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
193	C₂²	#41×Dic₄₇	1	2	yes	—
194	C₂²	#15689890	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	4	no	—
197	C₄	—	4	4	no	—
198	C₄	—	4	4	no	—
199	C₄	#125×D₄₇	1	1	yes	characteristic
200	C₄	#15689658	1	2	yes	—
201	C₄	#15689679	1	1	yes	characteristic
202	C₂	#125×Dic₄₇	1	1	yes	characteristic
203	C₂	#20252329	1	2	yes	—
204	C₂	#20252375	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#2402	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2402	D₄₇
hypocenter	C₄₇	#21677
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2402

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
#20244112	2
#20243954	2
#20243508	2
#20243263	2
#20243257	2
#20243075	2
C₄₇×#2402	2
#21677	47