Subgroups — #26137397

Order 12032.

1598 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241613	C₂	1	8	yes	maximal
4	C₄×C₄₇⋊#95	C₂	1	4	yes	maximal
5	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
9	#15688969	C₄	1	8	yes	complemented
10	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
11	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
12	C₄²×Dic₄₇	C₄	1	8	yes	complemented
13	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#376	C₄	1	8	yes	complemented
16	C₂×#95×C₄₇	C₂²	1	4	yes	—
17	C₂³×Dic₄₇	C₂³	1	4	yes	—
18	C₂³×Dic₄₇	C₂³	1	4	yes	—
19	C₂×C₄×Dic₄₇	—	2	8	no	—
20	C₂×C₄×Dic₄₇	—	2	8	no	—
21	C₂×C₄×Dic₄₇	—	2	16	no	—
22	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
23	C₂×C₄×Dic₄₇	C₂×C₄	1	16	yes	—
24	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
25	C₄₇⋊#95	—	2	32	no	—
26	C₄₇⋊#95	C₂×C₄	1	16	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	—	2	8	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
35	#95×C₄₇	C₂×C₄	1	16	yes	—
36	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	—	2	8	no	—
38	C₂²×C₄×C₄₇	—	2	8	no	—
39	C₂²×C₄×C₄₇	—	2	16	no	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
41	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	#41	1	8	yes	—
48	C₂²×Dic₄₇	—	2	8	no	—
49	C₂²×Dic₄₇	—	2	8	no	—
50	C₂²×Dic₄₇	—	2	16	no	—
51	C₂²×Dic₄₇	—	2	8	no	—
52	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
53	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
54	C₂²×Dic₄₇	#41	1	16	yes	—
55	C₄²×C₄₇	—	2	16	no	—
56	C₄²×C₄₇	C₄²	1	16	yes	—
57	C₄×Dic₄₇	—	2	16	no	—
58	C₄×Dic₄₇	—	2	16	no	—
59	C₄×Dic₄₇	—	2	32	no	—
60	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
61	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
62	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
63	C₂³×C₄₇	C₂×#41	1	8	yes	—
64	C₂³×C₄₇	C₂×#41	1	4	yes	—
65	C₂×C₄×C₄₇	—	2	8	no	—
66	C₂×C₄×C₄₇	—	2	8	no	—
67	C₂×C₄×C₄₇	—	2	8	no	—
68	C₂×C₄×C₄₇	—	2	8	no	—
69	C₂×C₄×C₄₇	—	2	16	no	—
70	C₂×C₄×C₄₇	—	4	16	no	—
71	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
72	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
73	C₂×C₄×C₄₇	#117	1	8	yes	—
74	C₂×C₄×C₄₇	#117	1	8	yes	—
75	C₂×C₄×C₄₇	#117	1	8	yes	—
76	C₂×Dic₄₇	—	2	8	no	—
77	C₂×Dic₄₇	—	2	8	no	—
78	C₂×Dic₄₇	—	2	32	no	—
79	C₂×Dic₄₇	—	2	8	no	—
80	C₂×Dic₄₇	—	2	16	no	—
81	C₂×Dic₄₇	—	4	16	no	—
82	C₂×Dic₄₇	#117	1	16	yes	—
83	#13757	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×#117	1	4	yes	—
86	C₂²×C₄₇	C₂×#117	1	2	yes	—
87	C₂²×C₄₇	C₂×#117	1	2	yes	—
88	C₂²×C₄₇	C₂×#117	1	2	yes	—
89	C₂²×C₄₇	C₄×#41	1	4	yes	—
90	C₂²×C₄₇	C₄×#41	1	4	yes	—
91	C₂²×C₄₇	#529	1	16	yes	—
92	C₄×C₄₇	—	2	8	no	—
93	C₄×C₄₇	—	4	8	no	—
94	C₄×C₄₇	#376	1	8	yes	—
95	Dic₄₇	—	2	16	no	—
96	Dic₄₇	—	4	8	no	—
97	C₂×#376	—	47	1	no	—
98	C₂×#376	D₄₇	1	1	yes	characteristic
99	C₂×C₄³	—	47	1	no	—
100	#1434	—	47	8	no	—
101	C₄×#95	—	47	4	no	—
102	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
103	C₂×C₄₇	C₄×#117	1	2	yes	—
104	C₂×C₄₇	#2283	1	4	yes	—
105	C₂×C₄₇	#2298	1	4	yes	—
106	C₂×C₄₇	#2298	1	4	yes	—
107	C₂²×C₄²	—	47	4	no	—
108	C₂²×C₄²	—	47	2	no	—
109	C₂²×C₄²	—	47	2	no	—
110	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
111	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
112	#376	—	47	8	no	—
113	#376	Dic₄₇	1	8	yes	complemented
114	C₂×#95	—	47	4	no	—
115	C₂×#95	—	47	16	no	—
116	C₂×#95	C₂×D₄₇	1	4	yes	—
117	C₄³	—	47	8	no	—
118	C₄₇	#13757	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	—	47	4	no	—
120	C₂³×C₄	—	47	4	no	—
121	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
122	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
123	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
124	C₂×C₄²	—	2	8	no	—
125	C₂×C₄²	—	47	16	no	—
126	C₂×C₄²	—	47	8	no	—
127	C₂×C₄²	—	47	8	no	—
128	C₂×C₄²	—	94	16	no	—
129	C₂×C₄²	—	94	8	no	—
130	C₂×C₄²	—	94	8	no	—
131	C₂×C₄²	C₄×D₄₇	1	4	yes	—
132	C₂×C₄²	C₄×D₄₇	1	4	yes	—
133	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
134	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
135	#95	—	47	16	no	—
136	#95	—	94	32	no	—
137	#95	C₂×Dic₄₇	1	16	yes	—
138	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	2	16	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	—	47	8	no	—
143	C₂²×C₄	—	47	16	no	—
144	C₂²×C₄	—	47	8	no	—
145	C₂²×C₄	—	94	8	no	—
146	C₂²×C₄	—	94	8	no	—
147	C₂²×C₄	—	94	16	no	—
148	C₂²×C₄	—	94	8	no	—
149	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
150	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
153	C₂²×C₄	#10625350	1	8	yes	—
154	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
155	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
156	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
157	C₄²	—	2	16	no	—
158	C₄²	—	94	32	no	—
159	C₄²	—	94	16	no	—
160	C₄²	—	94	16	no	—
161	C₄²	C₄×Dic₄₇	1	16	yes	—
162	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
163	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
164	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
165	C₂³	#41×D₄₇	1	8	yes	—
166	C₂³	C₂×#10625350	1	4	yes	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	16	no	—
172	C₂×C₄	—	4	16	no	—
173	C₂×C₄	—	47	16	no	—
174	C₂×C₄	—	94	8	no	—
175	C₂×C₄	—	94	8	no	—
176	C₂×C₄	—	94	32	no	—
177	C₂×C₄	—	94	8	no	—
178	C₂×C₄	—	94	16	no	—
179	C₂×C₄	—	188	16	no	—
180	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
181	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
182	C₂×C₄	#13093981	1	8	yes	—
183	C₂×C₄	#13093981	1	8	yes	—
184	C₂×C₄	#13093983	1	8	yes	—
185	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
186	C₂²	#117×D₄₇	1	4	yes	—
187	C₂²	C₂×#13093981	1	2	yes	—
188	C₂²	C₂×#13093981	1	2	yes	—
189	C₂²	C₂×#13093983	1	2	yes	—
190	C₂²	#15689591	1	4	yes	—
191	C₂²	#41×Dic₄₇	1	4	yes	—
192	C₂²	#15689651	1	16	yes	—
193	C₄	—	2	8	no	—
194	C₄	—	4	8	no	—
195	C₄	—	94	16	no	—
196	C₄	—	188	8	no	—
197	C₄	#15688969	1	8	yes	—
198	C₂	C₂×#15688969	1	1	yes	characteristic
199	C₂	#117×Dic₄₇	1	2	yes	—
200	C₂	#20245780	1	4	yes	—
201	C₂	#20245883	1	4	yes	—
202	C₂	#20245886	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂⁴	C₂³×D₄₇
Fitting	C₂×C₄₇×#376	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#13757
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
9. #15688969	C₄	semidirect
12. C₄²×Dic₄₇	C₄	semidirect
15. C₄₇×#376	C₄	semidirect
113. #376	Dic₄₇	semidirect
118. C₄₇	#13757	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688969	2
C₂×C₄²×Dic₄₇	2
#20241613	2
C₄×C₄₇⋊#95	2
C₂×C₄₇×#376	2
#13757	47