Subgroups — #26225148

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253912	C₂	1	1	yes	maximal, characteristic
2	#20252322	C₂	1	2	yes	maximal
3	#20244369	C₂	1	2	yes	maximal
4	#20244141	C₂	1	4	yes	maximal
5	#20243563	C₂	1	1	yes	maximal, characteristic
6	#20243267	C₂	1	4	yes	maximal
7	C₄₇×#1726	C₂	1	1	yes	maximal, characteristic
8	#15689680	C₂²	1	2	yes	—
9	#15689532	C₄	1	8	yes	complemented
10	#15689253	—	2	4	no	—
11	C₈×Dic₉₄	—	2	4	no	—
12	C₂×#13093960	C₂²	1	4	yes	—
13	#15689161	C₂²	1	4	yes	—
14	#15689149	C₂²	1	4	yes	—
15	#15689147	C₂²	1	2	yes	—
16	#15689078	C₂²	1	4	yes	—
17	#15688883	C₂²	1	8	yes	—
18	C₂×#97×C₄₇	C₂²	1	2	yes	—
19	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
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	C₂×#10625329	C₂³	1	4	yes	—
24	#13094024	C₂×C₄	1	4	yes	—
25	#13094016	C₂×C₄	1	4	yes	—
26	#13094003	—	2	4	no	—
27	C₄×Dic₉₄	—	2	4	no	—
28	#13093983	C₂³	1	4	yes	—
29	#13093960	—	2	8	no	—
30	#13093958	C₂×C₄	1	8	yes	—
31	#13093956	C₂×C₄	1	8	yes	—
32	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
33	Dic₄₇⋊C₈	—	2	8	no	—
34	#13093934	—	2	4	no	—
35	C₈×Dic₄₇	—	2	4	no	—
36	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	#97×C₄₇	—	2	4	no	—
42	#97×C₄₇	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₄₇	D₄	1	2	yes	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₄×C₈×C₄₇	Q₈	1	8	yes	complemented
46	#95×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	#41	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₉₄	—	2	4	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
57	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₄₇	#41	1	4	yes	—
59	C₄²×C₄₇	—	2	2	no	—
60	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
61	C₄²×C₄₇	C₂×D₄	1	2	yes	—
62	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
63	C₂²⋊Dic₄₇	—	2	4	no	—
64	C₄⋊Dic₄₇	—	2	4	no	—
65	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
66	Dic₄₇⋊C₄	—	2	8	no	—
67	C₄×Dic₄₇	—	2	4	no	—
68	#10625329	—	2	8	no	—
69	C₂×C₄₇⋊C₈	—	2	4	no	—
70	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	4	2	no	—
73	C₂×C₄×C₄₇	—	4	4	no	—
74	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
75	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
79	C₂×Dic₄₇	—	2	8	no	—
80	Dic₉₄	—	4	8	no	—
81	C₈×C₄₇	—	2	2	no	—
82	C₈×C₄₇	C₄×Q₈	1	8	yes	—
83	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
84	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
85	C₄₇⋊C₈	—	4	4	no	—
86	#21735	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	—	2	2	no	—
88	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
89	C₂²×C₄₇	#533	1	2	yes	—
90	C₄×C₄₇	—	2	4	no	—
91	C₄×C₄₇	—	4	4	no	—
92	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
93	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
94	C₄×C₄₇	#532	1	1	yes	characteristic
95	Dic₄₇	—	4	8	no	—
96	#3162	—	47	1	no	—
97	#2960	—	47	2	no	—
98	#1995	—	47	2	no	—
99	#1946	—	47	4	no	—
100	#1849	—	47	1	no	—
101	#1772	—	47	4	no	—
102	#1726	D₄₇	1	1	yes	characteristic
103	C₂×C₄₇	—	2	2	no	—
104	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄₇	#2955	1	2	yes	—
106	#533	—	47	2	no	—
107	#501	—	47	8	no	—
108	#446	—	94	4	no	—
109	C₈×Q₈	—	94	4	no	—
110	C₂×#108	—	47	4	no	—
111	#427	—	47	4	no	—
112	#424	—	47	4	no	—
113	#423	—	47	2	no	—
114	#405	—	47	4	no	—
115	C₂×#97	C₂×D₄₇	1	2	yes	—
116	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
118	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
119	#376	C₂×D₄₇	1	1	yes	characteristic
120	#339	—	47	8	no	—
121	C₄₇	#21735	1	1	yes	characteristic, complemented, Sylow
122	C₂×M₁₆	—	47	4	no	—
123	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
126	C₄⋊Q₈	—	47	4	no	—
127	#125	—	47	4	no	—
128	C₂²⋊Q₈	—	94	4	no	—
129	C₄×Q₈	—	94	4	no	—
130	#117	—	47	4	no	—
131	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
132	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
133	#108	—	94	8	no	—
134	#107	—	47	8	no	—
135	#106	—	47	8	no	—
136	C₄⋊C₈	—	94	8	no	—
137	C₄⋊C₈	—	94	4	no	—
138	#97	—	2	4	no	—
139	#97	—	94	4	no	—
140	#97	C₄×D₄₇	1	4	yes	—
141	C₄×C₈	—	94	4	no	—
142	C₄×C₈	Dic₉₄	1	8	yes	complemented
143	C₄×C₈	D₁₈₈	1	2	yes	—
144	C₄×C₈	D₁₈₈	1	2	yes	—
145	#95	—	2	2	no	—
146	C₂×Q₈	—	94	4	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	#10625350	1	2	yes	—
149	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
150	M₁₆	—	94	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	94	4	no	—
153	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
154	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
155	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
156	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
157	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
158	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
159	C₂×C₈	#10625350	1	4	yes	—
160	C₄⋊C₄	—	47	8	no	—
161	C₄⋊C₄	—	94	8	no	—
162	C₄⋊C₄	—	94	4	no	—
163	C₂²⋊C₄	—	94	4	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	94	4	no	—
166	C₄²	C₂×Dic₉₄	1	4	yes	—
167	C₄²	C₂×C₄×D₄₇	1	2	yes	—
168	C₄²	C₂×D₁₈₈	1	2	yes	—
169	C₂³	C₂×#10625350	1	1	yes	characteristic
170	Q₈	—	188	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	94	8	no	—
175	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
176	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
179	C₂×C₄	C₂×#10625350	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	188	4	no	—
182	C₈	C₄×Dic₉₄	1	8	yes	—
183	C₈	#13094024	1	2	yes	—
184	C₈	#13094024	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
187	C₂²	#15689680	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	4	4	no	—
190	C₄	—	188	8	no	—
191	C₄	C₂×C₄×Dic₉₄	1	2	yes	—
192	C₄	C₂×#13094024	1	1	yes	characteristic
193	C₄	#15689673	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₄×#13094024	1	1	yes	characteristic
196	C₂	#20252292	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1726	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1726	D₄₇
hypocenter	C₄₇	#21735
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
9. #15689532	C₄	semidirect
45. C₄×C₈×C₄₇	Q₈	semidirect
121. C₄₇	#21735	semidirect
142. C₄×C₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253912	2
#20252322	2
#20244369	2
#20244141	2
#20243563	2
#20243267	2
C₄₇×#1726	2
#21735	47