Subgroups — #25302681

Order 11008.

1990 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	#19142982	C₂	1	4	yes	maximal, complemented
2	#19142958	C₂	1	4	yes	maximal, complemented
3	#19142905	C₂	1	8	yes	maximal
4	C₄×#13063317	C₂	1	8	yes	maximal, complemented
5	#19137078	C₂	1	4	yes	maximal
6	#19137002	C₂	1	2	yes	maximal
7	C₂×C₄³×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063317	C₂²	1	8	yes	—
9	C₂×#13063284	C₂²	1	4	yes	—
10	#14582797	C₂²	1	16	yes	—
11	#14582797	C₂²	1	16	yes	—
12	#14582783	C₂²	1	32	yes	—
13	#14582772	C₂²	1	16	yes	—
14	#14582769	C₂²	1	4	yes	—
15	#14582765	C₂²	1	8	yes	—
16	#14582765	C₂²	1	4	yes	—
17	#14582759	C₂²	1	8	yes	—
18	C₄×Dic₄₃⋊C₄	C₂²	1	16	yes	—
19	C₄×C₄⋊Dic₄₃	C₂²	1	8	yes	—
20	C₂²×C₄²×C₄₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₃	C₂²	1	4	yes	—
22	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
23	C₄³×C₄₃	C₂²	1	2	yes	—
24	C₄³×C₄₃	C₂²	1	2	yes	—
25	C₄³×C₄₃	C₂²	1	4	yes	—
26	#13063317	—	2	32	no	—
27	#13063284	—	2	16	no	—
28	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
29	C₂×C₄⋊Dic₄₃	C₂³	1	16	yes	—
30	C₂×Dic₄₃⋊C₄	C₂³	1	32	yes	—
31	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
32	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
33	C₄₃⋊#95	C₂³	1	32	yes	—
34	C₄₃⋊#95	C₂³	1	16	yes	—
35	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
36	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	—	2	8	no	—
39	C₂×C₄²×C₄₃	—	2	16	no	—
40	C₂×C₄²×C₄₃	Q₈	1	8	yes	—
41	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₃	C₂³	1	8	yes	—
45	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
49	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	—	2	16	no	—
51	C₂²×C₄×C₄₃	—	2	8	no	—
52	C₂²×C₄×C₄₃	—	2	8	no	—
53	C₂²×C₄×C₄₃	C₂×Q₈	1	8	yes	—
54	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₃	#41	1	8	yes	—
56	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
62	C₄²×C₄₃	—	2	16	no	—
63	C₄²×C₄₃	—	2	8	no	—
64	C₄²×C₄₃	—	2	8	no	—
65	C₄²×C₄₃	—	2	8	no	—
66	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
67	C₄²×C₄₃	C₂×Q₈	1	8	yes	—
68	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
69	C₄²×C₄₃	#41	1	16	yes	—
70	C₂²⋊Dic₄₃	—	2	16	no	—
71	C₄⋊Dic₄₃	—	2	16	no	—
72	C₄⋊Dic₄₃	—	2	32	no	—
73	Dic₄₃⋊C₄	—	2	64	no	—
74	C₄×Dic₄₃	—	2	16	no	—
75	C₂³×C₄₃	—	2	4	no	—
76	C₂³×C₄₃	C₂²×Q₈	1	2	yes	—
77	C₂³×C₄₃	C₂×#41	1	4	yes	—
78	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
79	C₂×C₄×C₄₃	—	2	32	no	—
80	C₂×C₄×C₄₃	—	2	4	no	—
81	C₂×C₄×C₄₃	—	2	8	no	—
82	C₂×C₄×C₄₃	—	2	8	no	—
83	C₂×C₄×C₄₃	—	2	16	no	—
84	C₂×C₄×C₄₃	#125	1	16	yes	complemented
85	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
86	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
87	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
88	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
89	C₂×C₄×C₄₃	C₂×#41	1	16	yes	—
90	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
91	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
92	C₂×C₄×C₄₃	C₂×#41	1	8	yes	—
93	C₂×Dic₄₃	—	2	16	no	—
94	C₂×Dic₄₃	—	2	32	no	—
95	#36640	—	43	1	no	maximal, Sylow
96	C₂²×C₄₃	—	2	4	no	—
97	C₂²×C₄₃	—	2	8	no	—
98	C₂²×C₄₃	C₂×#125	1	4	yes	—
99	C₂²×C₄₃	C₂³×Q₈	1	1	yes	characteristic
100	C₂²×C₄₃	C₂²×#41	1	2	yes	—
101	C₂²×C₄₃	C₂²×#41	1	4	yes	—
102	C₄×C₄₃	—	2	4	no	—
103	C₄×C₄₃	—	2	16	no	—
104	C₄×C₄₃	C₂×#125	1	4	yes	—
105	C₄×C₄₃	C₂×#125	1	4	yes	—
106	C₄×C₄₃	#533	1	8	yes	—
107	Dic₄₃	—	4	16	no	—
108	#2864	—	43	4	no	—
109	#2859	—	43	4	no	—
110	#2851	—	43	8	no	—
111	C₄×#125	—	43	8	no	—
112	#2296	—	43	4	no	—
113	#2285	—	43	2	no	—
114	C₂×C₄³	D₄₃	1	1	yes	characteristic
115	C₂×C₄₃	—	2	4	no	—
116	C₂×C₄₃	C₂²×#125	1	1	yes	characteristic
117	C₂×C₄₃	C₂×#533	1	2	yes	—
118	C₂×C₄₃	#3444	1	4	yes	—
119	C₂×#125	—	43	8	no	—
120	C₂×#117	—	43	4	no	—
121	C₂²×C₄²	C₂×D₄₃	1	4	yes	—
122	C₂²×C₄²	C₂×D₄₃	1	2	yes	—
123	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
124	#389	—	43	16	no	—
125	#389	—	43	16	no	—
126	#387	—	43	32	no	—
127	#385	—	43	16	no	—
128	#384	—	43	4	no	—
129	#382	—	43	8	no	—
130	#382	—	43	4	no	—
131	#380	—	43	8	no	—
132	C₄×C₄⋊C₄	—	43	8	no	—
133	C₄×C₄⋊C₄	—	43	16	no	—
134	C₄³	C₂×D₄₃	1	4	yes	—
135	C₄³	C₂×D₄₃	1	2	yes	—
136	C₄³	C₂×D₄₃	1	2	yes	—
137	C₄₃	#36640	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
139	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
140	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
141	#125	—	86	32	no	—
142	#117	—	86	16	no	—
143	C₂×C₄⋊C₄	—	43	8	no	—
144	C₂×C₄⋊C₄	—	43	32	no	—
145	C₂×C₄⋊C₄	—	43	16	no	—
146	C₂×C₂²⋊C₄	—	43	8	no	—
147	C₂×C₄²	—	2	8	no	—
148	C₂×C₄²	—	2	16	no	—
149	C₂×C₄²	—	43	8	no	—
150	C₂×C₄²	Dic₈₆	1	8	yes	—
151	C₂×C₄²	C₂²×D₄₃	1	8	yes	—
152	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
153	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
154	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
155	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
156	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
158	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
159	#95	—	43	16	no	—
160	#95	—	43	32	no	—
161	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	16	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	43	16	no	—
166	C₂²×C₄	C₂×Dic₈₆	1	8	yes	—
167	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
168	C₂²×C₄	#10621898	1	8	yes	—
169	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
170	C₂²×C₄	C₂³×D₄₃	1	8	yes	—
171	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
173	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
174	C₄⋊C₄	—	86	16	no	—
175	C₄⋊C₄	—	86	64	no	—
176	C₄⋊C₄	—	86	32	no	—
177	C₂²⋊C₄	—	86	16	no	—
178	C₄²	—	2	16	no	—
179	C₄²	—	2	8	no	—
180	C₄²	—	2	8	no	—
181	C₄²	—	2	8	no	—
182	C₄²	—	86	16	no	—
183	C₄²	C₂×Dic₈₆	1	4	yes	—
184	C₄²	C₂×Dic₈₆	1	8	yes	—
185	C₄²	C₂×Dic₈₆	1	4	yes	—
186	C₄²	#10621898	1	16	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₂²×Dic₈₆	1	2	yes	—
189	C₂³	C₂×#10621898	1	4	yes	—
190	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
191	C₂×C₄	—	2	32	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	86	16	no	—
197	C₂×C₄	—	86	32	no	—
198	C₂×C₄	#13063317	1	16	yes	complemented
199	C₂×C₄	C₂²×Dic₈₆	1	4	yes	—
200	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
201	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
202	C₂×C₄	C₂²×Dic₈₆	1	4	yes	—
203	C₂×C₄	C₂×#10621898	1	16	yes	—
204	C₂×C₄	C₂×#10621898	1	4	yes	—
205	C₂×C₄	C₂×#10621898	1	4	yes	—
206	C₂×C₄	C₂×#10621898	1	8	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	C₂×#13063317	1	4	yes	—
210	C₂²	C₂³×Dic₈₆	1	1	yes	characteristic
211	C₂²	C₂²×#10621898	1	2	yes	—
212	C₂²	C₂²×#10621898	1	4	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	16	no	—
215	C₄	—	172	16	no	—
216	C₄	C₂×#13063317	1	4	yes	—
217	C₄	C₂×#13063317	1	4	yes	—
218	C₄	#14583456	1	8	yes	—
219	C₂	—	2	4	no	—
220	C₂	C₂²×#13063317	1	1	yes	characteristic
221	C₂	C₂×#14583456	1	2	yes	—
222	C₂	#19147891	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₃
commutator, socle, cosocle	C₂³×C₄₃	C₂⁵
Frattini	C₂³	C₂⁴×D₄₃
Fitting	C₂×C₄³×C₄₃	C₂
prime radical	C₂⁴×C₄₃	C₂⁴
hypercenter	C₂×C₄³	D₄₃
hypocenter	C₄₃	#36640
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. #19142982	C₂	semidirect
2. #19142958	C₂	semidirect
4. C₄×#13063317	C₂	semidirect
84. C₂×C₄×C₄₃	#125	semidirect
137. C₄₃	#36640	semidirect
198. C₂×C₄	#13063317	semidirect

Maximal subgroups

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

subgroup	index
#19142982	2
#19142958	2
#19142905	2
C₄×#13063317	2
#19137078	2
#19137002	2
C₂×C₄³×C₄₃	2
#36640	43