Subgroups — #26029318

Order 11392.

410 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152148	C₂	1	1	yes	maximal, characteristic
2	#19151677	C₂	1	2	yes	maximal, complemented
3	#19151675	C₂	1	2	yes	maximal, complemented
4	C₈×D₃₅₆	C₂	1	2	yes	maximal, complemented
5	#19151663	C₂	1	2	yes	maximal, complemented
6	#19151651	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151563	C₂	1	4	yes	maximal, complemented
8	#431×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13083574	C₂²	1	2	yes	—
10	C₂×#13083557	C₂²	1	4	yes	—
11	C₂×C₈×D₈₉	C₂²	1	4	yes	—
12	#15668012	C₄	1	1	yes	characteristic
13	#15668009	C₄	1	1	yes	characteristic
14	#15668000	C₄	1	2	yes	—
15	#15667986	C₄	1	2	yes	—
16	#15667986	C₄	1	2	yes	—
17	C₄×D₃₅₆	C₂²	1	2	yes	—
18	C₄×D₃₅₆	C₂²	1	2	yes	—
19	C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	C₄⋊#10623494	C₂²	1	1	yes	characteristic
21	C₂²⋊#10623494	C₂²	1	2	yes	—
22	#15667924	C₂²	1	4	yes	—
23	#15667924	C₂²	1	4	yes	—
24	C₄×#10623494	C₂²	1	2	yes	—
25	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
29	C₄×C₈×C₈₉	C₂²	1	2	yes	—
30	#13083574	—	4	2	no	—
31	#13083574	—	4	2	no	—
32	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
33	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
34	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
35	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊D₈₉	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₈₉	C₂³	1	4	yes	—
38	#13083557	—	4	4	no	—
39	C₂×#10623494	C₂³	1	2	yes	—
40	C₂×#10623494	C₂³	1	2	yes	—
41	C₈×D₈₉	—	4	4	no	—
42	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
44	#13083551	C₂×C₄	1	4	yes	—
45	#13083551	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂³	1	2	yes	—
50	C₂×C₈×C₈₉	C₂³	1	2	yes	—
51	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₈₉	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	1	no	—
60	C₂×C₄×C₈₉	—	2	1	no	—
61	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₂²⋊D₈₉	—	4	4	no	—
68	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
69	D₃₅₆	—	2	4	no	—
70	D₃₅₆	—	2	4	no	—
71	D₃₅₆	—	4	2	no	—
72	C₄×D₈₉	—	2	4	no	—
73	C₄×D₈₉	—	2	4	no	—
74	Dic₁₇₈	—	4	2	no	—
75	C₈×C₈₉	—	2	2	no	—
76	C₈×C₈₉	—	2	2	no	—
77	#10623494	—	2	2	no	—
78	#10623494	—	2	2	no	—
79	C₂²×C₈₉	—	2	1	no	—
80	C₂²×C₈₉	—	2	1	no	—
81	C₂²×C₈₉	—	2	1	no	—
82	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
83	C₂×D₈₉	—	2	4	no	—
84	C₂×D₈₉	—	2	4	no	—
85	C₂×D₈₉	—	2	4	no	—
86	C₄×C₈₉	—	2	1	no	—
87	C₄×C₈₉	—	2	1	no	—
88	C₄×C₈₉	—	2	1	no	—
89	C₄×C₈₉	#131	1	2	yes	—
90	C₄×C₈₉	#131	1	2	yes	—
91	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
92	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
93	Dic₈₉	—	2	4	no	—
94	C₂×C₈₉	—	4	1	no	—
95	C₂×C₈₉	#518	1	1	yes	characteristic
96	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
97	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
98	D₈₉	—	4	4	no	—
99	#2979	—	89	1	no	maximal, Sylow
100	C₈₉	#2979	1	1	yes	characteristic, complemented, Sylow
101	#532	—	89	1	no	—
102	#436	—	89	2	no	—
103	#436	—	89	2	no	—
104	C₈×D₄	—	89	2	no	—
105	C₈×D₄	—	89	2	no	—
106	#431	—	89	1	no	—
107	#431	D₈₉	1	1	yes	characteristic, complemented
108	#408	—	89	4	no	—
109	C₂×#41	—	89	2	no	—
110	C₂×M₁₆	—	89	4	no	—
111	C₂²×C₈	—	89	4	no	—
112	C₄⋊Q₈	—	89	1	no	—
113	#127	—	89	1	no	—
114	#124	—	89	2	no	—
115	#121	—	89	2	no	—
116	#121	—	89	2	no	—
117	C₄×D₄	—	89	2	no	—
118	C₄×D₄	—	89	2	no	—
119	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
120	C₄⋊C₈	—	89	1	no	—
121	C₄⋊C₈	—	89	1	no	—
122	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
124	C₂²⋊C₈	—	89	2	no	—
125	C₂²⋊C₈	—	89	4	no	—
126	C₂²⋊C₈	—	89	4	no	—
127	C₂²⋊C₈	C₂×D₈₉	1	2	yes	—
128	C₄×C₈	—	89	2	no	—
129	C₄×C₈	C₂×D₈₉	1	2	yes	—
130	#41	—	356	2	no	—
131	#41	—	356	2	no	—
132	C₂×Q₈	—	89	2	no	—
133	C₂×D₄	—	89	4	no	—
134	C₂×D₄	—	89	2	no	—
135	C₂×D₄	—	89	2	no	—
136	C₂×D₄	—	89	2	no	—
137	C₂²×C₄	—	89	4	no	—
138	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
139	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
141	M₁₆	—	356	4	no	—
142	C₂×C₈	—	89	2	no	—
143	C₂×C₈	—	89	2	no	—
144	C₂×C₈	—	356	4	no	—
145	C₂×C₈	C₂²×D₈₉	1	2	yes	—
146	C₂×C₈	C₂²×D₈₉	1	2	yes	—
147	C₄⋊C₄	—	89	2	no	—
148	C₄⋊C₄	—	89	2	no	—
149	C₂²⋊C₄	—	89	4	no	—
150	C₂²⋊C₄	—	89	4	no	—
151	C₄²	C₄×D₈₉	1	1	yes	characteristic
152	C₄²	C₄×D₈₉	1	1	yes	characteristic
153	C₄²	C₂²×D₈₉	1	1	yes	characteristic
154	C₄²	C₂²×D₈₉	1	1	yes	characteristic
155	C₂³	—	89	4	no	—
156	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
157	Q₈	—	356	2	no	—
158	D₄	—	356	2	no	—
159	D₄	—	356	4	no	—
160	D₄	—	178	4	no	—
161	D₄	—	178	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	89	4	no	—
167	C₂×C₄	—	178	4	no	—
168	C₂×C₄	—	178	4	no	—
169	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
174	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	—	178	2	no	—
178	C₈	—	178	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	178	4	no	—
183	C₂²	—	178	4	no	—
184	C₂²	—	178	4	no	—
185	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	178	4	no	—
190	C₄	#15668018	1	2	yes	—
191	C₄	#15668020	1	2	yes	—
192	C₄	#15668058	1	1	yes	characteristic
193	C₄	#15668058	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	356	4	no	—
196	C₂	#19152072	1	1	yes	characteristic
197	C₂	C₂×#15668018	1	1	yes	characteristic
198	C₂	C₂×#15668020	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#431×C₈₉	C₂
prime radical	#19152148	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#431	D₈₉
hypocenter	C₈₉	#2979
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #19151677	C₂	semidirect
3. #19151675	C₂	semidirect
4. C₈×D₃₅₆	C₂	semidirect
5. #19151663	C₂	semidirect
6. #19151651	C₂	semidirect
7. #19151563	C₂	semidirect
8. #431×C₈₉	C₂	semidirect
100. C₈₉	#2979	semidirect
107. #431	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152148	2
#19151677	2
#19151675	2
C₈×D₃₅₆	2
#19151663	2
#19151651	2
#19151563	2
#431×C₈₉	2
#2979	89