Subgroups — #26029365

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13083574	C₂	1	1	yes	maximal, characteristic
2	#19151728	C₂	1	2	yes	maximal, complemented
3	#19151677	C₂	1	2	yes	maximal, complemented
4	#19151666	C₂	1	4	yes	maximal, complemented
5	#19151654	C₂	1	2	yes	maximal, complemented
6	#19151620	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic, complemented
8	#431×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13083574	C₄	1	1	yes	characteristic
10	C₂×#13083557	C₂²	1	4	yes	—
11	C₂²×#10623494	C₂²	1	2	yes	—
12	C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
13	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	C₄×D₃₅₆	C₄	1	1	yes	characteristic
15	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₈₉	C₄	1	2	yes	—
17	C₄×C₂²⋊D₈₉	C₂²	1	2	yes	—
18	#15667971	C₄	1	1	yes	characteristic
19	#15667969	C₄	1	2	yes	—
20	C₄²×D₈₉	C₂²	1	2	yes	—
21	C₄⋊#10623494	C₂²	1	2	yes	—
22	C₄⋊#10623494	C₂²	1	2	yes	—
23	#15667941	C₂²	1	4	yes	—
24	#15667924	C₂²	1	4	yes	—
25	#15667922	C₂²	1	4	yes	—
26	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
30	C₄×C₈×C₈₉	C₂²	1	2	yes	—
31	#13083574	—	2	4	no	—
32	C₂×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
33	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₈₉	C₂³	1	2	yes	—
38	#13083557	—	4	4	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂×#10623494	C₂³	1	4	yes	—
41	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
43	#13083553	C₂×C₄	1	2	yes	—
44	#13083553	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
46	#13083551	C₂×C₄	1	2	yes	—
47	#13083551	C₂×C₄	1	2	yes	—
48	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
49	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
50	C₄×Dic₈₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂³	1	2	yes	—
55	C₂×C₈×C₈₉	C₂³	1	2	yes	—
56	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₈₉	—	2	1	no	—
63	C₂×C₄×C₈₉	—	2	1	no	—
64	C₂×C₄×C₈₉	—	4	1	no	—
65	C₂×C₄×C₈₉	M₁₆	1	4	yes	complemented
66	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₈₉	—	2	4	no	—
73	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
74	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
76	D₃₅₆	—	2	2	no	—
77	C₄×D₈₉	—	2	4	no	—
78	C₄×D₈₉	—	2	2	no	—
79	C₄×D₈₉	—	2	2	no	—
80	C₄×D₈₉	—	4	2	no	—
81	Dic₁₇₈	—	2	2	no	—
82	C₈×C₈₉	—	2	2	no	—
83	C₈×C₈₉	—	2	2	no	—
84	#10623494	—	2	4	no	—
85	C₂²×C₈₉	—	2	1	no	—
86	C₂²×C₈₉	—	2	1	no	—
87	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
88	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₈₉	—	2	2	no	—
90	C₂×D₈₉	—	2	2	no	—
91	C₂×D₈₉	—	2	2	no	—
92	C₄×C₈₉	—	2	1	no	—
93	C₄×C₈₉	—	2	1	no	—
94	C₄×C₈₉	—	2	1	no	—
95	C₄×C₈₉	C₂×M₁₆	1	2	yes	—
96	C₄×C₈₉	C₂×M₁₆	1	2	yes	—
97	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
99	Dic₈₉	—	2	2	no	—
100	Dic₈₉	—	2	2	no	—
101	Dic₈₉	—	2	2	no	—
102	C₂×C₈₉	—	2	2	no	—
103	C₂×C₈₉	#520	1	1	yes	characteristic
104	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
105	C₂×C₈₉	#568	1	1	yes	characteristic
106	D₈₉	—	4	2	no	—
107	#2989	—	89	1	no	maximal, Sylow
108	C₈₉	#2989	1	1	yes	characteristic, complemented, Sylow
109	C₄×#41	—	89	1	no	—
110	#446	—	89	2	no	—
111	#436	—	89	2	no	—
112	#435	—	89	4	no	—
113	#432	—	89	2	no	—
114	#431	D₈₉	1	1	yes	characteristic, complemented
115	#423	—	89	2	no	—
116	C₂×C₄⋊C₈	—	89	1	no	—
117	C₂×#41	—	89	1	no	—
118	C₂×M₁₆	—	89	4	no	—
119	C₂²×C₈	—	89	2	no	—
120	C₄×Q₈	—	89	1	no	—
121	C₄×Q₈	—	89	1	no	—
122	C₄×D₄	—	89	2	no	—
123	C₄×D₄	—	89	2	no	—
124	C₄×D₄	—	89	1	no	—
125	C₄×D₄	—	89	1	no	—
126	#117	—	89	1	no	—
127	#117	—	89	2	no	—
128	C₂×C₄²	—	89	2	no	—
129	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
130	C₄⋊C₈	—	89	4	no	—
131	C₄⋊C₈	—	89	2	no	—
132	C₄⋊C₈	—	89	2	no	—
133	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
135	C₂²⋊C₈	—	89	4	no	—
136	C₂²⋊C₈	C₂×D₈₉	1	2	yes	—
137	#97	—	89	4	no	—
138	C₄×C₈	C₂×D₈₉	1	2	yes	—
139	#41	—	178	4	no	—
140	C₂×Q₈	—	89	1	no	—
141	C₂×D₄	—	89	2	no	—
142	C₂×D₄	—	89	1	no	—
143	C₂²×C₄	—	89	2	no	—
144	C₂²×C₄	—	89	2	no	—
145	C₂²×C₄	—	89	2	no	—
146	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
149	M₁₆	—	356	4	no	—
150	C₂×C₈	—	89	4	no	—
151	C₂×C₈	—	356	2	no	—
152	C₂×C₈	C₂²×D₈₉	1	2	yes	—
153	C₂×C₈	C₂²×D₈₉	1	2	yes	—
154	C₄⋊C₄	—	89	2	no	—
155	C₄⋊C₄	—	89	2	no	—
156	C₄⋊C₄	—	89	1	no	—
157	C₄⋊C₄	—	89	1	no	—
158	C₂²⋊C₄	—	89	2	no	—
159	C₂²⋊C₄	—	89	2	no	—
160	C₂²⋊C₄	—	89	2	no	—
161	C₄²	—	89	2	no	—
162	C₄²	—	89	2	no	—
163	C₄²	—	89	2	no	—
164	C₄²	C₄×D₈₉	1	1	yes	characteristic
165	C₄²	C₄×D₈₉	1	1	yes	characteristic
166	C₄²	C₂²×D₈₉	1	1	yes	characteristic
167	C₄²	C₂²×D₈₉	1	1	yes	characteristic
168	C₂³	—	89	2	no	—
169	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
170	Q₈	—	178	2	no	—
171	D₄	—	178	4	no	—
172	D₄	—	178	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	89	2	no	—
177	C₂×C₄	—	89	2	no	—
178	C₂×C₄	—	89	2	no	—
179	C₂×C₄	—	356	2	no	—
180	C₂×C₄	—	178	4	no	—
181	C₂×C₄	—	178	2	no	—
182	C₂×C₄	—	178	2	no	—
183	C₂×C₄	#13083557	1	4	yes	complemented
184	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	178	4	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	178	2	no	—
196	C₂²	—	178	2	no	—
197	C₂²	—	178	2	no	—
198	C₂²	C₂×#13083557	1	2	yes	—
199	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	178	2	no	—
204	C₄	—	178	2	no	—
205	C₄	—	178	2	no	—
206	C₄	C₂×#13083557	1	2	yes	—
207	C₄	C₂×#13083557	1	2	yes	—
208	C₄	#15668058	1	1	yes	characteristic
209	C₄	#15668059	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	356	2	no	—
212	C₂	#19152078	1	1	yes	characteristic
213	C₂	C₂²×#13083557	1	1	yes	characteristic
214	C₂	#19152364	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	C₂×#13083574	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#431	D₈₉
hypocenter	C₈₉	#2989
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. #19151728	C₂	semidirect
3. #19151677	C₂	semidirect
4. #19151666	C₂	semidirect
5. #19151654	C₂	semidirect
6. #19151620	C₂	semidirect
7. C₂×C₄⋊#10623494	C₂	semidirect
8. #431×C₈₉	C₂	semidirect
65. C₂×C₄×C₈₉	M₁₆	semidirect
108. C₈₉	#2989	semidirect
114. #431	D₈₉	semidirect
183. C₂×C₄	#13083557	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13083574	2
#19151728	2
#19151677	2
#19151666	2
#19151654	2
#19151620	2
C₂×C₄⋊#10623494	2
#431×C₈₉	2
#2989	89