Subgroups — #36445628

Order 22784.

446 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×Dic₈₉	C₂	1	2	yes	maximal
2	C₄×#15667922	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
4	#26018086	C₂	1	2	yes	maximal
5	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×#15667922	C₄	1	2	yes	—
8	C₂×#15667922	C₄	1	2	yes	—
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₈₉	—	2	2	no	—
11	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
12	#19151352	—	2	2	no	—
13	#19151351	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#334×C₈₉	C₄	1	4	yes	complemented
17	C₂²×#10623494	—	2	2	no	—
18	C₂²×#10623494	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
21	C₂×C₄×Dic₈₉	Q₈	1	2	yes	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	C₂×C₄	1	4	yes	—
24	#15667922	—	2	4	no	—
25	#15667922	C₈	1	8	yes	complemented
26	C₄×#10623494	—	2	1	no	—
27	C₄×#10623494	—	2	1	no	—
28	C₈×Dic₈₉	—	4	4	no	—
29	C₂²×C₈×C₈₉	—	2	2	no	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	—	2	2	no	—
34	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
37	C₂²×Dic₈₉	C₄⋊C₄	1	4	yes	—
38	C₂×#10623494	—	2	2	no	—
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	—	2	4	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	C₂×C₈	1	4	yes	—
44	C₄×Dic₈₉	—	2	2	no	—
45	C₄×Dic₈₉	—	2	2	no	—
46	C₄×Dic₈₉	—	4	2	no	—
47	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
48	C₄×Dic₈₉	M₁₆	1	4	yes	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	2	4	no	—
54	C₂×C₈×C₈₉	—	4	2	no	—
55	C₂×C₈×C₈₉	—	4	2	no	—
56	C₂×C₈×C₈₉	C₄²	1	2	yes	—
57	C₂×C₈×C₈₉	C₄²	1	2	yes	—
58	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
59	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₈₉	#41	1	1	yes	characteristic
62	C₄²×C₈₉	#41	1	1	yes	characteristic
63	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₈₉	#137	1	2	yes	—
75	C₂×Dic₈₉	—	2	4	no	—
76	C₂×Dic₈₉	—	2	4	no	—
77	C₂×Dic₈₉	C₄⋊C₈	1	8	yes	—
78	C₈×C₈₉	—	2	2	no	—
79	C₈×C₈₉	—	4	4	no	—
80	C₈×C₈₉	C₄×C₈	1	4	yes	complemented
81	#10623494	—	2	4	no	—
82	#10623494	—	4	4	no	—
83	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
85	C₂²×C₈₉	#428	1	2	yes	—
86	C₂²×C₈₉	#430	1	2	yes	—
87	C₄×C₈₉	—	2	1	no	—
88	C₄×C₈₉	—	2	1	no	—
89	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
91	C₄×C₈₉	#431	1	2	yes	—
92	C₄×C₈₉	#440	1	2	yes	—
93	C₄×C₈₉	#441	1	1	yes	characteristic
94	C₄×C₈₉	#442	1	1	yes	characteristic
95	Dic₈₉	—	2	8	no	—
96	#8275	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	#1565	1	4	yes	—
98	C₂×C₈₉	C₄×C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₈₉	#1776	1	1	yes	characteristic
100	C₂×C₈₉	#1778	1	1	yes	characteristic
101	C₄×C₄⋊C₈	—	89	2	no	—
102	C₄×#97	—	89	1	no	—
103	C₂×#335	—	89	1	no	—
104	C₂×#334	D₈₉	1	1	yes	characteristic
105	#1277	—	89	2	no	—
106	C₈₉	#8275	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊C₈	—	89	2	no	—
108	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
109	C₂×#97	—	89	2	no	—
110	C₂×#97	—	89	2	no	—
111	C₂×C₄×C₈	—	89	1	no	—
112	C₂×C₄×C₈	—	178	2	no	—
113	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
114	C₄³	—	89	1	no	—
115	#336	—	178	2	no	—
116	#335	—	89	4	no	—
117	#334	Dic₈₉	1	4	yes	complemented
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	89	2	no	—
120	C₂²×C₈	—	178	2	no	—
121	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
123	C₂×C₄²	—	89	2	no	—
124	C₂×C₄²	—	89	2	no	—
125	C₂×C₄²	—	89	2	no	—
126	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	89	4	no	—
129	C₄⋊C₈	—	178	2	no	—
130	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
131	#97	—	89	8	no	—
132	#97	—	178	4	no	—
133	C₄×C₈	—	356	4	no	—
134	C₄×C₈	—	178	1	no	—
135	C₄×C₈	—	178	1	no	—
136	C₄×C₈	C₄×D₈₉	1	2	yes	—
137	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
138	C₂²×C₄	—	89	4	no	—
139	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
141	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	89	4	no	—
147	C₂×C₈	—	356	2	no	—
148	C₂×C₈	—	356	2	no	—
149	C₂×C₈	—	178	4	no	—
150	C₂×C₈	—	178	2	no	—
151	C₂×C₈	—	178	2	no	—
152	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
153	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
154	C₂×C₈	C₈×D₈₉	1	4	yes	—
155	C₄²	—	89	4	no	—
156	C₄²	—	89	4	no	—
157	C₄²	—	356	2	no	—
158	C₄²	—	178	2	no	—
159	C₄²	—	178	2	no	—
160	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
161	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
162	C₄²	#13083573	1	1	yes	characteristic
163	C₄²	#13083575	1	1	yes	characteristic
164	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	89	8	no	—
167	C₂×C₄	—	178	4	no	—
168	C₂×C₄	—	178	4	no	—
169	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
170	C₂×C₄	#15667970	1	1	yes	characteristic
171	C₂×C₄	#15667974	1	1	yes	characteristic
172	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
173	C₂×C₄	#15667977	1	1	yes	characteristic
174	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
175	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
176	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
177	C₂×C₄	#15668039	1	2	yes	—
178	C₂×C₄	#15668041	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	4	4	no	—
181	C₈	—	356	4	no	—
182	C₈	—	178	4	no	—
183	C₈	C₈×Dic₈₉	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
185	C₂²	C₄⋊C₈×D₈₉	1	2	yes	—
186	C₂²	#19151637	1	2	yes	—
187	C₂²	#19151645	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	178	8	no	—
191	C₄	C₂×C₈×Dic₈₉	1	1	yes	characteristic
192	C₄	M₁₆×Dic₈₉	1	1	yes	characteristic
193	C₄	#19151650	1	2	yes	—
194	C₄	#19151699	1	2	yes	—
195	C₄	#19151704	1	1	yes	characteristic
196	C₄	#19151709	1	1	yes	characteristic
197	C₂	#26019001	1	4	yes	—
198	C₂	C₄⋊C₈×Dic₈₉	1	1	yes	characteristic
199	C₂	#26020152	1	1	yes	characteristic
200	C₂	#26020160	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#334×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#334	D₈₉
hypocenter	C₈₉	#8275
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
13. #19151351	C₄	semidirect
16. #334×C₈₉	C₄	semidirect
25. #15667922	C₈	semidirect
80. C₈×C₈₉	C₄×C₈	semidirect
106. C₈₉	#8275	semidirect
117. #334	Dic₈₉	semidirect
183. C₈	C₈×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₈₉	2
C₄×#15667922	2
C₂×#19151351	2
#26018086	2
C₂×#334×C₈₉	2
#8275	89