Subgroups — #36445674

Order 22784.

430 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	C₄⋊C₄×#10623494	C₂	1	2	yes	maximal
2	C₂×#19151349	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
4	#26018093	C₂	1	2	yes	maximal
5	C₄×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	—	2	2	no	—
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151353	—	2	2	no	—
11	#19151349	C₄	1	4	yes	complemented
12	#19151319	C₄	1	4	yes	complemented
13	C₂×#107×C₈₉	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
16	#340×C₈₉	C₄	1	2	yes	—
17	C₂²×#10623494	—	2	2	no	—
18	C₂²×#10623494	—	2	1	no	—
19	C₂²×#10623494	D₄	1	2	yes	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	C₂×C₄	1	4	yes	—
22	C₄×#10623494	—	2	1	no	—
23	C₄×#10623494	—	4	4	no	—
24	C₄×#10623494	C₂×C₄	1	2	yes	—
25	C₄×#10623494	C₂×C₄	1	2	yes	—
26	C₄×#10623494	Q₈	1	2	yes	—
27	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₈₉	—	2	2	no	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	#107×C₈₉	C₈	1	8	yes	complemented
33	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	#95×C₈₉	—	2	2	no	—
36	C₂×#10623494	—	2	2	no	—
37	C₂×#10623494	—	2	2	no	—
38	C₂×#10623494	—	2	2	no	—
39	C₂×#10623494	—	2	4	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₈₉	—	2	4	no	—
45	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₈₉	—	2	2	no	—
49	C₂×C₈×C₈₉	C₄²	1	2	yes	—
50	C₂×C₈×C₈₉	C₄²	1	2	yes	—
51	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
52	C₄⋊C₄×C₈₉	—	2	4	no	—
53	C₄⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
54	C₄²×C₈₉	—	4	2	no	—
55	C₄²×C₈₉	—	4	2	no	—
56	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₈₉	#41	1	1	yes	characteristic
60	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	8	no	—
63	C₂×C₄×C₈₉	—	4	4	no	—
64	C₂×C₄×C₈₉	—	4	4	no	—
65	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#117	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	#131	1	2	yes	—
73	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₈₉	#137	1	2	yes	—
75	C₈×C₈₉	—	2	2	no	—
76	C₈×C₈₉	C₄×C₈	1	4	yes	complemented
77	#10623494	—	2	4	no	—
78	#10623494	—	4	2	no	—
79	#10623494	—	4	4	no	—
80	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈₉	C₈×D₄	1	2	yes	—
82	C₂²×C₈₉	#441	1	1	yes	characteristic
83	C₂²×C₈₉	#442	1	1	yes	characteristic
84	C₂²×C₈₉	#444	1	2	yes	—
85	C₄×C₈₉	—	2	1	no	—
86	C₄×C₈₉	—	2	1	no	—
87	C₄×C₈₉	—	4	8	no	—
88	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄×C₈₉	#405	1	1	yes	characteristic
90	C₄×C₈₉	#428	1	2	yes	—
91	C₄×C₈₉	#440	1	2	yes	—
92	C₄×C₈₉	C₈×Q₈	1	2	yes	—
93	#8284	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	#1579	1	2	yes	—
95	C₂×C₈₉	#1580	1	2	yes	—
96	C₂×C₈₉	C₈×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₈₉	#1773	1	1	yes	characteristic
98	C₂×C₈₉	#1776	1	1	yes	characteristic
99	C₄×#107	D₈₉	1	1	yes	characteristic
100	C₈×C₄⋊C₄	—	89	2	no	—
101	C₂×#334	—	89	1	no	—
102	C₂×#322	—	89	1	no	—
103	#1279	—	89	2	no	—
104	C₈₉	#8284	1	1	yes	characteristic, complemented, Sylow
105	C₂×#107	Dic₈₉	1	2	yes	—
106	C₂×C₄⋊C₈	—	89	2	no	—
107	C₂×C₄×C₈	—	89	1	no	—
108	C₂×C₄×C₈	—	89	1	no	—
109	C₂×C₄×C₈	—	178	2	no	—
110	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
111	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
112	#340	Dic₈₉	1	2	yes	—
113	#336	—	178	2	no	—
114	#334	—	89	4	no	—
115	#322	—	89	4	no	—
116	C₂²×C₈	—	89	2	no	—
117	C₂²×C₈	—	178	2	no	—
118	C₂²×C₈	—	178	1	no	—
119	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
120	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
122	C₂×C₄²	—	2	2	no	—
123	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
124	#107	#10623494	1	8	yes	complemented
125	C₄⋊C₈	—	89	4	no	—
126	C₄⋊C₈	—	178	2	no	—
127	C₄×C₈	—	89	2	no	—
128	C₄×C₈	—	89	2	no	—
129	C₄×C₈	—	89	2	no	—
130	C₄×C₈	—	356	4	no	—
131	C₄×C₈	—	178	1	no	—
132	C₄×C₈	C₄×D₈₉	1	2	yes	—
133	C₄×C₈	C₄×D₈₉	1	2	yes	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
137	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
138	C₂²×C₄	#13083573	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	89	4	no	—
141	C₂×C₈	—	356	2	no	—
142	C₂×C₈	—	356	2	no	—
143	C₂×C₈	—	356	2	no	—
144	C₂×C₈	—	178	2	no	—
145	C₂×C₈	—	178	4	no	—
146	C₂×C₈	—	178	2	no	—
147	C₂×C₈	—	178	2	no	—
148	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
149	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
150	C₂×C₈	C₂×#10623494	1	4	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	C₂×#10623494	1	8	yes	—
153	C₄²	—	4	2	no	—
154	C₄²	—	4	2	no	—
155	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
157	C₄²	Q₈×D₈₉	1	1	yes	characteristic
158	C₄²	#13083575	1	1	yes	characteristic
159	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
166	C₂×C₄	#15667970	1	1	yes	characteristic
167	C₂×C₄	#15667974	1	1	yes	characteristic
168	C₂×C₄	#15667977	1	1	yes	characteristic
169	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
170	C₂×C₄	C₂²×#10623494	1	2	yes	—
171	C₂×C₄	#15668019	1	2	yes	—
172	C₂×C₄	#15668036	1	2	yes	—
173	C₂×C₄	#15668044	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	—	356	2	no	—
176	C₈	—	356	4	no	—
177	C₈	—	178	4	no	—
178	C₈	C₄×#10623494	1	4	yes	complemented
179	C₂²	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
180	C₂²	D₄×#10623494	1	2	yes	—
181	C₂²	#19151705	1	1	yes	characteristic
182	C₂²	#19151710	1	1	yes	characteristic
183	C₂²	#19151720	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	8	no	—
187	C₄	C₂×C₄×#10623494	1	1	yes	characteristic
188	C₄	#19151554	1	1	yes	characteristic
189	C₄	#19151639	1	2	yes	—
190	C₄	#19151697	1	2	yes	—
191	C₄	Q₈×#10623494	1	2	yes	—
192	C₂	#26019071	1	2	yes	—
193	C₂	#26019076	1	2	yes	—
194	C₂	C₄⋊C₄×#10623494	1	1	yes	characteristic
195	C₂	#26020138	1	1	yes	characteristic
196	C₂	#26020151	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#107	D₈₉
hypocenter	C₈₉	#8284
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #19151349	C₄	semidirect
12. #19151319	C₄	semidirect
32. #107×C₈₉	C₈	semidirect
76. C₈×C₈₉	C₄×C₈	semidirect
104. C₈₉	#8284	semidirect
124. #107	#10623494	semidirect
178. C₈	C₄×#10623494	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#10623494	2
C₂×#19151349	2
C₂×#19151319	2
#26018093	2
C₄×#107×C₈₉	2
#8284	89