Subgroups — #36445737

Order 22784.

334 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019000	C₂	1	1	yes	maximal, characteristic
2	#26018539	C₂	1	2	yes	maximal
3	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
4	#26018090	C₂	1	2	yes	maximal
5	C₄×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	#19151351	C₄	1	2	yes	—
10	#19151349	C₄	1	2	yes	—
11	#19151320	—	2	4	no	—
12	#19151319	C₄	1	2	yes	—
13	#19151319	C₄	1	2	yes	—
14	C₂×#108×C₈₉	C₄	1	2	yes	—
15	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#341×C₈₉	C₄	1	2	yes	—
18	C₂²×#10623494	—	2	1	no	—
19	C₂²×#10623494	Q₈	1	2	yes	—
20	C₄⋊#10623494	C₂×C₄	1	4	yes	—
21	C₂²⋊#10623494	—	2	2	no	—
22	C₄×#10623494	—	2	2	no	—
23	C₄×#10623494	—	2	1	no	—
24	C₄×#10623494	C₂×C₄	1	2	yes	—
25	C₄×#10623494	C₂×C₄	1	2	yes	—
26	C₄×#10623494	D₄	1	2	yes	—
27	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	#108×C₈₉	—	2	4	no	—
32	#97×C₈₉	—	2	2	no	—
33	C₄×C₈×C₈₉	—	2	2	no	—
34	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
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	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
46	M₁₆×C₈₉	—	2	4	no	—
47	M₁₆×C₈₉	—	2	4	no	—
48	C₂×C₈×C₈₉	—	2	2	no	—
49	C₂×C₈×C₈₉	—	2	4	no	—
50	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
55	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₉	#41	1	1	yes	characteristic
59	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₈₉	#131	1	2	yes	—
70	C₈×C₈₉	—	2	1	no	—
71	C₈×C₈₉	—	2	1	no	—
72	C₈×C₈₉	—	4	4	no	—
73	C₈×C₈₉	#97	1	2	yes	—
74	C₈×C₈₉	#97	1	2	yes	—
75	#10623494	—	2	4	no	—
76	#10623494	—	4	2	no	—
77	#10623494	—	4	4	no	—
78	C₂²×C₈₉	—	2	1	no	—
79	C₂²×C₈₉	—	2	1	no	—
80	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈₉	#446	1	2	yes	—
82	C₄×C₈₉	—	2	1	no	—
83	C₄×C₈₉	—	2	1	no	—
84	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
85	C₄×C₈₉	#405	1	1	yes	characteristic
86	C₄×C₈₉	#430	1	2	yes	—
87	C₄×C₈₉	#436	1	2	yes	—
88	C₄×C₈₉	#444	1	2	yes	—
89	#8295	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	—	2	2	no	—
91	C₂×C₈₉	#1770	1	1	yes	characteristic
92	C₂×C₈₉	#1773	1	1	yes	characteristic
93	C₂×C₈₉	#1778	1	1	yes	characteristic
94	C₄×#108	D₈₉	1	1	yes	characteristic
95	#1564	—	89	1	no	—
96	#1451	—	89	2	no	—
97	C₂×#322	—	89	1	no	—
98	#1278	—	89	2	no	—
99	C₈₉	#8295	1	1	yes	characteristic, complemented, Sylow
100	#431	—	89	2	no	—
101	C₂×#108	Dic₈₉	1	2	yes	—
102	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
103	C₂×C₄×C₈	—	89	1	no	—
104	C₂×C₄×C₈	—	89	1	no	—
105	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
106	#341	Dic₈₉	1	2	yes	—
107	#335	—	89	2	no	—
108	#334	—	89	2	no	—
109	#322	—	89	2	no	—
110	#322	—	89	2	no	—
111	#322	—	178	4	no	—
112	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
113	C₂²×C₈	—	89	2	no	—
114	C₂²×C₈	—	178	1	no	—
115	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
117	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
118	#108	—	2	4	no	—
119	C₄⋊C₈	—	89	4	no	—
120	C₂²⋊C₈	—	178	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₄×C₈	—	89	2	no	—
124	C₄×C₈	—	89	2	no	—
125	C₄×C₈	—	89	2	no	—
126	C₄×C₈	—	178	2	no	—
127	C₄×C₈	—	178	1	no	—
128	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
129	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
130	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
131	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
132	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
134	C₂²×C₄	#13083575	1	1	yes	characteristic
135	M₁₆	—	2	4	no	—
136	M₁₆	—	2	4	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	89	4	no	—
140	C₂×C₈	—	356	2	no	—
141	C₂×C₈	—	356	2	no	—
142	C₂×C₈	—	178	2	no	—
143	C₂×C₈	—	178	4	no	—
144	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
145	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
146	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
147	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
148	C₂×C₈	#13083558	1	4	yes	—
149	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
150	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
151	C₄²	D₄×D₈₉	1	1	yes	characteristic
152	C₄²	#13083573	1	1	yes	characteristic
153	C₂³	Q₈×Dic₈₉	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
158	C₂×C₄	#15667970	1	1	yes	characteristic
159	C₂×C₄	#15667974	1	1	yes	characteristic
160	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
161	C₂×C₄	#15667977	1	1	yes	characteristic
162	C₂×C₄	C₂×#13083558	1	2	yes	—
163	C₂×C₄	#15668019	1	2	yes	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	4	no	—
167	C₈	—	356	2	no	—
168	C₈	—	356	4	no	—
169	C₈	—	178	4	no	—
170	C₈	#15667923	1	2	yes	—
171	C₈	#15667923	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
175	C₂²	#19151729	1	2	yes	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	C₂×#15667923	1	1	yes	characteristic
179	C₄	#19151554	1	1	yes	characteristic
180	C₄	#19151645	1	2	yes	—
181	C₄	#19151676	1	2	yes	—
182	C₄	#19151719	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	#26020121	1	1	yes	characteristic
185	C₂	#26020139	1	1	yes	characteristic
186	C₂	#26020160	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#108×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₄⋊C₄
prime radical	C₂³×C₈₉	C₄×Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#108	D₈₉
hypocenter	C₈₉	#8295
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
99. C₈₉	#8295	semidirect

Maximal subgroups

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

subgroup	index
#26019000	2
#26018539	2
C₂×#19151319	2
#26018090	2
C₄×#108×C₈₉	2
#8295	89