Subgroups — #36445728

Order 22784.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
2	#26018542	C₂	1	2	yes	maximal
3	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
4	#26018088	C₂	1	2	yes	maximal
5	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667948	C₄	1	2	yes	—
7	C₄×#13083558	C₂²	1	2	yes	—
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151363	C₄	1	2	yes	—
11	#19151320	—	2	2	no	—
12	#19151319	—	2	2	no	—
13	C₈×#10623494	C₄	1	2	yes	—
14	C₈×#10623494	C₄	1	2	yes	—
15	#431×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#335×C₈₉	C₄	1	2	yes	—
18	#334×C₈₉	C₄	1	2	yes	—
19	C₂×#13083558	C₂×C₄	1	4	yes	—
20	C₂²×#10623494	—	2	1	no	—
21	C₂²×#10623494	Q₈	1	2	yes	—
22	#15667948	—	2	4	no	—
23	#15667923	—	2	2	no	—
24	C₄×#10623494	—	2	2	no	—
25	C₄×#10623494	—	2	1	no	—
26	C₄×#10623494	C₂×C₄	1	2	yes	—
27	C₄×#10623494	C₂×C₄	1	2	yes	—
28	C₄×#10623494	D₄	1	1	yes	characteristic
29	C₄×#10623494	D₄	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₈₉	—	2	2	no	—
35	C₄×C₈×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	#13083558	—	2	4	no	—
41	#13083558	—	2	4	no	—
42	C₂×#10623494	—	2	4	no	—
43	C₂×#10623494	—	2	2	no	—
44	C₂×#10623494	—	2	4	no	—
45	C₂×#10623494	—	2	4	no	—
46	C₂×#10623494	C₄⋊C₄	1	2	yes	—
47	C₂×#10623494	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
51	C₂×C₈×C₈₉	—	2	2	no	—
52	C₂×C₈×C₈₉	—	2	4	no	—
53	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
58	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₈₉	#41	1	1	yes	characteristic
62	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	—	2	2	no	—
64	C₂×C₄×C₈₉	—	2	2	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×D₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₈₉	#131	1	2	yes	—
73	C₈×C₈₉	—	2	1	no	—
74	C₈×C₈₉	—	2	1	no	—
75	C₈×C₈₉	—	4	4	no	—
76	C₈×C₈₉	#97	1	2	yes	—
77	C₈×C₈₉	#97	1	2	yes	—
78	#10623494	—	2	4	no	—
79	#10623494	—	2	2	no	—
80	#10623494	—	4	4	no	—
81	#10623494	#108	1	4	yes	complemented
82	C₂²×C₈₉	—	2	1	no	—
83	C₂²×C₈₉	—	2	1	no	—
84	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₈₉	#446	1	2	yes	—
86	C₄×C₈₉	—	2	1	no	—
87	C₄×C₈₉	—	2	1	no	—
88	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
89	C₄×C₈₉	#405	1	1	yes	characteristic
90	C₄×C₈₉	C₂×#108	1	2	yes	—
91	C₄×C₈₉	#436	1	2	yes	—
92	C₄×C₈₉	#443	1	2	yes	—
93	#8293	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	—	2	2	no	—
95	C₂×C₈₉	#1770	1	1	yes	characteristic
96	C₂×C₈₉	#1772	1	1	yes	characteristic
97	C₂×C₈₉	C₄×#108	1	1	yes	characteristic
98	C₄×#108	—	89	1	no	—
99	#1564	D₈₉	1	1	yes	characteristic
100	#1451	—	89	2	no	—
101	C₂×C₈²	—	89	1	no	—
102	#1278	—	89	2	no	—
103	C₈₉	#8293	1	1	yes	characteristic, complemented, Sylow
104	#431	C₂×D₈₉	1	2	yes	—
105	C₂×#108	—	89	2	no	—
106	C₄×M₁₆	—	89	2	no	—
107	C₂×C₄×C₈	—	89	1	no	—
108	C₂×C₄×C₈	—	89	1	no	—
109	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
110	#341	—	89	2	no	—
111	#335	Dic₈₉	1	2	yes	—
112	#334	Dic₈₉	1	2	yes	—
113	#322	—	178	2	no	—
114	#322	—	178	2	no	—
115	C₈²	—	89	2	no	—
116	C₈²	—	89	2	no	—
117	C₂×M₁₆	—	89	4	no	—
118	C₂²×C₈	—	89	2	no	—
119	C₂²×C₈	—	178	1	no	—
120	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
123	#108	—	178	4	no	—
124	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	178	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	89	2	no	—
129	C₄×C₈	—	89	2	no	—
130	C₄×C₈	—	89	1	no	—
131	C₄×C₈	—	89	1	no	—
132	C₄×C₈	—	178	2	no	—
133	C₄×C₈	—	178	1	no	—
134	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
135	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
139	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
140	C₂²×C₄	#13083573	1	1	yes	characteristic
141	M₁₆	—	178	4	no	—
142	M₁₆	—	178	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	89	2	no	—
146	C₂×C₈	—	89	2	no	—
147	C₂×C₈	—	178	4	no	—
148	C₂×C₈	—	178	4	no	—
149	C₂×C₈	—	178	4	no	—
150	C₂×C₈	—	178	2	no	—
151	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
155	C₂×C₈	#13083557	1	4	yes	—
156	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
157	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
158	C₄²	D₄×D₈₉	1	1	yes	characteristic
159	C₄²	#13083575	1	1	yes	characteristic
160	C₂³	#15667977	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
165	C₂×C₄	#15667970	1	1	yes	characteristic
166	C₂×C₄	#15667974	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
169	C₂×C₄	C₂×#13083557	1	2	yes	—
170	C₂×C₄	#15668020	1	2	yes	—
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	4	4	no	—
174	C₈	—	89	4	no	—
175	C₈	—	356	4	no	—
176	C₈	—	178	4	no	—
177	C₈	—	178	2	no	—
178	C₈	#15667922	1	2	yes	—
179	C₈	#15667922	1	2	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
183	C₂²	#19151725	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×#15667922	1	1	yes	characteristic
187	C₄	#19151556	1	1	yes	characteristic
188	C₄	#108×D₈₉	1	2	yes	—
189	C₄	#19151675	1	2	yes	—
190	C₄	#19151715	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#26020122	1	1	yes	characteristic
193	C₂	#26020133	1	1	yes	characteristic
194	C₂	#108×Dic₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₄×C₄⋊C₄
prime radical	C₂³×C₈₉	C₄×Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#8293
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
81. #10623494	#108	semidirect
103. C₈₉	#8293	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667948	2
#26018542	2
C₂×C₈×#10623494	2
#26018088	2
C₈₉×#1564	2
#8293	89