Subgroups — #36441630

Order 22784.

230 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	#26022008	C₂	1	1	yes	maximal, characteristic
2	#26020416	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	—	2	1	no	—
5	C₂×#15667946	—	2	1	no	—
6	C₂×#15667944	C₄	1	2	yes	—
7	#19151464	—	2	1	no	—
8	C₄⋊#13083549	—	2	1	no	—
9	C₄⋊#13083549	C₄	1	2	yes	complemented
10	#431×C₈₉	—	2	1	no	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	#335×C₈₉	—	2	1	no	—
13	#334×C₈₉	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₈₉	—	2	2	no	—
15	#15667952	—	4	1	no	—
16	#15667952	—	4	1	no	—
17	C₂×#13083549	—	2	2	no	—
18	#15667946	—	4	1	no	—
19	#15667946	—	4	1	no	—
20	#15667944	—	4	2	no	—
21	#15667919	—	4	1	no	—
22	#15667919	—	4	1	no	—
23	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₈₉	—	2	2	no	—
27	C₂²⋊C₈×C₈₉	—	4	1	no	—
28	C₄×C₈×C₈₉	—	4	1	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
33	C₂²×Dic₈₉	—	4	2	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	#13083549	—	4	2	no	—
37	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₈×C₈₉	—	2	1	no	—
41	C₂×C₈×C₈₉	—	2	1	no	—
42	C₂×C₈×C₈₉	—	4	2	no	—
43	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
51	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
54	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
55	C₂³×C₈₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	4	1	no	—
57	C₂×C₄×C₈₉	—	4	1	no	—
58	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#101	1	1	yes	characteristic
61	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
68	C₂×Dic₈₉	—	8	2	no	—
69	C₂×Dic₈₉	—	8	2	no	—
70	C₂×Dic₈₉	—	8	2	no	—
71	C₈×C₈₉	—	2	1	no	—
72	C₈×C₈₉	—	2	1	no	—
73	C₈×C₈₉	—	8	2	no	—
74	C₈×C₈₉	#102	1	1	yes	characteristic
75	C₈×C₈₉	#103	1	1	yes	characteristic
76	C₈×C₈₉	#106	1	1	yes	characteristic
77	C₈×C₈₉	#107	1	1	yes	characteristic
78	C₂²×C₈₉	—	2	1	no	—
79	C₂²×C₈₉	—	2	1	no	—
80	C₂²×C₈₉	#331	1	1	yes	characteristic
81	C₂²×C₈₉	#332	1	1	yes	characteristic
82	C₂²×C₈₉	#340	1	1	yes	characteristic
83	C₄×C₈₉	—	2	1	no	—
84	C₄×C₈₉	—	2	1	no	—
85	C₄×C₈₉	—	2	1	no	—
86	C₄×C₈₉	—	2	1	no	—
87	C₄×C₈₉	#332	1	1	yes	characteristic
88	C₄×C₈₉	#333	1	1	yes	characteristic
89	C₄×C₈₉	#340	1	1	yes	characteristic
90	C₄×C₈₉	#343	1	1	yes	characteristic
91	Dic₈₉	—	16	2	no	—
92	#7279	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	—	2	1	no	—
94	C₂×C₈₉	—	2	1	no	—
95	C₂×C₈₉	#1296	1	1	yes	characteristic
96	C₂×C₈₉	#1384	1	1	yes	characteristic
97	C₂×C₈₉	#1389	1	1	yes	characteristic
98	#2150	—	89	1	no	—
99	#1846	—	89	1	no	—
100	#1564	D₈₉	1	1	yes	characteristic
101	C₈₉	#7279	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	—	178	1	no	—
103	#431	—	2	1	no	—
104	C₂×#107	—	178	1	no	—
105	C₂×#106	—	89	2	no	—
106	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
107	#382	—	178	1	no	—
108	C₄⋊C₁₆	—	89	2	no	—
109	C₄⋊C₁₆	—	178	1	no	—
110	#335	—	2	1	no	—
111	#334	Dic₈₉	1	2	yes	complemented
112	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
113	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	178	2	no	—
115	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
116	M₃₂	—	356	1	no	—
117	M₃₂	—	356	1	no	—
118	C₂×C₁₆	—	178	2	no	—
119	#107	—	356	1	no	—
120	#107	—	356	1	no	—
121	#106	—	356	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₂²⋊C₈	—	4	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
126	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
129	#95	—	356	1	no	—
130	#95	—	356	1	no	—
131	C₂²×C₄	—	356	2	no	—
132	C₂²×C₄	#13083551	1	1	yes	characteristic
133	C₂²×C₄	#13083559	1	1	yes	characteristic
134	C₂²×C₄	#13083561	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
139	C₂×C₈	#13083551	1	1	yes	characteristic
140	C₂×C₈	#13083553	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
142	C₂×C₈	D₇₁₂	1	1	yes	characteristic
143	C₂×C₈	#13083563	1	1	yes	characteristic
144	C₂×C₈	#13083563	1	1	yes	characteristic
145	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
146	C₄⋊C₄	—	356	2	no	—
147	C₄⋊C₄	—	356	2	no	—
148	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
149	C₄²	#13083553	1	1	yes	characteristic
150	C₄²	#13083562	1	1	yes	characteristic
151	C₄²	#13083564	1	1	yes	characteristic
152	C₁₆	—	356	2	no	—
153	C₂³	#15667932	1	1	yes	characteristic
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	712	2	no	—
157	C₂×C₄	—	712	2	no	—
158	C₂×C₄	—	712	2	no	—
159	C₂×C₄	#15667919	1	1	yes	characteristic
160	C₂×C₄	#15667928	1	1	yes	characteristic
161	C₂×C₄	#15667930	1	1	yes	characteristic
162	C₂×C₄	#15667933	1	1	yes	characteristic
163	C₂×C₄	#15667934	1	1	yes	characteristic
164	C₂×C₄	#15667935	1	1	yes	characteristic
165	C₂×C₄	#15667936	1	1	yes	characteristic
166	C₂×C₄	#15667937	1	1	yes	characteristic
167	C₂×C₄	#15667943	1	1	yes	characteristic
168	C₂×C₄	#15667945	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	8	2	no	—
172	C₈	#15667934	1	1	yes	characteristic
173	C₈	#15667937	1	1	yes	characteristic
174	C₈	#15667944	1	1	yes	characteristic
175	C₈	#15667946	1	1	yes	characteristic
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#19151341	1	1	yes	characteristic
179	C₂²	#19151345	1	1	yes	characteristic
180	C₂²	#19151360	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	1424	2	no	—
186	C₄	#19151343	1	1	yes	characteristic
187	C₄	#19151346	1	1	yes	characteristic
188	C₄	#19151361	1	1	yes	characteristic
189	C₄	#19151366	1	1	yes	characteristic
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	#26018136	1	1	yes	characteristic
193	C₂	#26018353	1	1	yes	characteristic
194	C₂	#26018366	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151360
commutator	C₂×C₈×C₈₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	#340
prime radical	C₂³×C₈₉	#102
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#7279
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
9. C₄⋊#13083549	C₄	semidirect
13. #334×C₈₉	C₄	semidirect
101. C₈₉	#7279	semidirect
111. #334	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26022008	2
#26020416	2
C₈₉×#1564	2
#7279	89