Subgroups — #36448122

Order 22784.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020416	C₂	1	1	yes	maximal, characteristic
2	#26018535	C₂	1	1	yes	maximal, characteristic
3	#26018112	C₂	1	4	yes	maximal
4	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	2	yes	—
6	C₂×#15667946	—	2	1	no	—
7	C₂×#15667944	—	2	1	no	—
8	#19151464	C₂²	1	2	yes	—
9	#19151319	C₄	1	2	yes	complemented
10	C₈×#10623494	C₄	1	2	yes	complemented
11	C₂×#108×C₈₉	—	2	1	no	—
12	C₂×#108×C₈₉	—	2	1	no	—
13	#423×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₉	—	2	2	no	—
16	#15667946	—	4	2	no	—
17	#15667944	—	4	2	no	—
18	C₄⋊#10623494	—	2	2	no	—
19	C₂²⋊#10623494	—	2	2	no	—
20	C₄×#10623494	C₂×C₄	1	4	yes	—
21	#15667919	—	2	4	no	—
22	C₂×M₁₆×C₈₉	—	2	2	no	—
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
26	#108×C₈₉	—	4	2	no	—
27	#108×C₈₉	—	4	2	no	—
28	C₄⋊C₈×C₈₉	—	2	4	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	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
33	C₂²×Dic₈₉	—	4	2	no	—
34	C₂×#10623494	—	2	4	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₄⋊Dic₈₉	—	4	2	no	—
37	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	#41	1	2	yes	—
39	M₁₆×C₈₉	—	4	2	no	—
40	M₁₆×C₈₉	—	4	2	no	—
41	C₂×C₈×C₈₉	—	4	2	no	—
42	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	D₈	1	2	yes	—
47	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
48	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
49	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
50	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₈₉	#41	1	2	yes	—
53	C₂³×C₈₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	4	2	no	—
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#117	1	2	yes	—
57	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#125	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₈₉	—	8	4	no	—
64	C₂×Dic₈₉	—	8	2	no	—
65	C₈×C₈₉	—	8	2	no	—
66	C₈×C₈₉	#106	1	2	yes	—
67	C₈×C₈₉	#106	1	2	yes	—
68	C₈×C₈₉	#107	1	2	yes	—
69	C₈×C₈₉	#107	1	2	yes	—
70	#10623494	—	4	4	no	—
71	C₂²×C₈₉	—	2	2	no	—
72	C₂²×C₈₉	#382	1	1	yes	characteristic
73	C₂²×C₈₉	#486	1	1	yes	characteristic
74	C₂²×C₈₉	#487	1	1	yes	characteristic
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	—	2	2	no	—
77	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
78	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
79	C₄×C₈₉	#486	1	1	yes	characteristic
80	C₄×C₈₉	#487	1	1	yes	characteristic
81	Dic₈₉	—	16	2	no	—
82	#8793	—	89	1	no	maximal, Sylow
83	C₂×C₈₉	—	2	1	no	—
84	C₂×C₈₉	—	2	1	no	—
85	C₂×C₈₉	#1833	1	2	yes	—
86	C₂×C₈₉	#1834	1	1	yes	characteristic
87	#1848	D₈₉	1	1	yes	characteristic
88	#1846	—	89	1	no	—
89	#1450	—	89	1	no	—
90	#1286	—	89	4	no	—
91	C₈₉	#8793	1	1	yes	characteristic, complemented, Sylow
92	#431	—	89	2	no	—
93	C₂×#108	—	2	1	no	—
94	C₂×#108	—	2	1	no	—
95	C₂×#107	—	178	1	no	—
96	C₂×#106	—	178	1	no	—
97	#423	C₂×D₈₉	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
99	#382	—	89	2	no	—
100	#322	—	89	2	no	—
101	C₈²	—	89	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	178	2	no	—
106	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
107	#108	—	4	2	no	—
108	#108	—	4	2	no	—
109	#107	—	356	2	no	—
110	#106	—	356	2	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	178	2	no	—
113	C₂²⋊C₈	—	178	2	no	—
114	C₄×C₈	—	89	4	no	—
115	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
116	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
117	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
118	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
119	#95	—	178	4	no	—
120	C₂²×C₄	—	356	2	no	—
121	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
122	C₂²×C₄	#13083575	1	2	yes	—
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	178	4	no	—
127	C₂×C₈	#13083553	1	1	yes	characteristic
128	C₂×C₈	#13083553	1	1	yes	characteristic
129	C₂×C₈	#13083553	1	1	yes	characteristic
130	C₂×C₈	#13083553	1	1	yes	characteristic
131	C₂×C₈	#13083559	1	2	yes	—
132	C₂×C₈	#13083561	1	2	yes	—
133	C₂×C₈	#13083562	1	2	yes	—
134	C₂×C₈	#13083564	1	2	yes	—
135	C₄⋊C₄	—	356	2	no	—
136	C₄⋊C₄	—	356	2	no	—
137	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
139	C₄²	#13083573	1	2	yes	—
140	C₂³	#15668001	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	712	4	no	—
143	C₂×C₄	—	712	2	no	—
144	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
145	C₂×C₄	#15667970	1	2	yes	—
146	C₂×C₄	#15667999	1	1	yes	characteristic
147	C₂×C₄	#15668003	1	2	yes	—
148	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
149	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
150	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
151	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
152	C₈	—	8	2	no	—
153	C₈	—	356	4	no	—
154	C₈	#15667943	1	2	yes	—
155	C₈	#15667943	1	2	yes	—
156	C₈	#15667945	1	2	yes	—
157	C₈	#15667945	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#19151463	1	1	yes	characteristic
160	C₂²	#19151946	1	1	yes	characteristic
161	C₂²	#19151950	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	1424	2	no	—
165	C₄	C₂×#15667943	1	1	yes	characteristic
166	C₄	C₂×#15667945	1	1	yes	characteristic
167	C₄	#19151944	1	1	yes	characteristic
168	C₄	#19151948	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#26020377	1	2	yes	—
172	C₂	#26020379	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151463
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	#382
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#8793
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
9. #19151319	C₄	semidirect
10. C₈×#10623494	C₄	semidirect
91. C₈₉	#8793	semidirect

Maximal subgroups

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

subgroup	index
#26020416	2
#26018535	2
#26018112	2
C₈₉×#1848	2
#8793	89