Subgroups — #36473033

Order 22784.

390 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	C₂×#19152003	C₂	1	1	yes	maximal, characteristic
2	#26020416	C₂	1	1	yes	maximal, characteristic
3	#26018344	C₂	1	4	yes	maximal
4	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	—	2	1	no	—
6	#19152003	—	2	2	no	—
7	#19152003	C₄	1	4	yes	complemented
8	C₂×#15667946	—	2	1	no	—
9	C₂×#15667946	C₂²	1	2	yes	—
10	C₂×#15667946	C₂²	1	2	yes	—
11	C₂×#15667944	—	2	1	no	—
12	#19151464	—	2	1	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#346×C₈₉	—	2	2	no	—
16	C₂²×Dic₁₇₈	—	4	1	no	—
17	#15668012	—	2	4	no	—
18	#15668012	—	4	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	—	4	1	no	—
22	#15667946	—	2	4	no	—
23	#15667946	—	2	2	no	—
24	#15667946	—	2	4	no	—
25	#15667946	C₂×C₄	1	4	yes	—
26	#15667944	—	4	2	no	—
27	#15667919	—	4	1	no	—
28	#15667919	—	4	1	no	—
29	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₈₉	—	2	2	no	—
33	C₂×C₁₆×C₈₉	—	2	2	no	—
34	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
35	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₈₉	D₄	1	2	yes	—
37	C₂×Dic₁₇₈	—	4	4	no	—
38	C₂×Dic₁₇₈	—	8	4	no	—
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₂²×Dic₈₉	—	4	2	no	—
41	C₄⋊Dic₈₉	—	2	4	no	—
42	C₄⋊Dic₈₉	—	4	4	no	—
43	C₄⋊Dic₈₉	—	4	2	no	—
44	C₄⋊Dic₈₉	—	4	2	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
51	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
52	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₈₉	#41	1	2	yes	—
56	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₈₉	SD₁₆	1	2	yes	—
58	C₁₆×C₈₉	—	2	4	no	—
59	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	#102	1	2	yes	—
63	C₂×C₄×C₈₉	#102	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₈₉	—	4	4	no	—
71	C₂×Dic₈₉	—	8	4	no	—
72	C₂×Dic₈₉	—	8	2	no	—
73	C₂×Dic₈₉	—	8	2	no	—
74	C₂×Dic₈₉	—	8	2	no	—
75	Dic₁₇₈	—	8	8	no	—
76	C₈×C₈₉	#103	1	4	yes	—
77	C₈×C₈₉	C₄×Q₈	1	2	yes	—
78	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
80	C₂²×C₈₉	#486	1	1	yes	characteristic
81	C₂²×C₈₉	#500	1	1	yes	characteristic
82	C₂²×C₈₉	#509	1	2	yes	—
83	C₂²×C₈₉	#510	1	2	yes	—
84	C₄×C₈₉	—	2	2	no	—
85	C₄×C₈₉	—	2	2	no	—
86	C₄×C₈₉	#391	1	1	yes	characteristic
87	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
88	C₄×C₈₉	#487	1	1	yes	characteristic
89	C₄×C₈₉	#498	1	1	yes	characteristic
90	Dic₈₉	—	8	4	no	—
91	Dic₈₉	—	16	2	no	—
92	#13283	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	#1990	1	1	yes	characteristic
94	C₂×C₈₉	#2140	1	1	yes	characteristic
95	C₂×C₈₉	#2141	1	1	yes	characteristic
96	C₂×C₈₉	#2243	1	2	yes	—
97	C₂×C₈₉	#2255	1	2	yes	—
98	C₂×#500	—	89	1	no	—
99	C₂×#346	D₈₉	1	1	yes	characteristic
100	#1846	—	89	1	no	—
101	#1380	—	89	4	no	—
102	C₈₉	#13283	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	178	1	no	—
104	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
105	#500	—	89	4	no	—
106	#500	—	178	2	no	—
107	C₂×#107	—	89	2	no	—
108	C₂×#107	—	89	2	no	—
109	C₂×#107	—	178	1	no	—
110	C₂×#106	—	178	1	no	—
111	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
112	#382	—	178	1	no	—
113	#346	—	2	2	no	—
114	C₂²×Q₈	—	356	1	no	—
115	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
117	C₄⋊Q₈	—	356	2	no	—
118	C₄⋊Q₈	—	178	4	no	—
119	C₂×C₄⋊C₄	—	356	1	no	—
120	C₂×C₄⋊C₄	—	178	2	no	—
121	C₂×C₄⋊C₄	—	178	2	no	—
122	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	Dic₁₇₈	1	4	yes	complemented
126	#107	—	89	4	no	—
127	#107	—	178	4	no	—
128	#107	—	178	4	no	—
129	#107	—	178	2	no	—
130	#106	—	356	2	no	—
131	C₄×C₈	C₄×D₈₉	1	2	yes	—
132	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
133	#95	—	356	1	no	—
134	#95	—	356	1	no	—
135	C₂×Q₈	—	356	4	no	—
136	C₂×Q₈	—	712	4	no	—
137	C₂²×C₄	—	356	2	no	—
138	C₂²×C₄	—	356	2	no	—
139	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
140	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
142	C₂×C₈	#13083551	1	2	yes	—
143	C₂×C₈	#13083563	1	2	yes	—
144	C₂×C₈	Dic₃₅₆	1	2	yes	—
145	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
146	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
148	C₂×C₈	#13083574	1	2	yes	—
149	C₄⋊C₄	—	356	2	no	—
150	C₄⋊C₄	—	356	4	no	—
151	C₄⋊C₄	—	356	2	no	—
152	C₄⋊C₄	—	356	2	no	—
153	C₄⋊C₄	—	178	4	no	—
154	C₄²	#13083551	1	2	yes	—
155	C₄²	#13083563	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
158	Q₈	—	712	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	356	4	no	—
162	C₂×C₄	—	712	2	no	—
163	C₂×C₄	—	712	2	no	—
164	C₂×C₄	—	712	2	no	—
165	C₂×C₄	—	712	4	no	—
166	C₂×C₄	#15667934	1	2	yes	—
167	C₂×C₄	#15667934	1	2	yes	—
168	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
169	C₂×C₄	#15668000	1	1	yes	characteristic
170	C₂×C₄	#15668012	1	1	yes	characteristic
171	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
172	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
174	C₈	#15667937	1	4	yes	—
175	C₈	C₄×Dic₁₇₈	1	2	yes	—
176	C₈	#15667991	1	2	yes	—
177	C₂²	C₂×#15667934	1	1	yes	characteristic
178	C₂²	#19151945	1	1	yes	characteristic
179	C₂²	#19152003	1	1	yes	characteristic
180	C₂²	#19152036	1	2	yes	—
181	C₂²	#19152041	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	1424	2	no	—
185	C₄	—	712	4	no	—
186	C₄	#19151504	1	1	yes	characteristic
187	C₄	C₂×#15667937	1	1	yes	characteristic
188	C₄	#19151949	1	1	yes	characteristic
189	C₄	#19151996	1	1	yes	characteristic
190	C₂	#26021213	1	1	yes	characteristic
191	C₂	#26021969	1	1	yes	characteristic
192	C₂	#26021973	1	1	yes	characteristic
193	C₂	#26022489	1	2	yes	—
194	C₂	#26022538	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#13283
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
7. #19152003	C₄	semidirect
34. C₂×C₁₆×C₈₉	Q₈	semidirect
102. C₈₉	#13283	semidirect
125. C₂×C₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152003	2
#26020416	2
#26018344	2
C₂×#346×C₈₉	2
#13283	89