Subgroups — #36470774

Order 22784.

366 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	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018329	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	C₂²	1	2	yes	—
6	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
8	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
9	C₄²×Dic₈₉	C₄	1	1	yes	characteristic
10	C₄⋊#13083549	—	2	2	no	—
11	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₈₉	—	2	2	no	—
14	C₂²×#10623494	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₈₉	C₈	1	2	yes	—
16	C₂×C₄×Dic₈₉	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₈₉	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
19	C₂×#13083549	—	2	2	no	—
20	C₂×#13083549	—	2	4	no	—
21	C₄×#10623494	—	2	2	no	—
22	C₈×Dic₈₉	—	2	4	no	—
23	C₈×Dic₈₉	—	2	2	no	—
24	C₈×Dic₈₉	—	2	2	no	—
25	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₉	—	2	2	no	—
29	C₂×C₁₆×C₈₉	—	2	4	no	—
30	C₄×C₈×C₈₉	D₄	1	2	yes	—
31	C₄×C₈×C₈₉	Q₈	1	2	yes	—
32	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
33	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
34	C₂×#10623494	—	2	2	no	—
35	C₂×#10623494	—	2	2	no	—
36	C₂×#10623494	M₁₆	1	4	yes	—
37	C₄×Dic₈₉	—	2	2	no	—
38	C₄×Dic₈₉	—	2	2	no	—
39	C₄×Dic₈₉	—	2	2	no	—
40	C₄×Dic₈₉	—	2	4	no	—
41	C₄×Dic₈₉	C₁₆	1	4	yes	complemented
42	#13083549	—	4	2	no	—
43	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	—	2	2	no	—
47	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₈₉	#41	1	2	yes	—
51	C₂×C₈×C₈₉	#41	1	2	yes	—
52	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
53	C₄²×C₈₉	M₁₆	1	2	yes	—
54	C₁₆×C₈₉	—	4	2	no	—
55	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
61	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#117	1	2	yes	—
63	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
65	C₂×Dic₈₉	—	2	2	no	—
66	C₂×Dic₈₉	—	2	2	no	—
67	C₂×Dic₈₉	—	2	2	no	—
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	C₂×C₁₆	1	4	yes	—
70	C₂×Dic₈₉	M₃₂	1	4	yes	—
71	C₈×C₈₉	—	2	2	no	—
72	C₈×C₈₉	#124	1	1	yes	characteristic
73	C₈×C₈₉	#124	1	1	yes	characteristic
74	C₈×C₈₉	#125	1	2	yes	—
75	#10623494	—	2	4	no	—
76	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂²×C₈₉	#431	1	2	yes	—
78	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
79	C₂²×C₈₉	C₂×M₃₂	1	1	yes	characteristic
80	C₂²×C₈₉	C₂×M₃₂	1	2	yes	—
81	C₄×C₈₉	—	2	2	no	—
82	C₄×C₈₉	C₄⋊C₁₆	1	4	yes	—
83	C₄×C₈₉	#382	1	1	yes	characteristic
84	C₄×C₈₉	#423	1	1	yes	characteristic
85	C₄×C₈₉	#432	1	2	yes	—
86	Dic₈₉	—	2	4	no	—
87	Dic₈₉	—	2	4	no	—
88	#12879	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	#1831	1	1	yes	characteristic
90	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
91	C₂×C₈₉	#2150	1	1	yes	characteristic
92	C₂×C₈₉	#2162	1	2	yes	—
93	C₂×C₈₉	#2163	1	2	yes	—
94	C₂×C₄⋊C₁₆	—	89	1	no	—
95	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
96	C₄²×C₈	—	89	1	no	—
97	#1374	—	89	4	no	—
98	C₈₉	#12879	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₁₆	—	89	2	no	—
100	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
101	C₂×C₄×C₈	—	89	2	no	—
102	C₂×C₄×C₈	—	89	2	no	—
103	C₂×C₄×C₈	—	89	1	no	—
104	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
105	C₄³	—	89	1	no	—
106	C₄⋊C₁₆	—	2	2	no	—
107	C₄⋊C₁₆	—	178	2	no	—
108	C₂²×C₈	—	89	2	no	—
109	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
111	C₂×C₄²	—	89	1	no	—
112	C₂×C₄²	—	89	2	no	—
113	C₂×C₄²	—	89	1	no	—
114	C₂×C₄²	—	89	2	no	—
115	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	178	2	no	—
119	C₂×C₁₆	—	178	4	no	—
120	C₄×C₈	—	178	4	no	—
121	C₄×C₈	—	178	2	no	—
122	C₄×C₈	—	178	2	no	—
123	C₄×C₈	—	178	2	no	—
124	C₄×C₈	Dic₁₇₈	1	2	yes	—
125	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
126	C₂²×C₄	—	89	2	no	—
127	C₂²×C₄	—	89	2	no	—
128	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
129	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	89	4	no	—
133	C₂×C₈	—	178	2	no	—
134	C₂×C₈	—	178	2	no	—
135	C₂×C₈	#13083553	1	2	yes	—
136	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
138	C₂×C₈	#13083573	1	2	yes	—
139	C₂×C₈	#13083575	1	2	yes	—
140	C₄²	—	89	4	no	—
141	C₄²	—	178	4	no	—
142	C₄²	—	178	2	no	—
143	C₄²	—	178	2	no	—
144	C₄²	—	178	2	no	—
145	C₄²	#13083553	1	2	yes	—
146	C₄²	#13083557	1	2	yes	—
147	C₁₆	—	4	2	no	—
148	C₁₆	—	356	2	no	—
149	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	89	4	no	—
152	C₂×C₄	—	89	4	no	—
153	C₂×C₄	—	178	2	no	—
154	C₂×C₄	—	178	2	no	—
155	C₂×C₄	—	178	2	no	—
156	C₂×C₄	—	178	2	no	—
157	C₂×C₄	#15667941	1	2	yes	—
158	C₂×C₄	#15667941	1	2	yes	—
159	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
160	C₂×C₄	#15667951	1	2	yes	—
161	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
162	C₂×C₄	#15667970	1	2	yes	—
163	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
164	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	178	4	no	—
167	C₈	#15667999	1	1	yes	characteristic
168	C₈	#15668001	1	1	yes	characteristic
169	C₈	#15668003	1	2	yes	—
170	C₂²	C₂×#15667941	1	1	yes	characteristic
171	C₂²	#19151650	1	2	yes	—
172	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
173	C₂²	M₃₂×D₈₉	1	2	yes	—
174	C₂²	C₂×#15667951	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	178	4	no	—
177	C₄	—	178	4	no	—
178	C₄	#19151416	1	4	yes	—
179	C₄	#19151463	1	1	yes	characteristic
180	C₄	#19151621	1	1	yes	characteristic
181	C₄	#19151654	1	2	yes	—
182	C₂	#26020368	1	1	yes	characteristic
183	C₂	C₂×#19151416	1	1	yes	characteristic
184	C₂	#26022007	1	1	yes	characteristic
185	C₂	#26022050	1	2	yes	—
186	C₂	#26022054	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₁₆
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#12879
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
41. C₄×Dic₈₉	C₁₆	semidirect
98. C₈₉	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#13083549	2
C₄×C₈×Dic₈₉	2
#26018329	2
C₂×C₄⋊C₁₆×C₈₉	2
#12879	89