Subgroups — #36441330

Order 22784.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 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	C₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083549	—	2	1	no	—
5	C₂×C₄⋊#10623494	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₈₉	—	2	1	no	—
7	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
8	C₄×#13083549	—	2	1	no	—
9	C₄×#13083549	C₄	1	2	yes	complemented
10	#19151352	—	2	1	no	—
11	C₂²×C₁₆×C₈₉	—	2	1	no	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₈₉	—	2	1	no	—
14	C₄⋊C₁₆×C₈₉	C₄	1	2	yes	complemented
15	C₂²×#10623494	—	2	1	no	—
16	C₂×C₄⋊Dic₈₉	C₈	1	2	yes	—
17	C₂×C₄×Dic₈₉	—	2	1	no	—
18	C₂×#13083549	—	2	2	no	—
19	C₂×#13083549	—	2	2	no	—
20	C₂×#13083549	—	2	2	no	—
21	C₄⋊#10623494	—	2	1	no	—
22	C₄⋊#10623494	C₈	1	2	yes	—
23	C₈×Dic₈₉	—	4	1	no	—
24	C₈×Dic₈₉	—	4	1	no	—
25	#15667919	—	2	1	no	—
26	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₈₉	—	2	2	no	—
30	C₂×C₁₆×C₈₉	—	4	1	no	—
31	C₂×C₁₆×C₈₉	—	4	1	no	—
32	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
35	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
36	C₂²×Dic₈₉	—	2	2	no	—
37	C₂×#10623494	—	2	2	no	—
38	C₂×#10623494	—	4	1	no	—
39	C₂×#10623494	—	4	1	no	—
40	C₄⋊Dic₈₉	—	2	2	no	—
41	C₄⋊Dic₈₉	C₁₆	1	4	yes	complemented
42	C₄×Dic₈₉	—	4	1	no	—
43	C₄×Dic₈₉	—	4	1	no	—
44	#13083549	—	2	4	no	—
45	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
48	C₂×C₈×C₈₉	—	2	1	no	—
49	C₂×C₈×C₈₉	—	2	1	no	—
50	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
55	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
58	C₄²×C₈₉	C₄²	1	1	yes	characteristic
59	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
62	C₁₆×C₈₉	—	4	2	no	—
63	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
70	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#104	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#104	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
77	C₂×Dic₈₉	—	2	4	no	—
78	C₂×Dic₈₉	—	4	2	no	—
79	C₂×Dic₈₉	—	4	2	no	—
80	C₈×C₈₉	—	2	1	no	—
81	C₈×C₈₉	—	2	1	no	—
82	C₈×C₈₉	#102	1	1	yes	characteristic
83	C₈×C₈₉	#103	1	1	yes	characteristic
84	C₈×C₈₉	#106	1	1	yes	characteristic
85	C₈×C₈₉	#107	1	1	yes	characteristic
86	#10623494	—	4	2	no	—
87	C₂²×C₈₉	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₈₉	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₈₉	#336	1	1	yes	characteristic
90	C₂²×C₈₉	C₂²⋊C₁₆	1	2	yes	—
91	C₂²×C₈₉	#350	1	2	yes	—
92	C₄×C₈₉	—	2	1	no	—
93	C₄×C₈₉	—	2	1	no	—
94	C₄×C₈₉	D₄⋊C₈	1	1	yes	characteristic
95	C₄×C₈₉	Q₈⋊C₈	1	1	yes	characteristic
96	C₄×C₈₉	#339	1	1	yes	characteristic
97	C₄×C₈₉	#340	1	1	yes	characteristic
98	C₄×C₈₉	C₄×C₁₆	1	1	yes	characteristic
99	C₄×C₈₉	#346	1	1	yes	characteristic
100	C₄×C₈₉	C₄⋊C₁₆	1	2	yes	—
101	Dic₈₉	—	4	4	no	—
102	#7167	—	89	1	no	maximal, Sylow
103	C₂×C₈₉	#1279	1	1	yes	characteristic
104	C₂×C₈₉	D₄⋊C₁₆	1	2	yes	—
105	C₂×C₈₉	Q₈⋊C₁₆	1	2	yes	—
106	C₂×C₈₉	#1374	1	1	yes	characteristic
107	C₂×C₈₉	#1379	1	1	yes	characteristic
108	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
109	C₂×C₄×C₁₆	—	89	1	no	—
110	C₈×C₄⋊C₄	—	89	1	no	—
111	C₈₉	#7167	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₁₆	—	2	1	no	—
113	C₂²×C₁₆	—	178	1	no	—
114	C₂×C₄⋊C₈	—	89	1	no	—
115	C₂×C₄×C₈	—	178	1	no	—
116	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	89	1	no	—
118	C₄⋊C₁₆	—	2	1	no	—
119	C₄⋊C₁₆	Dic₈₉	1	2	yes	complemented
120	C₄×C₁₆	—	89	2	no	—
121	C₄×C₁₆	—	178	1	no	—
122	#336	—	178	1	no	—
123	C₂²×C₈	—	178	1	no	—
124	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	89	2	no	—
127	C₂×C₄²	—	178	1	no	—
128	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	4	1	no	—
131	C₂×C₁₆	—	4	1	no	—
132	C₂×C₁₆	—	178	2	no	—
133	C₂×C₁₆	—	178	2	no	—
134	C₂×C₁₆	—	178	2	no	—
135	C₄⋊C₈	—	89	2	no	—
136	C₄⋊C₈	—	178	1	no	—
137	C₄×C₈	—	356	1	no	—
138	C₄×C₈	—	356	1	no	—
139	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
140	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
141	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
143	#95	—	178	1	no	—
144	C₂²×C₄	—	178	2	no	—
145	C₂²×C₄	#13083551	1	1	yes	characteristic
146	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
147	C₂²×C₄	#13083557	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	356	1	no	—
151	C₂×C₈	—	356	1	no	—
152	C₂×C₈	—	178	2	no	—
153	C₂×C₈	#13083551	1	1	yes	characteristic
154	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
155	C₂×C₈	#13083553	1	1	yes	characteristic
156	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
157	C₂×C₈	#13083559	1	1	yes	characteristic
158	C₂×C₈	#13083561	1	1	yes	characteristic
159	C₂×C₈	#13083562	1	1	yes	characteristic
160	C₂×C₈	#13083564	1	1	yes	characteristic
161	C₄⋊C₄	—	89	4	no	—
162	C₄⋊C₄	—	178	2	no	—
163	C₄²	—	356	1	no	—
164	C₄²	—	356	1	no	—
165	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
166	C₄²	#13083553	1	1	yes	characteristic
167	C₄²	C₈×D₈₉	1	1	yes	characteristic
168	C₄²	#13083557	1	1	yes	characteristic
169	C₁₆	—	4	2	no	—
170	C₁₆	—	178	4	no	—
171	C₂³	#15667924	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	356	2	no	—
174	C₂×C₄	—	356	2	no	—
175	C₂×C₄	—	178	4	no	—
176	C₂×C₄	#15667919	1	1	yes	characteristic
177	C₂×C₄	C₈×Dic₈₉	1	1	yes	characteristic
178	C₂×C₄	#15667922	1	1	yes	characteristic
179	C₂×C₄	#15667924	1	1	yes	characteristic
180	C₂×C₄	#15667932	1	1	yes	characteristic
181	C₂×C₄	#15667935	1	1	yes	characteristic
182	C₂×C₄	#15667938	1	1	yes	characteristic
183	C₂×C₄	#15667938	1	1	yes	characteristic
184	C₂×C₄	#15667941	1	1	yes	characteristic
185	C₂×C₄	#15667941	1	1	yes	characteristic
186	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
187	C₂×C₄	#15667951	1	2	yes	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	356	2	no	—
191	C₈	#15667933	1	1	yes	characteristic
192	C₈	#15667936	1	1	yes	characteristic
193	C₈	#15667943	1	1	yes	characteristic
194	C₈	#15667945	1	1	yes	characteristic
195	C₂²	#19151325	1	1	yes	characteristic
196	C₂²	#19151328	1	1	yes	characteristic
197	C₂²	#19151352	1	1	yes	characteristic
198	C₂²	#19151377	1	2	yes	—
199	C₂²	#19151383	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	356	4	no	—
203	C₄	#19151325	1	1	yes	characteristic
204	C₄	#19151328	1	1	yes	characteristic
205	C₄	#19151359	1	1	yes	characteristic
206	C₄	#19151360	1	1	yes	characteristic
207	C₄	C₁₆×Dic₈₉	1	1	yes	characteristic
208	C₄	#19151373	1	1	yes	characteristic
209	C₄	#19151416	1	2	yes	—
210	C₂	#26018091	1	1	yes	characteristic
211	C₂	#26018206	1	2	yes	—
212	C₂	#26018230	1	2	yes	—
213	C₂	#26018329	1	1	yes	characteristic
214	C₂	#26018342	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₈₉	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#7167
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×#13083549	C₄	semidirect
14. C₄⋊C₁₆×C₈₉	C₄	semidirect
41. C₄⋊Dic₈₉	C₁₆	semidirect
111. C₈₉	#7167	semidirect
119. C₄⋊C₁₆	Dic₈₉	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
C₈×C₄⋊Dic₈₉	2
C₂×C₄⋊C₁₆×C₈₉	2
#7167	89