Subgroups — #36441339

Order 22784.

262 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
2	#26020119	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×#15667922	—	2	1	no	—
7	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
8	#19151374	—	2	1	no	—
9	#19151374	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	#15667922	—	4	1	no	—
24	#15667922	—	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	—	4	2	no	—
41	C₄×Dic₈₉	—	4	1	no	—
42	C₄×Dic₈₉	—	4	1	no	—
43	#13083549	—	2	4	no	—
44	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
47	C₂×C₈×C₈₉	—	2	1	no	—
48	C₂×C₈×C₈₉	—	2	1	no	—
49	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
54	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
57	C₄²×C₈₉	C₄²	1	1	yes	characteristic
58	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
61	C₁₆×C₈₉	—	4	2	no	—
62	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	—	2	2	no	—
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×Dic₈₉	—	4	2	no	—
76	C₂×Dic₈₉	—	4	2	no	—
77	C₂×Dic₈₉	—	4	2	no	—
78	C₈×C₈₉	—	2	1	no	—
79	C₈×C₈₉	—	2	1	no	—
80	C₈×C₈₉	#102	1	1	yes	characteristic
81	C₈×C₈₉	#103	1	1	yes	characteristic
82	C₈×C₈₉	#106	1	1	yes	characteristic
83	C₈×C₈₉	#107	1	1	yes	characteristic
84	#10623494	—	4	2	no	—
85	C₂²×C₈₉	D₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₈₉	Q₈⋊C₈	1	1	yes	characteristic
87	C₂²×C₈₉	#336	1	1	yes	characteristic
88	C₂²×C₈₉	#349	1	2	yes	—
89	C₂²×C₈₉	#350	1	2	yes	—
90	C₄×C₈₉	—	2	1	no	—
91	C₄×C₈₉	—	2	1	no	—
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	Dic₈₉	—	8	2	no	—
99	#7170	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	#1279	1	1	yes	characteristic
101	C₂×C₈₉	#1330	1	2	yes	—
102	C₂×C₈₉	#1338	1	2	yes	—
103	C₂×C₈₉	#1377	1	1	yes	characteristic
104	C₂×C₈₉	#1379	1	1	yes	characteristic
105	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
106	C₂×#346	—	89	1	no	—
107	#1770	—	89	1	no	—
108	C₈₉	#7170	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₁₆	—	2	1	no	—
110	C₂²×C₁₆	—	178	1	no	—
111	C₂×C₄⋊C₈	—	89	1	no	—
112	C₂×#97	—	178	1	no	—
113	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	89	1	no	—
115	C₄⋊C₁₆	—	2	1	no	—
116	C₄⋊C₁₆	Dic₈₉	1	2	yes	complemented
117	#346	—	89	2	no	—
118	#346	—	178	1	no	—
119	#336	—	178	1	no	—
120	C₂²×C₈	—	178	1	no	—
121	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	89	2	no	—
124	C₂×C₄²	—	178	1	no	—
125	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	4	1	no	—
128	C₂×C₁₆	—	4	1	no	—
129	C₂×C₁₆	—	178	2	no	—
130	C₂×C₁₆	—	178	2	no	—
131	C₂×C₁₆	—	178	2	no	—
132	C₄⋊C₈	—	89	2	no	—
133	C₄⋊C₈	—	178	1	no	—
134	#97	—	356	1	no	—
135	#97	—	356	1	no	—
136	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
137	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
138	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
140	#95	—	178	1	no	—
141	C₂²×C₄	—	178	2	no	—
142	C₂²×C₄	#13083551	1	1	yes	characteristic
143	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
144	C₂²×C₄	#13083557	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	356	1	no	—
148	C₂×C₈	—	356	1	no	—
149	C₂×C₈	—	178	2	no	—
150	C₂×C₈	#13083551	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
152	C₂×C₈	#13083553	1	1	yes	characteristic
153	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
154	C₂×C₈	#13083559	1	1	yes	characteristic
155	C₂×C₈	#13083561	1	1	yes	characteristic
156	C₂×C₈	#13083562	1	1	yes	characteristic
157	C₂×C₈	#13083564	1	1	yes	characteristic
158	C₄⋊C₄	—	356	2	no	—
159	C₄²	—	356	1	no	—
160	C₄²	—	356	1	no	—
161	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
162	C₄²	#13083553	1	1	yes	characteristic
163	C₄²	C₈×D₈₉	1	1	yes	characteristic
164	C₄²	#13083557	1	1	yes	characteristic
165	C₁₆	—	4	2	no	—
166	C₁₆	—	178	4	no	—
167	C₂³	#15667924	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	356	2	no	—
171	C₂×C₄	—	356	2	no	—
172	C₂×C₄	—	356	2	no	—
173	C₂×C₄	#15667919	1	1	yes	characteristic
174	C₂×C₄	C₈×Dic₈₉	1	1	yes	characteristic
175	C₂×C₄	#15667922	1	1	yes	characteristic
176	C₂×C₄	#15667924	1	1	yes	characteristic
177	C₂×C₄	#15667932	1	1	yes	characteristic
178	C₂×C₄	#15667935	1	1	yes	characteristic
179	C₂×C₄	#15667938	1	1	yes	characteristic
180	C₂×C₄	#15667938	1	1	yes	characteristic
181	C₂×C₄	#15667941	1	1	yes	characteristic
182	C₂×C₄	#15667941	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	356	2	no	—
186	C₈	#15667933	1	1	yes	characteristic
187	C₈	#15667936	1	1	yes	characteristic
188	C₈	#15667943	1	1	yes	characteristic
189	C₈	#15667945	1	1	yes	characteristic
190	C₂²	#19151325	1	1	yes	characteristic
191	C₂²	#19151328	1	1	yes	characteristic
192	C₂²	#19151352	1	1	yes	characteristic
193	C₂²	#19151379	1	2	yes	—
194	C₂²	#19151381	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	712	2	no	—
200	C₄	#19151325	1	1	yes	characteristic
201	C₄	#19151328	1	1	yes	characteristic
202	C₄	#19151359	1	1	yes	characteristic
203	C₄	#19151360	1	1	yes	characteristic
204	C₂	#26018091	1	1	yes	characteristic
205	C₂	#26018209	1	2	yes	—
206	C₂	#26018233	1	2	yes	—
207	C₂	#26018335	1	1	yes	characteristic
208	C₂	#26018340	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#13083551
commutator	C₂×C₄×C₈₉	C₄×C₈
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	#7170
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. #19151374	C₄	semidirect
14. C₄⋊C₁₆×C₈₉	C₄	semidirect
108. C₈₉	#7170	semidirect
116. 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×#19151374	2
#26020119	2
C₂×C₄⋊C₁₆×C₈₉	2
#7170	89