Subgroups — #36453458

Order 22784.

570 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	#26028650	C₂	1	1	yes	maximal, characteristic
2	#26020404	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151346	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₄	1	4	yes	—
6	C₂×#15668012	C₂²	1	2	yes	—
7	C₂×C₄⋊#10623494	—	2	2	no	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	#19151468	—	2	2	no	—
10	#19151468	—	2	1	no	—
11	#19151468	—	2	1	no	—
12	#19151346	—	2	8	no	—
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	—	2	1	no	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	—	2	1	no	—
17	C₂²×Dic₁₇₈	C₂×C₄	1	4	yes	—
18	C₂²×#10623494	—	2	2	no	—
19	#15668012	—	2	4	no	—
20	#15668012	—	2	4	no	—
21	#15668012	—	4	8	no	—
22	C₂×C₄⋊Dic₈₉	—	2	4	no	—
23	C₂×C₄⋊Dic₈₉	—	2	4	no	—
24	C₂×C₄⋊Dic₈₉	—	2	2	no	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₄⋊#10623494	—	2	4	no	—
27	C₄⋊#10623494	—	4	4	no	—
28	C₂²×C₈×C₈₉	—	2	2	no	—
29	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₉	—	2	2	no	—
35	C₄⋊C₈×C₈₉	D₄	1	4	yes	—
36	C₄×C₈×C₈₉	—	4	2	no	—
37	C₂×Dic₁₇₈	—	2	8	no	—
38	C₂×Dic₁₇₈	—	4	8	no	—
39	C₂²×Dic₈₉	—	2	4	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×#10623494	—	4	4	no	—
42	C₄⋊Dic₈₉	—	4	8	no	—
43	C₄⋊Dic₈₉	—	4	8	no	—
44	C₄⋊Dic₈₉	—	4	4	no	—
45	C₄⋊Dic₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	4	2	no	—
47	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	4	no	—
53	C₂×C₈×C₈₉	—	4	2	no	—
54	C₂×C₈×C₈₉	—	4	2	no	—
55	C₄²×C₈₉	—	2	4	no	—
56	C₄²×C₈₉	—	2	2	no	—
57	C₄²×C₈₉	SD₁₆	1	2	yes	—
58	C₄²×C₈₉	SD₁₆	1	4	yes	—
59	C₄²×C₈₉	Q₁₆	1	2	yes	—
60	C₄²×C₈₉	Q₁₆	1	4	yes	—
61	C₄²×C₈₉	C₂×D₄	1	2	yes	—
62	C₄²×C₈₉	#41	1	2	yes	—
63	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	#100	1	4	yes	—
68	C₂×C₄×C₈₉	#103	1	4	yes	—
69	C₂×C₄×C₈₉	#103	1	4	yes	—
70	C₂×C₄×C₈₉	#103	1	4	yes	—
71	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₈₉	#121	1	2	yes	—
73	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
79	C₂×C₄×C₈₉	#137	1	2	yes	—
80	C₂×Dic₈₉	—	4	4	no	—
81	C₂×Dic₈₉	—	4	8	no	—
82	Dic₁₇₈	—	8	8	no	—
83	C₈×C₈₉	—	4	4	no	—
84	#10623494	—	8	2	no	—
85	C₂²×C₈₉	#390	1	1	yes	characteristic
86	C₂²×C₈₉	C₂×#100	1	2	yes	—
87	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
88	C₂²×C₈₉	C₂×#103	1	2	yes	—
89	C₂²×C₈₉	#419	1	1	yes	characteristic
90	C₄×C₈₉	—	2	4	no	—
91	C₄×C₈₉	—	2	2	no	—
92	C₄×C₈₉	#333	1	8	yes	complemented
93	C₄×C₈₉	#461	1	2	yes	—
94	C₄×C₈₉	#462	1	2	yes	—
95	C₄×C₈₉	#487	1	2	yes	—
96	C₄×C₈₉	#492	1	1	yes	characteristic
97	C₄×C₈₉	#494	1	1	yes	characteristic
98	Dic₈₉	—	8	4	no	—
99	#9691	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	C₂×#333	1	2	yes	—
101	C₂×C₈₉	#1548	1	2	yes	—
102	C₂×C₈₉	#1956	1	1	yes	characteristic
103	C₂×C₈₉	#1965	1	1	yes	characteristic
104	C₂×C₈₉	#1981	1	1	yes	characteristic
105	#2868	—	89	1	no	—
106	#1842	—	89	1	no	—
107	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
108	C₂×#333	—	89	4	no	—
109	C₈₉	#9691	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊Q₈	—	89	4	no	—
111	C₂×C₄⋊Q₈	—	89	2	no	—
112	C₂×C₄⋊C₈	—	89	2	no	—
113	C₂×C₄⋊C₈	—	178	2	no	—
114	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
115	C₂×C₄×C₈	—	2	1	no	—
116	#384	—	178	2	no	—
117	#384	—	178	1	no	—
118	#384	—	178	1	no	—
119	C₄³	C₂×D₈₉	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#333	—	178	8	no	—
122	C₂²×Q₈	—	89	4	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	178	2	no	—
125	C₄⋊Q₈	—	356	8	no	—
126	C₄⋊Q₈	—	178	4	no	—
127	C₄⋊Q₈	—	178	4	no	—
128	C₂×C₄⋊C₄	—	178	4	no	—
129	C₂×C₄⋊C₄	—	178	4	no	—
130	C₂×C₄⋊C₄	—	178	2	no	—
131	C₂×C₄⋊C₄	—	178	2	no	—
132	C₂×C₄²	C₄×D₈₉	1	2	yes	—
133	C₂×C₄²	D₃₅₆	1	2	yes	—
134	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	356	4	no	—
139	C₄⋊C₈	—	178	4	no	—
140	C₄⋊C₈	D₃₅₆	1	4	yes	—
141	C₄×C₈	—	4	2	no	—
142	C₂×Q₈	—	356	8	no	—
143	C₂×Q₈	—	178	8	no	—
144	C₂²×C₄	—	178	4	no	—
145	C₂²×C₄	#13083551	1	2	yes	—
146	C₂²×C₄	#13083551	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	356	2	no	—
154	C₂×C₈	—	356	4	no	—
155	C₄⋊C₄	—	356	8	no	—
156	C₄⋊C₄	—	356	4	no	—
157	C₄⋊C₄	—	356	2	no	—
158	C₄⋊C₄	—	356	2	no	—
159	C₄⋊C₄	—	356	8	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	4	no	—
162	C₄²	#13083562	1	2	yes	—
163	C₄²	#13083563	1	4	yes	—
164	C₄²	#13083564	1	2	yes	—
165	C₄²	Dic₃₅₆	1	4	yes	—
166	C₄²	C₂×D₃₅₆	1	2	yes	—
167	C₄²	#13083574	1	2	yes	—
168	C₂³	C₂×#13083551	1	1	yes	characteristic
169	Q₈	—	712	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	356	4	no	—
174	C₂×C₄	—	356	8	no	—
175	C₂×C₄	#15667928	1	4	yes	—
176	C₂×C₄	#15667935	1	4	yes	—
177	C₂×C₄	#15667937	1	4	yes	—
178	C₂×C₄	#15667937	1	4	yes	—
179	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
180	C₂×C₄	#15667986	1	2	yes	—
181	C₂×C₄	#15668000	1	1	yes	characteristic
182	C₂×C₄	#15668009	1	1	yes	characteristic
183	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
184	C₂×C₄	C₂×#13083563	1	2	yes	—
185	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
186	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
187	C₂×C₄	#15668042	1	2	yes	—
188	C₈	—	4	4	no	—
189	C₈	—	712	2	no	—
190	C₂²	#19151500	1	1	yes	characteristic
191	C₂²	C₂×#15667928	1	2	yes	—
192	C₂²	C₂×#15667935	1	1	yes	characteristic
193	C₂²	C₂×#15667937	1	2	yes	—
194	C₂²	#19151606	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	712	4	no	—
198	C₄	#19151346	1	8	yes	complemented
199	C₄	#19151803	1	2	yes	—
200	C₄	#19151808	1	2	yes	—
201	C₄	#19151949	1	2	yes	—
202	C₄	#19151971	1	1	yes	characteristic
203	C₄	#19151978	1	1	yes	characteristic
204	C₂	C₂×#19151346	1	2	yes	—
205	C₂	#26018932	1	2	yes	—
206	C₂	#26021054	1	1	yes	characteristic
207	C₂	#26021092	1	1	yes	characteristic
208	C₂	#26021175	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#9691
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
92. C₄×C₈₉	#333	semidirect
109. C₈₉	#9691	semidirect
198. C₄	#19151346	semidirect

Maximal subgroups

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

subgroup	index
#26028650	2
#26020404	2
C₂×#19151346	2
C₄×C₄⋊C₈×C₈₉	2
#9691	89