Subgroups — #36563129

Order 22784.

746 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×#19151728	C₂	1	2	yes	maximal
2	#26023545	C₂	1	1	yes	maximal, characteristic
3	#26021192	C₂	1	2	yes	maximal
4	#26020975	C₂	1	4	yes	maximal
5	#26020406	C₂	1	1	yes	maximal, characteristic
6	#26020119	C₂	1	4	yes	maximal
7	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
9	#19151728	—	2	8	no	—
10	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×#15667941	C₂²	1	4	yes	—
13	C₂×#15667922	C₂²	1	4	yes	—
14	#19151530	C₄	1	4	yes	—
15	#19151519	C₄	1	4	yes	—
16	C₄×C₄⋊Dic₈₉	C₂²	1	4	yes	—
17	#19151352	C₂²	1	8	yes	—
18	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#97×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
22	#376×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
24	C₂²×#10623494	C₂³	1	4	yes	—
25	C₄×Dic₁₇₈	—	2	4	no	—
26	C₄×Dic₁₇₈	—	2	4	no	—
27	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
29	C₂×#13083553	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₈₉	C₂³	1	4	yes	—
31	C₄⋊#10623494	—	2	4	no	—
32	C₄⋊#10623494	—	2	8	no	—
33	#15667941	—	2	8	no	—
34	#15667922	—	2	8	no	—
35	#15667919	C₂×C₄	1	4	yes	—
36	#15667919	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
42	#97×C₈₉	—	2	2	no	—
43	#97×C₈₉	—	2	4	no	—
44	C₄×C₈×C₈₉	D₄	1	4	yes	—
45	C₄×C₈×C₈₉	Q₈	1	8	yes	complemented
46	#95×C₈₉	C₂×C₄	1	4	yes	—
47	C₂×Dic₁₇₈	—	2	4	no	—
48	C₂×Dic₁₇₈	—	4	4	no	—
49	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
50	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
51	C₂×#10623494	—	2	4	no	—
52	C₂×#10623494	—	2	8	no	—
53	C₄⋊Dic₈₉	—	2	4	no	—
54	C₄⋊Dic₈₉	—	4	8	no	—
55	#13083553	—	2	8	no	—
56	C₄×Dic₈₉	—	2	4	no	—
57	C₄×Dic₈₉	—	2	4	no	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₈₉	—	2	2	no	—
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
67	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
68	C₄²×C₈₉	C₂×D₄	1	2	yes	—
69	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
70	C₄²×C₈₉	#41	1	2	yes	—
71	C₄²×C₈₉	#41	1	4	yes	—
72	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₉	—	2	8	no	—
74	C₂×C₄×C₈₉	—	2	4	no	—
75	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
76	C₂×C₄×C₈₉	C₄×Q₈	1	8	yes	—
77	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
81	C₂×Dic₈₉	—	2	4	no	—
82	C₂×Dic₈₉	—	2	4	no	—
83	C₂×Dic₈₉	—	2	4	no	—
84	C₂×Dic₈₉	—	2	8	no	—
85	C₂×Dic₈₉	—	2	4	no	—
86	Dic₁₇₈	—	4	8	no	—
87	C₈×C₈₉	—	2	2	no	—
88	C₈×C₈₉	—	2	4	no	—
89	C₈×C₈₉	C₄⋊Q₈	1	4	yes	—
90	#10623494	—	4	4	no	—
91	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
92	C₂²×C₈₉	C₂×C₄×Q₈	1	2	yes	—
93	C₂²×C₈₉	#568	1	4	yes	—
94	C₄×C₈₉	—	2	4	no	—
95	C₄×C₈₉	—	2	8	no	—
96	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₄×C₈₉	#532	1	1	yes	characteristic
98	C₄×C₈₉	#533	1	2	yes	—
99	Dic₈₉	—	4	4	no	—
100	Dic₈₉	—	4	4	no	—
101	#21716	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₈₉	#2930	1	2	yes	—
104	C₂×C₈₉	#2961	1	4	yes	—
105	C₂×#446	—	89	2	no	—
106	#2399	—	89	1	no	—
107	#1985	—	89	2	no	—
108	#1940	—	89	4	no	—
109	#1843	—	89	1	no	—
110	#1770	—	89	4	no	—
111	#1726	D₈₉	1	1	yes	characteristic
112	C₈₉	#21716	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄×Q₈	—	89	2	no	—
114	#446	—	178	8	no	—
115	C₂×C₄⋊C₈	—	89	4	no	—
116	C₂×C₄⋊C₈	—	89	2	no	—
117	C₂×C₄⋊C₈	—	89	4	no	—
118	C₂×#97	—	89	4	no	—
119	C₂×#97	C₂×D₈₉	1	2	yes	—
120	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
122	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
123	#398	—	89	4	no	—
124	#395	—	89	4	no	—
125	C₄×C₄⋊C₄	—	89	4	no	—
126	#376	C₂×D₈₉	1	1	yes	characteristic
127	#336	—	89	8	no	—
128	C₂²×Q₈	—	89	2	no	—
129	C₂²×C₈	—	89	4	no	—
130	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
132	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
133	C₄×Q₈	—	178	4	no	—
134	C₄×Q₈	—	178	4	no	—
135	C₂×C₄⋊C₄	—	89	8	no	—
136	C₂×C₄⋊C₄	—	89	4	no	—
137	C₂×C₄⋊C₄	—	89	2	no	—
138	C₂×C₄²	—	89	4	no	—
139	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
141	C₄⋊C₈	—	178	8	no	—
142	C₄⋊C₈	—	178	4	no	—
143	C₄⋊C₈	—	178	8	no	—
144	#97	—	2	2	no	—
145	#97	—	2	4	no	—
146	#97	—	178	8	no	—
147	C₄×C₈	Dic₁₇₈	1	8	yes	complemented
148	C₄×C₈	D₃₅₆	1	4	yes	—
149	#95	—	89	4	no	—
150	#95	—	89	4	no	—
151	#95	C₄×D₈₉	1	4	yes	—
152	C₂×Q₈	—	356	4	no	—
153	C₂×Q₈	—	178	4	no	—
154	C₂²×C₄	—	89	4	no	—
155	C₂²×C₄	—	89	4	no	—
156	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
157	C₂²×C₄	C₂×C₄×D₈₉	1	4	yes	—
158	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	178	4	no	—
163	C₂×C₈	—	178	8	no	—
164	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
165	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
166	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
167	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
168	C₄⋊C₄	—	356	8	no	—
169	C₄⋊C₄	—	178	8	no	—
170	C₄⋊C₄	—	178	4	no	—
171	C₄²	—	178	4	no	—
172	C₄²	—	178	4	no	—
173	C₄²	C₂×D₃₅₆	1	2	yes	—
174	C₄²	C₂×Dic₁₇₈	1	4	yes	—
175	C₄²	#13083574	1	2	yes	—
176	C₄²	#13083574	1	4	yes	—
177	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
178	Q₈	—	356	8	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	178	4	no	—
182	C₂×C₄	—	178	4	no	—
183	C₂×C₄	—	178	4	no	—
184	C₂×C₄	—	178	8	no	—
185	C₂×C₄	—	178	4	no	—
186	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
187	C₂×C₄	C₄×Dic₁₇₈	1	8	yes	—
188	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₁₇₈	1	2	yes	—
190	C₂×C₄	C₂×#13083574	1	2	yes	—
191	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	4	no	—
194	C₈	—	356	4	no	—
195	C₈	#15668012	1	4	yes	—
196	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
197	C₂²	C₂×C₄×Dic₁₇₈	1	2	yes	—
198	C₂²	#19152363	1	4	yes	—
199	C₄	—	2	8	no	—
200	C₄	—	2	4	no	—
201	C₄	—	356	4	no	—
202	C₄	—	356	4	no	—
203	C₄	C₂×#15668012	1	1	yes	characteristic
204	C₄	#19152148	1	1	yes	characteristic
205	C₄	#19152155	1	2	yes	—
206	C₂	C₄×#15668012	1	1	yes	characteristic
207	C₂	#26028957	1	2	yes	—
208	C₂	#26029190	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₈₉×#1726	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#21716
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1726

Complements

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

N	H ≅ G/N	type
45. C₄×C₈×C₈₉	Q₈	semidirect
112. C₈₉	#21716	semidirect
147. C₄×C₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151728	2
#26023545	2
#26021192	2
#26020975	2
#26020406	2
#26020119	2
C₈₉×#1726	2
#21716	89