Subgroups — #36447360

Order 22784.

398 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020248	C₂	1	2	yes	maximal
2	#26019011	C₂	1	2	yes	maximal
3	#26018127	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151622	C₂²	1	4	yes	—
6	C₂×#15667931	—	2	2	no	—
7	#19151367	—	2	2	no	—
8	#19151351	—	2	2	no	—
9	#19151351	—	2	2	no	—
10	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₈₉	—	2	2	no	—
12	C₈×Q₈×C₈₉	—	2	2	no	—
13	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083558	—	2	4	no	—
18	C₄⋊#10623494	—	2	4	no	—
19	C₄⋊#10623494	—	2	4	no	—
20	#15667931	—	4	4	no	—
21	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₈₉	—	2	2	no	—
26	C₄×Q₈×C₈₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
29	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₉	—	2	4	no	—
32	C₄⋊C₈×C₈₉	—	2	2	no	—
33	C₄×C₈×C₈₉	—	2	2	no	—
34	C₄×C₈×C₈₉	—	2	2	no	—
35	C₄×C₈×C₈₉	D₄	1	2	yes	—
36	C₄×C₈×C₈₉	D₄	1	2	yes	—
37	#13083558	—	4	4	no	—
38	#13083558	—	4	4	no	—
39	C₂×#10623494	—	4	4	no	—
40	C₂×Q₈×C₈₉	—	2	2	no	—
41	C₂×Q₈×C₈₉	—	2	2	no	—
42	C₂×Q₈×C₈₉	D₈	1	2	yes	—
43	C₂×Q₈×C₈₉	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₈₉	—	2	2	no	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	—	4	1	no	—
52	C₂×C₈×C₈₉	—	4	2	no	—
53	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₈₉	—	2	2	no	—
57	C₄⋊C₄×C₈₉	—	2	4	no	—
58	C₄²×C₈₉	—	2	2	no	—
59	C₄²×C₈₉	—	2	2	no	—
60	C₄²×C₈₉	C₂×D₄	1	2	yes	—
61	C₄²×C₈₉	#41	1	2	yes	—
62	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	Q₈×C₈₉	—	2	4	no	—
64	Q₈×C₈₉	—	2	4	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	—	2	2	no	—
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	—	4	1	no	—
71	C₂×C₄×C₈₉	—	4	2	no	—
72	C₂×C₄×C₈₉	#107	1	4	yes	—
73	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#121	1	2	yes	—
76	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#135	1	2	yes	—
80	C₈×C₈₉	—	2	1	no	—
81	C₈×C₈₉	—	2	2	no	—
82	C₈×C₈₉	—	4	2	no	—
83	C₈×C₈₉	#101	1	2	yes	complemented
84	#10623494	—	8	4	no	—
85	C₂²×C₈₉	—	2	1	no	—
86	C₂²×C₈₉	—	2	1	no	—
87	C₂²×C₈₉	#380	1	1	yes	characteristic
88	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
89	C₂²×C₈₉	#427	1	1	yes	characteristic
90	C₄×C₈₉	—	2	1	no	—
91	C₄×C₈₉	—	2	4	no	—
92	C₄×C₈₉	—	2	2	no	—
93	C₄×C₈₉	—	4	2	no	—
94	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
95	C₄×C₈₉	#413	1	1	yes	characteristic
96	C₄×C₈₉	#466	1	2	yes	—
97	C₄×C₈₉	#467	1	2	yes	—
98	#8617	—	89	1	no	maximal, Sylow
99	C₂×C₈₉	—	2	1	no	—
100	C₂×C₈₉	—	2	1	no	—
101	C₂×C₈₉	#1801	1	1	yes	characteristic
102	C₂×C₈₉	#1808	1	1	yes	characteristic
103	C₂×C₈₉	#1813	1	1	yes	characteristic
104	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
105	#1801	—	89	2	no	—
106	#1567	—	89	2	no	—
107	#1292	—	89	2	no	—
108	C₈₉	#8617	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
110	C₈×Q₈	—	2	2	no	—
111	C₈×Q₈	—	2	2	no	—
112	#423	—	89	4	no	—
113	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
114	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
115	C₂×#101	—	178	2	no	—
116	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
118	#343	—	178	2	no	—
119	#335	—	178	2	no	—
120	#335	—	178	2	no	—
121	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
122	C₂×M₁₆	—	178	4	no	—
123	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
125	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
126	C₄×Q₈	—	2	2	no	—
127	C₄×Q₈	—	2	2	no	—
128	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
129	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
130	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
131	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	C₄⋊C₈	—	2	4	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	178	4	no	—
135	C₄⋊C₈	—	178	4	no	—
136	#101	—	356	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
140	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	D₇₁₂	1	2	yes	—
144	C₂×Q₈	Dic₃₅₆	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
146	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
150	M₁₆	—	356	4	no	—
151	M₁₆	—	356	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	356	4	no	—
157	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
159	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
165	C₄²	#13083574	1	2	yes	—
166	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	#15667946	1	4	yes	—
177	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
178	C₂×C₄	#15667971	1	1	yes	characteristic
179	C₂×C₄	#15667986	1	2	yes	—
180	C₂×C₄	#15667991	1	2	yes	—
181	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
183	C₂×C₄	#15668034	1	2	yes	—
184	C₈	—	2	1	no	—
185	C₈	—	2	2	no	—
186	C₈	—	4	2	no	—
187	C₈	—	712	4	no	—
188	C₈	#15667931	1	2	yes	complemented
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	#19151458	1	1	yes	characteristic
192	C₂²	C₂×#15667946	1	1	yes	characteristic
193	C₂²	#19151636	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	C₂×#15667931	1	1	yes	characteristic
199	C₄	#19151582	1	1	yes	characteristic
200	C₄	#19151834	1	2	yes	—
201	C₄	#19151839	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	#26020248	1	1	yes	characteristic
205	C₂	#26020281	1	1	yes	characteristic
206	C₂	#26020302	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151458
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	#8617
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
83. C₈×C₈₉	#101	semidirect
108. C₈₉	#8617	semidirect
188. C₈	#15667931	semidirect

Maximal subgroups

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

subgroup	index
#26020248	2
#26019011	2
#26018127	2
C₂×C₈×Q₈×C₈₉	2
#8617	89