Subgroups — #36448089

Order 22784.

422 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	#26020404	C₂	1	2	yes	maximal
2	#26018087	C₂	1	4	yes	maximal
3	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10623494	—	2	2	no	—
5	C₂×C₄⋊#10623494	—	2	2	no	—
6	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
11	C₄³×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×#10623494	—	2	4	no	—
13	C₄⋊#10623494	—	2	8	no	—
14	C₄⋊#10623494	—	4	4	no	—
15	C₄⋊#10623494	—	4	4	no	—
16	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₈₉	—	2	4	no	—
26	C₄×C₈×C₈₉	—	2	2	no	—
27	C₄×C₈×C₈₉	—	2	4	no	—
28	C₄×C₈×C₈₉	C₈	1	4	yes	complemented
29	C₂×#10623494	—	4	4	no	—
30	C₂×#10623494	—	4	4	no	—
31	C₂×#10623494	—	4	4	no	—
32	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	—	2	2	no	—
43	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
45	C₄²×C₈₉	—	2	2	no	—
46	C₄²×C₈₉	—	2	2	no	—
47	C₄²×C₈₉	—	2	2	no	—
48	C₄²×C₈₉	C₂×C₈	1	2	yes	—
49	C₄²×C₈₉	M₁₆	1	2	yes	—
50	C₄²×C₈₉	D₈	1	2	yes	—
51	C₄²×C₈₉	SD₁₆	1	4	yes	—
52	C₄²×C₈₉	Q₁₆	1	2	yes	—
53	C₄²×C₈₉	#41	1	4	yes	—
54	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	1	no	—
59	C₂×C₄×C₈₉	—	2	1	no	—
60	C₂×C₄×C₈₉	—	2	1	no	—
61	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
63	C₂×C₄×C₈₉	#106	1	2	yes	—
64	C₂×C₄×C₈₉	#106	1	2	yes	—
65	C₂×C₄×C₈₉	#107	1	2	yes	—
66	C₂×C₄×C₈₉	#107	1	2	yes	—
67	C₂×C₄×C₈₉	#108	1	2	yes	—
68	C₂×C₄×C₈₉	#108	1	2	yes	—
69	C₂×C₄×C₈₉	#117	1	2	yes	—
70	C₂×C₄×C₈₉	#124	1	2	yes	—
71	C₂×C₄×C₈₉	#125	1	2	yes	—
72	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
78	C₈×C₈₉	—	2	4	no	—
79	C₈×C₈₉	—	2	4	no	—
80	#10623494	—	8	4	no	—
81	C₂²×C₈₉	#382	1	1	yes	characteristic
82	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₂²×C₈₉	#423	1	1	yes	characteristic
84	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
85	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
86	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
87	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
88	C₄×C₈₉	—	2	2	no	—
89	C₄×C₈₉	—	2	2	no	—
90	C₄×C₈₉	—	2	2	no	—
91	C₄×C₈₉	#334	1	4	yes	—
92	C₄×C₈₉	#335	1	4	yes	—
93	C₄×C₈₉	#431	1	2	yes	—
94	C₄×C₈₉	#432	1	2	yes	—
95	C₄×C₈₉	#486	1	2	yes	—
96	C₄×C₈₉	#487	1	2	yes	—
97	#8784	—	89	1	no	maximal, Sylow
98	C₂×C₈₉	C₂×#334	1	1	yes	characteristic
99	C₂×C₈₉	C₂×#335	1	1	yes	characteristic
100	C₂×C₈₉	#1566	1	1	yes	characteristic
101	C₂×C₈₉	#1567	1	1	yes	characteristic
102	C₂×C₈₉	#1831	1	1	yes	characteristic
103	C₂×C₈₉	#1834	1	1	yes	characteristic
104	C₂×C₈₉	#1837	1	1	yes	characteristic
105	#1842	—	89	2	no	—
106	C₄²×C₈	D₈₉	1	1	yes	characteristic
107	#1277	—	89	4	no	—
108	C₈₉	#8784	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄⋊C₈	—	89	4	no	—
110	C₂×C₄⋊C₈	—	178	2	no	—
111	C₂×C₄⋊C₈	—	178	2	no	—
112	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
113	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
114	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
116	C₄³	C₂×D₈₉	1	1	yes	characteristic
117	C₂²×C₈	—	178	4	no	—
118	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
119	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
120	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
121	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
122	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
123	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
127	C₄⋊C₈	—	356	4	no	—
128	C₄⋊C₈	—	356	4	no	—
129	C₄⋊C₈	—	178	8	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	4	no	—
133	C₄×C₈	#10623494	1	4	yes	complemented
134	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
136	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
137	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	356	4	no	—
146	C₂×C₈	—	356	4	no	—
147	C₂×C₈	—	356	4	no	—
148	C₂×C₈	C₂×#10623494	1	4	yes	—
149	C₂×C₈	#13083558	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	C₂×#10623494	1	2	yes	—
154	C₄²	#13083558	1	2	yes	—
155	C₄²	D₇₁₂	1	2	yes	—
156	C₄²	#13083563	1	4	yes	—
157	C₄²	Dic₃₅₆	1	2	yes	—
158	C₄²	#13083574	1	4	yes	—
159	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	C₄⋊#10623494	1	2	yes	—
167	C₂×C₄	C₄⋊#10623494	1	2	yes	—
168	C₂×C₄	#15667944	1	2	yes	—
169	C₂×C₄	#15667944	1	2	yes	—
170	C₂×C₄	#15667946	1	2	yes	—
171	C₂×C₄	#15667946	1	2	yes	—
172	C₂×C₄	#15667948	1	2	yes	—
173	C₂×C₄	#15667948	1	2	yes	—
174	C₂×C₄	#15667971	1	2	yes	—
175	C₂×C₄	#15668000	1	2	yes	—
176	C₂×C₄	#15668004	1	2	yes	—
177	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
178	C₂×C₄	C₂×#13083558	1	2	yes	—
179	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
180	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
181	C₂×C₄	C₂×#13083563	1	2	yes	—
182	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
183	C₈	—	2	4	no	—
184	C₈	—	2	4	no	—
185	C₈	—	712	4	no	—
186	C₂²	#19151464	1	1	yes	characteristic
187	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
188	C₂²	#19151622	1	1	yes	characteristic
189	C₂²	C₂×#15667944	1	1	yes	characteristic
190	C₂²	C₂×#15667946	1	1	yes	characteristic
191	C₂²	C₂×#15667948	1	1	yes	characteristic
192	C₂²	C₂×#15667948	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	#19151349	1	4	yes	—
197	C₄	#19151351	1	4	yes	—
198	C₄	#19151651	1	2	yes	—
199	C₄	#19151655	1	2	yes	—
200	C₄	#19151945	1	2	yes	—
201	C₄	#19151949	1	2	yes	—
202	C₂	C₂×#19151349	1	1	yes	characteristic
203	C₂	C₂×#19151351	1	1	yes	characteristic
204	C₂	#26019008	1	1	yes	characteristic
205	C₂	#26019011	1	1	yes	characteristic
206	C₂	#26020369	1	1	yes	characteristic
207	C₂	#26020380	1	1	yes	characteristic
208	C₂	#26020390	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#8784
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
28. C₄×C₈×C₈₉	C₈	semidirect
108. C₈₉	#8784	semidirect
133. C₄×C₈	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26020404	2
#26018087	2
C₄²×C₈×C₈₉	2
#8784	89