Subgroups — #36555067

Order 22784.

770 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028885	C₂	1	2	yes	maximal
2	#26020427	C₂	1	2	yes	maximal, complemented
3	#26020393	C₂	1	2	yes	maximal, complemented
4	#26020290	C₂	1	4	yes	maximal
5	#26020158	C₂	1	4	yes	maximal, complemented
6	C₂²×#97×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#19151655	—	2	4	no	—
8	#19151651	—	2	4	no	—
9	C₂×#15667948	C₂²	1	8	yes	—
10	#19151622	C₂²	1	4	yes	—
11	#19151562	C₂²	1	4	yes	—
12	C₄×#13083558	C₂²	1	4	yes	—
13	#19151363	C₂²	1	8	yes	—
14	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#97×C₈₉	C₄	1	8	yes	—
17	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#97×C₈₉	C₂²	1	2	yes	—
19	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083558	C₂³	1	8	yes	—
21	#15667948	—	2	16	no	—
22	C₄⋊#10623494	—	2	8	no	—
23	C₂²⋊#10623494	—	2	8	no	—
24	#15667923	—	2	4	no	—
25	C₄×#10623494	—	2	4	no	—
26	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
27	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	—	2	8	no	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₈₉	D₄	1	4	yes	—
34	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	#97×C₈₉	C₂×C₄	1	4	yes	—
38	#97×C₈₉	C₂×C₄	1	8	yes	—
39	#97×C₈₉	C₂×C₄	1	4	yes	—
40	#13083558	—	2	16	no	—
41	C₂×#10623494	—	2	8	no	—
42	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	—	2	4	no	—
44	C₂²×C₄×C₈₉	—	4	4	no	—
45	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₈₉	—	2	4	no	—
54	C₂×C₈×C₈₉	—	2	16	no	—
55	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₈₉	#41	1	8	yes	—
58	C₄²×C₈₉	C₄⋊C₄	1	8	yes	—
59	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
60	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
61	C₄²×C₈₉	C₂×D₄	1	2	yes	—
62	C₄²×C₈₉	C₂×D₄	1	2	yes	—
63	C₂³×C₈₉	—	2	2	no	—
64	C₂³×C₈₉	—	2	4	no	—
65	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
66	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	2	no	—
68	C₂×C₄×C₈₉	—	2	8	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	—	2	4	no	—
71	C₂×C₄×C₈₉	—	4	4	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	—	4	2	no	—
74	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
80	C₈×C₈₉	—	2	4	no	—
81	C₈×C₈₉	#117	1	8	yes	—
82	#10623494	—	4	8	no	—
83	C₂²×C₈₉	—	2	8	no	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	—	2	2	no	—
86	C₂²×C₈₉	—	2	2	no	—
87	C₂²×C₈₉	#430	1	4	yes	—
88	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈₉	#533	1	2	yes	—
90	C₄×C₈₉	—	2	4	no	—
91	C₄×C₈₉	—	2	4	no	—
92	C₄×C₈₉	—	2	2	no	—
93	C₄×C₈₉	C₂×#117	1	2	yes	—
94	C₄×C₈₉	#532	1	2	yes	—
95	#20830	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	—	2	2	no	—
97	C₂×C₈₉	—	2	4	no	—
98	C₂×C₈₉	#2285	1	1	yes	characteristic
99	C₂×C₈₉	C₂×#430	1	2	yes	—
100	#2920	—	89	2	no	—
101	C₂²×#97	D₈₉	1	1	yes	characteristic
102	#1849	—	89	2	no	—
103	#1838	—	89	2	no	—
104	#1810	—	89	4	no	—
105	#1778	—	89	4	no	—
106	C₈₉	#20830	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₈	C₂×D₈₉	1	2	yes	—
108	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
109	#432	—	178	4	no	—
110	#431	—	178	4	no	—
111	C₂×#108	—	89	8	no	—
112	#423	—	89	4	no	—
113	#407	—	89	4	no	—
114	C₄×M₁₆	—	89	4	no	—
115	C₂×#97	Dic₈₉	1	8	yes	—
116	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
117	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
118	C₂×#97	C₂×D₈₉	1	2	yes	—
119	#341	—	89	8	no	—
120	C₂³×C₄	Dic₁₇₈	1	2	yes	—
121	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
122	C₂×M₁₆	—	89	8	no	—
123	C₂²×C₈	—	2	8	no	—
124	C₂²×C₈	C₂×Dic₈₉	1	8	yes	—
125	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
126	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
127	C₂×C₄²	Dic₁₇₈	1	2	yes	—
128	C₂×C₄²	D₃₅₆	1	4	yes	—
129	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
130	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
131	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	#108	—	178	16	no	—
133	C₄⋊C₈	—	178	8	no	—
134	C₂²⋊C₈	—	178	8	no	—
135	#97	—	178	4	no	—
136	#97	C₂×Dic₈₉	1	4	yes	—
137	#97	C₂×Dic₈₉	1	8	yes	—
138	#97	C₂×Dic₈₉	1	4	yes	—
139	C₄×C₈	—	178	4	no	—
140	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
144	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
145	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
146	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
147	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
149	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
150	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
151	M₁₆	—	178	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	16	no	—
154	C₂×C₈	—	178	8	no	—
155	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
156	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
157	C₂×C₈	#13083574	1	8	yes	—
158	C₄²	C₄⋊Dic₈₉	1	8	yes	—
159	C₄²	C₂²×Dic₈₉	1	2	yes	—
160	C₄²	C₂²×Dic₈₉	1	2	yes	—
161	C₄²	C₂×D₃₅₆	1	2	yes	—
162	C₄²	C₂×D₃₅₆	1	2	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	4	no	—
165	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
166	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
177	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
178	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
179	C₂×C₄	C₂×#13083574	1	4	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	356	8	no	—
182	C₈	#15667971	1	8	yes	—
183	C₂²	—	2	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	#19151647	1	4	yes	—
188	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
189	C₂²	#19152155	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	C₂×#15667971	1	2	yes	—
194	C₄	#19152148	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	#26022659	1	1	yes	characteristic
198	C₂	C₂×#19151647	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂²×#97×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂²×#97	D₈₉
hypocenter	C₈₉	#20830
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #26020427	C₂	semidirect
3. #26020393	C₂	semidirect
5. #26020158	C₂	semidirect
106. C₈₉	#20830	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26020427	2
#26020393	2
#26020290	2
#26020158	2
C₂²×#97×C₈₉	2
#20830	89