Subgroups — #36542129

Order 22784.

922 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030831	C₂	1	4	yes	maximal, complemented
2	#26030680	C₂	1	2	yes	maximal, complemented
3	#26019870	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019867	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019858	C₂	1	2	yes	maximal, complemented
6	#26019844	C₂	1	4	yes	maximal, complemented
7	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	—	2	4	no	—
9	#19152148	C₂²	1	4	yes	—
10	#19152003	C₂²	1	4	yes	—
11	#19151982	—	2	4	no	—
12	#19151978	C₂²	1	2	yes	—
13	#19151975	C₂²	1	4	yes	—
14	#19151975	C₂²	1	2	yes	—
15	#19151971	C₂²	1	4	yes	—
16	#19151945	C₂²	1	8	yes	—
17	#19151834	—	2	8	no	—
18	C₄×D₇₁₂	—	2	8	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
21	C₈²×C₈₉	C₂²	1	2	yes	—
22	C₈²×C₈₉	C₂²	1	1	yes	characteristic
23	C₈²×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#13083574	—	2	4	no	—
25	#15668034	—	4	16	no	—
26	C₂×Dic₃₅₆	—	2	4	no	—
27	C₂×#13083563	—	2	8	no	—
28	C₂×D₇₁₂	—	2	8	no	—
29	C₂×D₇₁₂	—	2	4	no	—
30	#15668012	C₂³	1	4	yes	—
31	#15668009	C₂³	1	4	yes	—
32	#15668000	—	2	4	no	—
33	#15667986	—	2	8	no	—
34	C₄×D₃₅₆	—	2	8	no	—
35	#15667946	—	2	8	no	—
36	#15667934	—	2	16	no	—
37	C₂²×C₈×C₈₉	D₄	1	4	yes	—
38	C₂²×C₈×C₈₉	D₄	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₈₉	D₄	1	4	yes	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂³	1	2	yes	—
43	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₈₉	C₂³	1	2	yes	—
46	#13083574	—	4	16	no	—
47	C₂×Dic₁₇₈	—	2	4	no	—
48	C₂×D₃₅₆	—	2	8	no	—
49	C₂×D₃₅₆	—	2	4	no	—
50	C₂×C₂²⋊D₈₉	—	4	4	no	—
51	C₂×C₄×D₈₉	—	4	4	no	—
52	Dic₃₅₆	—	4	8	no	—
53	#13083563	—	4	16	no	—
54	D₇₁₂	—	4	8	no	—
55	D₇₁₂	—	4	16	no	—
56	C₄⋊Dic₈₉	—	2	8	no	—
57	#13083551	—	4	8	no	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	—	2	8	no	—
61	C₂×C₈×C₈₉	D₈	1	8	yes	complemented
62	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
66	C₄²×C₈₉	C₂×D₄	1	2	yes	—
67	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
69	C₂³×C₈₉	#127	1	1	yes	characteristic
70	C₂²×D₈₉	—	4	4	no	—
71	C₂×C₄×C₈₉	—	2	4	no	—
72	C₂×C₄×C₈₉	#127	1	2	yes	—
73	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
75	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
77	C₂²⋊D₈₉	—	8	8	no	—
78	C₂×Dic₈₉	—	4	4	no	—
79	D₃₅₆	—	4	8	no	—
80	D₃₅₆	—	4	16	no	—
81	C₄×D₈₉	—	8	8	no	—
82	Dic₁₇₈	—	4	8	no	—
83	C₈×C₈₉	—	2	4	no	—
84	C₈×C₈₉	C₂×D₈	1	4	yes	—
85	C₈×C₈₉	C₂×D₈	1	4	yes	—
86	C₈×C₈₉	#135	1	8	yes	complemented
87	C₂²×C₈₉	—	2	2	no	—
88	C₂²×C₈₉	#493	1	2	yes	—
89	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
90	C₂×D₈₉	—	8	4	no	—
91	C₂×D₈₉	—	8	8	no	—
92	C₄×C₈₉	—	2	2	no	—
93	C₄×C₈₉	#493	1	2	yes	—
94	C₄×C₈₉	C₂²×D₈	1	2	yes	—
95	C₄×C₈₉	C₂×#135	1	4	yes	—
96	Dic₈₉	—	8	4	no	—
97	#19603	—	89	1	no	maximal, Sylow
98	C₂×C₈₉	—	2	2	no	—
99	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
100	C₂×C₈₉	#3147	1	2	yes	—
101	D₈₉	—	16	4	no	—
102	#3169	—	89	4	no	—
103	#3147	—	89	2	no	—
104	#1713	—	89	1	no	—
105	#1712	—	89	1	no	—
106	#1709	—	89	2	no	—
107	#1706	—	89	4	no	—
108	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
109	C₈₉	#19603	1	1	yes	characteristic, complemented, Sylow
110	C₂×#135	—	178	4	no	—
111	#532	—	89	4	no	—
112	#500	—	89	4	no	—
113	#495	—	178	4	no	—
114	#494	—	89	2	no	—
115	#493	—	89	2	no	—
116	#493	—	89	4	no	—
117	#492	—	89	4	no	—
118	#486	—	89	8	no	—
119	#466	—	178	8	no	—
120	C₄×D₈	—	178	8	no	—
121	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
123	C₈²	C₂×D₈₉	1	2	yes	—
124	C₈²	C₂×D₈₉	1	1	yes	characteristic
125	C₈²	C₂×D₈₉	1	1	yes	characteristic
126	C₂×#41	—	178	4	no	—
127	#135	—	356	16	no	—
128	C₂×Q₁₆	—	178	4	no	—
129	C₂×SD₁₆	—	178	8	no	—
130	C₂×D₈	—	178	4	no	—
131	C₂×D₈	—	178	8	no	—
132	C₂²×C₈	D₃₅₆	1	4	yes	—
133	C₂²×C₈	D₃₅₆	1	2	yes	—
134	C₄⋊Q₈	—	89	4	no	—
135	#127	—	89	4	no	—
136	#124	—	178	4	no	—
137	#121	—	178	8	no	—
138	C₄×D₄	—	178	8	no	—
139	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
140	#107	—	178	8	no	—
141	#102	—	178	16	no	—
142	C₄×C₈	D₃₅₆	1	4	yes	—
143	C₄×C₈	D₃₅₆	1	2	yes	—
144	C₄×C₈	C₂²×D₈₉	1	2	yes	—
145	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
146	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
147	C₄×C₈	C₂²×D₈₉	1	2	yes	—
148	#41	—	356	16	no	—
149	C₂×Q₈	—	178	4	no	—
150	C₂×D₄	—	356	4	no	—
151	C₂×D₄	—	178	8	no	—
152	C₂×D₄	—	178	4	no	—
153	C₂²×C₄	—	356	4	no	—
154	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
155	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
156	Q₁₆	—	356	8	no	—
157	SD₁₆	—	356	16	no	—
158	D₈	—	356	8	no	—
159	D₈	—	356	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	D₇₁₂	1	8	yes	complemented
162	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
163	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
164	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
165	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
166	C₄⋊C₄	—	178	8	no	—
167	C₂²⋊C₄	—	356	8	no	—
168	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
169	C₄²	C₂×D₃₅₆	1	2	yes	—
170	C₄²	C₂³×D₈₉	1	1	yes	characteristic
171	C₂³	—	356	4	no	—
172	C₂³	#15668009	1	1	yes	characteristic
173	Q₈	—	356	8	no	—
174	D₄	—	356	8	no	—
175	D₄	—	356	16	no	—
176	D₄	—	712	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	356	4	no	—
179	C₂×C₄	—	712	8	no	—
180	C₂×C₄	#15668009	1	2	yes	—
181	C₂×C₄	#15668009	1	1	yes	characteristic
182	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
183	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
184	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	C₂×D₇₁₂	1	4	yes	—
187	C₈	C₂×D₇₁₂	1	4	yes	—
188	C₈	#15668034	1	8	yes	complemented
189	C₂²	—	2	2	no	—
190	C₂²	—	712	4	no	—
191	C₂²	—	712	8	no	—
192	C₂²	#19151975	1	2	yes	—
193	C₂²	C₂×#15668009	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	712	4	no	—
196	C₄	#19151975	1	2	yes	—
197	C₄	C₂²×D₇₁₂	1	2	yes	—
198	C₄	C₂×#15668034	1	4	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	1424	4	no	—
201	C₂	C₂×#19151975	1	1	yes	characteristic
202	C₂	#26030680	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#127
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#19603
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #26030831	C₂	semidirect
2. #26030680	C₂	semidirect
3. #26019870	C₂	semidirect
4. #26019867	C₂	semidirect
5. #26019858	C₂	semidirect
6. #26019844	C₂	semidirect
7. C₂×C₈²×C₈₉	C₂	semidirect
61. C₂×C₈×C₈₉	D₈	semidirect
86. C₈×C₈₉	#135	semidirect
108. C₂×C₈²	D₈₉	semidirect
109. C₈₉	#19603	semidirect
161. C₂×C₈	D₇₁₂	semidirect
188. C₈	#15668034	semidirect

Maximal subgroups

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

subgroup	index
#26030831	2
#26030680	2
#26019870	2
#26019867	2
#26019858	2
#26019844	2
C₂×C₈²×C₈₉	2
#19603	89