Subgroups — #36473041

Order 22784.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic
2	#26020424	C₂	1	1	yes	maximal, characteristic
3	#26018348	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19152155	—	2	1	no	—
6	#19152003	C₄	1	2	yes	—
7	#19152000	—	2	1	no	—
8	#19152000	C₄	1	2	yes	—
9	#19151996	—	2	1	no	—
10	C₂×#15667948	—	2	1	no	—
11	C₂×#15667948	C₂²	1	2	yes	—
12	#19151636	—	2	1	no	—
13	#19151636	C₂²	1	2	yes	—
14	#19151622	—	2	1	no	—
15	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₈₉	—	2	1	no	—
18	C₄×C₁₆×C₈₉	—	2	1	no	—
19	C₂×#13083558	—	2	2	no	—
20	#15668012	—	2	2	no	—
21	#15668004	—	2	2	no	—
22	#15667991	—	4	2	no	—
23	C₄×Dic₁₇₈	—	4	2	no	—
24	#15667971	—	2	2	no	—
25	#15667948	—	2	4	no	—
26	#15667948	—	4	2	no	—
27	#15667946	—	2	2	no	—
28	#15667946	C₂×C₄	1	4	yes	—
29	#15667944	—	2	2	no	—
30	#15667944	—	2	2	no	—
31	C₄⋊#10623494	—	4	2	no	—
32	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂×Dic₁₇₈	—	4	2	no	—
43	#13083558	—	4	2	no	—
44	#13083558	—	4	2	no	—
45	C₂×#10623494	—	4	2	no	—
46	C₄⋊Dic₈₉	—	2	4	no	—
47	C₄⋊Dic₈₉	—	4	2	no	—
48	#13083553	—	4	4	no	—
49	C₂²⋊Dic₈₉	—	4	2	no	—
50	C₄×Dic₈₉	—	4	2	no	—
51	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
56	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
57	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₈₉	#41	1	2	yes	—
61	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₈₉	D₈	1	2	yes	—
64	C₁₆×C₈₉	—	2	4	no	—
65	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₉	—	2	4	no	—
67	C₂×C₄×C₈₉	#102	1	2	yes	—
68	C₂×C₄×C₈₉	#102	1	2	yes	—
69	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₈₉	—	4	4	no	—
76	Dic₁₇₈	—	8	4	no	—
77	C₈×C₈₉	#103	1	2	yes	—
78	C₈×C₈₉	#103	1	2	yes	—
79	C₈×C₈₉	C₄×Q₈	1	2	yes	—
80	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
81	#10623494	—	8	2	no	—
82	C₂²×C₈₉	—	2	2	no	—
83	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
84	C₂²×C₈₉	#487	1	1	yes	characteristic
85	C₂²×C₈₉	#498	1	1	yes	characteristic
86	C₄×C₈₉	—	2	2	no	—
87	C₄×C₈₉	#359	1	4	yes	—
88	C₄×C₈₉	#391	1	1	yes	characteristic
89	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
90	C₄×C₈₉	#486	1	1	yes	characteristic
91	C₄×C₈₉	#500	1	1	yes	characteristic
92	Dic₈₉	—	8	4	no	—
93	#13285	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	—	2	1	no	—
95	C₂×C₈₉	—	2	1	no	—
96	C₂×C₈₉	#1990	1	1	yes	characteristic
97	C₂×C₈₉	C₂×#359	1	2	yes	—
98	#3160	—	89	1	no	—
99	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
100	#1848	—	89	1	no	—
101	#1382	—	89	4	no	—
102	C₈₉	#13285	1	1	yes	characteristic, complemented, Sylow
103	#533	—	178	1	no	—
104	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
105	#500	—	89	2	no	—
106	#499	—	89	2	no	—
107	#499	—	178	1	no	—
108	#498	—	178	1	no	—
109	C₂×#108	—	89	2	no	—
110	C₂×#108	—	178	1	no	—
111	#427	—	89	2	no	—
112	#427	—	178	1	no	—
113	#423	—	178	1	no	—
114	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	C₄×C₁₆	—	2	1	no	—
116	C₄×C₁₆	—	2	1	no	—
117	C₂×M₁₆	—	178	2	no	—
118	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
120	C₄⋊Q₈	—	178	2	no	—
121	#125	—	178	2	no	—
122	C₂²⋊Q₈	—	356	2	no	—
123	C₄×Q₈	—	356	2	no	—
124	#117	—	178	2	no	—
125	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	Dic₁₇₈	1	4	yes	complemented
129	#108	—	356	2	no	—
130	#108	—	178	4	no	—
131	#107	—	89	4	no	—
132	#107	—	178	2	no	—
133	#106	—	178	2	no	—
134	#106	—	178	2	no	—
135	C₄⋊C₈	—	356	2	no	—
136	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
137	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
140	C₂×Q₈	—	356	2	no	—
141	C₂²×C₄	#13083563	1	2	yes	—
142	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
143	M₁₆	—	356	2	no	—
144	M₁₆	—	356	2	no	—
145	C₂×C₈	—	356	2	no	—
146	C₂×C₈	#13083551	1	1	yes	characteristic
147	C₂×C₈	#13083551	1	1	yes	characteristic
148	C₂×C₈	#13083563	1	2	yes	—
149	C₂×C₈	Dic₃₅₆	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
153	C₂×C₈	#13083574	1	2	yes	—
154	C₄⋊C₄	—	356	4	no	—
155	C₄⋊C₄	—	356	2	no	—
156	C₄⋊C₄	—	178	4	no	—
157	C₂²⋊C₄	—	356	2	no	—
158	C₄²	—	356	2	no	—
159	C₄²	#13083551	1	1	yes	characteristic
160	C₄²	#13083551	1	1	yes	characteristic
161	C₄²	D₇₁₂	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×#13083563	1	1	yes	characteristic
164	Q₈	—	712	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	356	4	no	—
167	C₂×C₄	#15667934	1	2	yes	—
168	C₂×C₄	#15667934	1	2	yes	—
169	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
170	C₂×C₄	#15668000	1	1	yes	characteristic
171	C₂×C₄	#15668012	1	1	yes	characteristic
172	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
173	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
175	C₈	—	712	2	no	—
176	C₈	#15667937	1	2	yes	—
177	C₈	#15667937	1	2	yes	—
178	C₈	C₄×Dic₁₇₈	1	2	yes	—
179	C₈	#15667991	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#15667934	1	1	yes	characteristic
182	C₂²	#19151949	1	1	yes	characteristic
183	C₂²	#19151996	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	712	4	no	—
186	C₄	#19151406	1	4	yes	—
187	C₄	#19151504	1	1	yes	characteristic
188	C₄	C₂×#15667937	1	1	yes	characteristic
189	C₄	#19151945	1	1	yes	characteristic
190	C₄	#19152003	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#26021213	1	1	yes	characteristic
194	C₂	C₂×#19151406	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#13285
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
37. C₂×C₁₆×C₈₉	Q₈	semidirect
102. C₈₉	#13285	semidirect
128. C₂×C₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26020424	2
#26018348	2
C₂×C₄×C₁₆×C₈₉	2
#13285	89