Subgroups — #36445735

Order 22784.

334 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667948	C₂	1	1	yes	maximal, characteristic
2	#26019000	C₂	1	1	yes	maximal, characteristic
3	#26018541	C₂	1	2	yes	maximal
4	#26018089	C₂	1	2	yes	maximal
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×#15667948	C₄	1	2	yes	—
8	C₄×#13083558	C₂²	1	2	yes	—
9	#19151363	C₄	1	2	yes	—
10	#19151351	C₄	1	2	yes	—
11	#19151349	C₄	1	2	yes	—
12	#19151320	—	2	2	no	—
13	#19151319	—	2	2	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#322×C₈₉	C₄	1	2	yes	—
18	#322×C₈₉	C₄	1	2	yes	—
19	C₂×#13083558	C₂×C₄	1	4	yes	—
20	#15667948	—	2	4	no	—
21	C₄⋊#10623494	C₂×C₄	1	4	yes	—
22	C₂²⋊#10623494	—	2	2	no	—
23	#15667923	—	2	2	no	—
24	C₄×#10623494	—	2	2	no	—
25	C₄×#10623494	—	2	2	no	—
26	C₂²×C₈×C₈₉	—	2	1	no	—
27	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₈₉	—	2	1	no	—
32	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₉	D₄	1	2	yes	—
39	#13083558	—	2	4	no	—
40	#13083558	—	2	4	no	—
41	C₂×#10623494	—	2	4	no	—
42	C₂×#10623494	—	2	4	no	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
46	C₂×C₈×C₈₉	—	2	2	no	—
47	C₂×C₈×C₈₉	—	2	2	no	—
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₂×C₈×C₈₉	—	4	2	no	—
50	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
56	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₈₉	#41	1	1	yes	characteristic
60	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₈₉	#131	1	2	yes	—
71	C₈×C₈₉	—	2	1	no	—
72	C₈×C₈₉	—	2	1	no	—
73	C₈×C₈₉	—	2	4	no	—
74	C₈×C₈₉	—	4	2	no	—
75	C₈×C₈₉	#97	1	2	yes	—
76	C₈×C₈₉	#97	1	2	yes	—
77	#10623494	—	4	4	no	—
78	#10623494	—	4	4	no	—
79	C₂²×C₈₉	—	2	1	no	—
80	C₂²×C₈₉	—	2	1	no	—
81	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₈₉	#446	1	2	yes	—
83	C₄×C₈₉	—	2	1	no	—
84	C₄×C₈₉	—	2	1	no	—
85	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
86	C₄×C₈₉	#405	1	1	yes	characteristic
87	C₄×C₈₉	#430	1	2	yes	—
88	C₄×C₈₉	#436	1	2	yes	—
89	C₄×C₈₉	#444	1	2	yes	—
90	#8295	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	—	2	2	no	—
92	C₂×C₈₉	#1770	1	1	yes	characteristic
93	C₂×C₈₉	#1773	1	1	yes	characteristic
94	C₂×C₈₉	#1778	1	1	yes	characteristic
95	C₄×#108	—	89	1	no	—
96	#1564	—	89	1	no	—
97	#1451	—	89	2	no	—
98	C₂×#322	D₈₉	1	1	yes	characteristic
99	#1278	—	89	2	no	—
100	C₈₉	#8295	1	1	yes	characteristic, complemented, Sylow
101	#431	—	89	2	no	—
102	C₂×#108	—	89	2	no	—
103	C₄×M₁₆	—	89	2	no	—
104	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
107	#341	—	89	2	no	—
108	#335	—	89	2	no	—
109	#334	—	89	2	no	—
110	#322	—	178	2	no	—
111	#322	—	178	2	no	—
112	#322	Dic₈₉	1	2	yes	—
113	#322	Dic₈₉	1	2	yes	—
114	C₂×M₁₆	—	89	4	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	Dic₁₇₈	1	2	yes	—
117	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
120	#108	—	178	4	no	—
121	C₄⋊C₈	—	89	4	no	—
122	C₂²⋊C₈	—	178	2	no	—
123	#97	—	178	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	178	2	no	—
126	C₄×C₈	—	178	2	no	—
127	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
128	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
129	C₄×C₈	D₃₅₆	1	2	yes	—
130	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
131	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
133	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
136	C₂²×C₄	#13083574	1	1	yes	characteristic
137	M₁₆	—	178	4	no	—
138	M₁₆	—	178	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	178	4	no	—
144	C₂×C₈	—	178	4	no	—
145	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
146	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
147	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
150	C₂×C₈	#13083557	1	4	yes	—
151	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
152	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
153	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
154	C₄²	#13083574	1	1	yes	characteristic
155	C₂³	C₄×Dic₁₇₈	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
160	C₂×C₄	#15667971	1	1	yes	characteristic
161	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
162	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
163	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
164	C₂×C₄	C₂×#13083557	1	2	yes	—
165	C₂×C₄	#15668018	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	2	no	—
170	C₈	—	356	4	no	—
171	C₈	—	356	4	no	—
172	C₈	#15667922	1	2	yes	—
173	C₈	#15667922	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
177	C₂²	#19151728	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×#15667922	1	1	yes	characteristic
181	C₄	#19151553	1	1	yes	characteristic
182	C₄	#19151647	1	2	yes	—
183	C₄	#19151677	1	2	yes	—
184	C₄	#19151721	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	#26020119	1	1	yes	characteristic
187	C₂	#26020136	1	1	yes	characteristic
188	C₂	#26020158	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₄⋊C₄
prime radical	C₂³×C₈₉	C₄×Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#8295
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
100. C₈₉	#8295	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667948	2
#26019000	2
#26018541	2
#26018089	2
C₂×#322×C₈₉	2
#8295	89