Subgroups — #36445732

Order 22784.

358 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	#26019000	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#10623494	C₂	1	2	yes	maximal
3	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
4	#26018090	C₂	1	2	yes	maximal
5	C₄×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	#19151351	C₄	1	2	yes	—
10	#19151349	C₄	1	2	yes	—
11	#19151319	—	2	2	no	—
12	#19151319	C₄	1	2	yes	—
13	#19151319	C₄	1	2	yes	—
14	C₈×#10623494	—	2	2	no	—
15	C₂×#108×C₈₉	C₄	1	2	yes	—
16	C₄×M₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#341×C₈₉	C₄	1	2	yes	—
19	C₂²×#10623494	—	2	1	no	—
20	C₂²×#10623494	Q₈	1	2	yes	—
21	C₄⋊#10623494	C₂×C₄	1	4	yes	—
22	C₂²⋊#10623494	—	2	2	no	—
23	C₄×#10623494	—	2	2	no	—
24	C₄×#10623494	—	2	1	no	—
25	C₄×#10623494	C₂×C₄	1	2	yes	—
26	C₄×#10623494	C₂×C₄	1	2	yes	—
27	C₄×#10623494	D₄	1	2	yes	—
28	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	#108×C₈₉	C₈	1	8	yes	complemented
33	#97×C₈₉	—	2	2	no	—
34	C₄×C₈×C₈₉	—	2	2	no	—
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	1	yes	characteristic
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	—	2	4	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₂×#10623494	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
47	M₁₆×C₈₉	—	2	4	no	—
48	M₁₆×C₈₉	C₂×C₈	1	8	yes	—
49	C₂×C₈×C₈₉	—	2	4	no	—
50	C₂×C₈×C₈₉	—	2	2	no	—
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	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂×C₈	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	C₂²×C₈	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	8	no	—
74	C₈×C₈₉	C₄×C₈	1	2	yes	—
75	C₈×C₈₉	C₄×C₈	1	2	yes	—
76	#10623494	—	2	4	no	—
77	#10623494	—	4	2	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	C₈×Q₈	1	2	yes	—
83	C₄×C₈₉	—	2	1	no	—
84	C₄×C₈₉	—	2	1	no	—
85	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₈₉	#405	1	1	yes	characteristic
87	C₄×C₈₉	#430	1	2	yes	—
88	C₄×C₈₉	C₈×D₄	1	2	yes	—
89	C₄×C₈₉	#444	1	2	yes	—
90	#8294	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	—	2	2	no	—
92	C₂×C₈₉	C₈×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₈₉	#1773	1	1	yes	characteristic
94	C₂×C₈₉	#1778	1	1	yes	characteristic
95	C₄×#108	D₈₉	1	1	yes	characteristic
96	#1564	—	89	1	no	—
97	C₈×M₁₆	—	89	2	no	—
98	C₂×#322	—	89	1	no	—
99	#1278	—	89	2	no	—
100	C₈₉	#8294	1	1	yes	characteristic, complemented, Sylow
101	#431	—	89	2	no	—
102	C₂×#108	Dic₈₉	1	2	yes	—
103	C₄×M₁₆	C₂×D₈₉	1	2	yes	—
104	C₂×C₄×C₈	—	89	1	no	—
105	C₂×C₄×C₈	—	89	1	no	—
106	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
107	#341	Dic₈₉	1	2	yes	—
108	#335	—	89	2	no	—
109	#334	—	89	2	no	—
110	#322	—	89	2	no	—
111	#322	—	89	2	no	—
112	#322	—	178	2	no	—
113	C₈²	—	178	2	no	—
114	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
115	C₂²×C₈	—	89	2	no	—
116	C₂²×C₈	—	178	1	no	—
117	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
120	#108	#10623494	1	8	yes	complemented
121	C₄⋊C₈	—	89	4	no	—
122	C₂²⋊C₈	—	178	2	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	89	2	no	—
126	C₄×C₈	—	89	2	no	—
127	C₄×C₈	—	89	2	no	—
128	C₄×C₈	—	178	2	no	—
129	C₄×C₈	—	178	1	no	—
130	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
131	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
132	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
133	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
136	C₂²×C₄	#13083575	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×#10623494	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	89	4	no	—
142	C₂×C₈	—	356	2	no	—
143	C₂×C₈	—	356	2	no	—
144	C₂×C₈	—	178	2	no	—
145	C₂×C₈	—	178	4	no	—
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	1	yes	characteristic
150	C₂×C₈	C₂×#10623494	1	4	yes	—
151	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
153	C₄²	D₄×D₈₉	1	1	yes	characteristic
154	C₄²	#13083573	1	1	yes	characteristic
155	C₂³	Q₈×Dic₈₉	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	C₄⋊C₄×D₈₉	1	1	yes	characteristic
160	C₂×C₄	#15667970	1	1	yes	characteristic
161	C₂×C₄	#15667974	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
163	C₂×C₄	#15667977	1	1	yes	characteristic
164	C₂×C₄	C₂²×#10623494	1	2	yes	—
165	C₂×C₄	#15668019	1	2	yes	—
166	C₈	—	2	8	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	356	2	no	—
170	C₈	—	356	4	no	—
171	C₈	—	178	4	no	—
172	C₈	C₄×#10623494	1	2	yes	—
173	C₈	C₄×#10623494	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
177	C₂²	Q₈×#10623494	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×#10623494	1	1	yes	characteristic
181	C₄	#19151554	1	1	yes	characteristic
182	C₄	#19151645	1	2	yes	—
183	C₄	D₄×#10623494	1	2	yes	—
184	C₄	#19151719	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₄⋊C₄×#10623494	1	1	yes	characteristic
187	C₂	#26020139	1	1	yes	characteristic
188	C₂	#26020160	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#108×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₄⋊C₄
prime radical	C₂³×C₈₉	C₄×Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#108	D₈₉
hypocenter	C₈₉	#8294
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
32. #108×C₈₉	C₈	semidirect
100. C₈₉	#8294	semidirect
120. #108	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26019000	2
M₁₆×#10623494	2
C₂×#19151319	2
#26018090	2
C₄×#108×C₈₉	2
#8294	89