Subgroups — #36443817

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	2	yes	maximal
2	#26018079	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151651	—	2	4	no	—
5	#19151651	—	2	4	no	—
6	C₂×C₄⋊#10623494	C₄	1	4	yes	—
7	C₂×C₂²⋊#10623494	C₄	1	4	yes	—
8	C₂×C₄×#10623494	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₈₉	C₄	1	2	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₈₉	—	2	4	no	—
12	C₂×#97×C₈₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
14	C₂²×#10623494	C₂×C₄	1	8	yes	—
15	C₄⋊#10623494	—	2	8	no	—
16	C₂²⋊#10623494	—	2	8	no	—
17	C₄×#10623494	—	2	4	no	—
18	C₄×#10623494	—	2	4	no	—
19	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₈₉	—	2	4	no	—
22	C₂×M₁₆×C₈₉	—	2	8	no	—
23	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	2	yes	—
27	C₂×C₄²×C₈₉	D₄	1	4	yes	—
28	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
29	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	#97×C₈₉	—	2	4	no	—
31	C₄×C₈×C₈₉	—	2	4	no	—
32	C₂×#10623494	—	2	8	no	—
33	C₂×#10623494	—	2	16	no	—
34	C₂⁴×C₈₉	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	—	2	4	no	—
36	C₂²×C₄×C₈₉	—	2	8	no	—
37	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
39	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
45	M₁₆×C₈₉	—	2	16	no	—
46	C₂×C₈×C₈₉	—	2	4	no	—
47	C₂×C₈×C₈₉	—	2	4	no	—
48	C₂×C₈×C₈₉	M₁₆	1	8	yes	—
49	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
50	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
52	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
53	C₄²×C₈₉	C₂×D₄	1	2	yes	—
54	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₂³×C₈₉	—	2	2	no	—
57	C₂³×C₈₉	—	2	4	no	—
58	C₂³×C₈₉	#95	1	2	yes	—
59	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	4	no	—
62	C₂×C₄×C₈₉	—	2	4	no	—
63	C₂×C₄×C₈₉	—	2	8	no	—
64	C₂×C₄×C₈₉	—	2	4	no	—
65	C₂×C₄×C₈₉	—	2	8	no	—
66	C₂×C₄×C₈₉	#95	1	2	yes	—
67	C₂×C₄×C₈₉	#95	1	4	yes	—
68	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
73	C₂×C₄×C₈₉	#131	1	8	yes	—
74	C₈×C₈₉	—	2	8	no	—
75	#10623494	—	4	8	no	—
76	C₂²×C₈₉	—	2	2	no	—
77	C₂²×C₈₉	—	2	4	no	—
78	C₂²×C₈₉	—	4	4	no	—
79	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
80	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
81	C₂²×C₈₉	#405	1	4	yes	—
82	C₄×C₈₉	—	2	2	no	—
83	C₄×C₈₉	—	2	4	no	—
84	C₄×C₈₉	#407	1	2	yes	—
85	C₄×C₈₉	#408	1	4	yes	—
86	C₄×C₈₉	#423	1	2	yes	—
87	C₄×C₈₉	#424	1	4	yes	—
88	#7850	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	—	4	2	no	—
90	C₂×C₈₉	#1453	1	4	yes	—
91	C₂×C₈₉	#1728	1	1	yes	characteristic
92	C₂×C₈₉	#1729	1	2	yes	—
93	C₂×#431	—	89	2	no	—
94	C₂×C₄×M₁₆	D₈₉	1	1	yes	characteristic
95	#1273	—	89	4	no	—
96	C₈₉	#7850	1	1	yes	characteristic, complemented, Sylow
97	C₂²×M₁₆	Dic₈₉	1	2	yes	—
98	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
99	#431	—	178	4	no	—
100	#431	—	178	4	no	—
101	C₂×C₄⋊C₈	—	89	4	no	—
102	C₂×C₂²⋊C₈	—	89	4	no	—
103	C₄×M₁₆	—	2	4	no	—
104	C₂×#97	Dic₈₉	1	2	yes	—
105	C₂×C₄×C₈	—	89	4	no	—
106	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
107	C₂³×C₄	C₄×D₈₉	1	2	yes	—
108	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
109	C₂×M₁₆	—	2	4	no	—
110	C₂×M₁₆	—	2	8	no	—
111	C₂²×C₈	—	89	8	no	—
112	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
113	C₂×C₄²	Dic₁₇₈	1	2	yes	—
114	C₂×C₄²	C₄×D₈₉	1	2	yes	—
115	C₂×C₄²	D₃₅₆	1	2	yes	—
116	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₈₉	1	4	yes	—
118	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
119	C₄⋊C₈	—	178	8	no	—
120	C₂²⋊C₈	—	178	8	no	—
121	#97	—	2	4	no	—
122	C₄×C₈	—	2	4	no	—
123	C₄×C₈	—	178	4	no	—
124	C₄×C₈	—	178	4	no	—
125	C₂⁴	C₄×Dic₈₉	1	1	yes	characteristic
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	2	8	no	—
128	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
129	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
130	C₂²×C₄	#13083551	1	4	yes	—
131	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
132	C₂²×C₄	#13083553	1	4	yes	—
133	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
135	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
136	M₁₆	—	2	16	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	178	16	no	—
140	C₂×C₈	—	178	8	no	—
141	C₂×C₈	#13083558	1	8	yes	—
142	C₄²	#13083551	1	4	yes	—
143	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
144	C₄²	#13083553	1	4	yes	—
145	C₄²	C₄⋊Dic₈₉	1	2	yes	—
146	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
147	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
148	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
149	C₂³	—	2	2	no	—
150	C₂³	—	2	4	no	—
151	C₂³	#15667919	1	2	yes	—
152	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	#15667919	1	2	yes	—
160	C₂×C₄	#15667919	1	4	yes	—
161	C₂×C₄	C₂×#13083551	1	2	yes	—
162	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
163	C₂×C₄	C₂×#13083553	1	2	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
165	C₂×C₄	C₂×#13083558	1	4	yes	—
166	C₂×C₄	#15668020	1	8	yes	—
167	C₈	—	2	8	no	—
168	C₈	—	356	8	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	4	no	—
171	C₂²	—	4	4	no	—
172	C₂²	C₂×#15667919	1	1	yes	characteristic
173	C₂²	C₄×#13083558	1	2	yes	—
174	C₂²	#19151556	1	4	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	#19151562	1	2	yes	—
178	C₄	#19151566	1	4	yes	—
179	C₄	#19151622	1	2	yes	—
180	C₄	#19151626	1	4	yes	—
181	C₂	—	4	2	no	—
182	C₂	#26018549	1	4	yes	—
183	C₂	#26019946	1	1	yes	characteristic
184	C₂	#26019950	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₈₉
commutator, socle	C₂³×C₈₉	C₂×C₄²
Fitting	C₂×C₄×M₁₆×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₈₉
hypocenter	C₈₉	#7850
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #26018079	C₂	semidirect
96. C₈₉	#7850	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151651	2
#26018079	2
C₂×C₄×M₁₆×C₈₉	2
#7850	89