Subgroups — #36443813

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 202 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	C₂²×C₄×C₈×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	C₂³×C₈×C₈₉	C₄	1	2	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₉	—	2	4	no	—
12	C₂×C₄×C₈×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	C₂²×C₈×C₈₉	—	2	4	no	—
22	C₂²×C₈×C₈₉	—	2	4	no	—
23	C₂²×C₈×C₈₉	C₈	1	8	yes	complemented
24	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	D₄	1	2	yes	—
28	C₂×C₄²×C₈₉	D₄	1	4	yes	—
29	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₈₉	—	2	4	no	—
32	C₄×C₈×C₈₉	—	2	4	no	—
33	C₂×#10623494	—	2	8	no	—
34	C₂×#10623494	—	2	16	no	—
35	C₂⁴×C₈₉	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	—	2	8	no	—
37	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
38	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
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	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
45	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₈₉	—	2	16	no	—
48	C₂×C₈×C₈₉	—	2	4	no	—
49	C₂×C₈×C₈₉	—	2	4	no	—
50	C₂×C₈×C₈₉	C₂×C₈	1	8	yes	—
51	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
52	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
54	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₄²×C₈₉	C₂×D₄	1	2	yes	—
56	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
58	C₂³×C₈₉	—	2	4	no	—
59	C₂³×C₈₉	#95	1	2	yes	—
60	C₂³×C₈₉	C₄×C₈	1	2	yes	—
61	C₂³×C₈₉	#97	1	2	yes	—
62	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₈₉	—	2	4	no	—
64	C₂×C₄×C₈₉	—	2	8	no	—
65	C₂×C₄×C₈₉	—	2	8	no	—
66	C₂×C₄×C₈₉	#95	1	4	yes	—
67	C₂×C₄×C₈₉	#95	1	2	yes	—
68	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₈₉	#97	1	2	yes	—
70	C₂×C₄×C₈₉	C₂²⋊C₈	1	8	yes	—
71	C₂×C₄×C₈₉	C₄⋊C₈	1	8	yes	—
72	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₈₉	#131	1	8	yes	—
79	C₈×C₈₉	—	2	8	no	—
80	#10623494	—	4	8	no	—
81	C₂²×C₈₉	—	2	4	no	—
82	C₂²×C₈₉	—	2	4	no	—
83	C₂²×C₈₉	—	2	4	no	—
84	C₂²×C₈₉	#336	1	4	yes	—
85	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
86	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
87	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
88	C₂²×C₈₉	#405	1	4	yes	—
89	C₄×C₈₉	—	2	4	no	—
90	C₄×C₈₉	#336	1	4	yes	—
91	C₄×C₈₉	C₂×C₂²⋊C₈	1	2	yes	—
92	C₄×C₈₉	#408	1	4	yes	—
93	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
94	C₄×C₈₉	#424	1	4	yes	—
95	#7849	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	—	2	4	no	—
97	C₂×C₈₉	#1450	1	2	yes	—
98	C₂×C₈₉	#1452	1	2	yes	—
99	C₂×C₈₉	C₂×#336	1	1	yes	characteristic
100	C₂×C₈₉	#1729	1	2	yes	—
101	C₂×#431	—	89	2	no	—
102	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
103	#1273	—	89	4	no	—
104	C₈₉	#7849	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₈	Dic₈₉	1	2	yes	—
106	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
107	#431	—	178	4	no	—
108	#431	—	178	4	no	—
109	C₂×C₄⋊C₈	—	89	4	no	—
110	C₂×C₂²⋊C₈	—	89	4	no	—
111	C₂×C₄×C₈	—	2	4	no	—
112	C₂×C₄×C₈	—	89	4	no	—
113	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
115	C₂³×C₄	C₄×D₈₉	1	2	yes	—
116	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	89	8	no	—
120	C₂²×C₈	#10623494	1	8	yes	complemented
121	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
122	C₂×C₄²	Dic₁₇₈	1	2	yes	—
123	C₂×C₄²	C₄×D₈₉	1	2	yes	—
124	C₂×C₄²	D₃₅₆	1	2	yes	—
125	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₈₉	1	4	yes	—
127	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
128	C₄⋊C₈	—	178	8	no	—
129	C₂²⋊C₈	—	178	8	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	—	178	4	no	—
133	C₄×C₈	—	178	4	no	—
134	C₂⁴	C₄×Dic₈₉	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
137	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
138	C₂²×C₄	#13083551	1	4	yes	—
139	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
140	C₂²×C₄	#13083553	1	4	yes	—
141	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
142	C₂²×C₄	C₂×#10623494	1	4	yes	—
143	C₂²×C₄	#13083558	1	4	yes	—
144	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	2	16	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	178	16	no	—
150	C₂×C₈	—	178	8	no	—
151	C₂×C₈	C₂×#10623494	1	8	yes	—
152	C₄²	#13083551	1	4	yes	—
153	C₄²	C₂²⋊Dic₈₉	1	2	yes	—
154	C₄²	#13083553	1	4	yes	—
155	C₄²	C₄⋊Dic₈₉	1	2	yes	—
156	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
157	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
158	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
159	C₂³	—	2	4	no	—
160	C₂³	#15667919	1	2	yes	—
161	C₂³	C₄×#10623494	1	2	yes	—
162	C₂³	#15667923	1	2	yes	—
163	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	#15667919	1	4	yes	—
168	C₂×C₄	#15667919	1	2	yes	—
169	C₂×C₄	C₄×#10623494	1	2	yes	—
170	C₂×C₄	#15667923	1	2	yes	—
171	C₂×C₄	C₂²⋊#10623494	1	8	yes	—
172	C₂×C₄	C₄⋊#10623494	1	8	yes	—
173	C₂×C₄	C₂×#13083551	1	2	yes	—
174	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
175	C₂×C₄	C₂×#13083553	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
177	C₂×C₄	C₂²×#10623494	1	2	yes	—
178	C₂×C₄	C₂×#13083558	1	2	yes	—
179	C₂×C₄	#15668018	1	8	yes	—
180	C₈	—	2	8	no	—
181	C₈	—	356	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	2	4	no	—
185	C₂²	#19151353	1	4	yes	—
186	C₂²	C₂×#15667919	1	1	yes	characteristic
187	C₂²	C₂×C₄×#10623494	1	1	yes	characteristic
188	C₂²	C₂×#15667923	1	1	yes	characteristic
189	C₂²	#19151553	1	4	yes	—
190	C₄	—	2	4	no	—
191	C₄	#19151353	1	4	yes	—
192	C₄	C₂×C₂²⋊#10623494	1	2	yes	—
193	C₄	#19151564	1	4	yes	—
194	C₄	C₂×C₄⋊#10623494	1	2	yes	—
195	C₄	#19151624	1	4	yes	—
196	C₂	—	2	4	no	—
197	C₂	#26018535	1	2	yes	—
198	C₂	#26018544	1	2	yes	—
199	C₂	C₂×#19151353	1	1	yes	characteristic
200	C₂	#26019948	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄²
prime radical	C₂⁴×C₈₉	C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	#7849
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #26018079	C₂	semidirect
23. C₂²×C₈×C₈₉	C₈	semidirect
104. C₈₉	#7849	semidirect
120. C₂²×C₈	#10623494	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
C₂²×C₄×C₈×C₈₉	2
#7849	89