Subgroups — #36445682

Order 22784.

398 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	C₄×#15667944	C₂	1	1	yes	maximal, characteristic
2	#26020119	C₂	1	2	yes	maximal
3	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
4	#26018092	C₂	1	2	yes	maximal
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667944	C₄	1	2	yes	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×#15667922	—	2	2	no	—
9	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	#19151361	C₄	1	2	yes	—
11	#19151352	—	2	2	no	—
12	#19151351	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#322×C₈₉	C₄	1	4	yes	complemented
17	C₂²×#10623494	—	2	2	no	—
18	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₈₉	—	2	2	no	—
20	#15667944	—	2	4	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄⋊#10623494	C₂×C₄	1	4	yes	—
23	#15667922	—	4	4	no	—
24	#15667919	—	2	2	no	—
25	C₂²×C₈×C₈₉	—	2	1	no	—
26	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	D₄	1	2	yes	—
29	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₈₉	—	2	1	no	—
31	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₉	Q₈	1	2	yes	—
36	C₂²×Dic₈₉	—	2	4	no	—
37	C₂×#10623494	—	2	4	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₄⋊Dic₈₉	—	2	4	no	—
42	C₄×Dic₈₉	—	4	2	no	—
43	C₄×Dic₈₉	—	4	2	no	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
47	C₂×C₈×C₈₉	—	2	2	no	—
48	C₂×C₈×C₈₉	—	2	2	no	—
49	C₂×C₈×C₈₉	—	2	2	no	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	—	4	2	no	—
52	C₂×C₈×C₈₉	C₄²	1	2	yes	—
53	C₂×C₈×C₈₉	C₄²	1	2	yes	—
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×Q₈	1	1	yes	characteristic
59	C₄²×C₈₉	#41	1	1	yes	characteristic
60	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
69	C₂×C₄×C₈₉	#131	1	2	yes	—
70	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄×C₈₉	#137	1	2	yes	—
72	C₂×Dic₈₉	—	4	4	no	—
73	C₂×Dic₈₉	—	4	4	no	—
74	C₂×Dic₈₉	—	4	4	no	—
75	C₈×C₈₉	—	2	2	no	—
76	C₈×C₈₉	—	2	4	no	—
77	C₈×C₈₉	—	4	2	no	—
78	C₈×C₈₉	#97	1	4	yes	complemented
79	#10623494	—	4	4	no	—
80	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈₉	#436	1	2	yes	—
82	C₂²×C₈₉	#440	1	2	yes	—
83	C₂²×C₈₉	#444	1	2	yes	—
84	C₄×C₈₉	—	2	1	no	—
85	C₄×C₈₉	—	2	1	no	—
86	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
87	C₄×C₈₉	#405	1	1	yes	characteristic
88	C₄×C₈₉	#428	1	2	yes	—
89	C₄×C₈₉	#441	1	1	yes	characteristic
90	C₄×C₈₉	#442	1	1	yes	characteristic
91	C₄×C₈₉	#446	1	2	yes	—
92	Dic₈₉	—	8	4	no	—
93	#8286	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	#1592	1	4	yes	—
95	C₂×C₈₉	#1770	1	1	yes	characteristic
96	C₂×C₈₉	#1773	1	1	yes	characteristic
97	C₂×C₈₉	#1776	1	1	yes	characteristic
98	C₄×#106	—	89	1	no	—
99	#1770	—	89	2	no	—
100	C₂×#335	—	89	1	no	—
101	C₂×#322	D₈₉	1	1	yes	characteristic
102	#1279	—	89	2	no	—
103	C₈₉	#8286	1	1	yes	characteristic, complemented, Sylow
104	C₂×#106	—	89	2	no	—
105	C₂×C₄⋊C₈	—	89	2	no	—
106	C₂×#97	—	178	2	no	—
107	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	89	2	no	—
111	#340	—	89	2	no	—
112	#336	—	178	2	no	—
113	#335	—	89	4	no	—
114	#322	Dic₈₉	1	4	yes	complemented
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	178	2	no	—
117	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
119	C₂²×C₈	D₃₅₆	1	2	yes	—
120	C₂×C₄⋊C₄	—	89	4	no	—
121	C₂×C₄²	—	178	2	no	—
122	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
123	#106	—	178	4	no	—
124	C₄⋊C₈	—	89	4	no	—
125	C₄⋊C₈	—	178	2	no	—
126	#97	—	356	4	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	Dic₁₇₈	1	2	yes	—
129	C₄×C₈	C₄×D₈₉	1	2	yes	—
130	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
131	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
132	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
133	#95	—	178	2	no	—
134	C₂²×C₄	—	178	4	no	—
135	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
137	C₂²×C₄	#13083574	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	356	2	no	—
144	C₂×C₈	—	356	2	no	—
145	C₂×C₈	—	178	4	no	—
146	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
147	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
148	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
149	C₂×C₈	#13083557	1	4	yes	—
150	C₄⋊C₄	—	178	4	no	—
151	C₄⋊C₄	—	178	4	no	—
152	C₄²	—	356	2	no	—
153	C₄²	—	356	2	no	—
154	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
155	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
156	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
157	C₄²	#13083574	1	1	yes	characteristic
158	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	356	4	no	—
161	C₂×C₄	—	356	4	no	—
162	C₂×C₄	—	356	4	no	—
163	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
165	C₂×C₄	#15667971	1	1	yes	characteristic
166	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
167	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
169	C₂×C₄	C₂×#13083557	1	2	yes	—
170	C₂×C₄	#15668018	1	2	yes	—
171	C₂×C₄	#15668040	1	2	yes	—
172	C₂×C₄	#15668045	1	2	yes	—
173	C₈	—	2	4	no	—
174	C₈	—	2	2	no	—
175	C₈	—	4	2	no	—
176	C₈	—	356	4	no	—
177	C₈	#15667922	1	4	yes	complemented
178	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
179	C₂²	#19151677	1	2	yes	—
180	C₂²	#19151700	1	2	yes	—
181	C₂²	#19151721	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	712	4	no	—
185	C₄	C₂×#15667922	1	1	yes	characteristic
186	C₄	#19151553	1	1	yes	characteristic
187	C₄	#19151640	1	2	yes	—
188	C₄	#19151706	1	1	yes	characteristic
189	C₄	#19151711	1	1	yes	characteristic
190	C₄	#19151728	1	2	yes	—
191	C₂	#26019156	1	4	yes	—
192	C₂	#26020119	1	1	yes	characteristic
193	C₂	#26020136	1	1	yes	characteristic
194	C₂	#26020149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#8286
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
12. #19151351	C₄	semidirect
16. #322×C₈₉	C₄	semidirect
78. C₈×C₈₉	#97	semidirect
103. C₈₉	#8286	semidirect
114. #322	Dic₈₉	semidirect
177. C₈	#15667922	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667944	2
#26020119	2
C₂×#19151351	2
#26018092	2
C₂×#322×C₈₉	2
#8286	89