Subgroups — #36469744

Order 22784.

382 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	C₂×#19151416	C₂	1	2	yes	maximal
2	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
3	#26019938	C₂	1	1	yes	maximal, characteristic
4	#26018329	C₂	1	2	yes	maximal
5	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	C₂²	1	2	yes	—
7	C₂×#15667923	C₄	1	1	yes	characteristic
8	C₂×#15667922	C₄	1	1	yes	characteristic
9	C₂×#15667922	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
12	#19151444	C₄	1	1	yes	characteristic
13	#19151416	C₄	1	8	yes	complemented
14	C₄×#13083549	—	2	2	no	—
15	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#346×C₈₉	—	2	2	no	—
18	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
19	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×#13083549	—	2	2	no	—
23	C₂×#13083549	—	2	2	no	—
24	C₂×#13083549	C₂×C₄	1	4	yes	—
25	#15667923	—	2	2	no	—
26	#15667922	—	2	2	no	—
27	#15667922	C₈	1	4	yes	—
28	C₈×Dic₈₉	—	2	2	no	—
29	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
30	#15667919	C₈	1	4	yes	—
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₈₉	—	2	2	no	—
35	C₂×C₁₆×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₈₉	D₄	1	2	yes	—
38	C₄×C₈×C₈₉	Q₈	1	2	yes	—
39	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
40	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
41	C₂×#10623494	—	2	1	no	—
42	C₂×#10623494	—	2	1	no	—
43	C₂×#10623494	—	2	1	no	—
44	C₂×#10623494	—	2	1	no	—
45	C₂×#10623494	C₄²	1	2	yes	—
46	C₂×#10623494	C₂×C₈	1	2	yes	—
47	C₄×Dic₈₉	—	2	1	no	—
48	C₄×Dic₈₉	—	2	1	no	—
49	C₄×Dic₈₉	C₄²	1	2	yes	—
50	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
51	#13083549	—	2	4	no	—
52	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
57	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₈₉	#41	1	2	yes	—
61	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
62	C₄²×C₈₉	M₁₆	1	2	yes	—
63	C₁₆×C₈₉	—	2	4	no	—
64	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	—	2	4	no	—
66	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
67	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
73	C₂×Dic₈₉	—	2	4	no	—
74	C₂×Dic₈₉	—	2	2	no	—
75	C₂×Dic₈₉	—	2	4	no	—
76	C₂×Dic₈₉	C₄×C₈	1	4	yes	—
77	C₈×C₈₉	C₄⋊C₈	1	4	yes	—
78	C₈×C₈₉	C₄×D₄	1	2	yes	—
79	C₈×C₈₉	C₄×Q₈	1	2	yes	—
80	#10623494	—	2	2	no	—
81	#10623494	—	2	2	no	—
82	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₈₉	#431	1	1	yes	characteristic
85	C₂²×C₈₉	#504	1	4	yes	—
86	C₄×C₈₉	—	2	4	no	—
87	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
89	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₄×C₈₉	#431	1	1	yes	characteristic
91	Dic₈₉	—	2	4	no	—
92	Dic₈₉	—	4	2	no	—
93	#12654	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	C₄×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₈₉	#2108	1	2	yes	—
96	C₂×C₈₉	#2151	1	2	yes	—
97	C₂×C₈₉	#2164	1	2	yes	—
98	C₂×C₄⋊C₁₆	—	89	2	no	—
99	C₂×#346	D₈₉	1	1	yes	characteristic
100	C₂×C₄×C₁₆	—	89	1	no	—
101	#1726	—	89	1	no	—
102	#1374	—	89	2	no	—
103	C₈₉	#12654	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₁₆	—	89	2	no	—
105	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
106	C₂×#97	—	89	1	no	—
107	C₂×#97	—	89	1	no	—
108	C₂×#97	—	89	1	no	—
109	C₂×C₄×C₈	—	89	1	no	—
110	C₂×C₄×C₈	—	89	1	no	—
111	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
112	#376	—	89	1	no	—
113	C₄⋊C₁₆	—	89	8	no	—
114	#346	—	2	2	no	—
115	C₄×C₁₆	—	178	2	no	—
116	C₂²×C₈	—	89	1	no	—
117	C₂²×C₈	—	89	1	no	—
118	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
120	C₂×C₄²	—	89	1	no	—
121	C₂×C₄²	—	89	1	no	—
122	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	89	4	no	—
126	C₂×C₁₆	—	178	2	no	—
127	C₂×C₁₆	—	178	2	no	—
128	C₂×C₁₆	C₄×D₈₉	1	4	yes	—
129	#97	—	89	4	no	—
130	#97	—	178	2	no	—
131	#97	—	178	2	no	—
132	C₄×C₈	—	89	4	no	—
133	C₄×C₈	—	178	2	no	—
134	C₄×C₈	Dic₁₇₈	1	2	yes	—
135	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
136	#95	—	89	4	no	—
137	C₂²×C₄	—	89	2	no	—
138	C₂²×C₄	—	89	2	no	—
139	C₂²×C₄	C₈×D₈₉	1	2	yes	—
140	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
141	C₂×C₈	—	89	2	no	—
142	C₂×C₈	—	89	2	no	—
143	C₂×C₈	—	178	1	no	—
144	C₂×C₈	—	178	1	no	—
145	C₂×C₈	—	178	1	no	—
146	C₂×C₈	—	178	1	no	—
147	C₂×C₈	#13083553	1	2	yes	—
148	C₂×C₈	C₈×D₈₉	1	2	yes	—
149	C₂×C₈	#13083557	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
151	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
153	C₂×C₈	#13083574	1	2	yes	—
154	C₄²	—	89	2	no	—
155	C₄²	—	89	2	no	—
156	C₄²	—	178	1	no	—
157	C₄²	—	178	1	no	—
158	C₄²	#13083553	1	2	yes	—
159	C₄²	#13083557	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₁₆	—	178	4	no	—
162	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	89	4	no	—
165	C₂×C₄	—	178	2	no	—
166	C₂×C₄	—	178	4	no	—
167	C₂×C₄	—	178	4	no	—
168	C₂×C₄	#15667941	1	4	yes	—
169	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
170	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
171	C₂×C₄	#15667969	1	1	yes	characteristic
172	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
173	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
174	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
175	C₈	—	178	2	no	—
176	C₈	—	178	2	no	—
177	C₈	#15667941	1	4	yes	—
178	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
179	C₈	C₄×Dic₁₇₈	1	2	yes	—
180	C₂²	C₄×C₈×D₈₉	1	1	yes	characteristic
181	C₂²	C₂×#15667941	1	1	yes	characteristic
182	C₂²	#19151649	1	1	yes	characteristic
183	C₂²	#19152016	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	356	2	no	—
186	C₄	—	178	4	no	—
187	C₄	C₄×#13083553	1	1	yes	characteristic
188	C₄	C₄×#13083557	1	1	yes	characteristic
189	C₄	C₂×#15667941	1	1	yes	characteristic
190	C₄	#19151649	1	1	yes	characteristic
191	C₂	C₄×#15667941	1	1	yes	characteristic
192	C₂	#26021850	1	2	yes	—
193	C₂	#26022012	1	2	yes	—
194	C₂	#26022060	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄×C₈
Fitting	C₂×#346×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#12654
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. #19151416	C₄	semidirect
103. C₈₉	#12654	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151416	2
C₂×C₄×#13083549	2
#26019938	2
#26018329	2
C₂×#346×C₈₉	2
#12654	89