Subgroups — #36445643

Order 22784.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151350	C₂	1	2	yes	maximal
4	#26018086	C₂	1	2	yes	maximal
5	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	—	2	1	no	—
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151353	—	2	1	no	—
11	#19151353	C₄	1	4	yes	—
12	#19151350	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×#97×C₈₉	—	2	1	no	—
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	#336×C₈₉	—	2	1	no	—
17	C₂²×#10623494	—	2	2	no	—
18	C₂²×#10623494	C₂×C₄	1	2	yes	—
19	C₂²×#10623494	C₂×C₄	1	2	yes	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	C₂×C₄	1	4	yes	—
22	C₄×#10623494	—	2	2	no	—
23	C₄×#10623494	—	4	2	no	—
24	C₄×#10623494	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₈₉	—	2	2	no	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₈₉	—	2	2	no	—
32	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
33	#97×C₈₉	—	4	2	no	—
34	C₂×#10623494	—	2	4	no	—
35	C₂×#10623494	—	2	2	no	—
36	C₂×#10623494	—	2	4	no	—
37	C₂×#10623494	—	2	4	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₂×#10623494	C₄²	1	4	yes	—
41	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₈₉	—	2	4	no	—
47	C₂×C₈×C₈₉	—	4	2	no	—
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₄²×C₈₉	—	2	4	no	—
50	C₄²×C₈₉	—	2	2	no	—
51	C₄²×C₈₉	—	4	2	no	—
52	C₄²×C₈₉	D₈	1	2	yes	—
53	C₄²×C₈₉	Q₁₆	1	2	yes	—
54	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
55	C₄²×C₈₉	#41	1	2	yes	—
56	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	#107	1	4	yes	—
64	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₈₉	#131	1	4	yes	—
69	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#135	1	2	yes	—
72	C₈×C₈₉	—	4	4	no	—
73	#10623494	—	2	4	no	—
74	#10623494	—	4	2	no	—
75	#10623494	—	4	4	no	—
76	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₈₉	#424	1	2	yes	—
78	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
79	C₂²×C₈₉	#427	1	1	yes	characteristic
80	C₂²×C₈₉	#430	1	2	yes	—
81	C₄×C₈₉	—	2	4	no	—
82	C₄×C₈₉	—	2	2	no	—
83	C₄×C₈₉	—	4	2	no	—
84	C₄×C₈₉	#405	1	2	yes	—
85	C₄×C₈₉	#433	1	2	yes	—
86	C₄×C₈₉	C₄×D₈	1	2	yes	—
87	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
88	#8278	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	#1571	1	2	yes	—
90	C₂×C₈₉	#1572	1	2	yes	—
91	C₂×C₈₉	#1768	1	1	yes	characteristic
92	C₂×C₈₉	C₄×#107	1	1	yes	characteristic
93	C₂×C₈₉	#1778	1	1	yes	characteristic
94	#1767	D₈₉	1	1	yes	characteristic
95	C₄×C₄⋊C₈	—	89	1	no	—
96	#1744	—	89	1	no	—
97	C₂×#335	—	89	2	no	—
98	#1277	—	89	2	no	—
99	C₈₉	#8278	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄⋊C₈	—	89	2	no	—
101	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
102	C₂×#97	—	2	1	no	—
103	C₂×C₄×C₈	—	89	1	no	—
104	C₂×C₄×C₈	—	89	1	no	—
105	C₂×C₄×C₈	—	178	1	no	—
106	C₄³	C₂×D₈₉	1	1	yes	characteristic
107	#336	—	2	1	no	—
108	#336	—	89	4	no	—
109	#336	—	178	1	no	—
110	#335	—	89	8	no	—
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	89	2	no	—
113	C₂²×C₈	—	89	2	no	—
114	C₂²×C₈	—	178	2	no	—
115	C₂×C₄²	Dic₁₇₈	1	2	yes	—
116	C₂×C₄²	C₄×D₈₉	1	2	yes	—
117	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
118	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	89	4	no	—
122	C₄⋊C₈	—	178	2	no	—
123	C₄⋊C₈	C₄×D₈₉	1	4	yes	—
124	#97	—	4	2	no	—
125	C₄×C₈	—	89	4	no	—
126	C₄×C₈	—	356	2	no	—
127	C₄×C₈	—	178	2	no	—
128	C₂²×C₄	#13083553	1	2	yes	—
129	C₂²×C₄	#13083553	1	2	yes	—
130	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	89	4	no	—
137	C₂×C₈	—	356	2	no	—
138	C₂×C₈	—	356	2	no	—
139	C₂×C₈	—	178	2	no	—
140	C₂×C₈	—	178	4	no	—
141	C₂×C₈	—	178	4	no	—
142	C₂×C₈	—	178	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#13083559	1	2	yes	—
147	C₄²	#13083564	1	2	yes	—
148	C₄²	C₂×C₄×D₈₉	1	2	yes	—
149	C₄²	#13083574	1	2	yes	—
150	C₂³	C₂×#13083553	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#15667945	1	4	yes	—
158	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
159	C₂×C₄	#15667969	1	1	yes	characteristic
160	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
161	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
162	C₂×C₄	#15668020	1	4	yes	—
163	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
164	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
165	C₂×C₄	#15668032	1	2	yes	—
166	C₈	—	4	4	no	—
167	C₈	—	356	2	no	—
168	C₈	—	356	4	no	—
169	C₈	—	178	4	no	—
170	C₂²	C₄×#13083553	1	1	yes	characteristic
171	C₂²	#19151626	1	2	yes	—
172	C₂²	C₂×#15667945	1	1	yes	characteristic
173	C₂²	#19151634	1	1	yes	characteristic
174	C₂²	#19151646	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	#19151555	1	2	yes	—
179	C₄	#19151660	1	2	yes	—
180	C₄	C₄×#13083559	1	2	yes	—
181	C₄	C₄×#13083564	1	2	yes	—
182	C₂	#26019028	1	2	yes	—
183	C₂	#26019033	1	2	yes	—
184	C₂	#26020112	1	1	yes	characteristic
185	C₂	C₄×#15667945	1	1	yes	characteristic
186	C₂	#26020159	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1767	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1767	D₈₉
hypocenter	C₈₉	#8278
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
12. #19151350	C₄	semidirect
99. C₈₉	#8278	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10623494	2
#26019998	2
C₂×#19151350	2
#26018086	2
C₈₉×#1767	2
#8278	89