Subgroups — #36445586

Order 22784.

414 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019000	C₂	1	2	yes	maximal
2	#26018535	C₂	1	2	yes	maximal
3	#26018085	C₂	1	2	yes	maximal
4	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	4	yes	—
6	#19151351	C₄	1	4	yes	complemented
7	#19151349	C₄	1	4	yes	complemented
8	#19151319	—	2	2	no	—
9	C₈×#10623494	—	2	2	no	—
10	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₄³×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊#10623494	C₂×C₄	1	8	yes	—
16	C₂²⋊#10623494	—	2	4	no	—
17	C₄×#10623494	—	2	4	no	—
18	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₈₉	—	2	2	no	—
22	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₈₉	—	2	4	no	—
25	C₄×C₈×C₈₉	C₈	1	8	yes	complemented
26	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	Q₈	1	2	yes	—
31	C₂×#10623494	—	2	8	no	—
32	C₂²×C₄×C₈₉	—	2	2	no	—
33	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	#41	1	2	yes	—
35	C₂×C₈×C₈₉	—	2	2	no	—
36	C₂×C₈×C₈₉	C₄²	1	2	yes	—
37	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
44	C₄²×C₈₉	—	2	8	no	—
45	C₄²×C₈₉	—	2	2	no	—
46	C₄²×C₈₉	C₂×C₈	1	4	yes	—
47	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
48	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₈₉	#117	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	8	no	—
52	C₂×C₄×C₈₉	—	2	4	no	—
53	C₂×C₄×C₈₉	—	2	4	no	—
54	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
58	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
60	C₈×C₈₉	—	2	2	no	—
61	C₈×C₈₉	C₄×C₈	1	2	yes	—
62	C₈×C₈₉	C₄×C₈	1	2	yes	—
63	C₈×C₈₉	C₄⋊C₈	1	4	yes	—
64	C₈×C₈₉	C₄⋊C₈	1	4	yes	—
65	#10623494	—	4	8	no	—
66	C₂²×C₈₉	—	2	2	no	—
67	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₈₉	#431	1	2	yes	—
69	C₄×C₈₉	—	2	8	no	—
70	C₄×C₈₉	—	2	2	no	—
71	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
72	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
73	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
74	C₄×C₈₉	#443	1	4	yes	—
75	#8266	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	—	2	2	no	—
77	C₂×C₈₉	C₄×C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₈₉	#1772	1	2	yes	—
79	C₄²×C₈	D₈₉	1	1	yes	characteristic
80	#1564	—	89	2	no	—
81	#1450	—	89	2	no	—
82	#1276	—	89	2	no	—
83	C₈₉	#8266	1	1	yes	characteristic, complemented, Sylow
84	#431	—	89	4	no	—
85	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
86	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
87	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
89	C₄³	C₂×D₈₉	1	1	yes	characteristic
90	#335	—	89	4	no	—
91	#334	—	89	4	no	—
92	#322	—	178	2	no	—
93	C₈²	—	178	2	no	—
94	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
95	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
96	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
97	C₂×C₄²	—	2	2	no	—
98	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
99	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
100	C₄⋊C₈	—	89	8	no	—
101	C₂²⋊C₈	—	178	4	no	—
102	C₄×C₈	—	2	4	no	—
103	C₄×C₈	—	178	4	no	—
104	C₄×C₈	#10623494	1	8	yes	complemented
105	C₄×C₈	Dic₁₇₈	1	2	yes	—
106	C₄×C₈	C₄×D₈₉	1	2	yes	—
107	C₄×C₈	C₄×D₈₉	1	2	yes	—
108	C₄×C₈	D₃₅₆	1	1	yes	characteristic
109	C₄×C₈	D₃₅₆	1	1	yes	characteristic
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
112	C₂²×C₄	#13083574	1	2	yes	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	178	8	no	—
115	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
116	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
117	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
118	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
119	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
120	C₂×C₈	C₂×#10623494	1	4	yes	—
121	C₂×C₈	C₂×#10623494	1	4	yes	—
122	C₂×C₈	#13083558	1	4	yes	—
123	C₄²	—	2	8	no	—
124	C₄²	—	2	2	no	—
125	C₄²	C₂×#10623494	1	4	yes	—
126	C₄²	C₂×C₄×D₈₉	1	2	yes	—
127	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
128	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
129	C₂³	#15667971	1	1	yes	characteristic
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
135	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
136	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
137	C₂×C₄	C₂²×#10623494	1	2	yes	—
138	C₂×C₄	C₂×#13083558	1	2	yes	—
139	C₈	—	2	2	no	—
140	C₈	—	356	8	no	—
141	C₈	C₄×#10623494	1	2	yes	—
142	C₈	C₄×#10623494	1	2	yes	—
143	C₈	C₄⋊#10623494	1	4	yes	—
144	C₈	C₄⋊#10623494	1	4	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
147	C₂²	#19151651	1	2	yes	—
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	C₂×C₄×#10623494	1	1	yes	characteristic
151	C₄	C₄×#13083558	1	1	yes	characteristic
152	C₄	C₂×C₄⋊#10623494	1	2	yes	—
153	C₄	#19151716	1	4	yes	—
154	C₂	—	2	2	no	—
155	C₂	C₄×C₄⋊#10623494	1	1	yes	characteristic
156	C₂	#26020131	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₃₅₆
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₄⋊C₄
prime radical	C₂³×C₈₉	#117
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#8266
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #19151351	C₄	semidirect
7. #19151349	C₄	semidirect
25. C₄×C₈×C₈₉	C₈	semidirect
83. C₈₉	#8266	semidirect
104. C₄×C₈	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26019000	2
#26018535	2
#26018085	2
C₄²×C₈×C₈₉	2
#8266	89