Subgroups — #36469822

Order 22784.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151428	C₂	1	1	yes	maximal, characteristic
2	#26022036	C₂	1	1	yes	maximal, characteristic
3	#26020131	C₂	1	2	yes	maximal
4	#26018348	C₂	1	2	yes	maximal
5	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667948	C₂²	1	2	yes	—
7	#19151636	C₂²	1	2	yes	—
8	#19151553	—	2	2	no	—
9	#19151428	C₄	1	4	yes	complemented
10	#19151426	C₄	1	2	yes	complemented
11	#19151424	C₄	1	2	yes	complemented
12	#19151358	—	2	2	no	—
13	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#346×C₈₉	C₄	1	4	yes	complemented
17	C₂×#13083558	—	2	2	no	—
18	#15667971	—	2	2	no	—
19	#15667948	—	2	2	no	—
20	#15667948	C₂×C₄	1	4	yes	—
21	#15667946	C₂×C₄	1	4	yes	—
22	#15667944	—	2	2	no	—
23	#15667922	—	4	2	no	—
24	C₈×Dic₈₉	—	4	2	no	—
25	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₉	—	2	1	no	—
29	C₂×C₁₆×C₈₉	—	2	1	no	—
30	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₈₉	Q₈	1	2	yes	—
37	#13083558	—	2	4	no	—
38	#13083558	—	4	2	no	—
39	C₂×#10623494	—	4	2	no	—
40	C₄⋊Dic₈₉	—	2	4	no	—
41	C₂²⋊Dic₈₉	—	4	2	no	—
42	C₄×Dic₈₉	—	4	2	no	—
43	C₂²×C₄×C₈₉	—	2	1	no	—
44	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₈₉	D₈	1	2	yes	—
47	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
48	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
54	C₄²×C₈₉	—	2	1	no	—
55	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
56	C₁₆×C₈₉	—	2	2	no	—
57	C₁₆×C₈₉	C₄²	1	2	yes	—
58	C₁₆×C₈₉	C₄²	1	2	yes	—
59	C₂³×C₈₉	#135	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	4	2	no	—
62	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
68	C₂×Dic₈₉	—	4	4	no	—
69	C₈×C₈₉	#107	1	4	yes	—
70	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₈₉	#117	1	1	yes	characteristic
72	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
73	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
74	#10623494	—	4	4	no	—
75	C₂²×C₈₉	—	2	2	no	—
76	C₂²×C₈₉	#427	1	1	yes	characteristic
77	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
78	C₂²×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₈₉	—	4	2	no	—
80	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
82	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
83	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
84	Dic₈₉	—	8	4	no	—
85	#12671	—	89	1	no	maximal, Sylow
86	C₂×C₈₉	—	2	1	no	—
87	C₂×C₈₉	—	2	1	no	—
88	C₂×C₈₉	C₄×#107	1	1	yes	characteristic
89	C₂×C₈₉	#2179	1	2	yes	—
90	C₂×#368	—	89	1	no	—
91	#2158	—	89	1	no	—
92	C₂×#346	D₈₉	1	1	yes	characteristic
93	#1772	—	89	2	no	—
94	#1382	—	89	2	no	—
95	C₈₉	#12671	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
97	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
98	C₂×#108	—	89	2	no	—
99	#427	—	89	2	no	—
100	#405	—	178	2	no	—
101	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
102	#368	—	89	4	no	—
103	#367	—	89	2	no	—
104	#366	—	89	2	no	—
105	#346	Dic₈₉	1	4	yes	complemented
106	#339	—	178	2	no	—
107	C₂×M₁₆	—	178	2	no	—
108	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
109	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
110	#117	—	178	2	no	—
111	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
115	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
116	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
117	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
118	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
119	#108	—	89	4	no	—
120	#108	—	178	2	no	—
121	#107	—	89	4	no	—
122	#106	—	178	2	no	—
123	#97	—	356	2	no	—
124	C₄×C₈	—	356	2	no	—
125	C₄×C₈	Dic₁₇₈	1	2	yes	—
126	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
129	M₁₆	—	356	2	no	—
130	M₁₆	—	178	4	no	—
131	C₂×C₈	—	356	2	no	—
132	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
133	C₂×C₈	D₇₁₂	1	2	yes	—
134	C₂×C₈	Dic₃₅₆	1	2	yes	—
135	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
138	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
139	C₂×C₈	#13083574	1	1	yes	characteristic
140	C₂×C₈	#13083574	1	1	yes	characteristic
141	C₄⋊C₄	—	178	4	no	—
142	C₂²⋊C₄	—	356	2	no	—
143	C₄²	—	2	1	no	—
144	C₄²	—	356	2	no	—
145	C₄²	C₄⋊Dic₈₉	1	2	yes	—
146	C₁₆	—	2	2	no	—
147	C₁₆	C₄×Dic₈₉	1	2	yes	—
148	C₁₆	C₄×Dic₈₉	1	2	yes	—
149	C₂³	#15668034	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	356	4	no	—
153	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
154	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
155	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
156	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
158	C₂×C₄	#15668034	1	1	yes	characteristic
159	C₈	—	356	4	no	—
160	C₈	#15667946	1	4	yes	—
161	C₈	C₂×C₄×Dic₈₉	1	1	yes	characteristic
162	C₈	#15667971	1	1	yes	characteristic
163	C₈	C₄×Dic₁₇₈	1	1	yes	characteristic
164	C₈	C₄×Dic₁₇₈	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	#19151636	1	1	yes	characteristic
167	C₂²	C₄×D₇₁₂	1	1	yes	characteristic
168	C₂²	C₄×Dic₃₅₆	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	712	4	no	—
171	C₄	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
172	C₄	C₂×#15667946	1	1	yes	characteristic
173	C₄	C₄×D₇₁₂	1	1	yes	characteristic
174	C₄	C₄×Dic₃₅₆	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	C₄×#15667946	1	1	yes	characteristic
178	C₂	#26022139	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₄×D₈
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#12671
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
9. #19151428	C₄	semidirect
10. #19151426	C₄	semidirect
11. #19151424	C₄	semidirect
16. #346×C₈₉	C₄	semidirect
95. C₈₉	#12671	semidirect
105. #346	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151428	2
#26022036	2
#26020131	2
#26018348	2
C₂×#346×C₈₉	2
#12671	89