Subgroups — #36448228

Order 22784.

354 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020423	C₂	1	2	yes	maximal
2	#26019008	C₂	1	1	yes	maximal, characteristic
3	#26019007	C₂	1	2	yes	maximal
4	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667947	—	2	4	no	—
7	#19151622	C₂²	1	2	yes	—
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151349	—	2	2	no	—
11	#19151348	—	2	4	no	—
12	C₈×#10623494	C₄	1	1	yes	characteristic
13	C₈×#10623494	C₄	1	1	yes	characteristic
14	C₈×#10623494	C₄	1	1	yes	characteristic
15	C₈×#10623494	C₄	1	1	yes	characteristic
16	#423×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#334×C₈₉	—	2	2	no	—
19	C₂×#13083558	—	2	2	no	—
20	C₂²×#10623494	D₄	1	2	yes	—
21	C₂²×#10623494	Q₈	1	2	yes	—
22	#15667947	—	4	8	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
26	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
28	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
29	C₄×#10623494	D₄	1	2	yes	—
30	C₄×#10623494	Q₈	1	2	yes	—
31	C₂×M₁₆×C₈₉	—	2	2	no	—
32	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₈₉	—	2	2	no	—
35	C₄⋊C₈×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₈₉	D₄	1	2	yes	—
39	#13083558	—	4	4	no	—
40	C₂×#10623494	—	2	4	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₂×#10623494	C₄⋊C₄	1	2	yes	—
43	C₂×#10623494	C₄⋊C₄	1	2	yes	—
44	C₂×#10623494	C₄⋊C₄	1	2	yes	—
45	C₂×#10623494	C₄⋊C₄	1	2	yes	—
46	C₂×#10623494	SD₁₆	1	8	yes	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	M₁₆×C₈₉	—	4	4	no	—
51	C₂×C₈×C₈₉	—	2	4	no	—
52	C₂×C₈×C₈₉	—	4	2	no	—
53	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	2	no	—
61	C₂×C₄×C₈₉	—	2	2	no	—
62	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#127	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×SD₁₆	1	4	yes	—
69	C₈×C₈₉	—	2	2	no	—
70	C₈×C₈₉	—	8	2	no	—
71	C₈×C₈₉	#108	1	4	yes	complemented
72	#10623494	—	2	2	no	—
73	#10623494	—	8	2	no	—
74	#10623494	#106	1	4	yes	—
75	#10623494	#106	1	4	yes	—
76	#10623494	#108	1	4	yes	complemented
77	C₂²×C₈₉	—	2	1	no	—
78	C₂²×C₈₉	—	2	1	no	—
79	C₂²×C₈₉	#384	1	1	yes	characteristic
80	C₂²×C₈₉	#498	1	2	yes	—
81	C₄×C₈₉	—	2	1	no	—
82	C₄×C₈₉	—	2	1	no	—
83	C₄×C₈₉	C₂×#106	1	2	yes	—
84	C₄×C₈₉	C₂×#108	1	2	yes	—
85	C₄×C₈₉	C₂×#108	1	2	yes	—
86	C₄×C₈₉	#492	1	2	yes	—
87	#8822	—	89	1	no	maximal, Sylow
88	C₂×C₈₉	—	2	2	no	—
89	C₂×C₈₉	#1844	1	1	yes	characteristic
90	C₂×C₈₉	#1848	1	1	yes	characteristic
91	C₂×C₈₉	#1848	1	1	yes	characteristic
92	#1848	—	89	2	no	—
93	#1566	—	89	2	no	—
94	#1566	—	89	1	no	—
95	#1566	D₈₉	1	1	yes	characteristic
96	C₂×C₈²	—	89	1	no	—
97	C₈₉	#8822	1	1	yes	characteristic, complemented, Sylow
98	C₂×#108	—	178	4	no	—
99	#423	—	89	2	no	—
100	#423	C₂×D₈₉	1	2	yes	—
101	C₂×C₄×C₈	—	89	1	no	—
102	C₂×C₄×C₈	—	89	1	no	—
103	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
104	#334	—	2	2	no	—
105	#334	—	178	4	no	—
106	#334	—	178	2	no	—
107	C₈²	—	89	1	no	—
108	C₈²	—	89	1	no	—
109	C₈²	—	89	1	no	—
110	C₈²	—	89	1	no	—
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	178	2	no	—
113	C₂²×C₈	—	89	2	no	—
114	C₂²×C₈	—	89	2	no	—
115	C₂²×C₈	Dic₁₇₈	1	2	yes	—
116	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
117	#108	—	356	8	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	178	2	no	—
121	C₄⋊C₈	—	178	2	no	—
122	C₄×C₈	—	89	1	no	—
123	C₄×C₈	—	89	1	no	—
124	C₄×C₈	—	89	1	no	—
125	C₄×C₈	—	89	1	no	—
126	C₄×C₈	—	89	2	no	—
127	C₄×C₈	—	89	2	no	—
128	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
129	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
131	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
132	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
133	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
134	M₁₆	—	4	4	no	—
135	M₁₆	—	356	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	89	8	no	—
139	C₂×C₈	—	89	2	no	—
140	C₂×C₈	—	89	2	no	—
141	C₂×C₈	—	89	2	no	—
142	C₂×C₈	—	89	2	no	—
143	C₂×C₈	—	356	2	no	—
144	C₂×C₈	—	178	4	no	—
145	C₂×C₈	#13083553	1	2	yes	—
146	C₂×C₈	#13083553	1	2	yes	—
147	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
148	C₄²	D₄×D₈₉	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
150	C₄²	Q₈×D₈₉	1	1	yes	characteristic
151	C₂³	#15668011	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
155	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
156	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
157	C₂×C₄	#15668008	1	1	yes	characteristic
158	C₂×C₄	#15668010	1	1	yes	characteristic
159	C₂×C₄	#15668011	1	1	yes	characteristic
160	C₂×C₄	SD₁₆×D₈₉	1	4	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	8	2	no	—
163	C₈	—	89	4	no	—
164	C₈	—	89	4	no	—
165	C₈	—	89	4	no	—
166	C₈	—	178	2	no	—
167	C₈	—	712	2	no	—
168	C₈	#15667947	1	4	yes	complemented
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#19151467	1	1	yes	characteristic
172	C₂²	#19151995	1	2	yes	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#106×D₈₉	1	2	yes	—
176	C₄	#108×D₈₉	1	2	yes	—
177	C₄	C₂×#15667947	1	2	yes	—
178	C₄	#19151969	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#26020407	1	1	yes	characteristic
181	C₂	#26020422	1	1	yes	characteristic
182	C₂	#26020423	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	#384
prime radical	C₂³×C₈₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#8822
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
71. C₈×C₈₉	#108	semidirect
76. #10623494	#108	semidirect
97. C₈₉	#8822	semidirect
168. C₈	#15667947	semidirect

Maximal subgroups

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

subgroup	index
#26020423	2
#26019008	2
#26019007	2
C₂×C₈×#10623494	2
C₈₉×#1566	2
#8822	89