Subgroups — C₈×#15668019 #36520016

Order 22784.

922 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#10623494	C₂	1	1	yes	maximal, characteristic
2	C₄×#15668019	C₂	1	1	yes	maximal, characteristic
3	#26018536	C₂	1	3	yes	maximal, complemented
4	#26018533	C₂	1	3	yes	maximal, complemented
5	C₈×#13083558	C₂	1	3	yes	maximal, complemented
6	C₂×C₈×#10623494	C₂	1	3	yes	maximal, complemented
7	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668019	C₄	1	2	yes	—
9	Q₈×#10623494	C₄	1	2	yes	—
10	D₄×#10623494	C₄	1	6	yes	—
11	#19151651	C₂²	1	3	yes	—
12	#19151554	C₄	1	6	yes	—
13	C₄×#13083558	C₂²	1	3	yes	—
14	C₂×C₄×#10623494	C₂²	1	3	yes	—
15	C₂×C₄×#10623494	C₂²	1	3	yes	—
16	#19151320	C₂²	1	6	yes	—
17	C₈×#10623494	C₂²	1	1	yes	characteristic
18	C₈×#10623494	C₂²	1	3	yes	—
19	C₈×#10623494	C₂²	1	3	yes	—
20	C₈×#10623494	C₂²	1	3	yes	—
21	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
22	C₈×Q₈×C₈₉	C₄	1	2	yes	—
23	C₈×D₄×C₈₉	C₄	1	6	yes	—
24	#431×C₈₉	C₂²	1	3	yes	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
26	#15668019	C₈	1	16	yes	direct product factor
27	C₂×#13083558	C₂×C₄	1	6	yes	—
28	C₂²×#10623494	C₂×C₄	1	6	yes	—
29	C₂²×#10623494	C₂×C₄	1	6	yes	—
30	C₄⋊#10623494	C₂×C₄	1	6	yes	—
31	C₂²⋊#10623494	C₂×C₄	1	6	yes	—
32	#15667923	C₂×C₄	1	6	yes	—
33	C₄×#10623494	C₂×C₄	1	6	yes	—
34	C₄×#10623494	C₂×C₄	1	6	yes	—
35	C₄×#10623494	C₂³	1	3	yes	—
36	C₄×#10623494	C₂³	1	1	yes	characteristic
37	C₄×#10623494	C₂³	1	1	yes	characteristic
38	C₄×#10623494	C₂³	1	3	yes	—
39	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂×C₄	1	6	yes	—
41	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₈₉	C₂×C₄	1	3	yes	—
44	C₄×D₄×C₈₉	C₂×C₄	1	3	yes	—
45	#117×C₈₉	C₂×C₄	1	3	yes	—
46	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
47	C₄⋊C₈×C₈₉	C₂×C₄	1	6	yes	—
48	C₂²⋊C₈×C₈₉	C₂×C₄	1	6	yes	—
49	C₄×C₈×C₈₉	C₂×C₄	1	6	yes	—
50	C₄×C₈×C₈₉	C₂³	1	3	yes	—
51	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	#13083558	C₂×C₈	1	24	yes	complemented
53	C₂×#10623494	—	2	12	no	—
54	C₂×#10623494	C₂×C₈	1	24	yes	complemented
55	C₂×#10623494	C₂²×C₄	1	6	yes	—
56	C₂×#10623494	C₂²×C₄	1	2	yes	—
57	C₂×#10623494	C₂²×C₄	1	6	yes	—
58	C₂×#10623494	C₂²×C₄	1	2	yes	—
59	#41×C₈₉	C₂×C₈	1	8	yes	—
60	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
61	C₂×D₄×C₈₉	C₄²	1	3	yes	—
62	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
65	C₂×C₈×C₈₉	—	2	12	no	—
66	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
68	C₄⋊C₄×C₈₉	C₄²	1	3	yes	—
69	C₄⋊C₄×C₈₉	C₄²	1	3	yes	—
70	C₂²⋊C₄×C₈₉	C₄²	1	6	yes	—
71	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
72	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
73	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
74	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
75	C₂³×C₈₉	C₂×C₄²	1	3	yes	—
76	Q₈×C₈₉	C₄×C₈	1	4	yes	—
77	D₄×C₈₉	C₄×C₈	1	12	yes	—
78	C₂×C₄×C₈₉	—	2	6	no	—
79	C₂×C₄×C₈₉	—	2	6	no	—
80	C₂×C₄×C₈₉	C₂×C₄²	1	3	yes	—
81	C₂×C₄×C₈₉	C₂×C₄²	1	3	yes	—
82	C₂×C₄×C₈₉	C₂×C₄²	1	3	yes	—
83	C₂×C₄×C₈₉	C₂²×C₈	1	12	yes	—
84	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₈₉	—	2	6	no	—
88	C₈×C₈₉	#131	1	4	yes	complemented
89	#10623494	—	2	6	no	—
90	#10623494	C₂²×C₈	1	4	yes	—
91	#10623494	C₂²×C₈	1	12	yes	—
92	#10623494	#131	1	4	yes	complemented
93	C₂²×C₈₉	—	2	3	no	—
94	C₂²×C₈₉	—	2	3	no	—
95	C₂²×C₈₉	C₂×C₄×C₈	1	6	yes	—
96	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
97	C₄×C₈₉	—	2	3	no	—
98	C₄×C₈₉	—	2	3	no	—
99	C₄×C₈₉	C₂×C₄×C₈	1	6	yes	—
100	C₄×C₈₉	C₂³×C₈	1	2	yes	—
101	C₄×C₈₉	C₂×#131	1	2	yes	—
102	C₄×C₈₉	C₂×#131	1	2	yes	—
103	C₈×#131	—	89	1	no	maximal, Sylow
104	C₂×C₈₉	—	2	6	no	—
105	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
106	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
107	C₂×C₈₉	C₄×#131	1	1	yes	characteristic
108	C₈×#41	—	89	1	no	—
109	C₈×#41	D₈₉	1	1	yes	characteristic
110	C₄×#131	—	89	1	no	—
111	#1450	—	89	3	no	—
112	C₈×M₁₆	—	89	3	no	—
113	C₈×M₁₆	—	89	3	no	—
114	C₂×C₈²	—	89	3	no	—
115	C₈₉	C₈×#131	1	1	yes	characteristic, complemented, Sylow
116	C₂×#131	—	89	2	no	—
117	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
118	C₈×Q₈	—	89	2	no	—
119	C₈×Q₈	Dic₈₉	1	2	yes	—
120	C₈×D₄	—	89	6	no	—
121	C₈×D₄	Dic₈₉	1	6	yes	—
122	#431	—	89	3	no	—
123	#431	C₂×D₈₉	1	3	yes	—
124	#405	—	89	6	no	—
125	C₄×M₁₆	—	89	3	no	—
126	C₂×C₄×C₈	—	89	3	no	—
127	C₂×C₄×C₈	—	89	3	no	—
128	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
129	#322	—	89	6	no	—
130	C₈²	—	89	3	no	—
131	C₈²	—	89	3	no	—
132	C₈²	—	89	1	no	—
133	C₈²	—	89	3	no	—
134	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
135	#131	—	89	16	no	—
136	C₂×M₁₆	—	89	6	no	—
137	C₂²×C₈	—	89	6	no	—
138	C₂²×C₈	—	89	6	no	—
139	C₂²×C₈	C₂×Dic₈₉	1	6	yes	—
140	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
141	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
142	C₄×D₄	C₄×D₈₉	1	3	yes	—
143	C₄×D₄	C₂×Dic₈₉	1	3	yes	—
144	#117	C₄×D₈₉	1	3	yes	—
145	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
146	C₄⋊C₈	—	89	6	no	—
147	C₄⋊C₈	C₂×Dic₈₉	1	6	yes	—
148	C₂²⋊C₈	—	89	6	no	—
149	C₂²⋊C₈	C₂×Dic₈₉	1	6	yes	—
150	#97	—	89	6	no	—
151	C₄×C₈	—	89	6	no	—
152	C₄×C₈	—	89	6	no	—
153	C₄×C₈	—	89	1	no	—
154	C₄×C₈	—	89	1	no	—
155	C₄×C₈	—	89	3	no	—
156	C₄×C₈	—	89	3	no	—
157	C₄×C₈	C₂×Dic₈₉	1	6	yes	—
158	C₄×C₈	C₂²×D₈₉	1	3	yes	—
159	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
160	#41	C₈×D₈₉	1	8	yes	—
161	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
162	C₂×D₄	C₄×Dic₈₉	1	3	yes	—
163	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
164	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
165	C₂²×C₄	C₂²×Dic₈₉	1	3	yes	—
166	M₁₆	—	89	24	no	—
167	C₂×C₈	—	2	12	no	—
168	C₂×C₈	—	89	24	no	—
169	C₂×C₈	—	89	2	no	—
170	C₂×C₈	—	89	2	no	—
171	C₂×C₈	—	89	6	no	—
172	C₂×C₈	—	89	6	no	—
173	C₂×C₈	—	178	12	no	—
174	C₂×C₈	C₂²×Dic₈₉	1	6	yes	—
175	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
176	C₄⋊C₄	C₄×Dic₈₉	1	3	yes	—
177	C₄⋊C₄	C₄×Dic₈₉	1	3	yes	—
178	C₂²⋊C₄	C₄×Dic₈₉	1	6	yes	—
179	C₄²	C₂×C₄×D₈₉	1	3	yes	—
180	C₄²	C₂×C₄×D₈₉	1	3	yes	—
181	C₄²	C₂²×Dic₈₉	1	3	yes	—
182	C₄²	C₂³×D₈₉	1	1	yes	characteristic
183	C₂³	C₂×C₄×Dic₈₉	1	3	yes	—
184	Q₈	C₈×Dic₈₉	1	4	yes	—
185	D₄	C₈×Dic₈₉	1	12	yes	—
186	C₂×C₄	—	2	6	no	—
187	C₂×C₄	—	2	6	no	—
188	C₂×C₄	C₂×C₄×Dic₈₉	1	3	yes	—
189	C₂×C₄	C₂×C₄×Dic₈₉	1	3	yes	—
190	C₂×C₄	C₂×C₄×Dic₈₉	1	3	yes	—
191	C₂×C₄	C₂×C₈×D₈₉	1	12	yes	—
192	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
194	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
195	C₈	—	2	6	no	—
196	C₈	—	89	4	no	—
197	C₈	—	89	12	no	—
198	C₈	—	89	4	no	—
199	C₈	—	178	6	no	—
200	C₈	#15668019	1	4	yes	direct product factor
201	C₂²	—	2	3	no	—
202	C₂²	—	2	3	no	—
203	C₂²	C₂×C₈×Dic₈₉	1	6	yes	—
204	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
205	C₄	—	2	3	no	—
206	C₄	—	2	3	no	—
207	C₄	C₂×C₈×Dic₈₉	1	6	yes	—
208	C₄	C₂²×C₈×D₈₉	1	2	yes	—
209	C₄	#131×D₈₉	1	2	yes	—
210	C₄	C₂×#15668019	1	2	yes	—
211	C₂	—	2	6	no	—
212	C₂	C₂²×C₈×Dic₈₉	1	1	yes	characteristic
213	C₂	#131×Dic₈₉	1	1	yes	characteristic
214	C₂	C₄×#15668019	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×#41×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄²
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×#41	D₈₉
hypocenter	C₈₉	C₈×#131
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
3. #26018536	C₂	semidirect
4. #26018533	C₂	semidirect
5. C₈×#13083558	C₂	semidirect
6. C₂×C₈×#10623494	C₂	semidirect
26. #15668019	C₈	direct (N × H)
52. #13083558	C₂×C₈	semidirect
54. C₂×#10623494	C₂×C₈	semidirect
88. C₈×C₈₉	#131	semidirect
92. #10623494	#131	semidirect
115. C₈₉	C₈×#131	semidirect
200. C₈	#15668019	direct (N × H)

Maximal subgroups

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

subgroup	index
#41×#10623494	2
C₄×#15668019	2
#26018536	2
#26018533	2
C₈×#13083558	2
C₂×C₈×#10623494	2
C₈×#41×C₈₉	2
C₈×#131	89

Subgroups — C8×#15668019 #36520016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#15668019 #36520016