Subgroups — C₄×#19152076 #36712679

Order 22784.

3402 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152076	C₂	1	2	yes	maximal
2	#26022617	C₂	1	8	yes	maximal, complemented
3	#26022609	C₂	1	6	yes	maximal
4	C₄×Q₈×Dic₈₉	C₂	1	8	yes	maximal, complemented
5	C₄×#15667971	C₂	1	6	yes	maximal
6	C₂²×C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#19152076	C₄	1	32	yes	direct product factor
8	C₂×Q₈×Dic₈₉	C₂²	1	16	yes	—
9	C₂×#15667971	C₂²	1	12	yes	—
10	C₄⋊C₄×Dic₈₉	C₂²	1	48	yes	—
11	C₄×C₄⋊Dic₈₉	C₂²	1	12	yes	—
12	C₄×C₄⋊Dic₈₉	C₂²	1	12	yes	—
13	C₄×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
14	C₄×C₂²⋊Dic₈₉	C₂²	1	12	yes	—
15	#19151444	C₂²	1	12	yes	—
16	C₄²×Dic₈₉	C₂²	1	12	yes	—
17	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₈₉	C₄	1	16	yes	complemented
19	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
22	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
23	Q₈×Dic₈₉	C₂×C₄	1	128	yes	complemented
24	#15667971	C₂×C₄	1	96	yes	—
25	C₂×C₄⋊Dic₈₉	C₂³	1	24	yes	—
26	C₂×C₂²⋊Dic₈₉	C₂³	1	8	yes	—
27	C₂×C₄×Dic₈₉	C₂³	1	8	yes	—
28	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
29	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
30	#15667919	C₂³	1	8	yes	—
31	#15667919	C₂³	1	24	yes	—
32	C₂²×Q₈×C₈₉	C₂×C₄	1	16	yes	—
33	C₂²×Q₈×C₈₉	C₂×C₄	1	8	yes	—
34	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
38	C₄×Q₈×C₈₉	C₂×C₄	1	32	yes	—
39	C₄×Q₈×C₈₉	C₂×C₄	1	32	yes	—
40	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	24	yes	—
41	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
42	C₂×C₄²×C₈₉	C₂×C₄	1	24	yes	—
43	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
44	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
45	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
46	C₄⋊Dic₈₉	C₂²×C₄	1	96	yes	—
47	C₂²⋊Dic₈₉	C₂²×C₄	1	32	yes	—
48	C₄×Dic₈₉	—	2	16	no	—
49	C₄×Dic₈₉	—	4	24	no	—
50	C₄×Dic₈₉	C₂²×C₄	1	96	yes	—
51	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₈₉	C₄²	1	64	yes	complemented
53	C₂×Q₈×C₈₉	C₂²×C₄	1	32	yes	—
54	C₂×Q₈×C₈₉	C₂²×C₄	1	16	yes	—
55	C₂²×C₄×C₈₉	—	2	4	no	—
56	C₂²×C₄×C₈₉	—	4	6	no	—
57	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
63	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₈₉	C₂²×C₄	1	48	yes	—
66	C₄²×C₈₉	C₂²×C₄	1	24	yes	—
67	C₄²×C₈₉	C₂²×C₄	1	24	yes	—
68	C₂³×C₈₉	—	2	2	no	—
69	C₂³×C₈₉	—	2	2	no	—
70	C₂³×C₈₉	C₂³×C₄	1	4	yes	—
71	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
72	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
73	Q₈×C₈₉	C₂×C₄²	1	64	yes	—
74	C₂×C₄×C₈₉	—	2	2	no	—
75	C₂×C₄×C₈₉	—	2	12	no	—
76	C₂×C₄×C₈₉	—	2	8	no	—
77	C₂×C₄×C₈₉	—	4	12	no	—
78	C₂×C₄×C₈₉	C₂×C₄²	1	48	yes	—
79	C₂×C₄×C₈₉	C₂³×C₄	1	24	yes	—
80	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
82	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
83	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
84	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
85	C₂×Dic₈₉	—	2	16	no	—
86	C₂×Dic₈₉	—	2	16	no	—
87	C₂×Dic₈₉	C₂³×C₄	1	32	yes	—
88	C₂²×C₈₉	—	2	2	no	—
89	C₂²×C₈₉	—	2	2	no	—
90	C₂²×C₈₉	—	2	8	no	—
91	C₂²×C₈₉	C₂²×C₄²	1	4	yes	—
92	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
93	C₂²×C₈₉	C₂⁴×C₄	1	2	yes	—
94	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
95	C₂²×C₈₉	#585	1	2	yes	—
96	C₄×C₈₉	—	2	2	no	—
97	C₄×C₈₉	—	2	12	no	—
98	C₄×C₈₉	C₂²×C₄²	1	24	yes	—
99	C₄×C₈₉	#519	1	4	yes	complemented
100	Dic₈₉	—	2	32	no	—
101	C₄×#519	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	—	2	4	no	—
103	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
104	C₂×C₈₉	C₂×#519	1	1	yes	characteristic
105	C₂×C₈₉	#3428	1	1	yes	characteristic
106	C₂×C₈₉	C₄×2₋¹⁺⁴	1	4	yes	—
107	C₂×#519	—	89	2	no	—
108	C₂²×C₄×Q₈	D₈₉	1	1	yes	characteristic
109	#2276	—	89	8	no	—
110	#2274	—	89	6	no	—
111	C₄²×Q₈	—	89	8	no	—
112	C₄×#117	—	89	6	no	—
113	C₈₉	C₄×#519	1	1	yes	characteristic, complemented, Sylow
114	C₂³×Q₈	C₂×D₈₉	1	1	yes	characteristic
115	#519	—	89	32	no	—
116	C₂×C₄×Q₈	—	89	16	no	—
117	C₂×C₄×Q₈	Dic₈₉	1	16	yes	complemented
118	C₂×C₄×Q₈	C₂×D₈₉	1	4	yes	—
119	C₂×C₄×Q₈	C₂×D₈₉	1	4	yes	—
120	C₂×#117	—	89	12	no	—
121	C₂²×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
122	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
123	C₄×C₄⋊C₄	—	89	48	no	—
124	C₄×C₄⋊C₄	—	89	12	no	—
125	C₄×C₄⋊C₄	—	89	12	no	—
126	C₄×C₂²⋊C₄	—	89	4	no	—
127	C₄×C₂²⋊C₄	—	89	12	no	—
128	#376	—	89	12	no	—
129	C₄³	—	89	12	no	—
130	C₂²×Q₈	C₄×D₈₉	1	16	yes	—
131	C₂²×Q₈	C₂×Dic₈₉	1	8	yes	—
132	C₂²×Q₈	C₂²×D₈₉	1	4	yes	—
133	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
134	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
135	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
136	C₄×Q₈	—	89	128	no	—
137	C₄×Q₈	C₂×Dic₈₉	1	32	yes	—
138	C₄×Q₈	C₂×Dic₈₉	1	32	yes	—
139	#117	—	89	96	no	—
140	C₂×C₄⋊C₄	—	89	24	no	—
141	C₂×C₄⋊C₄	C₂×Dic₈₉	1	24	yes	—
142	C₂×C₄⋊C₄	C₂²×D₈₉	1	12	yes	—
143	C₂×C₂²⋊C₄	—	89	8	no	—
144	C₂×C₄²	—	89	24	no	—
145	C₂×C₄²	—	89	24	no	—
146	C₂×C₄²	—	89	8	no	—
147	C₂×C₄²	C₂×Dic₈₉	1	24	yes	—
148	C₂×C₄²	C₂²×D₈₉	1	6	yes	—
149	C₂×C₄²	C₂²×D₈₉	1	6	yes	—
150	#95	—	89	8	no	—
151	#95	—	89	24	no	—
152	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
153	C₂×Q₈	C₄×Dic₈₉	1	64	yes	complemented
154	C₂×Q₈	C₂×C₄×D₈₉	1	32	yes	—
155	C₂×Q₈	C₂²×Dic₈₉	1	16	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	6	no	—
158	C₂²×C₄	—	89	16	no	—
159	C₂²×C₄	C₂×C₄×D₈₉	1	24	yes	—
160	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
161	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
162	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
163	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
164	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
165	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
166	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
167	C₄⋊C₄	—	89	96	no	—
168	C₄⋊C₄	C₂²×Dic₈₉	1	48	yes	—
169	C₂²⋊C₄	—	89	32	no	—
170	C₄²	—	89	96	no	—
171	C₄²	—	356	24	no	—
172	C₄²	—	178	16	no	—
173	C₄²	C₂²×Dic₈₉	1	24	yes	—
174	C₄²	C₂²×Dic₈₉	1	24	yes	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	C₂²×C₄×D₈₉	1	4	yes	—
178	C₂³	C₂³×Dic₈₉	1	2	yes	—
179	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
180	Q₈	C₂×C₄×Dic₈₉	1	64	yes	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	12	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	4	12	no	—
185	C₂×C₄	—	89	32	no	—
186	C₂×C₄	—	178	16	no	—
187	C₂×C₄	—	178	16	no	—
188	C₂×C₄	C₂×C₄×Dic₈₉	1	48	yes	—
189	C₂×C₄	C₂²×C₄×D₈₉	1	24	yes	—
190	C₂×C₄	C₂³×Dic₈₉	1	2	yes	—
191	C₂×C₄	C₂³×Dic₈₉	1	12	yes	—
192	C₂×C₄	C₂³×Dic₈₉	1	12	yes	—
193	C₂×C₄	C₂³×Dic₈₉	1	2	yes	—
194	C₂×C₄	#15668060	1	4	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	C₂²×C₄×Dic₈₉	1	4	yes	—
199	C₂²	C₂³×C₄×D₈₉	1	2	yes	—
200	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
201	C₂²	C₂×#15668060	1	2	yes	—
202	C₂²	#19152434	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	12	no	—
205	C₄	—	178	32	no	—
206	C₄	C₂²×C₄×Dic₈₉	1	24	yes	—
207	C₄	#19152076	1	4	yes	direct product factor
208	C₂	—	2	4	no	—
209	C₂	C₂³×C₄×Dic₈₉	1	1	yes	characteristic
210	C₂	C₂×#19152076	1	1	yes	characteristic
211	C₂	#26033455	1	1	yes	characteristic
212	C₂	C₄×#15668060	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂²×C₄×Q₈×C₈₉	C₂
socle, cosocle	C₂³×C₈₉	C₂⁵
prime radical	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₈₉
hypocenter	C₈₉	C₄×#519
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #26022617	C₂	semidirect
4. C₄×Q₈×Dic₈₉	C₂	semidirect
7. #19152076	C₄	direct (N × H)
18. C₂×C₄×Q₈×C₈₉	C₄	semidirect
23. Q₈×Dic₈₉	C₂×C₄	semidirect
52. C₂×Q₈×C₈₉	C₄²	semidirect
99. C₄×C₈₉	#519	semidirect
113. C₈₉	C₄×#519	semidirect
117. C₂×C₄×Q₈	Dic₈₉	semidirect
153. C₂×Q₈	C₄×Dic₈₉	semidirect
207. C₄	#19152076	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19152076	2
#26022617	2
#26022609	2
C₄×Q₈×Dic₈₉	2
C₄×#15667971	2
C₂²×C₄×Q₈×C₈₉	2
C₄×#519	89

Subgroups — C4×#19152076 #36712679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19152076 #36712679