Subgroups — C₂³×#15667919 #36712595

Order 22784.

14334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×Dic₈₉	C₂	1	2	yes	maximal
2	C₂²×#15667919	C₂	1	28	yes	maximal, direct product factor
3	C₂⁵×C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Dic₈₉	—	2	16	no	—
5	C₂⁴×Dic₈₉	—	2	16	no	—
6	C₂⁴×Dic₈₉	C₄	1	32	yes	complemented
7	C₂⁴×Dic₈₉	C₂²	1	28	yes	—
8	C₂×#15667919	C₂²	1	112	yes	direct product factor
9	C₂⁶×C₈₉	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₈₉	—	2	16	no	—
11	C₂⁴×C₄×C₈₉	C₄	1	16	yes	complemented
12	C₂⁴×C₄×C₈₉	C₂²	1	14	yes	—
13	C₂³×Dic₈₉	—	2	256	no	—
14	C₂³×Dic₈₉	—	2	112	no	—
15	C₂³×Dic₈₉	—	2	112	no	—
16	C₂³×Dic₈₉	C₂×C₄	1	224	yes	complemented
17	C₂³×Dic₈₉	C₂³	1	56	yes	—
18	#15667919	C₂³	1	64	yes	direct product factor
19	C₂⁵×C₈₉	C₂×C₄	1	16	yes	—
20	C₂⁵×C₈₉	C₂×C₄	1	8	yes	—
21	C₂⁵×C₈₉	D₄	1	8	yes	—
22	C₂⁵×C₈₉	D₄	1	16	yes	—
23	C₂⁵×C₈₉	Q₈	1	8	yes	—
24	C₂⁵×C₈₉	C₂³	1	7	yes	—
25	C₂³×C₄×C₈₉	—	2	128	no	—
26	C₂³×C₄×C₈₉	—	2	112	no	—
27	C₂³×C₄×C₈₉	C₂×C₄	1	112	yes	complemented
28	C₂³×C₄×C₈₉	C₂³	1	28	yes	—
29	C₂²×Dic₈₉	—	2	896	no	—
30	C₂²×Dic₈₉	—	2	112	no	—
31	C₂²×Dic₈₉	—	2	112	no	—
32	C₂²×Dic₈₉	C₂²×C₄	1	224	yes	complemented
33	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
34	C₂⁴×C₈₉	C₄²	1	64	yes	—
35	C₂⁴×C₈₉	C₂²⋊C₄	1	128	yes	—
36	C₂⁴×C₈₉	C₂²⋊C₄	1	64	yes	—
37	C₂⁴×C₈₉	C₄⋊C₄	1	128	yes	—
38	C₂⁴×C₈₉	C₄⋊C₄	1	64	yes	—
39	C₂⁴×C₈₉	C₂²×C₄	1	56	yes	—
40	C₂⁴×C₈₉	C₂²×C₄	1	28	yes	—
41	C₂⁴×C₈₉	C₂×D₄	1	56	yes	—
42	C₂⁴×C₈₉	C₂×D₄	1	28	yes	—
43	C₂⁴×C₈₉	C₂×Q₈	1	28	yes	—
44	C₂⁴×C₈₉	C₂⁴	1	7	yes	—
45	C₂²×C₄×C₈₉	—	2	448	no	—
46	C₂²×C₄×C₈₉	—	2	112	no	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	112	yes	complemented
48	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
49	C₂³×C₈₉	#95	1	512	yes	complemented
50	C₂³×C₈₉	C₂×C₄²	1	112	yes	—
51	C₂³×C₈₉	C₂×C₂²⋊C₄	1	224	yes	—
52	C₂³×C₈₉	C₂×C₂²⋊C₄	1	112	yes	—
53	C₂³×C₈₉	C₂×C₄⋊C₄	1	224	yes	—
54	C₂³×C₈₉	C₂×C₄⋊C₄	1	112	yes	—
55	C₂³×C₈₉	C₂³×C₄	1	28	yes	—
56	C₂³×C₈₉	C₂³×C₄	1	14	yes	—
57	C₂³×C₈₉	C₂²×D₄	1	28	yes	—
58	C₂³×C₈₉	C₂²×D₄	1	14	yes	—
59	C₂³×C₈₉	C₂²×Q₈	1	14	yes	—
60	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
61	C₂×C₄×C₈₉	—	2	224	no	—
62	C₂×C₄×C₈₉	—	2	16	no	—
63	C₂×C₄×C₈₉	C₂³×C₄	1	16	yes	complemented
64	C₂×Dic₈₉	—	2	448	no	—
65	C₂×Dic₈₉	—	2	16	no	—
66	C₂×Dic₈₉	—	2	16	no	—
67	C₂×Dic₈₉	C₂³×C₄	1	32	yes	complemented
68	C₂²×C₈₉	C₂×#95	1	448	yes	complemented
69	C₂²×C₈₉	C₂²×C₄²	1	28	yes	—
70	C₂²×C₈₉	C₂²×C₂²⋊C₄	1	56	yes	—
71	C₂²×C₈₉	C₂²×C₂²⋊C₄	1	28	yes	—
72	C₂²×C₈₉	C₂²×C₄⋊C₄	1	56	yes	—
73	C₂²×C₈₉	C₂²×C₄⋊C₄	1	28	yes	—
74	C₂²×C₈₉	C₂⁴×C₄	1	2	yes	—
75	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
76	C₂²×C₈₉	C₂³×D₄	1	2	yes	—
77	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
78	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
79	C₄×C₈₉	—	2	16	no	—
80	Dic₈₉	—	2	32	no	—
81	C₂³×#95	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	C₂²×#95	1	56	yes	complemented
83	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
84	C₂×C₈₉	C₂³×C₂²⋊C₄	1	2	yes	—
85	C₂×C₈₉	C₂³×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈₉	C₂³×C₄⋊C₄	1	2	yes	—
87	C₂×C₈₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
88	C₂⁵×C₄	—	89	2	no	—
89	C₂⁵×C₄	D₈₉	1	1	yes	characteristic
90	C₂²×#95	—	89	28	no	—
91	C₈₉	C₂³×#95	1	1	yes	characteristic, complemented, Sylow
92	C₂⁶	C₂×D₈₉	1	1	yes	characteristic
93	C₂⁴×C₄	—	2	16	no	—
94	C₂⁴×C₄	—	89	32	no	—
95	C₂⁴×C₄	—	89	28	no	—
96	C₂⁴×C₄	—	178	16	no	—
97	C₂⁴×C₄	—	178	16	no	—
98	C₂⁴×C₄	Dic₈₉	1	16	yes	complemented
99	C₂⁴×C₄	C₂×D₈₉	1	14	yes	—
100	C₂×#95	—	89	112	no	—
101	C₂⁵	Dic₁₇₈	1	8	yes	—
102	C₂⁵	C₄×D₈₉	1	16	yes	—
103	C₂⁵	D₃₅₆	1	8	yes	—
104	C₂⁵	C₂×Dic₈₉	1	8	yes	—
105	C₂⁵	C₂²⋊D₈₉	1	16	yes	—
106	C₂⁵	C₂²×D₈₉	1	7	yes	—
107	C₂³×C₄	—	2	128	no	—
108	C₂³×C₄	—	2	112	no	—
109	C₂³×C₄	—	89	224	no	—
110	C₂³×C₄	—	89	56	no	—
111	C₂³×C₄	—	178	256	no	—
112	C₂³×C₄	—	178	112	no	—
113	C₂³×C₄	—	178	112	no	—
114	C₂³×C₄	C₂×Dic₈₉	1	112	yes	complemented
115	C₂³×C₄	C₂²×D₈₉	1	28	yes	—
116	#95	—	89	64	no	—
117	C₂⁴	C₄×Dic₈₉	1	64	yes	—
118	C₂⁴	#13083551	1	128	yes	—
119	C₂⁴	C₂²⋊Dic₈₉	1	64	yes	—
120	C₂⁴	#13083553	1	128	yes	—
121	C₂⁴	C₄⋊Dic₈₉	1	64	yes	—
122	C₂⁴	C₂×C₄×D₈₉	1	56	yes	—
123	C₂⁴	C₂²×Dic₈₉	1	28	yes	—
124	C₂⁴	C₂×C₂²⋊D₈₉	1	56	yes	—
125	C₂⁴	C₂×D₃₅₆	1	28	yes	—
126	C₂⁴	C₂×Dic₁₇₈	1	28	yes	—
127	C₂⁴	C₂³×D₈₉	1	7	yes	—
128	C₂²×C₄	—	2	112	no	—
129	C₂²×C₄	—	2	448	no	—
130	C₂²×C₄	—	89	224	no	—
131	C₂²×C₄	—	89	16	no	—
132	C₂²×C₄	—	178	112	no	—
133	C₂²×C₄	—	178	112	no	—
134	C₂²×C₄	—	178	896	no	—
135	C₂²×C₄	C₂²×Dic₈₉	1	112	yes	complemented
136	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
137	C₂³	#15667919	1	512	yes	direct product factor
138	C₂³	C₂×C₄×Dic₈₉	1	112	yes	—
139	C₂³	C₂×#13083551	1	224	yes	—
140	C₂³	C₂×C₂²⋊Dic₈₉	1	112	yes	—
141	C₂³	C₂×#13083553	1	224	yes	—
142	C₂³	C₂×C₄⋊Dic₈₉	1	112	yes	—
143	C₂³	C₂²×C₄×D₈₉	1	28	yes	—
144	C₂³	C₂³×Dic₈₉	1	14	yes	—
145	C₂³	C₂²×C₂²⋊D₈₉	1	28	yes	—
146	C₂³	C₂²×D₃₅₆	1	14	yes	—
147	C₂³	C₂²×Dic₁₇₈	1	14	yes	—
148	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
149	C₂×C₄	—	2	16	no	—
150	C₂×C₄	—	2	224	no	—
151	C₂×C₄	—	89	32	no	—
152	C₂×C₄	—	178	16	no	—
153	C₂×C₄	—	178	16	no	—
154	C₂×C₄	—	178	448	no	—
155	C₂×C₄	C₂³×Dic₈₉	1	16	yes	complemented
156	C₂²	C₂×#15667919	1	448	yes	direct product factor
157	C₂²	C₂²×C₄×Dic₈₉	1	28	yes	—
158	C₂²	C₂²×#13083551	1	56	yes	—
159	C₂²	C₂²×C₂²⋊Dic₈₉	1	28	yes	—
160	C₂²	C₂²×#13083553	1	56	yes	—
161	C₂²	C₂²×C₄⋊Dic₈₉	1	28	yes	—
162	C₂²	C₂³×C₄×D₈₉	1	2	yes	—
163	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
164	C₂²	C₂³×C₂²⋊D₈₉	1	2	yes	—
165	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
166	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
167	C₄	—	2	16	no	—
168	C₄	—	178	32	no	—
169	C₂	C₂²×#15667919	1	56	yes	direct product factor
170	C₂	C₂³×C₄×Dic₈₉	1	1	yes	characteristic
171	C₂	C₂³×#13083551	1	2	yes	—
172	C₂	C₂³×C₂²⋊Dic₈₉	1	1	yes	characteristic
173	C₂	C₂³×#13083553	1	2	yes	—
174	C₂	C₂³×C₄⋊Dic₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁶	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂⁵×C₄×C₈₉	C₂
socle, prime radical	C₂⁶×C₈₉	C₂²
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂⁵×C₄	D₈₉
hypocenter	C₈₉	C₂³×#95
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁶ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#15667919	C₂	direct (N × H)
6. C₂⁴×Dic₈₉	C₄	semidirect
8. C₂×#15667919	C₂²	direct (N × H)
11. C₂⁴×C₄×C₈₉	C₄	semidirect
16. C₂³×Dic₈₉	C₂×C₄	semidirect
18. #15667919	C₂³	direct (N × H)
27. C₂³×C₄×C₈₉	C₂×C₄	semidirect
32. C₂²×Dic₈₉	C₂²×C₄	semidirect
47. C₂²×C₄×C₈₉	C₂²×C₄	semidirect
49. C₂³×C₈₉	#95	semidirect
63. C₂×C₄×C₈₉	C₂³×C₄	semidirect
67. C₂×Dic₈₉	C₂³×C₄	semidirect
68. C₂²×C₈₉	C₂×#95	semidirect
82. C₂×C₈₉	C₂²×#95	semidirect
91. C₈₉	C₂³×#95	semidirect
98. C₂⁴×C₄	Dic₈₉	semidirect
114. C₂³×C₄	C₂×Dic₈₉	semidirect
135. C₂²×C₄	C₂²×Dic₈₉	semidirect
137. C₂³	#15667919	direct (N × H)
155. C₂×C₄	C₂³×Dic₈₉	semidirect
156. C₂²	C₂×#15667919	direct (N × H)
169. C₂	C₂²×#15667919	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×Dic₈₉	2
C₂²×#15667919	2
C₂⁵×C₄×C₈₉	2
C₂³×#95	89

Subgroups — C23×#15667919 #36712595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#15667919 #36712595