Subgroups — C₂×#26023433 #36717063

Order 22784.

2326 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26023433	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19151505	C₂	1	4	yes	maximal
3	C₂×#19151484	C₂	1	8	yes	maximal
4	C₂×#19151466	C₂	1	2	yes	maximal
5	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
7	C₂²×C₄×Dic₈₉	C₂²	1	4	yes	—
8	#19151505	C₂²	1	32	yes	—
9	#19151484	C₂²	1	64	yes	—
10	#19151466	C₂²	1	16	yes	—
11	C₂×#15667919	C₂²	1	8	yes	—
12	C₂×#15667919	C₂²	1	8	yes	—
13	C₂×#15667919	C₂²	1	4	yes	—
14	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₈₉	C₄	1	16	yes	complemented
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
18	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₂³×Dic₈₉	C₂³	1	8	yes	—
20	C₂×C₄⋊Dic₈₉	—	4	8	no	—
21	C₂×C₄⋊Dic₈₉	C₂³	1	16	yes	—
22	C₂×C₄×Dic₈₉	—	4	8	no	—
23	C₂×C₄×Dic₈₉	C₂³	1	16	yes	—
24	#15667919	C₂³	1	32	yes	—
25	#15667919	C₂³	1	16	yes	—
26	#15667919	C₂³	1	32	yes	—
27	C₂²×Q₈×C₈₉	—	4	4	no	—
28	C₂²×Q₈×C₈₉	—	4	4	no	—
29	C₂²×Q₈×C₈₉	C₂×C₄	1	16	yes	—
30	C₂²×Q₈×C₈₉	C₂³	1	8	yes	—
31	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
32	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₈₉	C₂×C₄	1	64	yes	complemented
35	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	32	yes	—
36	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
37	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	—
38	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
39	C₂²×Dic₈₉	—	2	16	no	—
40	C₂²×Dic₈₉	—	2	32	no	—
41	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
42	C₄⋊Dic₈₉	—	4	16	no	—
43	C₄×Dic₈₉	—	4	16	no	—
44	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₈₉	—	4	32	no	—
46	C₂×Q₈×C₈₉	—	4	8	no	—
47	C₂×Q₈×C₈₉	—	4	8	no	—
48	C₂×Q₈×C₈₉	C₂²×C₄	1	32	yes	—
49	C₂²×C₄×C₈₉	—	2	4	no	—
50	C₂²×C₄×C₈₉	—	2	8	no	—
51	C₂²×C₄×C₈₉	—	2	8	no	—
52	C₂²×C₄×C₈₉	—	2	8	no	—
53	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
54	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₈₉	C₂²×C₄	1	16	yes	—
56	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
57	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
59	C₄⋊C₄×C₈₉	C₂²×C₄	1	64	yes	—
60	C₄²×C₈₉	C₂²×C₄	1	16	yes	—
61	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
62	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
63	C₂³×C₈₉	2₋¹⁺⁴	1	8	yes	—
64	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
65	Q₈×C₈₉	—	4	32	no	—
66	C₂×C₄×C₈₉	—	2	32	no	—
67	C₂×C₄×C₈₉	—	2	8	no	—
68	C₂×C₄×C₈₉	—	2	16	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	—	2	8	no	—
71	C₂×C₄×C₈₉	—	2	8	no	—
72	C₂×C₄×C₈₉	—	2	8	no	—
73	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
74	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
75	C₂×C₄×C₈₉	C₂³×C₄	1	16	yes	—
76	C₂×Dic₈₉	—	2	16	no	—
77	C₂×Dic₈₉	—	2	32	no	—
78	C₂×Dic₈₉	—	4	32	no	—
79	C₂²×C₈₉	#518	1	4	yes	—
80	C₂²×C₈₉	#519	1	8	yes	—
81	C₂²×C₈₉	#563	1	16	yes	—
82	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
83	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
84	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	4	yes	—
85	C₄×C₈₉	—	2	16	no	—
86	C₄×C₈₉	—	2	4	no	—
87	C₄×C₈₉	—	2	8	no	—
88	Dic₈₉	—	4	16	no	—
89	C₂×#2381	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	#2381	1	8	yes	complemented
91	C₂×C₈₉	C₂×#518	1	1	yes	characteristic
92	C₂×C₈₉	C₂×#519	1	2	yes	—
93	C₂×C₈₉	C₂×#563	1	4	yes	—
94	C₂²×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
95	#2381	—	89	16	no	—
96	C₂×#391	—	89	4	no	—
97	C₂×#387	—	89	8	no	—
98	C₂×#383	—	89	2	no	—
99	C₈₉	C₂×#2381	1	1	yes	characteristic, complemented, Sylow
100	C₂³×Q₈	C₂×D₈₉	1	2	yes	—
101	C₂×C₄⋊Q₈	Dic₈₉	1	16	yes	complemented
102	C₂×C₄⋊Q₈	C₂×D₈₉	1	8	yes	—
103	C₂²×C₄⋊C₄	—	89	4	no	—
104	C₂²×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
105	C₂²×C₄²	—	89	4	no	—
106	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
107	#391	—	89	32	no	—
108	#387	—	89	64	no	—
109	#383	—	89	16	no	—
110	C₂×#95	—	89	8	no	—
111	C₂×#95	—	89	4	no	—
112	C₂×#95	—	89	8	no	—
113	C₂²×Q₈	—	4	4	no	—
114	C₂²×Q₈	—	4	4	no	—
115	C₂²×Q₈	C₂×Dic₈₉	1	16	yes	—
116	C₂²×Q₈	C₂²×D₈₉	1	8	yes	—
117	C₂³×C₄	—	89	8	no	—
118	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
119	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
120	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×Dic₈₉	1	64	yes	complemented
122	C₂×C₄⋊C₄	—	89	16	no	—
123	C₂×C₄⋊C₄	—	356	8	no	—
124	C₂×C₄⋊C₄	C₂×Dic₈₉	1	32	yes	—
125	C₂×C₄⋊C₄	C₂²×D₈₉	1	16	yes	—
126	C₂×C₄²	—	89	16	no	—
127	C₂×C₄²	—	356	8	no	—
128	C₂×C₄²	C₂×Dic₈₉	1	8	yes	—
129	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
130	#95	—	89	32	no	—
131	#95	—	89	16	no	—
132	#95	—	89	32	no	—
133	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
134	C₂×Q₈	—	4	32	no	—
135	C₂×Q₈	—	4	8	no	—
136	C₂×Q₈	—	4	8	no	—
137	C₂×Q₈	C₂²×Dic₈₉	1	32	yes	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	—	89	16	no	—
143	C₂²×C₄	—	178	16	no	—
144	C₂²×C₄	—	178	32	no	—
145	C₂²×C₄	C₂²×Dic₈₉	1	8	yes	—
146	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
147	C₂²×C₄	C₂²×Dic₈₉	1	16	yes	—
148	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
149	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
150	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
151	C₄⋊C₄	—	356	16	no	—
152	C₄⋊C₄	C₂²×Dic₈₉	1	64	yes	—
153	C₄²	—	356	16	no	—
154	C₄²	C₂²×Dic₈₉	1	16	yes	—
155	C₂³	C₂³×Dic₈₉	1	2	yes	—
156	C₂³	#15668057	1	4	yes	—
157	C₂³	#15668060	1	8	yes	—
158	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
159	Q₈	—	4	32	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	32	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	16	no	—
167	C₂×C₄	—	356	32	no	—
168	C₂×C₄	—	178	16	no	—
169	C₂×C₄	—	178	32	no	—
170	C₂×C₄	C₂³×Dic₈₉	1	8	yes	—
171	C₂×C₄	C₂³×Dic₈₉	1	4	yes	—
172	C₂×C₄	C₂³×Dic₈₉	1	16	yes	—
173	C₂²	#19152073	1	4	yes	—
174	C₂²	#19152076	1	8	yes	—
175	C₂²	#19152347	1	16	yes	—
176	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
177	C₂²	C₂×#15668057	1	2	yes	—
178	C₂²	C₂×#15668060	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	16	no	—
181	C₄	—	2	8	no	—
182	C₄	—	356	16	no	—
183	C₂	#26023433	1	8	yes	direct product factor
184	C₂	C₂×#19152073	1	1	yes	characteristic
185	C₂	C₂×#19152076	1	2	yes	—
186	C₂	C₂×#19152347	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂²×C₄⋊Q₈×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂⁴
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂²×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	C₂×#2381
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #26023433	C₂	direct (N × H)
15. C₂×C₄⋊Q₈×C₈₉	C₄	semidirect
34. C₄⋊Q₈×C₈₉	C₂×C₄	semidirect
90. C₂×C₈₉	#2381	semidirect
99. C₈₉	C₂×#2381	semidirect
101. C₂×C₄⋊Q₈	Dic₈₉	semidirect
121. C₄⋊Q₈	C₂×Dic₈₉	semidirect
183. C₂	#26023433	direct (N × H)

Maximal subgroups

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

subgroup	index
#26023433	2
C₂×#19151505	2
C₂×#19151484	2
C₂×#19151466	2
C₂²×C₄⋊Q₈×C₈₉	2
C₂×#2381	89

Subgroups — C2×#26023433 #36717063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26023433 #36717063