Subgroups — C₂×#26028639 #36760452

Order 22784.

2078 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028639	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19151496	C₂	1	4	yes	maximal
3	C₂×#19151482	C₂	1	8	yes	maximal
4	C₂×#19151464	C₂	1	2	yes	maximal
5	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
7	C₂²×#13083553	C₂²	1	8	yes	—
8	#19151496	C₂²	1	32	yes	—
9	#19151482	C₂²	1	64	yes	—
10	#19151464	C₂²	1	16	yes	—
11	C₂×#15667919	C₂²	1	8	yes	—
12	C₂×#15667919	C₂²	1	8	yes	—
13	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
14	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₄³×C₈₉	C₂²	1	8	yes	—
17	C₂³×Dic₈₉	C₂³	1	8	yes	—
18	C₂×C₄⋊Dic₈₉	—	2	16	no	—
19	C₂×C₄⋊Dic₈₉	C₂³	1	16	yes	—
20	C₂×#13083553	—	2	32	no	—
21	C₂×#13083553	C₂³	1	32	yes	—
22	#15667919	C₂³	1	32	yes	—
23	#15667919	C₂³	1	32	yes	—
24	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
25	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
26	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	—	2	16	no	—
28	C₂×C₄²×C₈₉	—	2	8	no	—
29	C₂×C₄²×C₈₉	Q₈	1	8	yes	—
30	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
32	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
33	C₂²×Dic₈₉	—	2	32	no	—
34	C₂²×Dic₈₉	—	2	16	no	—
35	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
36	C₄⋊Dic₈₉	—	2	32	no	—
37	#13083553	—	2	64	no	—
38	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	—	2	8	no	—
40	C₂²×C₄×C₈₉	—	2	8	no	—
41	C₂²×C₄×C₈₉	—	2	8	no	—
42	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
43	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₈₉	#41	1	16	yes	—
45	C₂²×C₄×C₈₉	#41	1	8	yes	—
46	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
47	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
48	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
49	C₄²×C₈₉	—	2	32	no	—
50	C₄²×C₈₉	—	2	16	no	—
51	C₄²×C₈₉	C₂×Q₈	1	16	yes	—
52	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
53	C₂³×C₈₉	C₂×#41	1	4	yes	—
54	C₂³×C₈₉	C₂×#41	1	8	yes	—
55	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	32	no	—
57	C₂×C₄×C₈₉	—	2	16	no	—
58	C₂×C₄×C₈₉	—	2	8	no	—
59	C₂×C₄×C₈₉	—	2	8	no	—
60	C₂×C₄×C₈₉	—	2	8	no	—
61	C₂×C₄×C₈₉	#125	1	16	yes	—
62	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
63	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
64	C₂×C₄×C₈₉	C₂×#41	1	16	yes	—
65	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
66	C₂×Dic₈₉	—	2	32	no	—
67	C₂×Dic₈₉	—	2	16	no	—
68	C₂×Dic₈₉	—	4	32	no	—
69	C₂²×C₈₉	C₂×#125	1	4	yes	—
70	C₂²×C₈₉	#529	1	16	yes	—
71	C₂²×C₈₉	#533	1	8	yes	—
72	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
73	C₂²×C₈₉	C₂²×#41	1	2	yes	—
74	C₂²×C₈₉	C₂²×#41	1	4	yes	—
75	C₄×C₈₉	—	2	16	no	—
76	C₄×C₈₉	—	2	8	no	—
77	C₄×C₈₉	C₂×#125	1	8	yes	—
78	Dic₈₉	—	4	16	no	—
79	C₂×#2864	—	89	1	no	maximal, Sylow
80	C₂×C₈₉	#2864	1	8	yes	complemented
81	C₂×C₈₉	C₂²×#125	1	1	yes	characteristic
82	C₂×C₈₉	C₂×#529	1	4	yes	—
83	C₂×C₈₉	C₂×#533	1	2	yes	—
84	#2864	—	89	16	no	—
85	C₂×#389	—	89	4	no	—
86	C₂×#387	—	89	8	no	—
87	C₂×#382	—	89	2	no	—
88	C₂×C₄³	D₈₉	1	1	yes	characteristic
89	C₈₉	C₂×#2864	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄⋊C₄	—	89	8	no	—
91	C₂²×C₄⋊C₄	—	89	4	no	—
92	C₂²×C₄²	C₂×D₈₉	1	4	yes	—
93	C₂²×C₄²	C₂×D₈₉	1	2	yes	—
94	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
95	#389	—	89	32	no	—
96	#387	—	89	64	no	—
97	#382	—	89	16	no	—
98	C₂×#95	—	89	8	no	—
99	C₂×#95	—	89	8	no	—
100	C₄³	C₂×D₈₉	1	8	yes	—
101	C₂³×C₄	—	89	8	no	—
102	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
103	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
104	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	89	16	no	—
106	C₂×C₄⋊C₄	—	89	32	no	—
107	C₂×C₄⋊C₄	—	178	32	no	—
108	C₂×C₄⋊C₄	—	178	16	no	—
109	C₂×C₄²	—	2	16	no	—
110	C₂×C₄²	—	2	8	no	—
111	C₂×C₄²	Dic₁₇₈	1	8	yes	—
112	C₂×C₄²	C₂²×D₈₉	1	16	yes	—
113	C₂×C₄²	C₂²×D₈₉	1	8	yes	—
114	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
115	#95	—	89	32	no	—
116	#95	—	89	32	no	—
117	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
118	C₂²×C₄	—	2	8	no	—
119	C₂²×C₄	—	2	8	no	—
120	C₂²×C₄	—	2	8	no	—
121	C₂²×C₄	—	89	16	no	—
122	C₂²×C₄	—	178	32	no	—
123	C₂²×C₄	—	178	16	no	—
124	C₂²×C₄	C₂×Dic₁₇₈	1	8	yes	—
125	C₂²×C₄	C₂×Dic₁₇₈	1	4	yes	—
126	C₂²×C₄	#13083574	1	16	yes	—
127	C₂²×C₄	#13083574	1	8	yes	—
128	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
129	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
130	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
131	C₄⋊C₄	—	178	64	no	—
132	C₄⋊C₄	—	178	32	no	—
133	C₄²	—	2	32	no	—
134	C₄²	—	2	16	no	—
135	C₄²	C₂×Dic₁₇₈	1	16	yes	—
136	C₂³	C₂²×Dic₁₇₈	1	2	yes	—
137	C₂³	C₂×#13083574	1	8	yes	—
138	C₂³	C₂×#13083574	1	4	yes	—
139	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
140	C₂×C₄	—	2	32	no	—
141	C₂×C₄	—	2	16	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	356	32	no	—
146	C₂×C₄	—	178	16	no	—
147	C₂×C₄	—	178	32	no	—
148	C₂×C₄	#15668004	1	16	yes	—
149	C₂×C₄	C₂²×Dic₁₇₈	1	8	yes	—
150	C₂×C₄	C₂²×Dic₁₇₈	1	4	yes	—
151	C₂×C₄	C₂×#13083574	1	16	yes	—
152	C₂×C₄	C₂×#13083574	1	8	yes	—
153	C₂²	C₂×#15668004	1	4	yes	—
154	C₂²	#19152132	1	16	yes	—
155	C₂²	#19152155	1	8	yes	—
156	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
157	C₂²	C₂²×#13083574	1	2	yes	—
158	C₂²	C₂²×#13083574	1	4	yes	—
159	C₄	—	2	16	no	—
160	C₄	—	2	8	no	—
161	C₄	—	356	16	no	—
162	C₄	C₂×#15668004	1	8	yes	—
163	C₂	#26028639	1	8	yes	direct product factor
164	C₂	C₂²×#15668004	1	1	yes	characteristic
165	C₂	C₂×#19152132	1	4	yes	—
166	C₂	C₂×#19152155	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator, cosocle	C₂³×C₈₉	C₂⁵
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	C₂×#2864
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. #26028639	C₂	direct (N × H)
80. C₂×C₈₉	#2864	semidirect
89. C₈₉	C₂×#2864	semidirect
163. C₂	#26028639	direct (N × H)

Maximal subgroups

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

subgroup	index
#26028639	2
C₂×#19151496	2
C₂×#19151482	2
C₂×#19151464	2
C₂×C₄³×C₈₉	2
C₂×#2864	89

Subgroups — C2×#26028639 #36760452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26028639 #36760452