Subgroups — C₂×#26022528 #36709907

Order 22784.

494 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	C₂×#19152000	C₂	1	1	yes	maximal, characteristic
2	#26022528	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151425	C₂	1	2	yes	maximal
4	C₂×#19151423	C₂	1	2	yes	maximal
5	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
6	C₂×#499×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668004	—	2	1	no	—
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	#19152000	—	2	2	no	—
11	#19152000	C₂²	1	4	yes	—
12	C₂×#15667946	C₂²	1	2	yes	—
13	C₂×#15667944	—	2	1	no	—
14	#19151425	C₂²	1	8	yes	—
15	#19151423	C₂²	1	8	yes	—
16	C₄×#13083549	C₂²	1	4	yes	—
17	C₂×#125×C₈₉	—	2	1	no	—
18	#499×C₈₉	—	2	2	no	—
19	#499×C₈₉	C₂²	1	4	yes	—
20	C₂×#107×C₈₉	C₂²	1	2	yes	—
21	C₂×#106×C₈₉	—	2	1	no	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#15668004	—	2	4	no	—
24	#15668004	—	4	2	no	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₂×#13083553	—	4	2	no	—
27	C₂×#13083549	Q₈	1	4	yes	—
28	C₂×#13083549	Q₈	1	4	yes	—
29	C₂×#13083549	C₂³	1	2	yes	—
30	C₂×#13083549	C₂³	1	2	yes	—
31	#15667946	—	2	2	no	—
32	#15667946	C₂³	1	4	yes	—
33	#15667944	—	2	4	no	—
34	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	#125×C₈₉	—	2	4	no	—
37	#125×C₈₉	—	4	2	no	—
38	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	#107×C₈₉	—	2	2	no	—
42	#107×C₈₉	C₂³	1	4	yes	—
43	#106×C₈₉	—	2	4	no	—
44	C₄×C₈×C₈₉	D₄	1	2	yes	—
45	C₄×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂²×Dic₈₉	—	4	2	no	—
47	C₄⋊Dic₈₉	—	2	4	no	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	#13083553	—	4	8	no	—
50	#13083549	C₂×Q₈	1	4	yes	—
51	#13083549	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₈₉	—	4	2	no	—
53	C₂²×C₄×C₈₉	D₈	1	2	yes	—
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
56	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₈₉	—	2	4	no	—
62	C₄⋊C₄×C₈₉	—	4	2	no	—
63	C₄⋊C₄×C₈₉	—	4	8	no	—
64	C₄²×C₈₉	D₈	1	2	yes	—
65	C₄²×C₈₉	C₂×D₄	1	2	yes	—
66	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	—	2	4	no	—
68	C₂×C₄×C₈₉	—	4	4	no	—
69	C₂×C₄×C₈₉	—	8	4	no	—
70	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
72	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₈₉	—	4	4	no	—
76	C₂×Dic₈₉	—	8	4	no	—
77	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
78	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
79	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
80	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₈₉	#500	1	2	yes	—
82	C₂²×C₈₉	#508	1	4	yes	—
83	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
84	C₄×C₈₉	—	2	4	no	—
85	C₄×C₈₉	—	8	4	no	—
86	C₄×C₈₉	#500	1	2	yes	—
87	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
88	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
89	Dic₈₉	—	8	4	no	—
90	C₂×#2253	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	#2253	1	4	yes	complemented
92	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
93	C₂×C₈₉	C₂×#508	1	2	yes	—
94	C₂×#499	—	89	1	no	—
95	C₂×#499	D₈₉	1	1	yes	characteristic
96	#2253	—	89	8	no	—
97	C₂×#367	—	89	2	no	—
98	C₂×#366	—	89	2	no	—
99	C₂×C₄×C₁₆	—	89	1	no	—
100	C₈₉	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
101	C₂×#125	—	2	1	no	—
102	C₂×#125	—	178	1	no	—
103	C₂²×C₁₆	—	89	1	no	—
104	C₂²×C₁₆	—	89	1	no	—
105	#499	—	2	2	no	—
106	#499	—	89	4	no	—
107	#499	—	178	2	no	—
108	#499	C₂×D₈₉	1	4	yes	—
109	C₂×#107	—	89	2	no	—
110	C₂×#107	C₂×D₈₉	1	2	yes	—
111	C₂×#106	—	2	1	no	—
112	C₂×#106	—	178	1	no	—
113	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
114	#367	—	89	8	no	—
115	#366	—	89	8	no	—
116	C₄×C₁₆	—	89	4	no	—
117	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
119	#125	—	2	4	no	—
120	#125	—	4	2	no	—
121	#125	—	356	2	no	—
122	#125	—	178	4	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	4	2	no	—
125	C₂×C₄⋊C₄	—	356	2	no	—
126	C₂×C₄⋊C₄	—	178	2	no	—
127	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
128	C₂×C₁₆	—	89	4	no	—
129	C₂×C₁₆	—	89	4	no	—
130	C₂×C₁₆	—	89	2	no	—
131	C₂×C₁₆	—	89	2	no	—
132	#107	—	2	2	no	—
133	#107	—	89	4	no	—
134	#107	—	178	2	no	—
135	#107	C₂²×D₈₉	1	4	yes	—
136	#106	—	2	4	no	—
137	#106	—	178	4	no	—
138	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
139	C₄×C₈	C₂²×D₈₉	1	2	yes	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	356	2	no	—
142	C₂²×C₄	#13083559	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
144	C₂×C₈	#13083564	1	4	yes	—
145	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
146	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
147	C₂×C₈	Q₈×D₈₉	1	2	yes	—
148	C₂×C₈	Q₈×D₈₉	1	2	yes	—
149	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	356	8	no	—
154	C₄⋊C₄	—	356	2	no	—
155	C₄⋊C₄	—	178	4	no	—
156	C₄²	#13083559	1	2	yes	—
157	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
158	C₁₆	—	89	4	no	—
159	C₁₆	—	89	4	no	—
160	C₂³	C₂×#13083559	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	356	4	no	—
165	C₂×C₄	—	712	4	no	—
166	C₂×C₄	#15668010	1	2	yes	—
167	C₂×C₄	C₂×#13083559	1	4	yes	—
168	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
169	C₂×C₄	C₂×#13083564	1	2	yes	—
170	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
171	C₈	#15668010	1	2	yes	—
172	C₈	C₂×#13083564	1	4	yes	—
173	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
174	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
175	C₂²	#19152001	1	2	yes	—
176	C₂²	#19152028	1	4	yes	—
177	C₂²	C₂²×#13083559	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	712	4	no	—
181	C₄	#19152001	1	2	yes	—
182	C₄	C₂×#15668010	1	1	yes	characteristic
183	C₄	C₂²×#13083564	1	1	yes	characteristic
184	C₂	#26022528	1	4	yes	direct product factor
185	C₂	C₂×#19152001	1	1	yes	characteristic
186	C₂	C₂×#19152028	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#499×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#499	D₈₉
hypocenter	C₈₉	C₂×#2253
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #26022528	C₂	direct (N × H)
91. C₂×C₈₉	#2253	semidirect
100. C₈₉	C₂×#2253	semidirect
184. C₂	#26022528	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×#19152000	2
#26022528	2
C₂×#19151425	2
C₂×#19151423	2
C₂×C₄×#13083549	2
C₂×#499×C₈₉	2
C₂×#2253	89

Subgroups — C2×#26022528 #36709907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022528 #36709907