Subgroups — C₂×#26022525 #36709903

Order 22784.

558 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152003	C₂	1	1	yes	maximal, characteristic
2	#26022525	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151423	C₂	1	4	yes	maximal
4	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	—	2	1	no	—
7	C₂²×#13083549	C₂²	1	2	yes	—
8	#19152003	—	2	2	no	—
9	#19152003	C₂²	1	4	yes	—
10	C₂×#15667946	—	2	1	no	—
11	C₂×#15667946	C₂²	1	2	yes	—
12	#19151423	C₂²	1	16	yes	—
13	C₄×#13083549	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
15	#500×C₈₉	—	2	2	no	—
16	#500×C₈₉	C₂²	1	4	yes	—
17	C₂×#107×C₈₉	—	2	1	no	—
18	C₂×#107×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₇₈	—	4	1	no	—
21	#15668012	—	2	4	no	—
22	#15668012	—	4	2	no	—
23	C₂×C₄⋊Dic₈₉	—	2	2	no	—
24	C₂×C₄⋊Dic₈₉	—	4	1	no	—
25	C₂×#13083549	Q₈	1	8	yes	complemented
26	C₂×#13083549	C₂³	1	4	yes	—
27	#15667946	—	2	4	no	—
28	#15667946	—	2	2	no	—
29	#15667946	C₂³	1	4	yes	—
30	C₂²×Q₈×C₈₉	—	4	1	no	—
31	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₈₉	—	2	4	no	—
34	C₄⋊Q₈×C₈₉	—	4	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
37	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
38	#107×C₈₉	—	2	4	no	—
39	#107×C₈₉	—	2	2	no	—
40	#107×C₈₉	C₂³	1	4	yes	—
41	C₄×C₈×C₈₉	D₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂³	1	2	yes	—
43	C₂×Dic₁₇₈	—	4	4	no	—
44	C₂×Dic₁₇₈	—	8	4	no	—
45	C₂²×Dic₈₉	—	4	2	no	—
46	C₄⋊Dic₈₉	—	2	4	no	—
47	C₄⋊Dic₈₉	—	4	4	no	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	#13083549	C₂×Q₈	1	8	yes	complemented
50	C₂×Q₈×C₈₉	—	4	4	no	—
51	C₂×Q₈×C₈₉	—	8	4	no	—
52	C₂²×C₄×C₈₉	—	4	2	no	—
53	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
57	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₈₉	C₂×Q₈	1	4	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	4	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	Q₈×C₈₉	—	8	8	no	—
68	C₂×C₄×C₈₉	—	4	4	no	—
69	C₂×C₄×C₈₉	—	8	4	no	—
70	C₂×C₄×C₈₉	D₁₆	1	4	yes	—
71	C₂×C₄×C₈₉	Q₃₂	1	4	yes	—
72	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
77	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₈₉	—	4	4	no	—
79	C₂×Dic₈₉	—	8	4	no	—
80	Dic₁₇₈	—	8	8	no	—
81	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
82	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
83	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
84	C₂²×C₈₉	#500	1	2	yes	—
85	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
86	C₂²×C₈₉	C₂×Q₃₂	1	2	yes	—
87	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₈₉	—	8	4	no	—
89	C₄×C₈₉	#500	1	2	yes	—
90	C₄×C₈₉	C₂×D₁₆	1	4	yes	—
91	C₄×C₈₉	C₂×Q₃₂	1	4	yes	—
92	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
94	Dic₈₉	—	8	4	no	—
95	C₂×#2252	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	#2252	1	4	yes	complemented
97	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
98	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
99	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
100	C₂×#500	—	89	1	no	—
101	C₂×#500	D₈₉	1	1	yes	characteristic
102	#2252	—	89	8	no	—
103	C₂×#366	—	89	4	no	—
104	C₂×C₄×C₁₆	—	89	1	no	—
105	C₈₉	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×C₄⋊Q₈	—	178	1	no	—
108	C₂²×C₁₆	—	89	2	no	—
109	#500	—	2	2	no	—
110	#500	—	89	4	no	—
111	#500	—	178	2	no	—
112	#500	C₂×D₈₉	1	4	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	—	89	2	no	—
115	C₂×#107	—	178	1	no	—
116	C₂×#107	C₂×D₈₉	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
118	#366	—	89	16	no	—
119	C₄×C₁₆	—	89	4	no	—
120	C₂²×Q₈	—	4	1	no	—
121	C₂²×Q₈	—	356	1	no	—
122	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	C₄⋊Q₈	—	356	2	no	—
127	C₄⋊Q₈	—	178	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	356	1	no	—
131	C₂×C₄⋊C₄	—	178	2	no	—
132	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
133	C₂×C₁₆	—	89	8	no	—
134	C₂×C₁₆	—	89	4	no	—
135	#107	—	2	4	no	—
136	#107	—	2	2	no	—
137	#107	—	89	4	no	—
138	#107	—	178	2	no	—
139	#107	—	178	4	no	—
140	#107	C₂²×D₈₉	1	4	yes	—
141	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
142	C₄×C₈	C₂²×D₈₉	1	2	yes	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	8	4	no	—
145	C₂×Q₈	—	356	4	no	—
146	C₂×Q₈	—	712	4	no	—
147	C₂²×C₄	—	4	2	no	—
148	C₂²×C₄	—	356	2	no	—
149	C₂²×C₄	#13083559	1	1	yes	characteristic
150	C₂²×C₄	#13083559	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
152	C₂×C₈	#13083564	1	4	yes	—
153	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
154	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
155	C₂×C₈	Q₈×D₈₉	1	4	yes	—
156	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	356	4	no	—
161	C₄⋊C₄	—	356	2	no	—
162	C₄⋊C₄	—	178	4	no	—
163	C₄²	#13083559	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
165	C₁₆	—	89	8	no	—
166	C₂³	C₂×#13083559	1	1	yes	characteristic
167	Q₈	—	8	8	no	—
168	Q₈	—	712	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	—	356	4	no	—
172	C₂×C₄	—	712	4	no	—
173	C₂×C₄	#15667953	1	4	yes	—
174	C₂×C₄	#15667958	1	4	yes	—
175	C₂×C₄	#15668010	1	2	yes	—
176	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
177	C₂×C₄	C₂×#13083559	1	2	yes	—
178	C₂×C₄	C₂×#13083559	1	2	yes	—
179	C₂×C₄	C₂×#13083564	1	2	yes	—
180	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
181	C₈	#15668010	1	2	yes	—
182	C₈	C₂×#13083564	1	4	yes	—
183	C₈	C₂×Q₈×D₈₉	1	2	yes	—
184	C₂²	#19152001	1	2	yes	—
185	C₂²	C₂×#15667953	1	2	yes	—
186	C₂²	C₂×#15667958	1	2	yes	—
187	C₂²	C₂²×#13083559	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	712	4	no	—
190	C₄	#19152001	1	2	yes	—
191	C₄	C₂×#15667953	1	4	yes	—
192	C₄	C₂×#15667958	1	4	yes	—
193	C₄	C₂×#15668010	1	1	yes	characteristic
194	C₄	C₂²×#13083564	1	1	yes	characteristic
195	C₂	#26022525	1	4	yes	direct product factor
196	C₂	C₂×#19152001	1	1	yes	characteristic
197	C₂	C₂²×#15667953	1	1	yes	characteristic
198	C₂	C₂²×#15667958	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#500×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#500	D₈₉
hypocenter	C₈₉	C₂×#2252
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #26022525	C₂	direct (N × H)
25. C₂×#13083549	Q₈	semidirect
49. #13083549	C₂×Q₈	semidirect
96. C₂×C₈₉	#2252	semidirect
105. C₈₉	C₂×#2252	semidirect
195. C₂	#26022525	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×#19152003	2
#26022525	2
C₂×#19151423	2
C₂×C₄×#13083549	2
C₂×#500×C₈₉	2
C₂×#2252	89

Subgroups — C2×#26022525 #36709903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022525 #36709903