Subgroups — C₂×#26022538 #36709925

Order 22784.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 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	C₂×#19152000	C₂	1	1	yes	maximal, characteristic
3	#26022538	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151426	C₂	1	2	yes	maximal
5	C₂×#19151424	C₂	1	2	yes	maximal
6	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668012	—	2	1	no	—
8	C₂×#15668004	—	2	1	no	—
9	#19152003	—	2	2	no	—
10	#19152003	C₂²	1	4	yes	—
11	#19152000	—	2	2	no	—
12	#19152000	C₂²	1	4	yes	—
13	C₂×#15667946	—	2	1	no	—
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667946	C₂²	1	2	yes	—
16	C₂×#15667946	C₂²	1	1	yes	characteristic
17	C₂×#15667944	—	2	1	no	—
18	#19151426	C₂²	1	8	yes	—
19	#19151424	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#346×C₈₉	C₂²	1	4	yes	—
23	C₂²×Dic₁₇₈	—	4	1	no	—
24	#15668012	—	2	4	no	—
25	#15668012	—	4	2	no	—
26	#15668004	—	2	4	no	—
27	#15668004	—	4	2	no	—
28	C₂×C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	—	2	2	no	—
31	C₂×C₄⋊Dic₈₉	—	4	1	no	—
32	C₂×#13083553	—	4	1	no	—
33	C₂×#13083553	—	4	1	no	—
34	#15667946	—	2	4	no	—
35	#15667946	—	2	1	no	—
36	#15667946	—	2	1	no	—
37	#15667946	—	2	2	no	—
38	#15667946	C₂³	1	2	yes	—
39	#15667946	C₂³	1	4	yes	—
40	#15667946	C₂³	1	2	yes	—
41	#15667944	—	2	4	no	—
42	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₈₉	Q₈	1	8	yes	complemented
46	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
47	C₄×C₈×C₈₉	D₄	1	2	yes	—
48	C₄×C₈×C₈₉	C₂³	1	2	yes	—
49	C₂×Dic₁₇₈	—	4	4	no	—
50	C₂×Dic₁₇₈	—	8	4	no	—
51	C₂²×Dic₈₉	—	4	2	no	—
52	C₂²×Dic₈₉	—	4	1	no	—
53	C₂²×Dic₈₉	—	4	1	no	—
54	C₄⋊Dic₈₉	—	2	2	no	—
55	C₄⋊Dic₈₉	—	2	2	no	—
56	C₄⋊Dic₈₉	—	2	4	no	—
57	C₄⋊Dic₈₉	—	4	1	no	—
58	C₄⋊Dic₈₉	—	4	1	no	—
59	C₄⋊Dic₈₉	—	4	4	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	#13083553	—	4	4	no	—
62	#13083553	—	4	4	no	—
63	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
71	C₄²×C₈₉	D₈	1	2	yes	—
72	C₄²×C₈₉	C₂×D₄	1	2	yes	—
73	C₁₆×C₈₉	C₂×Q₈	1	8	yes	complemented
74	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	—	2	2	no	—
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₈₉	—	4	2	no	—
84	C₂×Dic₈₉	—	4	2	no	—
85	C₂×Dic₈₉	—	4	4	no	—
86	C₂×Dic₈₉	—	8	2	no	—
87	C₂×Dic₈₉	—	8	4	no	—
88	C₂×Dic₈₉	—	8	2	no	—
89	Dic₁₇₈	—	8	8	no	—
90	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
91	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
92	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
93	C₂²×C₈₉	#500	1	2	yes	—
94	C₂²×C₈₉	#509	1	2	yes	—
95	C₂²×C₈₉	#510	1	2	yes	—
96	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	—	2	2	no	—
99	C₄×C₈₉	#500	1	2	yes	—
100	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₈₉	—	8	4	no	—
103	Dic₈₉	—	8	2	no	—
104	Dic₈₉	—	8	2	no	—
105	C₂×#2255	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	#2255	1	4	yes	complemented
107	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
108	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
109	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
110	C₂×#500	—	89	1	no	—
111	C₂×#499	—	89	1	no	—
112	#2255	—	89	8	no	—
113	C₂×#367	—	89	2	no	—
114	C₂×#366	—	89	2	no	—
115	C₂×#346	D₈₉	1	1	yes	characteristic
116	C₈₉	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊Q₈	—	178	1	no	—
118	C₂×#125	—	178	1	no	—
119	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
120	#500	—	89	4	no	—
121	#500	—	178	2	no	—
122	#499	—	89	4	no	—
123	#499	—	178	2	no	—
124	C₂×#107	—	89	1	no	—
125	C₂×#107	—	89	1	no	—
126	C₂×#107	—	89	2	no	—
127	C₂×#107	—	178	1	no	—
128	C₂×#106	—	178	1	no	—
129	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
130	#367	—	89	8	no	—
131	#366	—	89	8	no	—
132	#346	C₂×D₈₉	1	4	yes	—
133	C₂²×Q₈	—	356	1	no	—
134	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	356	2	no	—
137	C₄⋊Q₈	—	178	4	no	—
138	#125	—	356	2	no	—
139	#125	—	178	4	no	—
140	C₂×C₄⋊C₄	—	356	1	no	—
141	C₂×C₄⋊C₄	—	356	1	no	—
142	C₂×C₄⋊C₄	—	356	1	no	—
143	C₂×C₄⋊C₄	—	178	1	no	—
144	C₂×C₄⋊C₄	—	178	1	no	—
145	C₂×C₄⋊C₄	—	178	2	no	—
146	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
147	C₂×C₁₆	Dic₁₇₈	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
149	#107	—	89	2	no	—
150	#107	—	89	4	no	—
151	#107	—	89	2	no	—
152	#107	—	178	1	no	—
153	#107	—	178	1	no	—
154	#107	—	178	2	no	—
155	#107	—	178	4	no	—
156	#106	—	178	4	no	—
157	C₄×C₈	D₃₅₆	1	2	yes	—
158	C₄×C₈	C₂²×D₈₉	1	2	yes	—
159	C₂×Q₈	—	356	4	no	—
160	C₂×Q₈	—	712	4	no	—
161	C₂²×C₄	—	356	1	no	—
162	C₂²×C₄	—	356	1	no	—
163	C₂²×C₄	—	356	2	no	—
164	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
165	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
167	C₂×C₈	Dic₃₅₆	1	4	yes	—
168	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
169	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
170	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
171	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
172	C₄⋊C₄	—	356	4	no	—
173	C₄⋊C₄	—	356	4	no	—
174	C₄⋊C₄	—	356	4	no	—
175	C₄⋊C₄	—	356	1	no	—
176	C₄⋊C₄	—	356	1	no	—
177	C₄⋊C₄	—	356	2	no	—
178	C₄⋊C₄	—	178	2	no	—
179	C₄⋊C₄	—	178	2	no	—
180	C₄⋊C₄	—	178	4	no	—
181	C₄²	D₇₁₂	1	2	yes	—
182	C₄²	C₂×D₃₅₆	1	2	yes	—
183	C₁₆	C₂×Dic₁₇₈	1	8	yes	complemented
184	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
185	Q₈	—	712	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	356	2	no	—
189	C₂×C₄	—	356	2	no	—
190	C₂×C₄	—	356	4	no	—
191	C₂×C₄	—	712	2	no	—
192	C₂×C₄	—	712	4	no	—
193	C₂×C₄	—	712	2	no	—
194	C₂×C₄	#15668012	1	2	yes	—
195	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
196	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
197	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
199	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
200	C₈	#15668012	1	2	yes	—
201	C₈	C₂×Dic₃₅₆	1	4	yes	—
202	C₈	C₂²×Dic₁₇₈	1	2	yes	—
203	C₂²	#19152003	1	2	yes	—
204	C₂²	#19152036	1	2	yes	—
205	C₂²	#19152041	1	2	yes	—
206	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	712	4	no	—
210	C₄	—	712	2	no	—
211	C₄	—	712	2	no	—
212	C₄	#19152003	1	2	yes	—
213	C₄	C₂×#15668012	1	1	yes	characteristic
214	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
215	C₂	#26022538	1	4	yes	direct product factor
216	C₂	C₂×#19152003	1	1	yes	characteristic
217	C₂	C₂×#19152036	1	1	yes	characteristic
218	C₂	C₂×#19152041	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	C₂×#2255
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #26022538	C₂	direct (N × H)
45. C₂×C₁₆×C₈₉	Q₈	semidirect
73. C₁₆×C₈₉	C₂×Q₈	semidirect
106. C₂×C₈₉	#2255	semidirect
116. C₈₉	C₂×#2255	semidirect
147. C₂×C₁₆	Dic₁₇₈	semidirect
183. C₁₆	C₂×Dic₁₇₈	semidirect
215. C₂	#26022538	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
C₂×#19152000	2
#26022538	2
C₂×#19151426	2
C₂×#19151424	2
C₂×#346×C₈₉	2
C₂×#2255	89

Subgroups — C2×#26022538 #36709925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022538 #36709925