Subgroups — C₂×#26022539 #36709926

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	#26022539	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×#19151374	C₂	1	1	yes	maximal, characteristic
6	C₂×#499×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668012	—	2	1	no	—
8	C₂²×#13083549	C₂²	1	2	yes	—
9	#19152003	—	2	2	no	—
10	#19152003	C₂²	1	4	yes	—
11	C₂×#15667946	—	2	1	no	—
12	C₂×#15667946	C₂²	1	2	yes	—
13	#19151425	C₂²	1	8	yes	—
14	#19151423	C₂²	1	8	yes	—
15	#19151374	C₂²	1	4	yes	—
16	C₂×#125×C₈₉	—	2	1	no	—
17	#499×C₈₉	—	2	2	no	—
18	#499×C₈₉	C₂²	1	4	yes	—
19	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#106×C₈₉	—	2	1	no	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₇₈	—	4	1	no	—
24	#15668012	—	2	4	no	—
25	#15668012	—	4	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	—	4	1	no	—
28	C₂×#13083549	Q₈	1	8	yes	complemented
29	C₂×#13083549	C₂³	1	4	yes	—
30	#15667946	—	2	4	no	—
31	#15667946	—	2	2	no	—
32	#15667946	C₂³	1	4	yes	—
33	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	#125×C₈₉	—	2	4	no	—
36	#125×C₈₉	—	4	2	no	—
37	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
40	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	#107×C₈₉	—	2	1	no	—
43	#107×C₈₉	—	2	1	no	—
44	#107×C₈₉	C₂³	1	2	yes	—
45	#107×C₈₉	C₂³	1	2	yes	—
46	#106×C₈₉	—	2	4	no	—
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	C₄⋊Dic₈₉	—	2	4	no	—
53	C₄⋊Dic₈₉	—	4	4	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	#13083549	C₂×Q₈	1	8	yes	complemented
56	C₂²×C₄×C₈₉	—	4	1	no	—
57	C₂²×C₄×C₈₉	—	4	1	no	—
58	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
62	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₈₉	—	2	2	no	—
67	C₄⋊C₄×C₈₉	—	2	2	no	—
68	C₄⋊C₄×C₈₉	—	4	1	no	—
69	C₄⋊C₄×C₈₉	—	4	1	no	—
70	C₄⋊C₄×C₈₉	—	4	4	no	—
71	C₄⋊C₄×C₈₉	—	4	4	no	—
72	C₄²×C₈₉	D₈	1	2	yes	—
73	C₄²×C₈₉	C₂×D₄	1	2	yes	—
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	—	4	2	no	—
78	C₂×C₄×C₈₉	—	4	2	no	—
79	C₂×C₄×C₈₉	—	8	2	no	—
80	C₂×C₄×C₈₉	—	8	2	no	—
81	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
87	C₂×Dic₈₉	—	4	4	no	—
88	C₂×Dic₈₉	—	8	4	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	—	8	2	no	—
100	C₄×C₈₉	—	8	2	no	—
101	C₄×C₈₉	#500	1	2	yes	—
102	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₈₉	—	8	4	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	D₈₉	1	1	yes	characteristic
112	#2255	—	89	8	no	—
113	C₂×#367	—	89	2	no	—
114	C₂×#366	—	89	2	no	—
115	C₂×#346	—	89	1	no	—
116	C₈₉	C₂×#2255	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊Q₈	—	178	1	no	—
118	C₂×#125	—	2	1	no	—
119	C₂²×C₁₆	—	89	2	no	—
120	#500	—	89	4	no	—
121	#500	—	178	2	no	—
122	#499	—	2	2	no	—
123	#499	C₂×D₈₉	1	4	yes	—
124	C₂×#107	—	89	2	no	—
125	C₂×#107	—	178	1	no	—
126	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
127	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
128	C₂×#106	—	2	1	no	—
129	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
130	#367	—	89	8	no	—
131	#366	—	89	8	no	—
132	#346	—	89	4	no	—
133	C₂²×Q₈	—	356	1	no	—
134	C₂²×C₈	C₂²⋊D₈₉	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	—	2	4	no	—
139	#125	—	4	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄⋊C₄	—	356	1	no	—
145	C₂×C₄⋊C₄	—	178	2	no	—
146	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂×C₁₆	—	89	8	no	—
148	C₂×C₁₆	—	89	4	no	—
149	#107	—	2	1	no	—
150	#107	—	2	1	no	—
151	#107	—	89	4	no	—
152	#107	—	178	2	no	—
153	#107	—	178	4	no	—
154	#107	C₂²×D₈₉	1	2	yes	—
155	#107	C₂²×D₈₉	1	2	yes	—
156	#106	—	2	4	no	—
157	C₄×C₈	C₂²⋊D₈₉	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	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	—	356	2	no	—
164	C₂²×C₄	#13083559	1	1	yes	characteristic
165	C₂²×C₄	#13083559	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
167	C₂×C₈	#13083564	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
170	C₂×C₈	Q₈×D₈₉	1	4	yes	—
171	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	356	4	no	—
179	C₄⋊C₄	—	356	2	no	—
180	C₄⋊C₄	—	178	4	no	—
181	C₄²	#13083559	1	2	yes	—
182	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
183	C₁₆	—	89	8	no	—
184	C₂³	C₂×#13083559	1	1	yes	characteristic
185	Q₈	—	712	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	356	4	no	—
193	C₂×C₄	—	712	4	no	—
194	C₂×C₄	#15668010	1	2	yes	—
195	C₂×C₄	C₂×#13083559	1	2	yes	—
196	C₂×C₄	C₂×#13083559	1	2	yes	—
197	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
198	C₂×C₄	C₂×#13083564	1	2	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
200	C₈	#15668010	1	2	yes	—
201	C₈	C₂×#13083564	1	4	yes	—
202	C₈	C₂×Q₈×D₈₉	1	2	yes	—
203	C₂²	#19152001	1	2	yes	—
204	C₂²	#19152033	1	2	yes	—
205	C₂²	#19152038	1	2	yes	—
206	C₂²	C₂²×#13083559	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	2	no	—
211	C₄	—	712	4	no	—
212	C₄	#19152001	1	2	yes	—
213	C₄	C₂×#15668010	1	1	yes	characteristic
214	C₄	C₂²×#13083564	1	1	yes	characteristic
215	C₂	#26022539	1	4	yes	direct product factor
216	C₂	C₂×#19152001	1	1	yes	characteristic
217	C₂	C₂×#19152033	1	1	yes	characteristic
218	C₂	C₂×#19152038	1	1	yes	characteristic

	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×#2255
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. #26022539	C₂	direct (N × H)
28. C₂×#13083549	Q₈	semidirect
55. #13083549	C₂×Q₈	semidirect
106. C₂×C₈₉	#2255	semidirect
116. C₈₉	C₂×#2255	semidirect
215. C₂	#26022539	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
#26022539	2
C₂×#19151425	2
C₂×#19151423	2
C₂×#19151374	2
C₂×#499×C₈₉	2
C₂×#2255	89

Subgroups — C2×#26022539 #36709926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022539 #36709926