Subgroups — C₂×#26022614 #36712648

Order 22784.

4694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃₅₆	C₂	1	3	yes	maximal
2	#26022614	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊Dic₈₉	C₂	1	3	yes	maximal, complemented
4	C₂×#19151445	C₂	1	6	yes	maximal, complemented
5	C₂×#95×D₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#376×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₅₆	C₂²	1	1	yes	characteristic
8	C₂³×C₄×D₈₉	C₂²	1	6	yes	—
9	C₂×C₄×D₃₅₆	C₄	1	48	yes	complemented
10	C₂×C₄×D₃₅₆	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₈₉	C₂²	1	3	yes	—
12	C₂²×#13083551	C₂²	1	6	yes	—
13	C₂²×C₄×Dic₈₉	C₂²	1	6	yes	—
14	C₄×C₄⋊Dic₈₉	C₂²	1	24	yes	complemented
15	#19151445	C₂²	1	48	yes	complemented
16	C₂×#15667919	C₂²	1	6	yes	—
17	#95×D₈₉	C₂²	1	16	yes	complemented
18	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
19	#376×C₈₉	C₂²	1	8	yes	complemented
20	C₂×#95×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#95×C₈₉	C₂²	1	3	yes	—
22	C₂⁴×D₈₉	C₂³	1	2	yes	—
23	C₂²×D₃₅₆	C₂×C₄	1	24	yes	—
24	C₂²×D₃₅₆	C₂³	1	4	yes	—
25	C₂³×Dic₈₉	C₂³	1	6	yes	—
26	C₂²×C₄×D₈₉	—	4	24	no	—
27	C₂²×C₄×D₈₉	C₂×C₄	1	48	yes	complemented
28	C₂²×C₄×D₈₉	C₂³	1	24	yes	—
29	C₄×D₃₅₆	C₂×C₄	1	192	yes	complemented
30	C₂×C₄⋊Dic₈₉	C₂×C₄	1	24	yes	complemented
31	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	—
32	C₂×#13083551	C₂×C₄	1	48	yes	complemented
33	C₂×#13083551	C₂³	1	24	yes	—
34	C₂×C₄×Dic₈₉	—	4	12	no	—
35	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
36	#15667919	C₂³	1	24	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
38	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
39	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂×C₄	1	24	yes	complemented
41	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
42	#95×C₈₉	C₂³	1	4	yes	—
43	#95×C₈₉	C₂³	1	12	yes	—
44	C₂³×D₈₉	—	2	12	no	—
45	C₂³×D₈₉	—	2	4	no	—
46	C₂³×D₈₉	C₂²×C₄	1	24	yes	—
47	C₂³×D₈₉	C₂⁴	1	4	yes	—
48	C₂×D₃₅₆	C₄²	1	64	yes	complemented
49	C₂×D₃₅₆	C₂²×C₄	1	48	yes	—
50	C₂²×Dic₈₉	—	2	24	no	—
51	C₂²×Dic₈₉	C₂²×C₄	1	24	yes	—
52	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
53	C₂×C₄×D₈₉	—	4	96	no	—
54	C₂×C₄×D₈₉	—	4	48	no	—
55	C₂×C₄×D₈₉	C₂²×C₄	1	96	yes	complemented
56	C₄⋊Dic₈₉	C₂²×C₄	1	48	yes	complemented
57	#13083551	C₂²×C₄	1	96	yes	complemented
58	C₄×Dic₈₉	—	4	24	no	—
59	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	—	2	12	no	—
61	C₂²×C₄×C₈₉	—	2	6	no	—
62	C₂²×C₄×C₈₉	—	2	6	no	—
63	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
66	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
69	C₄²×C₈₉	C₂²×C₄	1	48	yes	complemented
70	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
71	C₂³×C₈₉	H(5,𝔽₂)	1	6	yes	—
72	C₂³×C₈₉	2₋¹⁺⁴	1	2	yes	—
73	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
74	C₂²×D₈₉	—	2	48	no	—
75	C₂²×D₈₉	—	2	48	no	—
76	C₂²×D₈₉	—	2	12	no	—
77	C₂²×D₈₉	—	2	4	no	—
78	C₂²×D₈₉	C₂×C₄²	1	32	yes	—
79	C₂²×D₈₉	C₂³×C₄	1	24	yes	—
80	C₂×C₄×C₈₉	—	2	24	no	—
81	C₂×C₄×C₈₉	—	2	24	no	—
82	C₂×C₄×C₈₉	—	2	6	no	—
83	C₂×C₄×C₈₉	—	2	12	no	—
84	C₂×C₄×C₈₉	—	2	6	no	—
85	C₂×C₄×C₈₉	C₂×C₄²	1	16	yes	—
86	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
87	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
89	C₂×Dic₈₉	—	2	48	no	—
90	C₂×Dic₈₉	—	2	24	no	—
91	C₂×Dic₈₉	C₂³×C₄	1	24	yes	—
92	D₃₅₆	C₂×C₄²	1	64	yes	complemented
93	C₄×D₈₉	—	4	96	no	—
94	C₂²×C₈₉	C₂²×C₄²	1	4	yes	—
95	C₂²×C₈₉	#518	1	12	yes	—
96	C₂²×C₈₉	#520	1	12	yes	—
97	C₂²×C₈₉	C₂⁴×C₄	1	3	yes	—
98	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	3	yes	—
99	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
100	C₂×D₈₉	—	2	24	no	—
101	C₂×D₈₉	—	2	24	no	—
102	C₂×D₈₉	—	2	64	no	—
103	C₂×D₈₉	C₂²×C₄²	1	16	yes	—
104	C₄×C₈₉	—	2	12	no	—
105	C₄×C₈₉	—	2	12	no	—
106	C₄×C₈₉	C₂²×C₄²	1	8	yes	—
107	Dic₈₉	—	2	24	no	—
108	C₂×#2275	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	#2275	1	8	yes	complemented
110	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
111	C₂×C₈₉	C₂×#518	1	3	yes	—
112	C₂×C₈₉	C₂×#520	1	3	yes	—
113	D₈₉	—	2	16	no	—
114	C₂²×C₄×D₄	—	89	3	no	—
115	#2275	—	89	16	no	—
116	C₂×C₄×C₄⋊C₄	—	89	3	no	—
117	C₂×C₄×C₂²⋊C₄	—	89	6	no	—
118	C₂×#376	D₈₉	1	1	yes	characteristic, complemented
119	C₂²×#95	—	89	2	no	—
120	C₈₉	C₂×#2275	1	1	yes	characteristic, complemented, Sylow
121	C₂³×D₄	—	89	1	no	—
122	C₂⁴×C₄	—	89	6	no	—
123	C₂×C₄×D₄	—	89	48	no	—
124	C₂×C₄×D₄	—	89	24	no	—
125	C₂²×C₄⋊C₄	—	89	3	no	—
126	C₂²×C₂²⋊C₄	—	89	6	no	—
127	C₂²×C₄²	—	89	6	no	—
128	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
129	C₄×C₄⋊C₄	—	89	24	no	—
130	C₄×C₂²⋊C₄	—	89	48	no	—
131	#376	C₂×D₈₉	1	8	yes	complemented
132	C₂×#95	—	89	6	no	—
133	C₂×#95	—	89	16	no	—
134	C₂×#95	C₂×D₈₉	1	1	yes	characteristic
135	C₂×#95	C₂×D₈₉	1	3	yes	—
136	C₂⁵	—	89	2	no	—
137	C₂²×D₄	—	89	24	no	—
138	C₂²×D₄	—	89	4	no	—
139	C₂³×C₄	—	89	6	no	—
140	C₂³×C₄	—	89	48	no	—
141	C₂³×C₄	—	89	24	no	—
142	C₂³×C₄	—	356	24	no	—
143	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
144	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
145	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
146	C₄×D₄	—	89	192	no	—
147	C₂×C₄⋊C₄	—	89	24	no	—
148	C₂×C₄⋊C₄	—	89	12	no	—
149	C₂×C₂²⋊C₄	—	89	48	no	—
150	C₂×C₂²⋊C₄	—	89	24	no	—
151	C₂×C₄²	—	89	24	no	—
152	C₂×C₄²	—	356	12	no	—
153	C₂×C₄²	C₄×D₈₉	1	24	yes	complemented
154	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
155	#95	—	89	24	no	—
156	#95	C₂²×D₈₉	1	4	yes	—
157	#95	C₂²×D₈₉	1	12	yes	—
158	C₂⁴	—	89	24	no	—
159	C₂⁴	—	89	4	no	—
160	C₂⁴	—	178	12	no	—
161	C₂⁴	—	178	4	no	—
162	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
163	C₂×D₄	—	89	48	no	—
164	C₂×D₄	—	89	64	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	89	24	no	—
169	C₂²×C₄	—	89	12	no	—
170	C₂²×C₄	—	89	96	no	—
171	C₂²×C₄	—	356	48	no	—
172	C₂²×C₄	—	356	96	no	—
173	C₂²×C₄	—	178	24	no	—
174	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
175	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
176	C₂²×C₄	C₂×C₄×D₈₉	1	12	yes	—
177	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
178	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
179	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
180	C₄⋊C₄	—	89	48	no	—
181	C₂²⋊C₄	—	89	96	no	—
182	C₄²	—	356	24	no	—
183	C₄²	C₂×C₄×D₈₉	1	48	yes	complemented
184	C₂³	—	89	24	no	—
185	C₂³	—	89	32	no	—
186	C₂³	—	178	12	no	—
187	C₂³	—	178	4	no	—
188	C₂³	—	178	48	no	—
189	C₂³	—	178	48	no	—
190	C₂³	C₂²×C₄×D₈₉	1	6	yes	—
191	C₂³	#15668058	1	6	yes	—
192	C₂³	#15668059	1	2	yes	—
193	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
194	D₄	—	89	64	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	2	12	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	24	no	—
199	C₂×C₄	—	2	24	no	—
200	C₂×C₄	—	89	24	no	—
201	C₂×C₄	—	356	96	no	—
202	C₂×C₄	—	178	24	no	—
203	C₂×C₄	—	178	48	no	—
204	C₂×C₄	C₄²×D₈₉	1	16	yes	—
205	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
206	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
207	C₂×C₄	C₂²×C₄×D₈₉	1	12	yes	—
208	C₂²	—	89	16	no	—
209	C₂²	—	178	24	no	—
210	C₂²	—	178	24	no	—
211	C₂²	—	178	64	no	—
212	C₂²	C₂×C₄²×D₈₉	1	4	yes	—
213	C₂²	#19152072	1	12	yes	—
214	C₂²	#19152078	1	12	yes	—
215	C₂²	C₂³×C₄×D₈₉	1	3	yes	—
216	C₂²	C₂×#15668058	1	3	yes	—
217	C₂²	C₂×#15668059	1	1	yes	characteristic
218	C₄	—	2	12	no	—
219	C₄	—	2	12	no	—
220	C₄	—	178	24	no	—
221	C₄	C₂×C₄²×D₈₉	1	8	yes	—
222	C₂	—	178	16	no	—
223	C₂	#26022614	1	8	yes	direct product factor
224	C₂	C₂²×C₄²×D₈₉	1	1	yes	characteristic
225	C₂	C₂×#19152072	1	3	yes	—
226	C₂	C₂×#19152078	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×#376×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂⁴
prime radical	C₂³×D₃₅₆	C₂²
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂×#376	D₈₉
hypocenter	C₈₉	C₂×#2275
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
2. #26022614	C₂	direct (N × H)
3. C₂×C₄×C₄⋊Dic₈₉	C₂	semidirect
4. C₂×#19151445	C₂	semidirect
5. C₂×#95×D₈₉	C₂	semidirect
6. C₂×#376×C₈₉	C₂	semidirect
9. C₂×C₄×D₃₅₆	C₄	semidirect
14. C₄×C₄⋊Dic₈₉	C₂²	semidirect
15. #19151445	C₂²	semidirect
17. #95×D₈₉	C₂²	semidirect
19. #376×C₈₉	C₂²	semidirect
27. C₂²×C₄×D₈₉	C₂×C₄	semidirect
29. C₄×D₃₅₆	C₂×C₄	semidirect
30. C₂×C₄⋊Dic₈₉	C₂×C₄	semidirect
32. C₂×#13083551	C₂×C₄	semidirect
40. C₂×C₄²×C₈₉	C₂×C₄	semidirect
48. C₂×D₃₅₆	C₄²	semidirect
55. C₂×C₄×D₈₉	C₂²×C₄	semidirect
56. C₄⋊Dic₈₉	C₂²×C₄	semidirect
57. #13083551	C₂²×C₄	semidirect
69. C₄²×C₈₉	C₂²×C₄	semidirect
92. D₃₅₆	C₂×C₄²	semidirect
109. C₂×C₈₉	#2275	semidirect
118. C₂×#376	D₈₉	semidirect
120. C₈₉	C₂×#2275	semidirect
131. #376	C₂×D₈₉	semidirect
153. C₂×C₄²	C₄×D₈₉	semidirect
183. C₄²	C₂×C₄×D₈₉	semidirect
223. C₂	#26022614	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₃₅₆	2
#26022614	2
C₂×C₄×C₄⋊Dic₈₉	2
C₂×#19151445	2
C₂×#95×D₈₉	2
C₂×#376×C₈₉	2
C₂×#2275	89

Subgroups — C2×#26022614 #36712648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022614 #36712648