Subgroups — C₂×#26022012 #36707412

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#13083549	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151553	C₂	1	1	yes	maximal, characteristic
3	#26022012	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151416	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₄	1	1	yes	characteristic
7	C₂×#15667971	C₄	1	1	yes	characteristic
8	C₂²×#13083549	—	2	4	no	—
9	C₂²×#13083549	C₂²	1	2	yes	—
10	C₂²×#13083549	C₂²	1	4	yes	—
11	#19151553	C₄	1	4	yes	—
12	#19151553	C₂²	1	4	yes	—
13	C₂×#15667922	C₄	1	2	yes	—
14	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
15	#19151416	C₂²	1	16	yes	complemented
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₈₉	—	2	4	no	—
18	C₂×M₃₂×C₈₉	C₂²	1	4	yes	—
19	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂×#13083558	—	2	4	no	—
21	C₂×#13083558	C₂×C₄	1	4	yes	—
22	C₂²×#10623494	C₂×C₄	1	2	yes	—
23	#15667971	C₈	1	4	yes	—
24	#15667971	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₈₉	C₈	1	2	yes	—
26	C₂×C₂²⋊Dic₈₉	C₈	1	2	yes	—
27	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
28	C₂×#13083549	—	2	4	no	—
29	C₂×#13083549	—	2	8	no	—
30	C₂×#13083549	C₂³	1	4	yes	—
31	#15667922	C₂×C₄	1	8	yes	—
32	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
33	C₈×Dic₈₉	C₂³	1	4	yes	—
34	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₈₉	D₄	1	4	yes	—
38	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
39	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	M₃₂×C₈₉	—	2	8	no	—
42	C₂×C₁₆×C₈₉	—	2	4	no	—
43	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
44	C₂²×Dic₈₉	C₂×C₈	1	4	yes	—
45	#13083558	—	2	8	no	—
46	C₂×#10623494	—	2	4	no	—
47	C₂×#10623494	C₂²×C₄	1	4	yes	—
48	C₄⋊Dic₈₉	C₂×C₈	1	8	yes	—
49	C₂²⋊Dic₈₉	C₂×C₈	1	8	yes	—
50	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
51	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
52	#13083549	—	2	4	no	—
53	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
61	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
62	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
67	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
69	C₁₆×C₈₉	—	2	4	no	—
70	C₂³×C₈₉	—	2	4	no	—
71	C₂³×C₈₉	C₄⋊C₈	1	4	yes	—
72	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
73	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₄⋊C₈	1	8	yes	—
75	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
76	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
83	C₂×Dic₈₉	—	2	8	no	—
84	C₂×Dic₈₉	C₂²×C₈	1	8	yes	—
85	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
86	C₈×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
87	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
89	#10623494	—	2	4	no	—
90	C₂²×C₈₉	—	2	4	no	—
91	C₂²×C₈₉	—	2	8	no	—
92	C₂²×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
93	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
94	C₂²×C₈₉	#504	1	4	yes	—
95	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
97	C₄×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
98	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
99	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
100	Dic₈₉	—	2	8	no	—
101	C₂×#2151	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	—	2	4	no	—
103	C₂×C₈₉	#2151	1	4	yes	complemented
104	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₈₉	C₂×#504	1	2	yes	—
106	C₂²×M₃₂	D₈₉	1	1	yes	characteristic
107	C₂³×C₁₆	—	89	1	no	—
108	C₂×#405	—	89	1	no	—
109	#2151	—	89	8	no	—
110	C₂×C₄⋊C₁₆	—	89	4	no	—
111	C₈₉	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
112	C₂²×M₁₆	—	89	1	no	—
113	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
114	C₂×#117	—	89	1	no	—
115	C₂×M₃₂	—	2	4	no	—
116	C₂×M₃₂	C₂×D₈₉	1	4	yes	—
117	C₂²×C₁₆	—	89	4	no	—
118	C₂²×C₁₆	—	89	2	no	—
119	C₂²×C₁₆	—	178	4	no	—
120	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
121	#405	—	89	4	no	—
122	#405	—	89	4	no	—
123	C₂×#97	—	89	2	no	—
124	C₂×C₄×C₈	—	89	2	no	—
125	C₄⋊C₁₆	—	89	16	no	—
126	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
127	C₂×M₁₆	—	89	4	no	—
128	C₂×M₁₆	—	178	4	no	—
129	C₂²×C₈	—	89	2	no	—
130	C₂²×C₈	Dic₁₇₈	1	4	yes	—
131	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₈₉	1	2	yes	—
133	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
134	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
135	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
136	#117	—	89	4	no	—
137	#117	—	89	4	no	—
138	C₂×C₄⋊C₄	—	89	2	no	—
139	C₂×C₂²⋊C₄	—	89	2	no	—
140	C₂×C₄²	—	89	2	no	—
141	M₃₂	—	2	8	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	—	89	4	no	—
144	C₂×C₁₆	—	178	4	no	—
145	C₂×C₁₆	—	178	8	no	—
146	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
147	#97	—	89	8	no	—
148	C₄×C₈	—	89	4	no	—
149	C₄×C₈	—	89	4	no	—
150	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
151	C₂²×C₄	—	89	4	no	—
152	C₂²×C₄	#13083553	1	4	yes	—
153	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
154	C₂²×C₄	C₈×D₈₉	1	2	yes	—
155	C₂²×C₄	#13083557	1	4	yes	—
156	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
158	M₁₆	—	178	8	no	—
159	C₂×C₈	—	89	4	no	—
160	C₂×C₈	—	178	4	no	—
161	C₂×C₈	#13083553	1	4	yes	—
162	C₂×C₈	#13083553	1	8	yes	—
163	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
164	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
165	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
166	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
167	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
168	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
169	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
170	C₄⋊C₄	—	89	8	no	—
171	C₂²⋊C₄	—	89	8	no	—
172	C₄²	—	89	4	no	—
173	C₄²	—	89	4	no	—
174	C₁₆	—	2	4	no	—
175	C₁₆	—	178	4	no	—
176	C₂³	—	2	4	no	—
177	C₂³	#15667941	1	4	yes	—
178	C₂³	C₂×C₈×D₈₉	1	2	yes	—
179	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
180	C₂×C₄	—	89	8	no	—
181	C₂×C₄	—	178	8	no	—
182	C₂×C₄	#15667941	1	8	yes	—
183	C₂×C₄	#15667941	1	4	yes	—
184	C₂×C₄	C₂×#13083553	1	4	yes	—
185	C₂×C₄	C₂×#13083553	1	2	yes	—
186	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
188	C₂×C₄	C₂×#13083557	1	2	yes	—
189	C₂×C₄	C₂×#13083557	1	4	yes	—
190	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
191	C₈	—	178	4	no	—
192	C₈	C₂×#13083553	1	2	yes	—
193	C₈	C₂×#13083553	1	4	yes	—
194	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₂×#15667941	1	4	yes	—
199	C₂²	C₂×#15667941	1	2	yes	—
200	C₂²	#19152016	1	4	yes	—
201	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
202	C₄	—	178	8	no	—
203	C₄	C₂×#15667941	1	2	yes	—
204	C₄	C₂×#15667941	1	4	yes	—
205	C₄	C₂²×#13083553	1	1	yes	characteristic
206	C₄	C₂²×#13083557	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#26022012	1	4	yes	direct product factor
209	C₂	C₂²×#15667941	1	1	yes	characteristic
210	C₂	C₂×#19152016	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂²×M₃₂×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×C₈
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂²×M₃₂	D₈₉
hypocenter	C₈₉	C₂×#2151
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #26022012	C₂	direct (N × H)
4. C₂×#19151416	C₂	semidirect
15. #19151416	C₂²	semidirect
103. C₂×C₈₉	#2151	semidirect
111. C₈₉	C₂×#2151	semidirect
208. C₂	#26022012	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#13083549	2
C₂×#19151553	2
#26022012	2
C₂×#19151416	2
C₂²×M₃₂×C₈₉	2
C₂×#2151	89

Subgroups — C2×#26022012 #36707412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022012 #36707412