Subgroups — C₂×#26022481 #36709718

Order 22784.

622 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151978	C₂	1	1	yes	maximal, characteristic
2	#26022481	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151401	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	C₂²×Dic₃₅₆	—	2	1	no	—
7	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
8	C₂×#15668012	—	2	1	no	—
9	C₂²×#13083549	C₂²	1	2	yes	—
10	#19151978	—	2	2	no	—
11	#19151978	C₂²	1	4	yes	—
12	#19151401	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	—	2	2	no	—
21	C₂×Dic₃₅₆	—	2	2	no	—
22	C₂×Dic₃₅₆	—	2	4	no	—
23	C₂×Dic₃₅₆	—	2	8	no	—
24	C₂×Dic₃₅₆	—	4	4	no	—
25	C₂×Dic₃₅₆	C₂³	1	4	yes	—
26	#15668012	—	2	4	no	—
27	#15668012	—	4	2	no	—
28	C₂×C₄⋊Dic₈₉	—	4	1	no	—
29	C₂×C₄⋊Dic₈₉	—	4	1	no	—
30	C₂×#13083549	—	2	4	no	—
31	C₂×#13083549	C₂³	1	4	yes	—
32	C₂²×Q₈×C₈₉	—	4	1	no	—
33	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₈₉	—	2	4	no	—
36	C₄⋊Q₈×C₈₉	—	4	2	no	—
37	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	#107×C₈₉	—	2	4	no	—
41	#107×C₈₉	—	2	2	no	—
42	#107×C₈₉	C₂³	1	4	yes	—
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	C₄×C₈×C₈₉	C₂³	1	2	yes	—
45	C₂×Dic₁₇₈	—	2	4	no	—
46	C₂×Dic₁₇₈	—	4	4	no	—
47	C₂×Dic₁₇₈	—	4	4	no	—
48	C₂×Dic₁₇₈	—	4	2	no	—
49	C₂²×Dic₈₉	—	4	2	no	—
50	Dic₃₅₆	—	2	8	no	—
51	Dic₃₅₆	—	4	4	no	—
52	Dic₃₅₆	—	4	8	no	—
53	C₄⋊Dic₈₉	—	4	4	no	—
54	C₄⋊Dic₈₉	—	4	4	no	—
55	#13083549	—	2	4	no	—
56	C₂×Q₈×C₈₉	—	4	4	no	—
57	C₂×Q₈×C₈₉	—	8	4	no	—
58	C₂²×C₄×C₈₉	—	4	2	no	—
59	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
63	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₈₉	#41	1	4	yes	—
66	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₈₉	—	2	4	no	—
68	C₄⋊C₄×C₈₉	—	4	2	no	—
69	C₄⋊C₄×C₈₉	—	4	4	no	—
70	C₄²×C₈₉	D₈	1	2	yes	—
71	C₄²×C₈₉	C₂×D₄	1	2	yes	—
72	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₈₉	—	8	8	no	—
74	C₂×C₄×C₈₉	—	4	4	no	—
75	C₂×C₄×C₈₉	—	8	4	no	—
76	C₂×C₄×C₈₉	SD₃₂	1	4	yes	—
77	C₂×C₄×C₈₉	Q₃₂	1	4	yes	—
78	C₂×C₄×C₈₉	#124	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₈₉	—	4	4	no	—
85	C₂×Dic₈₉	—	8	4	no	—
86	Dic₁₇₈	—	4	4	no	—
87	Dic₁₇₈	—	4	4	no	—
88	Dic₁₇₈	—	8	4	no	—
89	C₈×C₈₉	#124	1	2	yes	—
90	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
91	C₈×C₈₉	C₂×#41	1	2	yes	—
92	C₂²×C₈₉	#486	1	2	yes	—
93	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
94	C₂²×C₈₉	C₂×Q₃₂	1	2	yes	—
95	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
96	C₄×C₈₉	—	8	4	no	—
97	C₄×C₈₉	#486	1	2	yes	—
98	C₄×C₈₉	C₂×SD₃₂	1	4	yes	—
99	C₄×C₈₉	C₂×Q₃₂	1	4	yes	—
100	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
101	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
102	Dic₈₉	—	8	4	no	—
103	C₂×#2241	—	89	1	no	maximal, Sylow
104	C₂×C₈₉	#2241	1	4	yes	complemented
105	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
106	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
107	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
108	C₂×#500	D₈₉	1	1	yes	characteristic
109	C₂×#494	—	89	1	no	—
110	#2241	—	89	8	no	—
111	C₂×#358	—	89	4	no	—
112	C₂×C₄×C₁₆	—	89	1	no	—
113	C₈₉	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₁₆	—	89	2	no	—
115	C₂²×Q₁₆	—	178	1	no	—
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×C₄⋊Q₈	—	178	1	no	—
118	C₂²×C₁₆	—	89	2	no	—
119	#500	—	2	2	no	—
120	#500	C₂×D₈₉	1	4	yes	—
121	#494	—	89	4	no	—
122	#494	—	178	2	no	—
123	C₂×#107	—	2	1	no	—
124	C₂×#107	C₂×D₈₉	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
126	#358	—	89	16	no	—
127	C₄×C₁₆	—	89	4	no	—
128	C₂²×Q₈	—	4	1	no	—
129	C₂²×Q₈	—	178	2	no	—
130	C₂×Q₁₆	—	89	4	no	—
131	C₂×Q₁₆	—	356	4	no	—
132	C₂×Q₁₆	—	178	2	no	—
133	C₂×Q₁₆	—	178	4	no	—
134	C₂×Q₁₆	—	178	8	no	—
135	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	4	no	—
138	C₄⋊Q₈	—	4	2	no	—
139	C₄⋊Q₈	—	356	2	no	—
140	C₄⋊Q₈	—	178	4	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	356	1	no	—
144	C₂×C₄⋊C₄	—	356	1	no	—
145	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
146	C₂×C₁₆	—	89	4	no	—
147	C₂×C₁₆	—	178	4	no	—
148	#107	—	2	4	no	—
149	#107	—	2	2	no	—
150	#107	C₂²×D₈₉	1	4	yes	—
151	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
152	C₄×C₈	C₂²×D₈₉	1	2	yes	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂×Q₈	—	356	2	no	—
156	C₂×Q₈	—	356	4	no	—
157	C₂×Q₈	—	356	4	no	—
158	C₂×Q₈	—	178	4	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	356	2	no	—
161	C₂²×C₄	#13083559	1	1	yes	characteristic
162	C₂²×C₄	#13083559	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
164	Q₁₆	—	356	4	no	—
165	Q₁₆	—	356	8	no	—
166	Q₁₆	—	178	8	no	—
167	C₂×C₈	#13083561	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	#13083575	1	4	yes	—
171	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	356	4	no	—
176	C₄⋊C₄	—	356	4	no	—
177	C₄²	#13083559	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
179	C₁₆	—	178	4	no	—
180	C₂³	C₂×#13083559	1	1	yes	characteristic
181	Q₈	—	8	8	no	—
182	Q₈	—	356	4	no	—
183	Q₈	—	356	4	no	—
184	Q₈	—	712	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	356	4	no	—
188	C₂×C₄	—	712	4	no	—
189	C₂×C₄	#15667956	1	4	yes	—
190	C₂×C₄	#15667958	1	4	yes	—
191	C₂×C₄	#15668001	1	2	yes	—
192	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
193	C₂×C₄	C₂×#13083559	1	2	yes	—
194	C₂×C₄	C₂×#13083559	1	2	yes	—
195	C₂×C₄	C₂×#13083561	1	2	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
197	C₈	#15668001	1	2	yes	—
198	C₈	C₂×#13083561	1	4	yes	—
199	C₈	C₂×#13083575	1	2	yes	—
200	C₂²	#19151946	1	2	yes	—
201	C₂²	C₂×#15667956	1	2	yes	—
202	C₂²	C₂×#15667958	1	2	yes	—
203	C₂²	C₂²×#13083559	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	—	712	4	no	—
206	C₄	#19151946	1	2	yes	—
207	C₄	C₂×#15667956	1	4	yes	—
208	C₄	C₂×#15667958	1	4	yes	—
209	C₄	C₂×#15668001	1	1	yes	characteristic
210	C₄	C₂²×#13083561	1	1	yes	characteristic
211	C₂	#26022481	1	4	yes	direct product factor
212	C₂	C₂×#19151946	1	1	yes	characteristic
213	C₂	C₂²×#15667956	1	1	yes	characteristic
214	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×#2241
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. #26022481	C₂	direct (N × H)
104. C₂×C₈₉	#2241	semidirect
113. C₈₉	C₂×#2241	semidirect
211. C₂	#26022481	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×#19151978	2
#26022481	2
C₂×#19151401	2
C₂×C₄×#13083549	2
C₂×#500×C₈₉	2
C₂×#2241	89

Subgroups — C2×#26022481 #36709718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022481 #36709718