Subgroups — C₂×#26018306 #36465705

Order 22784.

350 conjugacy classes of subgroups.

The subgroup lattice has 218 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×#19151351	C₂	1	1	yes	maximal, characteristic
3	#26018306	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	—	2	1	no	—
6	#19152003	—	2	1	no	—
7	#19152003	—	2	1	no	—
8	#19152003	C₄	1	2	yes	—
9	#19152003	C₂²	1	2	yes	—
10	C₂×#15667946	—	2	1	no	—
11	C₂×#15667946	C₄	1	2	yes	—
12	C₂×C₄⋊#10623494	—	2	1	no	—
13	#19151351	—	2	1	no	—
14	#19151351	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₈₉	—	2	1	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₈₉	—	2	1	no	—
18	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂²×Dic₁₇₈	—	4	1	no	—
20	C₂²×#10623494	—	4	1	no	—
21	#15668012	—	2	2	no	—
22	#15668012	—	2	2	no	—
23	#15668012	—	4	1	no	—
24	#15668012	—	4	1	no	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	—	4	1	no	—
27	#15667946	—	2	2	no	—
28	#15667946	—	2	2	no	—
29	#15667946	—	2	2	no	—
30	#15667946	C₂×C₄	1	4	yes	—
31	C₄⋊#10623494	—	2	2	no	—
32	C₄⋊#10623494	—	4	1	no	—
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₂×C₁₆×C₈₉	—	4	2	no	—
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×Dic₁₇₈	—	4	2	no	—
43	C₂×Dic₁₇₈	—	4	2	no	—
44	C₂×Dic₁₇₈	—	8	2	no	—
45	C₂×Dic₁₇₈	—	8	2	no	—
46	C₂²×Dic₈₉	—	4	2	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₂×#10623494	—	8	2	no	—
49	C₄⋊Dic₈₉	—	2	4	no	—
50	C₄⋊Dic₈₉	—	4	2	no	—
51	C₄⋊Dic₈₉	—	4	2	no	—
52	C₄⋊Dic₈₉	—	4	2	no	—
53	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
60	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
61	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₈₉	D₈	1	1	yes	characteristic
65	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₁₆×C₈₉	—	4	2	no	—
68	C₂³×C₈₉	#102	1	1	yes	characteristic
69	C₂×C₄×C₈₉	—	2	2	no	—
70	C₂×C₄×C₈₉	#101	1	2	yes	—
71	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
72	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
74	C₂×C₄×C₈₉	#103	1	2	yes	—
75	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
76	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
77	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×Dic₈₉	—	4	4	no	—
81	C₂×Dic₈₉	—	8	4	no	—
82	Dic₁₇₈	—	8	4	no	—
83	Dic₁₇₈	—	8	2	no	—
84	Dic₁₇₈	—	8	2	no	—
85	C₈×C₈₉	—	2	1	no	—
86	C₈×C₈₉	—	2	1	no	—
87	C₈×C₈₉	#103	1	2	yes	—
88	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
90	#10623494	—	8	2	no	—
91	C₂²×C₈₉	#332	1	2	yes	—
92	C₂²×C₈₉	#357	1	2	yes	—
93	C₂²×C₈₉	#362	1	2	yes	—
94	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
95	C₄×C₈₉	—	2	1	no	—
96	C₄×C₈₉	—	2	1	no	—
97	C₄×C₈₉	#332	1	2	yes	—
98	C₄×C₈₉	#357	1	2	yes	—
99	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
100	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
101	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
102	C₄×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
103	Dic₈₉	—	8	4	no	—
104	C₂×#1364	—	89	1	no	maximal, Sylow
105	C₂×C₈₉	#1364	1	4	yes	complemented
106	C₂×C₈₉	C₂×#332	1	1	yes	characteristic
107	C₂×C₈₉	C₂×#357	1	1	yes	characteristic
108	C₂×C₈₉	C₂×#362	1	1	yes	characteristic
109	C₂×#500	—	89	1	no	—
110	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
111	C₂×#335	—	89	1	no	—
112	#1364	—	89	4	no	—
113	C₈₉	C₂×#1364	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄⋊Q₈	—	178	1	no	—
115	C₂²×C₁₆	—	2	1	no	—
116	#500	—	89	2	no	—
117	#500	—	89	2	no	—
118	#500	—	178	1	no	—
119	#500	—	178	1	no	—
120	C₂×#107	—	89	2	no	—
121	C₂×#107	—	178	1	no	—
122	C₂×C₄⋊C₈	—	178	1	no	—
123	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
124	C₄⋊C₁₆	—	2	1	no	—
125	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
126	#335	—	89	2	no	—
127	#335	—	178	1	no	—
128	C₂²×Q₈	—	356	1	no	—
129	C₂²×C₈	—	356	1	no	—
130	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
132	C₄⋊Q₈	—	356	1	no	—
133	C₄⋊Q₈	—	356	1	no	—
134	C₄⋊Q₈	—	178	2	no	—
135	C₄⋊Q₈	—	178	2	no	—
136	C₂×C₄⋊C₄	—	356	1	no	—
137	C₂×C₄⋊C₄	—	178	2	no	—
138	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	4	2	no	—
141	#107	—	89	4	no	—
142	#107	—	178	2	no	—
143	#107	—	178	2	no	—
144	#107	—	178	2	no	—
145	C₄⋊C₈	—	356	1	no	—
146	C₄⋊C₈	—	178	2	no	—
147	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
148	C₄×C₈	D₃₅₆	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
150	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
151	C₂×Q₈	—	356	2	no	—
152	C₂×Q₈	—	356	2	no	—
153	C₂×Q₈	—	712	2	no	—
154	C₂×Q₈	—	712	2	no	—
155	C₂²×C₄	—	356	2	no	—
156	C₂²×C₄	#13083551	1	1	yes	characteristic
157	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
158	C₂²×C₄	#13083563	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	356	2	no	—
161	C₂×C₈	—	712	2	no	—
162	C₂×C₈	#13083551	1	1	yes	characteristic
163	C₂×C₈	#13083551	1	1	yes	characteristic
164	C₂×C₈	#13083562	1	2	yes	—
165	C₂×C₈	#13083564	1	2	yes	—
166	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
168	C₄⋊C₄	—	356	2	no	—
169	C₄⋊C₄	—	356	2	no	—
170	C₄⋊C₄	—	356	2	no	—
171	C₄⋊C₄	—	178	4	no	—
172	C₄²	#13083551	1	1	yes	characteristic
173	C₄²	D₇₁₂	1	1	yes	characteristic
174	C₄²	#13083563	1	1	yes	characteristic
175	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
176	C₁₆	—	4	2	no	—
177	C₂³	#15667934	1	1	yes	characteristic
178	Q₈	—	712	4	no	—
179	Q₈	—	712	2	no	—
180	Q₈	—	712	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	356	4	no	—
183	C₂×C₄	—	712	4	no	—
184	C₂×C₄	#15667930	1	2	yes	—
185	C₂×C₄	#15667934	1	1	yes	characteristic
186	C₂×C₄	#15667934	1	1	yes	characteristic
187	C₂×C₄	#15667934	1	1	yes	characteristic
188	C₂×C₄	#15667935	1	2	yes	—
189	C₂×C₄	D₁₄₂₄	1	2	yes	—
190	C₂×C₄	#15667957	1	2	yes	—
191	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
192	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
193	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	712	2	no	—
197	C₈	#15667935	1	2	yes	—
198	C₈	C₂×#13083562	1	1	yes	characteristic
199	C₈	C₂×#13083564	1	1	yes	characteristic
200	C₂²	#19151343	1	2	yes	—
201	C₂²	#19151400	1	2	yes	—
202	C₂²	#19151415	1	2	yes	—
203	C₂²	C₂×#15667934	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	712	4	no	—
207	C₄	#19151343	1	2	yes	—
208	C₄	#19151400	1	2	yes	—
209	C₄	C₂×#15667930	1	1	yes	characteristic
210	C₄	C₂×#15667935	1	1	yes	characteristic
211	C₄	C₂×D₁₄₂₄	1	1	yes	characteristic
212	C₄	C₂×#15667957	1	1	yes	characteristic
213	C₂	#26018306	1	4	yes	direct product factor
214	C₂	C₂×#19151343	1	1	yes	characteristic
215	C₂	C₂×#19151400	1	1	yes	characteristic
216	C₂	C₂×#19151415	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667934
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#102
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	C₂×#1364
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #26018306	C₂	direct (N × H)
105. C₂×C₈₉	#1364	semidirect
113. C₈₉	C₂×#1364	semidirect
213. C₂	#26018306	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×#19151351	2
#26018306	2
C₂×C₄⋊C₁₆×C₈₉	2
C₂×#1364	89

Subgroups — C2×#26018306 #36465705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018306 #36465705