Subgroups — C₂×#26019850 #36542078

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151994	C₂	1	2	yes	maximal
2	C₂×#19151978	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151948	C₂	1	2	yes	maximal
4	#26019850	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
6	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₅₆	—	2	2	no	—
8	C₂×#15668012	C₂²	1	2	yes	—
9	#19151994	—	2	4	no	—
10	#19151994	C₂²	1	8	yes	—
11	#19151978	—	2	2	no	—
12	#19151978	C₂²	1	4	yes	—
13	#19151948	—	2	4	no	—
14	#19151948	C₂²	1	8	yes	—
15	C₂×#15667943	—	2	4	no	—
16	C₂×#15667936	—	2	4	no	—
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
19	C₈×#10623494	C₂²	1	4	yes	—
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
21	#500×C₈₉	—	2	2	no	—
22	#500×C₈₉	C₂²	1	4	yes	—
23	C₂×#107×C₈₉	—	2	2	no	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₇₈	—	2	2	no	—
26	C₂×Dic₃₅₆	—	2	8	no	—
27	C₂×Dic₃₅₆	—	4	8	no	—
28	C₂²×#10623494	D₄	1	2	yes	—
29	C₂²×#10623494	D₄	1	2	yes	—
30	#15668012	—	2	2	no	—
31	#15668012	—	2	2	no	—
32	#15668012	—	2	2	no	—
33	#15668012	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₈₉	—	2	2	no	—
35	C₂×C₄⋊Dic₈₉	—	2	2	no	—
36	#15667943	—	2	16	no	—
37	#15667936	—	2	16	no	—
38	C₄×#10623494	D₄	1	2	yes	—
39	C₄×#10623494	D₄	1	2	yes	—
40	C₄×#10623494	C₂³	1	2	yes	—
41	C₄×#10623494	C₂³	1	2	yes	—
42	C₂²×Q₈×C₈₉	—	2	2	no	—
43	C₂²×C₈×C₈₉	D₄	1	2	yes	—
44	C₄⋊Q₈×C₈₉	—	2	2	no	—
45	C₄⋊Q₈×C₈₉	—	2	2	no	—
46	C₄⋊Q₈×C₈₉	—	2	2	no	—
47	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	#107×C₈₉	—	2	8	no	—
52	C₄×C₈×C₈₉	D₄	1	2	yes	—
53	C₄×C₈×C₈₉	C₂³	1	2	yes	—
54	C₂×Dic₁₇₈	—	2	4	no	—
55	C₂×Dic₁₇₈	—	4	8	no	—
56	C₂×Dic₁₇₈	—	4	2	no	—
57	C₂²×Dic₈₉	—	4	2	no	—
58	Dic₃₅₆	—	4	16	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	Q₁₆	1	8	yes	complemented
61	C₂×#10623494	C₂×D₄	1	4	yes	—
62	C₂×#10623494	C₂×D₄	1	4	yes	—
63	C₄⋊Dic₈₉	—	2	4	no	—
64	C₄⋊Dic₈₉	—	2	4	no	—
65	C₄⋊Dic₈₉	—	4	2	no	—
66	C₄⋊Dic₈₉	—	4	2	no	—
67	C₂×Q₈×C₈₉	—	2	4	no	—
68	C₂×Q₈×C₈₉	—	4	2	no	—
69	C₂×Q₈×C₈₉	—	4	8	no	—
70	C₂²×C₄×C₈₉	—	4	2	no	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	SD₁₆	1	8	yes	—
75	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
76	C₄⋊C₄×C₈₉	—	2	4	no	—
77	C₄⋊C₄×C₈₉	—	2	4	no	—
78	C₄⋊C₄×C₈₉	—	4	2	no	—
79	C₄⋊C₄×C₈₉	—	4	2	no	—
80	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₂³×C₈₉	#127	1	1	yes	characteristic
85	Q₈×C₈₉	—	4	8	no	—
86	Q₈×C₈₉	—	8	4	no	—
87	C₂×C₄×C₈₉	—	4	4	no	—
88	C₂×C₄×C₈₉	—	8	4	no	—
89	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
93	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
94	C₂×C₄×C₈₉	#135	1	4	yes	—
95	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₈₉	—	4	4	no	—
99	C₂×Dic₈₉	—	8	4	no	—
100	Dic₁₇₈	—	4	8	no	—
101	Dic₁₇₈	—	8	4	no	—
102	C₈×C₈₉	C₂×SD₁₆	1	8	yes	—
103	#10623494	—	2	4	no	—
104	#10623494	C₂×Q₁₆	1	8	yes	complemented
105	C₂²×C₈₉	#492	1	2	yes	—
106	C₂²×C₈₉	#494	1	2	yes	—
107	C₂²×C₈₉	#495	1	2	yes	—
108	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
109	C₄×C₈₉	—	8	4	no	—
110	C₄×C₈₉	#492	1	2	yes	—
111	C₄×C₈₉	#494	1	2	yes	—
112	C₄×C₈₉	#495	1	2	yes	—
113	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
114	C₄×C₈₉	C₂²×Q₁₆	1	2	yes	—
115	C₄×C₈₉	C₂×#135	1	2	yes	—
116	Dic₈₉	—	8	4	no	—
117	C₂×#1707	—	89	1	no	maximal, Sylow
118	C₂×C₈₉	#1707	1	4	yes	complemented
119	C₂×C₈₉	C₂×#492	1	1	yes	characteristic
120	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
121	C₂×C₈₉	C₂×#495	1	1	yes	characteristic
122	C₂×#500	D₈₉	1	1	yes	characteristic
123	C₂×#498	—	89	2	no	—
124	C₂×#494	—	89	1	no	—
125	C₂×#487	—	89	2	no	—
126	#1707	—	89	8	no	—
127	C₂×C₈²	—	89	1	no	—
128	C₈₉	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	—	178	2	no	—
130	C₂×C₄⋊Q₈	—	89	2	no	—
131	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
132	#500	—	2	2	no	—
133	#500	C₂×D₈₉	1	4	yes	—
134	#498	—	89	8	no	—
135	#498	—	178	4	no	—
136	#494	—	89	4	no	—
137	#494	—	178	2	no	—
138	#487	—	89	8	no	—
139	#487	—	178	4	no	—
140	C₂×#107	—	2	2	no	—
141	C₂×#106	—	178	4	no	—
142	C₂×#103	—	178	4	no	—
143	C₂×C₄×C₈	—	89	1	no	—
144	C₂×C₄×C₈	—	89	1	no	—
145	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
146	C₈²	—	89	4	no	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	178	2	no	—
149	C₂×Q₁₆	—	356	8	no	—
150	C₂×Q₁₆	—	178	8	no	—
151	C₂²×C₈	—	89	2	no	—
152	C₂²×C₈	—	89	2	no	—
153	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	89	4	no	—
158	C₄⋊Q₈	—	178	2	no	—
159	C₄⋊Q₈	—	178	2	no	—
160	C₄⋊Q₈	—	178	2	no	—
161	C₄⋊Q₈	C₂²×D₈₉	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	178	2	no	—
165	C₂×C₄⋊C₄	—	178	2	no	—
166	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
167	#107	—	2	8	no	—
168	#106	—	178	16	no	—
169	#103	—	178	16	no	—
170	C₄×C₈	—	89	2	no	—
171	C₄×C₈	—	89	2	no	—
172	C₄×C₈	—	89	2	no	—
173	C₄×C₈	—	89	2	no	—
174	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
175	C₄×C₈	C₂²×D₈₉	1	2	yes	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	8	no	—
179	C₂×Q₈	—	356	2	no	—
180	C₂×Q₈	—	356	8	no	—
181	C₂×Q₈	—	178	4	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	356	2	no	—
184	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
187	Q₁₆	—	356	16	no	—
188	C₂×C₈	—	89	8	no	—
189	C₂×C₈	—	89	4	no	—
190	C₂×C₈	—	89	4	no	—
191	C₂×C₈	—	178	4	no	—
192	C₂×C₈	#13083561	1	8	yes	—
193	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	356	2	no	—
199	C₄⋊C₄	—	356	2	no	—
200	C₄⋊C₄	—	178	4	no	—
201	C₄⋊C₄	—	178	4	no	—
202	C₄²	D₄×D₈₉	1	1	yes	characteristic
203	C₄²	D₄×D₈₉	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
205	C₄²	C₂³×D₈₉	1	1	yes	characteristic
206	C₂³	#15668008	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	8	4	no	—
209	Q₈	—	356	8	no	—
210	Q₈	—	712	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	356	4	no	—
214	C₂×C₄	—	712	4	no	—
215	C₂×C₄	#15668008	1	1	yes	characteristic
216	C₂×C₄	#15668008	1	1	yes	characteristic
217	C₂×C₄	#15668008	1	1	yes	characteristic
218	C₂×C₄	C₂×#13083561	1	4	yes	—
219	C₂×C₄	Q₁₆×D₈₉	1	4	yes	—
220	C₂×C₄	#15668031	1	4	yes	—
221	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
224	C₈	—	89	8	no	—
225	C₈	—	178	4	no	—
226	C₈	C₂×#13083561	1	8	yes	—
227	C₂²	#19151972	1	2	yes	—
228	C₂²	#19151976	1	2	yes	—
229	C₂²	#19151981	1	2	yes	—
230	C₂²	C₂×#15668008	1	1	yes	characteristic
231	C₄	—	8	4	no	—
232	C₄	—	712	4	no	—
233	C₄	#19151972	1	2	yes	—
234	C₄	#19151976	1	2	yes	—
235	C₄	#19151981	1	2	yes	—
236	C₄	C₂²×#13083561	1	2	yes	—
237	C₄	C₂×Q₁₆×D₈₉	1	2	yes	—
238	C₄	C₂×#15668031	1	2	yes	—
239	C₂	#26019850	1	4	yes	direct product factor
240	C₂	C₂×#19151972	1	1	yes	characteristic
241	C₂	C₂×#19151976	1	1	yes	characteristic
242	C₂	C₂×#19151981	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#500×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#127
hypercenter	C₂×#500	D₈₉
hypocenter	C₈₉	C₂×#1707
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #26019850	C₂	direct (N × H)
60. C₂×#10623494	Q₁₆	semidirect
104. #10623494	C₂×Q₁₆	semidirect
118. C₂×C₈₉	#1707	semidirect
128. C₈₉	C₂×#1707	semidirect
239. C₂	#26019850	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×#19151994	2
C₂×#19151978	2
C₂×#19151948	2
#26019850	2
C₂×C₈×#10623494	2
C₂×#500×C₈₉	2
C₂×#1707	89

Subgroups — C2×#26019850 #36542078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019850 #36542078