Subgroups — C₂×#26020122 #36552373

Order 22784.

1310 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667922	C₂	1	2	yes	maximal
2	C₂²×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#26020122	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151352	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10623494	C₂²	1	1	yes	characteristic
8	C₂³×#10623494	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₈₉	C₄	1	2	yes	—
10	C₂²×C₄×Dic₈₉	C₂²	1	2	yes	—
11	C₂×#15667922	—	2	8	no	—
12	C₂×#15667922	C₂²	1	8	yes	—
13	C₂×C₄×#10623494	C₄	1	8	yes	complemented
14	C₂×C₄×#10623494	C₂²	1	4	yes	—
15	C₄×C₄⋊Dic₈₉	C₄	1	4	yes	—
16	C₄×C₄⋊Dic₈₉	C₂²	1	4	yes	—
17	C₂×#15667919	C₄	1	2	yes	—
18	#19151352	C₂²	1	8	yes	—
19	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₈₉	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
23	C₂³×Dic₈₉	C₂×C₄	1	4	yes	—
24	C₂²×#10623494	—	2	2	no	—
25	C₂²×#10623494	—	2	2	no	—
26	C₂²×#10623494	C₂×C₄	1	4	yes	—
27	C₂²×#10623494	C₂×C₄	1	4	yes	—
28	C₂²×#10623494	D₄	1	4	yes	—
29	C₂²×#10623494	Q₈	1	4	yes	—
30	C₂²×#10623494	C₂³	1	2	yes	—
31	C₂²×#10623494	C₂³	1	2	yes	—
32	C₂×C₄⋊Dic₈₉	—	2	8	no	—
33	C₂×C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
34	C₂×C₄×Dic₈₉	—	2	4	no	—
35	C₂×C₄×Dic₈₉	—	2	4	no	—
36	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₈₉	C₂³	1	4	yes	—
38	#15667922	—	2	16	no	—
39	C₄×#10623494	C₂×C₄	1	16	yes	complemented
40	#15667919	C₂×C₄	1	8	yes	—
41	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	—	2	4	no	—
45	C₂²×C₈×C₈₉	—	2	4	no	—
46	C₂²×C₈×C₈₉	C₂×C₄	1	8	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
52	C₄⋊C₈×C₈₉	C₂×C₄	1	16	yes	complemented
53	C₂²×Dic₈₉	—	2	8	no	—
54	C₂²×Dic₈₉	—	2	8	no	—
55	C₂²×Dic₈₉	—	2	8	no	—
56	C₂²×Dic₈₉	C₂²×C₄	1	8	yes	—
57	C₂×#10623494	—	2	8	no	—
58	C₂×#10623494	—	2	2	no	—
59	C₂×#10623494	—	2	2	no	—
60	C₂×#10623494	C₄⋊C₄	1	16	yes	complemented
61	C₂×#10623494	C₂²×C₄	1	4	yes	—
62	C₂×#10623494	C₂²×C₄	1	4	yes	—
63	C₂×#10623494	C₂×D₄	1	4	yes	—
64	C₂×#10623494	C₂×Q₈	1	4	yes	—
65	C₄⋊Dic₈₉	—	2	16	no	—
66	C₄×Dic₈₉	—	2	8	no	—
67	C₄×Dic₈₉	—	2	4	no	—
68	C₄×Dic₈₉	—	2	4	no	—
69	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	—	2	4	no	—
71	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
72	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
73	C₂²×C₄×C₈₉	M₁₆	1	8	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₈₉	—	2	16	no	—
85	C₂×C₈×C₈₉	—	2	4	no	—
86	C₂×C₈×C₈₉	—	2	4	no	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
88	C₄²×C₈₉	C₄²	1	8	yes	—
89	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
90	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
91	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
92	C₂³×C₈₉	C₄×D₄	1	2	yes	—
93	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
94	C₂³×C₈₉	C₂×M₁₆	1	4	yes	—
95	C₂³×C₈₉	#131	1	4	yes	—
96	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	2	8	no	—
99	C₂×C₄×C₈₉	#97	1	16	yes	complemented
100	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄²	1	4	yes	—
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₈₉	#117	1	4	yes	—
105	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
107	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₈₉	C₂×M₁₆	1	8	yes	—
110	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₈₉	—	2	8	no	—
117	C₂×Dic₈₉	—	2	8	no	—
118	C₂×Dic₈₉	—	2	8	no	—
119	C₂×Dic₈₉	—	4	16	no	—
120	C₈×C₈₉	—	2	8	no	—
121	#10623494	—	2	4	no	—
122	#10623494	C₂×C₄⋊C₄	1	8	yes	complemented
123	C₂²×C₈₉	C₄×C₄⋊C₄	1	4	yes	—
124	C₂²×C₈₉	C₂×#97	1	4	yes	—
125	C₂²×C₈₉	#405	1	4	yes	—
126	C₂²×C₈₉	#436	1	8	yes	—
127	C₂²×C₈₉	#446	1	8	yes	—
128	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
129	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₂²×C₈₉	C₂²×M₁₆	1	2	yes	—
132	C₂²×C₈₉	C₂×#131	1	2	yes	—
133	C₄×C₈₉	—	2	4	no	—
134	C₄×C₈₉	C₄×C₄⋊C₄	1	4	yes	—
135	C₄×C₈₉	C₂×#97	1	8	yes	complemented
136	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
138	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
140	Dic₈₉	—	4	8	no	—
141	C₂×#1770	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	#1770	1	8	yes	complemented
143	C₂×C₈₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₈₉	C₂²×#97	1	1	yes	characteristic
145	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
146	C₂×C₈₉	C₂×#436	1	2	yes	—
147	C₂×C₈₉	C₂×#446	1	2	yes	—
148	C₂²×C₄⋊C₈	D₈₉	1	1	yes	characteristic
149	C₂²×#97	—	89	2	no	—
150	C₂²×C₄×C₈	—	89	1	no	—
151	C₂×C₄×C₄⋊C₄	—	89	1	no	—
152	#1770	—	89	8	no	—
153	C₂×#336	—	89	2	no	—
154	C₈₉	C₂×#1770	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₈	—	89	1	no	—
156	C₂³×C₈	—	89	1	no	—
157	C₂³×C₈	C₂×D₈₉	1	2	yes	—
158	C₂²×C₄⋊C₄	—	89	2	no	—
159	C₂²×C₄²	—	89	2	no	—
160	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
161	C₂×C₄⋊C₈	Dic₈₉	1	8	yes	complemented
162	C₂×C₄⋊C₈	C₂×D₈₉	1	4	yes	—
163	C₂×#97	—	89	8	no	—
164	C₂×#97	—	178	8	no	—
165	C₂×C₄×C₈	—	89	8	no	—
166	C₂×C₄×C₈	—	89	4	no	—
167	C₄×C₄⋊C₄	—	89	4	no	—
168	C₄×C₄⋊C₄	—	89	4	no	—
169	C₂×#95	—	89	2	no	—
170	#336	—	89	8	no	—
171	C₂³×C₄	—	89	4	no	—
172	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
173	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	89	4	no	—
178	C₂²×C₈	—	89	4	no	—
179	C₂²×C₈	—	89	4	no	—
180	C₂²×C₈	—	89	2	no	—
181	C₂²×C₈	—	89	4	no	—
182	C₂²×C₈	—	89	2	no	—
183	C₂²×C₈	—	178	2	no	—
184	C₂²×C₈	—	178	2	no	—
185	C₂²×C₈	C₂×Dic₈₉	1	8	yes	—
186	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
187	C₂×C₄⋊C₄	—	89	8	no	—
188	C₂×C₄⋊C₄	—	178	8	no	—
189	C₂×C₄²	—	89	4	no	—
190	C₂×C₄²	—	89	4	no	—
191	C₂×C₄²	—	178	4	no	—
192	C₂×C₄²	—	178	4	no	—
193	C₂×C₄²	C₄×D₈₉	1	2	yes	—
194	C₂×C₄²	C₄×D₈₉	1	4	yes	—
195	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
196	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
197	C₄⋊C₈	C₂×Dic₈₉	1	16	yes	complemented
198	#97	—	178	16	no	—
199	C₄×C₈	—	89	16	no	—
200	#95	—	89	8	no	—
201	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	89	8	no	—
204	C₂²×C₄	—	178	8	no	—
205	C₂²×C₄	—	178	8	no	—
206	C₂²×C₄	—	178	8	no	—
207	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
208	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
209	C₂²×C₄	#13083557	1	8	yes	—
210	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
215	C₂²×C₄	D₄×D₈₉	1	2	yes	—
216	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
217	C₂²×C₄	#13083573	1	2	yes	—
218	C₂²×C₄	#13083575	1	2	yes	—
219	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
220	C₂×C₈	—	2	16	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	89	4	no	—
224	C₂×C₈	—	89	4	no	—
225	C₂×C₈	—	89	4	no	—
226	C₂×C₈	—	89	16	no	—
227	C₂×C₈	—	89	4	no	—
228	C₂×C₈	—	178	8	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	C₂²×Dic₈₉	1	8	yes	—
232	C₄⋊C₄	—	178	16	no	—
233	C₄²	—	178	8	no	—
234	C₄²	—	178	4	no	—
235	C₄²	—	178	4	no	—
236	C₄²	C₄×Dic₈₉	1	8	yes	—
237	C₄²	C₂×C₄×D₈₉	1	4	yes	—
238	C₄²	C₂²×Dic₈₉	1	4	yes	—
239	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
240	C₂³	#15667974	1	2	yes	—
241	C₂³	#15667977	1	2	yes	—
242	C₂³	C₂×#13083557	1	4	yes	—
243	C₂³	#15668020	1	4	yes	—
244	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	356	16	no	—
248	C₂×C₄	—	178	8	no	—
249	C₂×C₄	—	178	8	no	—
250	C₂×C₄	—	178	8	no	—
251	C₂×C₄	#15667922	1	16	yes	complemented
252	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₈₉	1	4	yes	—
255	C₂×C₄	C₄⋊C₄×D₈₉	1	4	yes	—
256	C₂×C₄	#15667970	1	4	yes	—
257	C₂×C₄	#15667974	1	2	yes	—
258	C₂×C₄	D₄×Dic₈₉	1	4	yes	—
259	C₂×C₄	#15667977	1	2	yes	—
260	C₂×C₄	Q₈×Dic₈₉	1	4	yes	—
261	C₂×C₄	C₂×#13083557	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
267	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	89	8	no	—
270	C₈	—	178	4	no	—
271	C₂²	C₄⋊C₄×Dic₈₉	1	4	yes	—
272	C₂²	C₂×#15667922	1	4	yes	—
273	C₂²	#19151556	1	4	yes	—
274	C₂²	#19151675	1	8	yes	—
275	C₂²	#19151725	1	8	yes	—
276	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
277	C₂²	C₂×#15667974	1	1	yes	characteristic
278	C₂²	C₂×#15667977	1	1	yes	characteristic
279	C₂²	C₂²×#13083557	1	2	yes	—
280	C₂²	C₂×#15668020	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	356	8	no	—
283	C₄	C₄⋊C₄×Dic₈₉	1	4	yes	—
284	C₄	C₂×#15667922	1	8	yes	complemented
285	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
286	C₄	C₂×#15667970	1	1	yes	characteristic
287	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
288	C₄	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
289	C₂	#26020122	1	8	yes	direct product factor
290	C₂	C₂×C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
291	C₂	C₂²×#15667922	1	1	yes	characteristic
292	C₂	C₂×#19151556	1	1	yes	characteristic
293	C₂	C₂×#19151675	1	2	yes	—
294	C₂	C₂×#19151725	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂²×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₈₉
hypocenter	C₈₉	C₂×#1770
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #26020122	C₂	direct (N × H)
13. C₂×C₄×#10623494	C₄	semidirect
21. C₂×C₄⋊C₈×C₈₉	C₄	semidirect
39. C₄×#10623494	C₂×C₄	semidirect
52. C₄⋊C₈×C₈₉	C₂×C₄	semidirect
60. C₂×#10623494	C₄⋊C₄	semidirect
99. C₂×C₄×C₈₉	#97	semidirect
122. #10623494	C₂×C₄⋊C₄	semidirect
135. C₄×C₈₉	C₂×#97	semidirect
142. C₂×C₈₉	#1770	semidirect
154. C₈₉	C₂×#1770	semidirect
161. C₂×C₄⋊C₈	Dic₈₉	semidirect
197. C₄⋊C₈	C₂×Dic₈₉	semidirect
251. C₂×C₄	#15667922	semidirect
284. C₄	C₂×#15667922	semidirect
289. C₂	#26020122	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×#15667922	2
C₂²×C₄×#10623494	2
C₂×C₄×C₄⋊Dic₈₉	2
#26020122	2
C₂×#19151352	2
C₂²×C₄⋊C₈×C₈₉	2
C₂×#1770	89

Subgroups — C2×#26020122 #36552373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26020122 #36552373