Subgroups — C₄×#19151720 #36553003

Order 22784.

1022 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151720	C₂	1	2	yes	maximal
2	C₄×#15668019	C₂	1	2	yes	maximal, complemented
3	C₄×#15667948	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020138	C₂	1	2	yes	maximal, complemented
5	#26020099	C₂	1	4	yes	maximal, complemented
6	C₄×#15667939	C₂	1	2	yes	maximal, complemented
7	C₄×#15667922	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#15668019	C₂²	1	4	yes	—
10	#19151720	C₄	1	16	yes	direct product factor
11	C₂×#15667948	C₂²	1	2	yes	—
12	C₂×#15667939	C₂²	1	4	yes	—
13	#19151554	—	2	4	no	—
14	#19151554	C₂²	1	4	yes	—
15	C₄×#13083558	—	2	2	no	—
16	C₄×#13083558	C₂²	1	2	yes	—
17	C₂×#15667922	C₂²	1	2	yes	—
18	C₂×#15667922	C₂²	1	2	yes	—
19	C₂×C₄×#10623494	—	2	2	no	—
20	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
21	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
22	#19151363	C₂²	1	2	yes	—
23	#19151357	C₂²	1	4	yes	—
24	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
25	C₄×#41×C₈₉	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₈₉	C₄	1	2	yes	complemented
27	C₄×SD₁₆×C₈₉	C₄	1	4	yes	complemented
28	C₄×D₈×C₈₉	C₄	1	2	yes	complemented
29	#427×C₈₉	C₂²	1	1	yes	characteristic
30	#416×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#15668019	—	2	8	no	—
33	#15668019	C₂×C₄	1	16	yes	complemented
34	C₂×#13083558	—	2	4	no	—
35	C₂×#13083558	C₂³	1	4	yes	—
36	C₂²×#10623494	—	2	4	no	—
37	C₂²×#10623494	C₂³	1	2	yes	—
38	C₂×C₄×Dic₈₉	—	2	2	no	—
39	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
40	#15667948	C₂×C₄	1	8	yes	complemented
41	#15667939	C₂×C₄	1	16	yes	complemented
42	#15667923	—	2	4	no	—
43	#15667923	—	2	2	no	—
44	#15667922	—	2	4	no	—
45	#15667922	C₂×C₄	1	8	yes	complemented
46	C₄×#10623494	—	2	4	no	—
47	C₄×#10623494	—	2	2	no	—
48	C₄×#10623494	D₄	1	2	yes	—
49	C₄×#10623494	D₄	1	2	yes	—
50	C₂×#41×C₈₉	C₂³	1	2	yes	—
51	#135×C₈₉	C₂×C₄	1	8	yes	—
52	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₉	C₂×C₄	1	2	yes	—
54	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
58	C₄×D₄×C₈₉	—	2	4	no	—
59	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
60	#117×C₈₉	—	2	2	no	—
61	#117×C₈₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₈₉	—	2	2	no	—
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	#107×C₈₉	C₂×C₄	1	2	yes	—
65	#106×C₈₉	C₂×C₄	1	2	yes	—
66	#103×C₈₉	C₂×C₄	1	4	yes	—
67	#102×C₈₉	C₂×C₄	1	4	yes	—
68	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
70	C₂²×Dic₈₉	—	2	2	no	—
71	#13083558	—	2	4	no	—
72	#13083558	—	2	16	no	—
73	#13083558	C₂²×C₄	1	8	yes	—
74	C₂×#10623494	—	2	4	no	—
75	C₂×#10623494	—	2	16	no	—
76	C₂×#10623494	C₂²×C₄	1	4	yes	—
77	C₂×#10623494	C₂×D₄	1	4	yes	—
78	C₂×#10623494	#41	1	4	yes	—
79	C₄×Dic₈₉	—	2	2	no	—
80	C₄×Dic₈₉	—	4	4	no	—
81	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
82	#41×C₈₉	—	2	4	no	—
83	#41×C₈₉	C₂²×C₄	1	8	yes	—
84	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂×D₄×C₈₉	—	2	2	no	—
86	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₈₉	—	2	2	no	—
88	C₂²×C₄×C₈₉	—	2	4	no	—
89	C₂²×C₄×C₈₉	#41	1	2	yes	—
90	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
91	Q₁₆×C₈₉	C₄²	1	4	yes	complemented
92	SD₁₆×C₈₉	C₄²	1	8	yes	complemented
93	D₈×C₈₉	C₄²	1	4	yes	complemented
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
97	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	—	2	4	no	—
100	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
101	C₂²⋊C₄×C₈₉	—	2	2	no	—
102	C₂²⋊C₄×C₈₉	—	2	4	no	—
103	C₄²×C₈₉	—	2	4	no	—
104	C₄²×C₈₉	—	2	2	no	—
105	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₈₉	—	2	2	no	—
109	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
110	Q₈×C₈₉	—	2	2	no	—
111	Q₈×C₈₉	C₂×C₄²	1	4	yes	—
112	D₄×C₈₉	—	2	8	no	—
113	D₄×C₈₉	—	2	2	no	—
114	D₄×C₈₉	C₂×C₄²	1	4	yes	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	8	no	—
117	C₂×C₄×C₈₉	—	2	4	no	—
118	C₂×C₄×C₈₉	—	2	4	no	—
119	C₂×C₄×C₈₉	—	4	8	no	—
120	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
126	C₂×Dic₈₉	—	2	4	no	—
127	C₂×Dic₈₉	—	4	4	no	—
128	C₈×C₈₉	—	2	2	no	—
129	C₈×C₈₉	C₂×C₄²	1	2	yes	—
130	C₈×C₈₉	C₂×C₄²	1	2	yes	—
131	#10623494	—	2	8	no	—
132	#10623494	C₄×D₄	1	8	yes	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	4	no	—
135	C₂²×C₈₉	—	4	4	no	—
136	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₈₉	C₄×#41	1	2	yes	—
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	—	2	4	no	—
140	C₄×C₈₉	—	4	4	no	—
141	C₄×C₈₉	#444	1	4	yes	complemented
142	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
143	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
144	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
145	Dic₈₉	—	4	4	no	—
146	C₄×#444	—	89	1	no	maximal, Sylow
147	C₂×C₈₉	—	2	2	no	—
148	C₂×C₈₉	—	4	4	no	—
149	C₂×C₈₉	C₄²×D₄	1	1	yes	characteristic
150	C₂×C₈₉	C₂×#444	1	2	yes	—
151	C₂×#444	—	89	2	no	—
152	C₄×#135	D₈₉	1	1	yes	characteristic
153	C₄×#131	—	89	2	no	—
154	C₄×#108	—	89	1	no	—
155	#1773	—	89	2	no	—
156	#1765	—	89	4	no	—
157	C₄×#104	—	89	2	no	—
158	C₄×#97	—	89	1	no	—
159	C₈₉	C₄×#444	1	1	yes	characteristic, complemented, Sylow
160	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
161	C₂×#131	—	89	4	no	—
162	C₄×#41	C₂×D₈₉	1	2	yes	—
163	#444	—	89	16	no	—
164	C₄×Q₁₆	Dic₈₉	1	2	yes	complemented
165	C₄×SD₁₆	Dic₈₉	1	4	yes	complemented
166	C₄×D₈	Dic₈₉	1	2	yes	complemented
167	C₂×#108	—	89	2	no	—
168	#427	C₂×D₈₉	1	1	yes	characteristic
169	C₂×#104	—	89	4	no	—
170	#416	C₂×D₈₉	1	2	yes	—
171	#405	—	89	4	no	—
172	#405	—	178	4	no	—
173	C₄×M₁₆	—	89	2	no	—
174	C₄×M₁₆	—	178	2	no	—
175	C₂×#97	—	89	2	no	—
176	C₂×#97	—	89	2	no	—
177	C₂×C₄×C₈	—	89	1	no	—
178	C₂×C₄×C₈	—	178	2	no	—
179	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
180	C₄³	—	89	1	no	—
181	#341	—	89	2	no	—
182	#339	—	89	4	no	—
183	C₂×#41	C₂²×D₈₉	1	2	yes	—
184	#135	C₄×D₈₉	1	8	yes	—
185	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
186	C₂×SD₁₆	C₂×Dic₈₉	1	2	yes	—
187	C₂×D₈	C₂×Dic₈₉	1	1	yes	characteristic
188	#131	—	89	16	no	—
189	#131	—	178	8	no	—
190	C₂×M₁₆	—	89	4	no	—
191	C₂×M₁₆	—	178	4	no	—
192	C₂²×C₈	—	89	2	no	—
193	C₂²×C₈	—	178	4	no	—
194	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
196	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
197	C₄×D₄	—	2	4	no	—
198	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
199	#117	—	2	2	no	—
200	#117	C₂²×D₈₉	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	89	2	no	—
203	C₂×C₄²	—	178	2	no	—
204	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
205	#108	—	89	8	no	—
206	#107	C₂×Dic₈₉	1	2	yes	—
207	#106	C₂×Dic₈₉	1	2	yes	—
208	#104	—	89	16	no	—
209	#103	C₂×Dic₈₉	1	4	yes	—
210	#102	C₂×Dic₈₉	1	4	yes	—
211	#97	—	89	8	no	—
212	#97	—	178	4	no	—
213	#97	—	178	4	no	—
214	#97	—	178	2	no	—
215	C₄×C₈	—	89	2	no	—
216	C₄×C₈	—	89	2	no	—
217	C₄×C₈	—	178	4	no	—
218	C₄×C₈	—	178	2	no	—
219	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
220	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
221	#41	—	2	4	no	—
222	#41	C₂×C₄×D₈₉	1	8	yes	—
223	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	178	2	no	—
229	C₂²×C₄	#13083573	1	2	yes	—
230	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
231	Q₁₆	C₄×Dic₈₉	1	4	yes	complemented
232	SD₁₆	C₄×Dic₈₉	1	8	yes	complemented
233	D₈	C₄×Dic₈₉	1	4	yes	complemented
234	M₁₆	—	89	8	no	—
235	M₁₆	—	178	16	no	—
236	M₁₆	—	178	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	89	4	no	—
239	C₂×C₈	—	89	4	no	—
240	C₂×C₈	—	89	4	no	—
241	C₂×C₈	—	178	16	no	—
242	C₂×C₈	—	178	4	no	—
243	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
244	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
245	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	89	4	no	—
254	C₄²	—	356	4	no	—
255	C₄²	—	178	2	no	—
256	C₄²	C₂²×Dic₈₉	1	2	yes	—
257	C₄²	D₄×D₈₉	1	1	yes	characteristic
258	C₄²	D₄×D₈₉	1	1	yes	characteristic
259	C₂³	—	2	2	no	—
260	C₂³	C₂×#13083573	1	1	yes	characteristic
261	Q₈	—	2	2	no	—
262	Q₈	C₂×C₄×Dic₈₉	1	4	yes	—
263	D₄	—	2	8	no	—
264	D₄	—	2	2	no	—
265	D₄	C₂×C₄×Dic₈₉	1	4	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	356	4	no	—
272	C₂×C₄	—	178	4	no	—
273	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
274	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
275	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
276	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
278	C₂×C₄	#41×D₈₉	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	89	8	no	—
281	C₈	—	178	8	no	—
282	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
283	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
288	C₂²	C₄×#13083573	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	4	4	no	—
292	C₄	—	356	4	no	—
293	C₄	#19151720	1	4	yes	direct product factor
294	C₄	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
295	C₄	C₄×D₄×D₈₉	1	2	yes	—
296	C₄	#41×Dic₈₉	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	4	4	no	—
299	C₂	C₄×D₄×Dic₈₉	1	1	yes	characteristic
300	C₂	C₂×#19151720	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₄×C₈₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#135×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#135×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×#135	D₈₉
hypocenter	C₈₉	C₄×#444
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₄×#15668019	C₂	semidirect
3. C₄×#15667948	C₂	semidirect
4. #26020138	C₂	semidirect
5. #26020099	C₂	semidirect
6. C₄×#15667939	C₂	semidirect
7. C₄×#15667922	C₂	semidirect
10. #19151720	C₄	direct (N × H)
26. C₄×Q₁₆×C₈₉	C₄	semidirect
27. C₄×SD₁₆×C₈₉	C₄	semidirect
28. C₄×D₈×C₈₉	C₄	semidirect
33. #15668019	C₂×C₄	semidirect
40. #15667948	C₂×C₄	semidirect
41. #15667939	C₂×C₄	semidirect
45. #15667922	C₂×C₄	semidirect
91. Q₁₆×C₈₉	C₄²	semidirect
92. SD₁₆×C₈₉	C₄²	semidirect
93. D₈×C₈₉	C₄²	semidirect
141. C₄×C₈₉	#444	semidirect
159. C₈₉	C₄×#444	semidirect
164. C₄×Q₁₆	Dic₈₉	semidirect
165. C₄×SD₁₆	Dic₈₉	semidirect
166. C₄×D₈	Dic₈₉	semidirect
231. Q₁₆	C₄×Dic₈₉	semidirect
232. SD₁₆	C₄×Dic₈₉	semidirect
233. D₈	C₄×Dic₈₉	semidirect
293. C₄	#19151720	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×#19151720	2
C₄×#15668019	2
C₄×#15667948	2
#26020138	2
#26020099	2
C₄×#15667939	2
C₄×#15667922	2
C₄×#135×C₈₉	2
C₄×#444	89

Subgroups — C4×#19151720 #36553003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151720 #36553003