Subgroups — C₄×#19151715 #36552984

Order 22784.

1046 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151715	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	#26020133	C₂	1	2	yes	maximal, complemented
5	#26020092	C₂	1	4	yes	maximal, complemented
6	C₄×#15667939	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₈₉	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	#19151715	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×C₄×#10623494	—	2	2	no	—
18	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
20	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
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	C₄×#10623494	—	2	4	no	—
45	C₄×#10623494	—	2	2	no	—
46	C₄×#10623494	D₄	1	1	yes	characteristic
47	C₄×#10623494	D₄	1	1	yes	characteristic
48	C₄×#10623494	D₄	1	2	yes	—
49	C₈×Dic₈₉	—	2	4	no	—
50	C₈×Dic₈₉	C₂×C₄	1	8	yes	complemented
51	C₂×#41×C₈₉	C₂³	1	2	yes	—
52	#135×C₈₉	C₂×C₄	1	8	yes	—
53	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
59	C₄×D₄×C₈₉	—	2	4	no	—
60	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
61	#117×C₈₉	—	2	2	no	—
62	#117×C₈₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₈₉	—	2	2	no	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	#107×C₈₉	C₂×C₄	1	2	yes	—
66	#106×C₈₉	C₂×C₄	1	2	yes	—
67	#103×C₈₉	C₂×C₄	1	4	yes	—
68	#102×C₈₉	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
71	C₂²×Dic₈₉	—	2	2	no	—
72	#13083558	—	2	4	no	—
73	#13083558	—	2	16	no	—
74	#13083558	C₂²×C₄	1	8	yes	—
75	C₂×#10623494	—	2	4	no	—
76	C₂×#10623494	—	2	16	no	—
77	C₂×#10623494	C₂²×C₄	1	4	yes	—
78	C₂×#10623494	C₂×D₄	1	2	yes	—
79	C₂×#10623494	C₂×D₄	1	2	yes	—
80	C₂×#10623494	#41	1	4	yes	—
81	C₄×Dic₈₉	—	2	8	no	—
82	C₄×Dic₈₉	—	2	2	no	—
83	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
84	#41×C₈₉	—	2	4	no	—
85	#41×C₈₉	C₂²×C₄	1	8	yes	—
86	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂×D₄×C₈₉	—	2	2	no	—
88	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₈₉	—	2	2	no	—
90	C₂²×C₄×C₈₉	—	2	4	no	—
91	C₂²×C₄×C₈₉	#41	1	2	yes	—
92	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₈₉	C₄²	1	4	yes	complemented
94	SD₁₆×C₈₉	C₄²	1	8	yes	complemented
95	D₈×C₈₉	C₄²	1	4	yes	complemented
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	4	no	—
102	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
103	C₂²⋊C₄×C₈₉	—	2	2	no	—
104	C₂²⋊C₄×C₈₉	—	2	4	no	—
105	C₄²×C₈₉	—	2	4	no	—
106	C₄²×C₈₉	—	2	2	no	—
107	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₈₉	—	2	2	no	—
111	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
112	Q₈×C₈₉	—	2	2	no	—
113	Q₈×C₈₉	C₂×C₄²	1	4	yes	—
114	D₄×C₈₉	—	2	8	no	—
115	D₄×C₈₉	—	2	2	no	—
116	D₄×C₈₉	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	8	no	—
119	C₂×C₄×C₈₉	—	2	4	no	—
120	C₂×C₄×C₈₉	—	2	4	no	—
121	C₂×C₄×C₈₉	—	4	8	no	—
122	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
128	C₂×Dic₈₉	—	2	8	no	—
129	C₂×Dic₈₉	—	2	4	no	—
130	C₈×C₈₉	—	2	2	no	—
131	C₈×C₈₉	C₂×C₄²	1	2	yes	—
132	C₈×C₈₉	C₂×C₄²	1	2	yes	—
133	#10623494	—	2	8	no	—
134	#10623494	C₄×D₄	1	4	yes	—
135	#10623494	C₄×D₄	1	4	yes	—
136	C₂²×C₈₉	—	2	2	no	—
137	C₂²×C₈₉	—	2	4	no	—
138	C₂²×C₈₉	—	4	4	no	—
139	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₈₉	C₄×#41	1	2	yes	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	2	4	no	—
143	C₄×C₈₉	—	4	4	no	—
144	C₄×C₈₉	#443	1	4	yes	complemented
145	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
147	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
148	Dic₈₉	—	2	8	no	—
149	C₄×#443	—	89	1	no	maximal, Sylow
150	C₂×C₈₉	—	2	2	no	—
151	C₂×C₈₉	—	4	4	no	—
152	C₂×C₈₉	C₄²×D₄	1	1	yes	characteristic
153	C₂×C₈₉	C₂×#443	1	2	yes	—
154	C₂×#443	—	89	2	no	—
155	C₄×#135	D₈₉	1	1	yes	characteristic
156	C₄×#131	—	89	2	no	—
157	C₄×#108	—	89	1	no	—
158	#1772	—	89	2	no	—
159	#1764	—	89	4	no	—
160	C₄×#104	—	89	2	no	—
161	C₄²×C₈	—	89	1	no	—
162	C₈₉	C₄×#443	1	1	yes	characteristic, complemented, Sylow
163	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
164	C₂×#131	—	89	4	no	—
165	C₄×#41	C₂×D₈₉	1	2	yes	—
166	#443	—	89	16	no	—
167	C₄×Q₁₆	Dic₈₉	1	2	yes	complemented
168	C₄×SD₁₆	Dic₈₉	1	4	yes	complemented
169	C₄×D₈	Dic₈₉	1	2	yes	complemented
170	C₂×#108	—	89	2	no	—
171	#427	C₂×D₈₉	1	1	yes	characteristic
172	C₂×#104	—	89	4	no	—
173	#416	C₂×D₈₉	1	2	yes	—
174	#405	—	89	4	no	—
175	#405	—	178	4	no	—
176	C₄×M₁₆	—	89	2	no	—
177	C₄×M₁₆	—	178	2	no	—
178	C₂×C₄×C₈	—	89	2	no	—
179	C₂×C₄×C₈	—	89	1	no	—
180	C₂×C₄×C₈	—	89	2	no	—
181	C₂×C₄×C₈	—	178	2	no	—
182	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
183	C₄³	—	89	1	no	—
184	#341	—	89	2	no	—
185	#339	—	89	4	no	—
186	C₂×#41	C₂²×D₈₉	1	2	yes	—
187	#135	C₄×D₈₉	1	8	yes	—
188	C₂×Q₁₆	C₂×Dic₈₉	1	1	yes	characteristic
189	C₂×SD₁₆	C₂×Dic₈₉	1	2	yes	—
190	C₂×D₈	C₂×Dic₈₉	1	1	yes	characteristic
191	#131	—	89	16	no	—
192	#131	—	178	8	no	—
193	C₂×M₁₆	—	89	4	no	—
194	C₂×M₁₆	—	178	4	no	—
195	C₂²×C₈	—	89	2	no	—
196	C₂²×C₈	—	178	4	no	—
197	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
202	#117	—	2	2	no	—
203	#117	C₂²×D₈₉	1	2	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	89	2	no	—
206	C₂×C₄²	—	178	2	no	—
207	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
208	#108	—	89	8	no	—
209	#107	C₂×Dic₈₉	1	2	yes	—
210	#106	C₂×Dic₈₉	1	2	yes	—
211	#104	—	89	16	no	—
212	#103	C₂×Dic₈₉	1	4	yes	—
213	#102	C₂×Dic₈₉	1	4	yes	—
214	#97	—	178	4	no	—
215	#97	—	178	2	no	—
216	C₄×C₈	—	89	8	no	—
217	C₄×C₈	—	89	2	no	—
218	C₄×C₈	—	89	1	no	—
219	C₄×C₈	—	89	1	no	—
220	C₄×C₈	—	178	4	no	—
221	C₄×C₈	—	178	4	no	—
222	C₄×C₈	—	178	2	no	—
223	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
224	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
225	#41	—	2	4	no	—
226	#41	C₂×C₄×D₈₉	1	8	yes	—
227	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	178	2	no	—
233	C₂²×C₄	#13083573	1	2	yes	—
234	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
235	Q₁₆	C₄×Dic₈₉	1	4	yes	complemented
236	SD₁₆	C₄×Dic₈₉	1	8	yes	complemented
237	D₈	C₄×Dic₈₉	1	4	yes	complemented
238	M₁₆	—	89	8	no	—
239	M₁₆	—	178	16	no	—
240	M₁₆	—	178	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	89	4	no	—
243	C₂×C₈	—	89	2	no	—
244	C₂×C₈	—	89	2	no	—
245	C₂×C₈	—	89	4	no	—
246	C₂×C₈	—	178	16	no	—
247	C₂×C₈	—	178	4	no	—
248	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
249	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
250	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	89	4	no	—
259	C₄²	—	178	8	no	—
260	C₄²	—	178	2	no	—
261	C₄²	C₂²×Dic₈₉	1	2	yes	—
262	C₄²	D₄×D₈₉	1	1	yes	characteristic
263	C₄²	D₄×D₈₉	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	C₂×#13083573	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	C₂×C₄×Dic₈₉	1	4	yes	—
268	D₄	—	2	8	no	—
269	D₄	—	2	2	no	—
270	D₄	C₂×C₄×Dic₈₉	1	4	yes	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	—	178	8	no	—
277	C₂×C₄	—	178	4	no	—
278	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
279	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
281	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
283	C₂×C₄	#41×D₈₉	1	2	yes	—
284	C₈	—	2	2	no	—
285	C₈	—	89	4	no	—
286	C₈	—	89	4	no	—
287	C₈	—	178	8	no	—
288	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
289	C₈	C₂×C₄×Dic₈₉	1	2	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
294	C₂²	C₄×#13083573	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	4	no	—
298	C₄	—	178	8	no	—
299	C₄	#19151715	1	4	yes	direct product factor
300	C₄	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
301	C₄	C₄×D₄×D₈₉	1	2	yes	—
302	C₄	#41×Dic₈₉	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	4	4	no	—
305	C₂	C₄×D₄×Dic₈₉	1	1	yes	characteristic
306	C₂	C₂×#19151715	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×#443
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. #26020133	C₂	semidirect
5. #26020092	C₂	semidirect
6. C₄×#15667939	C₂	semidirect
7. C₄×C₈×Dic₈₉	C₂	semidirect
10. #19151715	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
50. C₈×Dic₈₉	C₂×C₄	semidirect
93. Q₁₆×C₈₉	C₄²	semidirect
94. SD₁₆×C₈₉	C₄²	semidirect
95. D₈×C₈₉	C₄²	semidirect
144. C₄×C₈₉	#443	semidirect
162. C₈₉	C₄×#443	semidirect
167. C₄×Q₁₆	Dic₈₉	semidirect
168. C₄×SD₁₆	Dic₈₉	semidirect
169. C₄×D₈	Dic₈₉	semidirect
235. Q₁₆	C₄×Dic₈₉	semidirect
236. SD₁₆	C₄×Dic₈₉	semidirect
237. D₈	C₄×Dic₈₉	semidirect
299. C₄	#19151715	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×#19151715	2
C₄×#15668019	2
C₄×#15667948	2
#26020133	2
#26020092	2
C₄×#15667939	2
C₄×C₈×Dic₈₉	2
C₄×#135×C₈₉	2
C₄×#443	89

Subgroups — C4×#19151715 #36552984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19151715 #36552984