Subgroups — C₄×#19098346 #34203318

Order 20224.

490 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	C₄×#14579167	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098346	C₂	1	2	yes	maximal
3	C₄×#14579101	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761488	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579167	C₄	1	1	yes	characteristic
7	C₂×#14579167	C₂²	1	1	yes	characteristic
8	C₄×#11929624	C₄	1	1	yes	characteristic
9	C₂×#14579101	C₂²	1	2	yes	—
10	#19098518	C₄	1	1	yes	characteristic
11	#19098518	C₂²	1	1	yes	characteristic
12	#19098517	—	2	1	no	—
13	#19098517	—	2	1	no	—
14	C₄×#11929608	C₄	1	1	yes	characteristic
15	C₄×#11929607	—	2	1	no	—
16	C₄×C₈×D₇₉	—	2	1	no	—
17	#19098346	C₄	1	8	yes	direct product factor
18	#19098339	—	2	1	no	—
19	C₄×C₇₉⋊C₁₆	—	2	1	no	—
20	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₇₉	—	2	1	no	—
23	C₄×C₁₆×C₇₉	—	2	1	no	—
24	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
25	#14579167	—	2	2	no	—
26	#14579167	—	2	2	no	—
27	#14579167	C₈	1	4	yes	—
28	#14579167	C₂×C₄	1	4	yes	—
29	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
30	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
31	C₂×#11929607	—	2	1	no	—
32	C₂×#11929607	—	2	1	no	—
33	C₂×C₈×D₇₉	—	2	1	no	—
34	C₂×C₈×D₇₉	—	2	1	no	—
35	C₄×Dic₁₅₈	C₈	1	2	yes	—
36	C₄×D₃₁₆	C₈	1	2	yes	—
37	C₄×C₂²⋊D₇₉	—	2	2	no	—
38	#14579120	C₂×C₄	1	1	yes	characteristic
39	#14579118	—	2	1	no	—
40	C₄²×D₇₉	—	2	1	no	—
41	#14579101	—	2	2	no	—
42	#14579101	C₂×C₄	1	4	yes	complemented
43	C₂×C₇₉⋊C₁₆	—	2	2	no	—
44	#14579072	—	2	1	no	—
45	#14579071	—	2	1	no	—
46	#14579071	—	2	1	no	—
47	C₄×C₇₉⋊C₈	—	2	1	no	—
48	C₈×Dic₇₉	—	2	1	no	—
49	C₈×Dic₇₉	—	2	1	no	—
50	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₇₉	—	2	2	no	—
54	C₂×C₁₆×C₇₉	—	2	2	no	—
55	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	complemented
56	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
57	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
60	#11929624	—	2	1	no	—
61	#11929624	—	2	1	no	—
62	#11929624	C₄²	1	2	yes	—
63	#11929624	C₂×C₈	1	2	yes	—
64	C₂×Dic₁₅₈	C₂×C₈	1	1	yes	characteristic
65	C₂×D₃₁₆	C₂×C₈	1	1	yes	characteristic
66	C₂×C₂²⋊D₇₉	—	2	1	no	—
67	C₂×C₄×D₇₉	—	2	2	no	—
68	C₂×C₄×D₇₉	—	2	1	no	—
69	#11929608	—	2	1	no	—
70	#11929608	—	2	1	no	—
71	#11929608	C₄²	1	2	yes	—
72	#11929608	C₂×C₈	1	2	yes	—
73	#11929607	—	2	4	no	—
74	#11929607	—	2	4	no	—
75	C₂×C₇₉⋊C₈	—	2	1	no	—
76	C₂×C₇₉⋊C₈	—	2	1	no	—
77	C₈×D₇₉	—	2	4	no	—
78	C₈×D₇₉	—	2	4	no	—
79	C₄⋊Dic₇₉	C₂×C₈	1	2	yes	—
80	Dic₇₉⋊C₄	—	2	2	no	—
81	C₂²⋊Dic₇₉	—	2	1	no	—
82	(C₂×D₇₉)⋊C₄	—	2	2	no	—
83	C₄×Dic₇₉	—	2	2	no	—
84	C₄×Dic₇₉	—	2	1	no	—
85	C₇₉⋊C₁₆	—	2	4	no	—
86	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
87	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
91	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
92	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	#41	1	2	yes	—
96	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₇₉	C₂×C₈	1	2	yes	—
99	C₁₆×C₇₉	—	2	4	no	—
100	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₇₉	—	2	1	no	—
102	C₂×C₄×C₇₉	—	2	4	no	—
103	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂²⋊D₇₉	—	2	2	no	—
112	C₂²⋊D₇₉	—	2	2	no	—
113	C₂×Dic₇₉	—	2	1	no	—
114	C₂×Dic₇₉	—	2	2	no	—
115	D₃₁₆	—	2	1	no	—
116	D₃₁₆	C₄×C₈	1	2	yes	—
117	C₄×D₇₉	—	2	2	no	—
118	C₄×D₇₉	—	2	2	no	—
119	C₄×D₇₉	—	4	4	no	—
120	Dic₁₅₈	—	2	1	no	—
121	Dic₁₅₈	C₄×C₈	1	2	yes	—
122	C₈×C₇₉	C₂²⋊C₈	1	2	yes	—
123	C₈×C₇₉	C₂²⋊C₈	1	2	yes	—
124	C₈×C₇₉	C₄×D₄	1	2	yes	—
125	C₈×C₇₉	C₄×D₄	1	2	yes	—
126	C₇₉⋊C₈	—	2	2	no	—
127	C₇₉⋊C₈	—	2	2	no	—
128	C₂²×C₇₉	—	2	2	no	—
129	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
130	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₇₉	#431	1	1	yes	characteristic
132	C₂×D₇₉	—	2	2	no	—
133	C₂×D₇₉	—	4	2	no	—
134	C₄×C₇₉	—	2	2	no	—
135	C₄×C₇₉	#350	1	4	yes	complemented
136	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₇₉	#431	1	1	yes	characteristic
140	Dic₇₉	—	2	2	no	—
141	Dic₇₉	—	4	2	no	—
142	C₄×#350	—	79	1	no	maximal, Sylow
143	C₂×C₇₉	—	2	1	no	—
144	C₂×C₇₉	—	2	1	no	—
145	C₂×C₇₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₇₉	C₂×#350	1	2	yes	—
147	D₇₉	—	4	2	no	—
148	C₄×#131	—	79	1	no	—
149	C₂×#350	—	79	2	no	—
150	C₄×M₃₂	—	79	1	no	—
151	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic, complemented
152	#1379	—	79	2	no	—
153	C₇₉	C₄×#350	1	1	yes	characteristic, complemented, Sylow
154	C₂×#131	—	79	1	no	—
155	C₂×#131	—	79	1	no	—
156	C₄×#41	—	79	1	no	—
157	C₂×M₃₂	—	79	2	no	—
158	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
159	#405	—	79	1	no	—
160	#405	—	79	1	no	—
161	#405	—	158	1	no	—
162	#405	—	158	1	no	—
163	C₄×M₁₆	—	79	1	no	—
164	C₄×M₁₆	—	158	1	no	—
165	C₂×C₄×C₈	—	158	1	no	—
166	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
167	#350	—	79	8	no	—
168	#346	—	158	1	no	—
169	C₄×C₁₆	—	2	1	no	—
170	C₄×C₁₆	—	2	1	no	—
171	C₄×C₁₆	—	158	1	no	—
172	C₂×#41	—	79	1	no	—
173	#131	—	79	4	no	—
174	#131	—	79	4	no	—
175	#131	—	158	2	no	—
176	#131	—	158	2	no	—
177	C₂×M₁₆	—	79	1	no	—
178	C₂×M₁₆	—	79	1	no	—
179	C₂×M₁₆	—	158	1	no	—
180	C₂×M₁₆	—	158	1	no	—
181	C₂²×C₈	—	158	1	no	—
182	C₂²×C₈	—	158	1	no	—
183	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
185	C₄×Q₈	—	79	2	no	—
186	C₄×D₄	—	79	2	no	—
187	C₄×D₄	—	158	2	no	—
188	#117	—	79	1	no	—
189	#117	—	158	1	no	—
190	C₂×C₄²	—	158	1	no	—
191	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
192	M₃₂	—	79	4	no	—
193	M₃₂	—	158	2	no	—
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	158	2	no	—
197	C₂×C₁₆	C₄×D₇₉	1	4	yes	complemented
198	#97	—	158	1	no	—
199	#97	—	158	1	no	—
200	#97	—	158	1	no	—
201	C₄×C₈	—	158	1	no	—
202	C₄×C₈	—	158	1	no	—
203	C₄×C₈	—	158	1	no	—
204	C₄×C₈	D₃₁₆	1	1	yes	characteristic
205	C₄×C₈	D₃₁₆	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
208	#41	—	79	2	no	—
209	#41	—	79	2	no	—
210	#41	—	158	1	no	—
211	#41	—	158	1	no	—
212	C₂×Q₈	—	79	1	no	—
213	C₂×D₄	—	79	1	no	—
214	C₂×D₄	—	158	1	no	—
215	C₂²×C₄	—	158	2	no	—
216	C₂²×C₄	—	158	1	no	—
217	C₂²×C₄	#11929607	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
219	M₁₆	—	79	2	no	—
220	M₁₆	—	79	2	no	—
221	M₁₆	—	158	1	no	—
222	M₁₆	—	158	1	no	—
223	M₁₆	—	158	4	no	—
224	M₁₆	—	158	4	no	—
225	C₂×C₈	—	158	1	no	—
226	C₂×C₈	—	158	1	no	—
227	C₂×C₈	—	158	4	no	—
228	C₂×C₈	—	158	4	no	—
229	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
230	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
231	C₂×C₈	C₈×D₇₉	1	2	yes	—
232	C₂×C₈	#11929607	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
234	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
235	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
236	C₂×C₈	#11929624	1	2	yes	—
237	C₄⋊C₄	—	79	2	no	—
238	C₄⋊C₄	—	158	2	no	—
239	C₂²⋊C₄	—	158	1	no	—
240	C₂²⋊C₄	—	158	2	no	—
241	C₄²	—	158	2	no	—
242	C₄²	—	158	1	no	—
243	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
244	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
245	C₄²	C₈×D₇₉	1	2	yes	—
246	C₁₆	—	2	4	no	—
247	C₁₆	—	158	4	no	—
248	C₂³	—	158	1	no	—
249	C₂³	C₂×#11929607	1	1	yes	characteristic
250	Q₈	—	79	2	no	—
251	Q₈	—	158	1	no	—
252	D₄	—	79	2	no	—
253	D₄	—	158	2	no	—
254	D₄	—	158	2	no	—
255	D₄	—	158	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	316	4	no	—
258	C₂×C₄	—	158	2	no	—
259	C₂×C₄	—	158	1	no	—
260	C₂×C₄	—	158	2	no	—
261	C₂×C₄	—	158	2	no	—
262	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
263	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
264	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
265	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	#14579118	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
269	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
270	C₈	—	158	2	no	—
271	C₈	—	158	2	no	—
272	C₈	(C₂×D₇₉)⋊C₈	1	2	yes	—
273	C₈	(C₂×D₇₉)⋊C₈	1	2	yes	—
274	C₈	C₄×C₂²⋊D₇₉	1	2	yes	—
275	C₈	C₄×D₃₁₆	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	316	2	no	—
278	C₂²	—	158	2	no	—
279	C₂²	C₄×#11929607	1	1	yes	characteristic
280	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
281	C₂²	#19098614	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	316	2	no	—
284	C₄	—	158	2	no	—
285	C₄	#19098346	1	4	yes	direct product factor
286	C₄	C₄×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
287	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
288	C₄	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
289	C₄	#19098614	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	316	2	no	—
293	C₂	C₄×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
294	C₂	C₂×#19098346	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₃₁₆
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	C₄×#350
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#14579101	C₂	semidirect
4. #23761488	C₂	semidirect
5. C₂×C₄×C₁₆×C₇₉	C₂	semidirect
17. #19098346	C₄	direct (N × H)
42. #14579101	C₂×C₄	semidirect
55. C₂×C₁₆×C₇₉	C₂×C₄	semidirect
135. C₄×C₇₉	#350	semidirect
151. C₂×C₄×C₁₆	D₇₉	semidirect
153. C₇₉	C₄×#350	semidirect
197. C₂×C₁₆	C₄×D₇₉	semidirect
285. C₄	#19098346	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#14579167	2
C₂×#19098346	2
C₄×#14579101	2
#23761488	2
C₂×C₄×C₁₆×C₇₉	2
C₄×#350	79

Subgroups — C4×#19098346 #34203318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098346 #34203318