Subgroups — C₄×#19098624 #34286413

Order 20224.

970 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098624	C₂	1	2	yes	maximal
2	#23763257	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#23763166	C₂	1	2	yes	maximal
5	C₄×C₂²⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal
6	#23763152	C₂	1	2	yes	maximal, complemented
7	C₄×#14579072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#117×C₇₉	C₂	1	1	yes	maximal, characteristic
10	#19098624	C₄	1	16	yes	direct product factor
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
12	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
13	C₂×#14579072	C₂²	1	2	yes	—
14	C₂×#14579072	C₂²	1	2	yes	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
19	#19098318	C₂²	1	4	yes	—
20	#19098318	C₂²	1	4	yes	—
21	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₇₉	C₄	1	4	yes	—
25	C₄×C₂²⋊C₄×C₇₉	C₄	1	4	yes	—
26	#376×C₇₉	C₂²	1	1	yes	characteristic
27	#376×C₇₉	C₂²	1	1	yes	characteristic
28	C₄³×C₇₉	C₂²	1	1	yes	characteristic
29	C₄³×C₇₉	C₂²	1	1	yes	characteristic
30	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
31	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
32	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	16	yes	complemented
33	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	16	yes	—
34	#14579072	—	2	4	no	—
35	#14579072	C₂×C₄	1	8	yes	complemented
36	C₄×C₇₉⋊C₈	—	2	4	no	—
37	C₄×C₇₉⋊C₈	—	2	4	no	—
38	C₄×C₇₉⋊C₈	—	2	4	no	—
39	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	complemented
40	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	#117×C₇₉	—	2	4	no	—
43	#117×C₇₉	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₇₉	—	2	4	no	—
49	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
58	#95×C₇₉	C₂×C₄	1	4	yes	—
59	#95×C₇₉	C₂×C₄	1	4	yes	—
60	C₂×C₇₉⋊C₈	—	2	4	no	—
61	C₂×C₇₉⋊C₈	—	2	4	no	—
62	C₂×C₇₉⋊C₈	—	2	4	no	—
63	C₂×C₇₉⋊C₈	—	2	4	no	—
64	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
65	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
66	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	—	2	2	no	—
68	C₂²×C₄×C₇₉	—	2	2	no	—
69	C₂²×C₄×C₇₉	—	2	4	no	—
70	C₂²×C₄×C₇₉	—	2	8	no	—
71	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₇₉	—	2	4	no	—
81	C₄⋊C₄×C₇₉	C₄²	1	8	yes	—
82	C₂²⋊C₄×C₇₉	—	2	4	no	—
83	C₂²⋊C₄×C₇₉	C₄²	1	8	yes	—
84	C₄²×C₇₉	—	2	4	no	—
85	C₄²×C₇₉	—	2	8	no	—
86	C₄²×C₇₉	—	2	8	no	—
87	C₄²×C₇₉	—	2	2	no	—
88	C₄²×C₇₉	—	2	2	no	—
89	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
90	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
91	C₄²×C₇₉	#41	1	4	yes	—
92	C₄²×C₇₉	#41	1	4	yes	—
93	C₂³×C₇₉	—	2	1	no	—
94	C₂³×C₇₉	—	2	1	no	—
95	C₂³×C₇₉	—	2	4	no	—
96	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
97	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	—	2	1	no	—
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	4	no	—
101	C₂×C₄×C₇₉	—	2	4	no	—
102	C₂×C₄×C₇₉	—	2	4	no	—
103	C₂×C₄×C₇₉	—	2	8	no	—
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	—	2	2	no	—
106	C₂×C₄×C₇₉	—	2	4	no	—
107	C₂×C₄×C₇₉	—	2	4	no	—
108	C₂×C₄×C₇₉	—	4	8	no	—
109	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
110	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₇₉	#117	1	4	yes	—
113	C₂×C₄×C₇₉	#117	1	4	yes	—
114	C₂×C₄×C₇₉	#131	1	8	yes	—
115	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
117	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
118	C₇₉⋊C₈	—	2	8	no	—
119	C₇₉⋊C₈	—	2	8	no	—
120	C₂²×C₇₉	—	2	4	no	—
121	C₂²×C₇₉	—	2	2	no	—
122	C₂²×C₇₉	—	4	2	no	—
123	C₂²×C₇₉	—	4	2	no	—
124	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
125	C₂²×C₇₉	C₂×#117	1	1	yes	characteristic
126	C₂²×C₇₉	C₂×#117	1	1	yes	characteristic
127	C₂²×C₇₉	C₂×#131	1	4	yes	—
128	C₄×C₇₉	—	2	4	no	—
129	C₄×C₇₉	—	2	4	no	—
130	C₄×C₇₉	—	2	4	no	—
131	C₄×C₇₉	—	2	2	no	—
132	C₄×C₇₉	—	4	2	no	—
133	C₄×C₇₉	—	4	2	no	—
134	C₄×C₇₉	#433	1	8	yes	complemented
135	C₄×C₇₉	C₄×#41	1	4	yes	—
136	C₄×#433	—	79	1	no	maximal, Sylow
137	C₂×C₇₉	—	4	2	no	—
138	C₂×C₇₉	C₄×#117	1	1	yes	characteristic
139	C₂×C₇₉	C₄×#131	1	2	yes	—
140	C₂×C₇₉	C₂×#433	1	2	yes	—
141	C₂×C₇₉	#2922	1	2	yes	—
142	C₂×#433	—	79	2	no	—
143	C₄×#117	D₇₉	1	1	yes	characteristic
144	#1768	—	79	2	no	—
145	C₄×C₄⋊C₈	—	79	2	no	—
146	#1750	—	79	2	no	—
147	C₄×C₂²⋊C₈	—	79	2	no	—
148	#1746	—	79	2	no	—
149	C₄×#97	—	79	1	no	—
150	C₄²×C₈	—	79	1	no	—
151	C₇₉	C₄×#433	1	1	yes	characteristic, complemented, Sylow
152	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
153	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
154	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
155	#433	—	79	16	no	—
156	C₂×C₄⋊C₈	—	79	4	no	—
157	C₂×C₂²⋊C₈	—	79	4	no	—
158	C₂×#97	—	79	2	no	—
159	C₂×#97	—	79	2	no	—
160	C₂×C₄×C₈	—	79	2	no	—
161	C₂×C₄×C₈	—	79	2	no	—
162	C₂×C₄×C₈	—	79	2	no	—
163	C₂×C₄×C₈	—	79	2	no	—
164	C₄×C₄⋊C₄	Dic₇₉	1	4	yes	—
165	C₄×C₂²⋊C₄	Dic₇₉	1	4	yes	—
166	#376	C₂×D₇₉	1	1	yes	characteristic
167	#376	C₂×D₇₉	1	1	yes	characteristic
168	C₄³	C₂×D₇₉	1	1	yes	characteristic
169	C₄³	C₂×D₇₉	1	1	yes	characteristic
170	#336	—	79	4	no	—
171	#336	—	79	4	no	—
172	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
173	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
174	C₂²×C₈	—	79	4	no	—
175	C₂²×C₈	—	79	4	no	—
176	#117	—	2	4	no	—
177	#117	C₄×D₇₉	1	8	yes	—
178	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	2	yes	—
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
184	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
185	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
186	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
192	C₄⋊C₈	—	79	16	no	—
193	C₂²⋊C₈	—	79	16	no	—
194	#97	—	79	8	no	—
195	#97	—	158	4	no	—
196	C₄×C₈	—	79	8	no	—
197	C₄×C₈	—	158	4	no	—
198	C₄×C₈	—	158	4	no	—
199	C₄×C₈	—	158	4	no	—
200	#95	C₂×Dic₇₉	1	4	yes	—
201	#95	C₂×Dic₇₉	1	4	yes	—
202	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	C₂×C₄×D₇₉	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
215	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
216	C₂×C₈	—	79	8	no	—
217	C₂×C₈	—	79	8	no	—
218	C₂×C₈	—	158	4	no	—
219	C₂×C₈	—	158	4	no	—
220	C₂×C₈	—	158	4	no	—
221	C₂×C₈	—	158	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	C₄×Dic₇₉	1	8	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	C₄×Dic₇₉	1	8	yes	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	8	no	—
230	C₄²	—	2	8	no	—
231	C₄²	C₂×C₄×D₇₉	1	4	yes	—
232	C₄²	C₂×C₄×D₇₉	1	4	yes	—
233	C₄²	#11929624	1	4	yes	—
234	C₄²	#11929624	1	4	yes	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
239	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
252	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
254	C₂×C₄	#14579120	1	4	yes	—
255	C₂×C₄	#14579120	1	4	yes	—
256	C₂×C₄	#14579168	1	8	yes	—
257	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
258	C₂×C₄	C₂×#11929624	1	2	yes	—
259	C₂×C₄	C₂×#11929624	1	2	yes	—
260	C₈	—	158	8	no	—
261	C₈	—	158	8	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
267	C₂²	C₂×#14579120	1	1	yes	characteristic
268	C₂²	C₂×#14579120	1	1	yes	characteristic
269	C₂²	C₂×#14579168	1	4	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	#19098624	1	8	yes	direct product factor
277	C₄	C₄×#11929624	1	4	yes	—
278	C₂	—	4	2	no	—
279	C₂	C₄×#14579120	1	1	yes	characteristic
280	C₂	C₄×#14579168	1	2	yes	—
281	C₂	C₂×#19098624	1	2	yes	—
282	C₂	#23772044	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₄×#117×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄×#117	D₇₉
hypocenter	C₇₉	C₄×#433
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #23763257	C₂	semidirect
3. C₄×C₄⋊(C₇₉⋊C₈)	C₂	semidirect
6. #23763152	C₂	semidirect
7. C₄×#14579072	C₂	semidirect
8. C₄²×C₇₉⋊C₈	C₂	semidirect
10. #19098624	C₄	direct (N × H)
32. C₄⋊(C₇₉⋊C₈)	C₂×C₄	semidirect
35. #14579072	C₂×C₄	semidirect
39. C₄×C₇₉⋊C₈	C₂×C₄	semidirect
134. C₄×C₇₉	#433	semidirect
151. C₇₉	C₄×#433	semidirect
276. C₄	#19098624	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×#19098624	2
#23763257	2
C₄×C₄⋊(C₇₉⋊C₈)	2
#23763166	2
C₄×C₂²⋊(C₇₉⋊C₈)	2
#23763152	2
C₄×#14579072	2
C₄²×C₇₉⋊C₈	2
C₄×#117×C₇₉	2
C₄×#433	79

Subgroups — C4×#19098624 #34286413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098624 #34286413