Subgroups — #34178384

Order 20224.

630 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×#19098620	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098616	C₂	1	1	yes	maximal, characteristic
3	#23761283	C₂	1	4	yes	maximal, complemented
4	C₇₉×#2285	C₂	1	1	yes	maximal, characteristic
5	#19098620	—	2	2	no	—
6	#19098620	C₄	1	4	yes	complemented
7	#19098616	—	2	2	no	—
8	#19098616	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
12	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
13	C₂×#14579072	—	2	1	no	—
14	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
15	C₂×#117×C₇₉	—	2	1	no	—
16	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	#384×C₇₉	C₂²	1	2	yes	—
18	#382×C₇₉	—	2	1	no	—
19	#380×C₇₉	—	2	1	no	—
20	#380×C₇₉	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
22	C₄×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	C₂²×C₇₉⋊C₈	—	2	2	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
28	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
29	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
30	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
31	#14579072	—	2	4	no	—
32	C₄×C₇₉⋊C₈	—	2	4	no	—
33	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
36	#117×C₇₉	—	4	4	no	—
37	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
41	C₂×C₄²×C₇₉	—	2	2	no	—
42	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₉	D₄	1	2	yes	—
48	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
49	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	#95×C₇₉	—	2	2	no	—
51	#95×C₇₉	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	2	4	no	—
54	C₂×C₇₉⋊C₈	—	4	4	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	4	2	no	—
57	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	—	2	4	no	—
59	C₂²×C₄×C₇₉	—	2	2	no	—
60	C₂²×C₄×C₇₉	—	2	2	no	—
61	C₂²×C₄×C₇₉	—	2	4	no	—
62	C₂²×C₄×C₇₉	—	4	2	no	—
63	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₇₉	—	4	4	no	—
76	C₄⋊C₄×C₇₉	—	4	4	no	—
77	C₄⋊C₄×C₇₉	—	4	4	no	—
78	C₂²⋊C₄×C₇₉	—	4	4	no	—
79	C₄²×C₇₉	—	4	4	no	—
80	C₄²×C₇₉	C₄²	1	2	yes	—
81	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
85	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
86	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
90	C₂³×C₇₉	—	2	2	no	—
91	C₂³×C₇₉	—	2	2	no	—
92	C₂³×C₇₉	—	2	1	no	—
93	C₂³×C₇₉	—	2	1	no	—
94	C₂³×C₇₉	#95	1	2	yes	—
95	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₉	—	2	2	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	—	2	2	no	—
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	1	no	—
101	C₂×C₄×C₇₉	—	2	4	no	—
102	C₂×C₄×C₇₉	—	2	4	no	—
103	C₂×C₄×C₇₉	—	2	4	no	—
104	C₂×C₄×C₇₉	—	2	4	no	—
105	C₂×C₄×C₇₉	—	4	2	no	—
106	C₂×C₄×C₇₉	—	4	2	no	—
107	C₂×C₄×C₇₉	—	4	4	no	—
108	C₂×C₄×C₇₉	—	4	4	no	—
109	C₂×C₄×C₇₉	—	4	4	no	—
110	C₂×C₄×C₇₉	#95	1	2	yes	—
111	C₂×C₄×C₇₉	#95	1	2	yes	—
112	C₂×C₄×C₇₉	#95	1	2	yes	—
113	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	#135	1	4	yes	—
120	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₇₉	#137	1	2	yes	—
122	C₇₉⋊C₈	—	4	4	no	—
123	C₇₉⋊C₈	—	4	4	no	—
124	C₂²×C₇₉	—	2	2	no	—
125	C₂²×C₇₉	—	2	2	no	—
126	C₂²×C₇₉	—	2	2	no	—
127	C₂²×C₇₉	—	4	2	no	—
128	C₂²×C₇₉	—	4	2	no	—
129	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
130	C₂²×C₇₉	#413	1	2	yes	—
131	C₂²×C₇₉	#416	1	2	yes	—
132	C₂²×C₇₉	#418	1	1	yes	characteristic
133	C₂²×C₇₉	#419	1	1	yes	characteristic
134	C₄×C₇₉	—	2	2	no	—
135	C₄×C₇₉	—	2	2	no	—
136	C₄×C₇₉	—	2	2	no	—
137	C₄×C₇₉	—	2	1	no	—
138	C₄×C₇₉	—	2	1	no	—
139	C₄×C₇₉	—	8	4	no	—
140	C₄×C₇₉	#416	1	2	yes	—
141	C₄×C₇₉	#417	1	2	yes	—
142	C₄×C₇₉	#427	1	2	yes	—
143	C₄×C₇₉	#428	1	2	yes	—
144	#8046	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	—	4	2	no	—
146	C₂×C₇₉	#1553	1	2	yes	—
147	C₂×C₇₉	#1554	1	2	yes	—
148	C₂×C₇₉	#1735	1	1	yes	characteristic
149	C₂×C₇₉	#1736	1	1	yes	characteristic
150	C₂×C₇₉	#1742	1	1	yes	characteristic
151	C₂×#432	—	79	1	no	—
152	C₂×#431	—	79	1	no	—
153	#2285	D₇₉	1	1	yes	characteristic
154	#1296	—	79	4	no	—
155	C₇₉	#8046	1	1	yes	characteristic, complemented, Sylow
156	C₂×#117	—	2	1	no	—
157	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
158	#432	—	79	4	no	—
159	#432	—	158	2	no	—
160	#431	—	79	4	no	—
161	#431	—	158	2	no	—
162	C₂×C₄⋊C₈	—	79	2	no	—
163	C₂×C₄⋊C₈	—	79	2	no	—
164	C₂×C₂²⋊C₈	—	158	1	no	—
165	C₂×C₂²⋊C₈	—	158	1	no	—
166	C₂×#97	—	158	1	no	—
167	C₂×C₄×C₈	—	158	1	no	—
168	#384	C₂×D₇₉	1	2	yes	—
169	#382	—	2	1	no	—
170	#380	—	2	1	no	—
171	#380	Dic₇₉	1	2	yes	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	Dic₇₉	1	2	yes	—
174	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
177	C₂²×C₈	—	158	2	no	—
178	C₂²×C₈	—	158	2	no	—
179	#117	—	4	4	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	Dic₁₅₈	1	2	yes	—
186	C₂×C₄²	C₄×D₇₉	1	2	yes	—
187	C₂×C₄²	C₄×D₇₉	1	2	yes	—
188	C₂×C₄²	D₃₁₆	1	2	yes	—
189	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
193	C₄⋊C₈	—	79	4	no	—
194	C₄⋊C₈	—	79	4	no	—
195	C₄⋊C₈	—	158	2	no	—
196	C₄⋊C₈	—	158	2	no	—
197	C₂²⋊C₈	—	158	4	no	—
198	C₂²⋊C₈	—	158	4	no	—
199	#97	—	158	4	no	—
200	C₄×C₈	—	158	4	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
210	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
211	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
212	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
215	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
216	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
217	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
221	C₂×C₈	—	316	4	no	—
222	C₂×C₈	—	316	2	no	—
223	C₂×C₈	—	316	2	no	—
224	C₂×C₈	—	158	4	no	—
225	C₂×C₈	—	158	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	4	4	no	—
231	C₄²	C₄×Dic₇₉	1	2	yes	—
232	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
233	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
234	C₄²	Dic₇₉⋊C₄	1	2	yes	—
235	C₄²	Dic₇₉⋊C₄	1	2	yes	—
236	C₄²	C₄⋊Dic₇₉	1	2	yes	—
237	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
239	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
240	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	C₇₉⋊#95	1	2	yes	—
246	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	C₇₉⋊#95	1	2	yes	—
262	C₂×C₄	C₇₉⋊#95	1	2	yes	—
263	C₂×C₄	C₇₉⋊#95	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
265	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
270	C₂×C₄	#14579181	1	4	yes	—
271	C₂×C₄	#14579187	1	2	yes	—
272	C₂×C₄	#14579192	1	2	yes	—
273	C₈	—	316	4	no	—
274	C₈	—	316	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
281	C₂²	#19098547	1	2	yes	—
282	C₂²	#19098559	1	2	yes	—
283	C₂²	#19098568	1	1	yes	characteristic
284	C₂²	#19098572	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	4	no	—
291	C₄	#19098557	1	2	yes	—
292	C₄	#19098562	1	2	yes	—
293	C₄	#19098599	1	2	yes	—
294	C₄	#19098603	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	#23762105	1	2	yes	—
297	C₂	#23762110	1	2	yes	—
298	C₂	#23763116	1	1	yes	characteristic
299	C₂	#23763119	1	1	yes	characteristic
300	C₂	#23763139	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2285	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2285	D₇₉
hypocenter	C₇₉	#8046
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #23761283	C₂	semidirect
6. #19098620	C₄	semidirect
8. #19098616	C₄	semidirect
155. C₇₉	#8046	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098620	2
C₂×#19098616	2
#23761283	2
C₇₉×#2285	2
#8046	79