Subgroups — #34177751

Order 20224.

630 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098619	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098614	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×D₇₉	C₂	1	1	yes	maximal, characteristic
4	#23761235	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761234	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761234	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₉	—	2	1	no	—
9	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
10	#19098619	—	2	2	no	—
11	#19098619	C₄	1	4	yes	—
12	#19098614	—	2	2	no	—
13	#19098614	—	2	2	no	—
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×Dic₇₉⋊C₈	—	2	1	no	—
17	C₂×Dic₇₉⋊C₈	C₄	1	2	yes	—
18	C₄⋊C₈×D₇₉	—	2	2	no	—
19	C₄⋊C₈×D₇₉	C₄	1	4	yes	—
20	C₂×(C₂×D₇₉)⋊C₈	—	2	1	no	—
21	C₂×(C₂×D₇₉)⋊C₈	C₄	1	2	yes	—
22	C₂×#14579071	—	2	1	no	—
23	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₇₉	Q₈	1	1	yes	characteristic
30	C₂²×C₇₉⋊C₈	—	2	1	no	—
31	C₂²×C₇₉⋊C₈	—	2	1	no	—
32	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₇₉	—	2	4	no	—
34	C₂×C₈×D₇₉	—	4	2	no	—
35	C₂×C₈×D₇₉	—	4	2	no	—
36	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₇₉	Q₈	1	1	yes	characteristic
39	C₄²×D₇₉	C₂×C₄	1	2	yes	—
40	C₄²×D₇₉	C₂×C₄	1	2	yes	—
41	C₄²×D₇₉	D₄	1	2	yes	—
42	C₄²×D₇₉	D₄	1	2	yes	—
43	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
44	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
46	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
47	Dic₇₉⋊C₈	—	2	4	no	—
48	Dic₇₉⋊C₈	—	2	4	no	—
49	(C₂×D₇₉)⋊C₈	—	2	4	no	—
50	(C₂×D₇₉)⋊C₈	—	2	4	no	—
51	#14579071	—	2	4	no	—
52	C₄×C₇₉⋊C₈	—	2	1	no	—
53	C₄×C₇₉⋊C₈	—	2	1	no	—
54	C₂²×C₈×C₇₉	—	2	1	no	—
55	C₂²×C₈×C₇₉	—	2	1	no	—
56	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₇₉	—	2	1	no	—
59	C₄⋊C₈×C₇₉	—	2	1	no	—
60	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
61	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₇₉	—	2	1	no	—
63	C₄×C₈×C₇₉	—	2	1	no	—
64	C₂³×D₇₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×D₇₉	—	2	4	no	—
69	C₂×C₄×D₇₉	—	2	2	no	—
70	C₂×C₄×D₇₉	—	2	2	no	—
71	C₂×C₄×D₇₉	—	2	1	no	—
72	C₂×C₄×D₇₉	—	2	1	no	—
73	C₂×C₄×D₇₉	—	2	1	no	—
74	C₂×C₄×D₇₉	—	2	1	no	—
75	C₂×C₄×D₇₉	C₄²	1	2	yes	—
76	C₂×C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×D₇₉	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×D₇₉	C₄⋊C₄	1	2	yes	—
81	C₂×C₇₉⋊C₈	—	2	2	no	—
82	C₂×C₇₉⋊C₈	—	2	2	no	—
83	C₂×C₇₉⋊C₈	—	2	2	no	—
84	C₂×C₇₉⋊C₈	—	2	2	no	—
85	C₂×C₇₉⋊C₈	—	2	2	no	—
86	C₂×C₇₉⋊C₈	—	4	1	no	—
87	C₂×C₇₉⋊C₈	—	4	1	no	—
88	C₂×C₇₉⋊C₈	—	4	1	no	—
89	C₂×C₇₉⋊C₈	—	4	1	no	—
90	C₈×D₇₉	—	4	8	no	—
91	C₄×Dic₇₉	C₄²	1	2	yes	—
92	C₄×Dic₇₉	C₂²⋊C₄	1	2	yes	—
93	C₄×Dic₇₉	C₂²⋊C₄	1	2	yes	—
94	C₄×Dic₇₉	C₂²⋊C₄	1	2	yes	—
95	C₄×Dic₇₉	C₄⋊C₄	1	2	yes	—
96	C₄×Dic₇₉	C₄⋊C₄	1	2	yes	—
97	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₇₉	—	2	2	no	—
101	C₂×C₈×C₇₉	—	2	2	no	—
102	C₂×C₈×C₇₉	—	2	2	no	—
103	C₂×C₈×C₇₉	—	2	2	no	—
104	C₂×C₈×C₇₉	—	2	2	no	—
105	C₂×C₈×C₇₉	—	4	1	no	—
106	C₂×C₈×C₇₉	—	4	1	no	—
107	C₂×C₈×C₇₉	—	4	1	no	—
108	C₂×C₈×C₇₉	—	4	1	no	—
109	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₇₉	—	2	2	no	—
115	C₂²×D₇₉	—	2	1	no	—
116	C₂²×D₇₉	—	2	1	no	—
117	C₂²×D₇₉	#95	1	2	yes	—
118	C₂²×D₇₉	#108	1	4	yes	—
119	C₂×C₄×C₇₉	—	2	2	no	—
120	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#131	1	2	yes	—
127	C₂×C₄×C₇₉	#131	1	2	yes	—
128	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
132	C₂×Dic₇₉	—	2	2	no	—
133	C₂×Dic₇₉	—	2	1	no	—
134	C₂×Dic₇₉	—	2	1	no	—
135	C₂×Dic₇₉	—	2	2	no	—
136	C₂×Dic₇₉	#95	1	2	yes	—
137	C₂×Dic₇₉	#95	1	2	yes	—
138	C₂×Dic₇₉	#95	1	2	yes	—
139	C₄×D₇₉	—	2	4	no	—
140	C₄×D₇₉	—	2	4	no	—
141	C₄×D₇₉	—	2	2	no	—
142	C₄×D₇₉	—	2	2	no	—
143	C₄×D₇₉	—	2	4	no	—
144	C₄×D₇₉	—	2	2	no	—
145	C₄×D₇₉	—	2	2	no	—
146	C₄×D₇₉	—	2	2	no	—
147	C₄×D₇₉	—	2	2	no	—
148	C₈×C₇₉	—	4	2	no	—
149	C₈×C₇₉	—	4	2	no	—
150	C₈×C₇₉	—	4	2	no	—
151	C₇₉⋊C₈	—	4	2	no	—
152	C₇₉⋊C₈	—	4	2	no	—
153	C₇₉⋊C₈	—	4	2	no	—
154	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
155	C₂²×C₇₉	#424	1	2	yes	—
156	C₂²×C₇₉	#428	1	1	yes	characteristic
157	C₂²×C₇₉	#428	1	1	yes	characteristic
158	C₂²×C₇₉	C₂×#108	1	2	yes	—
159	C₂×D₇₉	—	2	4	no	—
160	C₂×D₇₉	—	2	4	no	—
161	C₂×D₇₉	—	2	2	no	—
162	C₂×D₇₉	—	2	2	no	—
163	C₂×D₇₉	#341	1	4	yes	—
164	C₄×C₇₉	—	2	1	no	—
165	C₄×C₇₉	—	2	1	no	—
166	C₄×C₇₉	#405	1	1	yes	characteristic
167	C₄×C₇₉	#405	1	1	yes	characteristic
168	C₄×C₇₉	#408	1	2	yes	—
169	C₄×C₇₉	#417	1	1	yes	characteristic
170	C₄×C₇₉	#417	1	1	yes	characteristic
171	C₄×C₇₉	#418	1	1	yes	characteristic
172	C₄×C₇₉	#419	1	1	yes	characteristic
173	Dic₇₉	—	2	2	no	—
174	Dic₇₉	—	2	2	no	—
175	Dic₇₉	—	2	2	no	—
176	#7879	—	79	1	no	maximal, Sylow
177	C₂×C₇₉	#1573	1	2	yes	—
178	C₂×C₇₉	#1573	1	2	yes	—
179	C₂×C₇₉	#1729	1	1	yes	characteristic
180	C₂×C₇₉	#1736	1	1	yes	characteristic
181	C₂×C₇₉	C₂×#341	1	1	yes	characteristic
182	D₇₉	—	2	4	no	—
183	C₂×#432	—	79	1	no	—
184	C₂×#431	—	79	1	no	—
185	C₂²×C₄⋊C₈	—	79	1	no	—
186	#1277	—	79	1	no	—
187	#1277	—	79	1	no	—
188	#1277	—	79	1	no	—
189	#1277	D₇₉	1	1	yes	characteristic, complemented
190	C₇₉	#7879	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₈	—	158	1	no	—
192	C₂²×C₄²	—	79	1	no	—
193	#432	—	79	4	no	—
194	#432	—	158	2	no	—
195	#431	—	158	2	no	—
196	#431	—	158	2	no	—
197	C₂×C₄⋊C₈	—	79	1	no	—
198	C₂×C₄⋊C₈	—	79	1	no	—
199	C₂×C₄⋊C₈	—	79	4	no	—
200	C₂×C₄⋊C₈	—	79	2	no	—
201	C₂×C₄⋊C₈	—	158	2	no	—
202	C₂×C₄⋊C₈	—	158	1	no	—
203	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
205	C₂×C₂²⋊C₈	—	79	2	no	—
206	C₂×C₂²⋊C₈	—	158	1	no	—
207	C₂×#97	—	158	1	no	—
208	C₂×C₄×C₈	—	79	1	no	—
209	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
210	C₂³×C₄	—	79	1	no	—
211	C₂³×C₄	—	79	1	no	—
212	C₂³×C₄	—	79	1	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	2	1	no	—
215	C₂²×C₈	—	316	2	no	—
216	C₂²×C₈	—	316	2	no	—
217	C₂²×C₈	—	79	2	no	—
218	C₂²×C₈	—	158	4	no	—
219	C₂²×C₈	—	158	1	no	—
220	C₂²×C₈	—	158	1	no	—
221	C₂²×C₈	C₄×D₇₉	1	2	yes	—
222	C₂×C₄²	—	79	1	no	—
223	C₂×C₄²	—	79	1	no	—
224	C₂×C₄²	—	79	1	no	—
225	C₂×C₄²	—	79	2	no	—
226	C₂×C₄²	—	79	2	no	—
227	C₂×C₄²	—	79	2	no	—
228	C₂×C₄²	—	79	2	no	—
229	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	79	2	no	—
233	C₄⋊C₈	—	79	2	no	—
234	C₄⋊C₈	—	158	4	no	—
235	C₄⋊C₈	—	158	4	no	—
236	C₄⋊C₈	—	158	1	no	—
237	C₄⋊C₈	—	158	1	no	—
238	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
239	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
240	C₂²⋊C₈	—	158	4	no	—
241	C₂²⋊C₈	—	158	4	no	—
242	#97	—	158	4	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	158	1	no	—
246	C₄×C₈	—	158	1	no	—
247	C₂⁴	—	79	1	no	—
248	C₂²×C₄	—	79	1	no	—
249	C₂²×C₄	—	79	1	no	—
250	C₂²×C₄	—	79	1	no	—
251	C₂²×C₄	—	79	2	no	—
252	C₂²×C₄	—	79	2	no	—
253	C₂²×C₄	—	79	2	no	—
254	C₂²×C₄	—	79	2	no	—
255	C₂²×C₄	—	79	2	no	—
256	C₂²×C₄	—	79	2	no	—
257	C₂²×C₄	—	158	4	no	—
258	C₂²×C₄	—	158	2	no	—
259	C₂²×C₄	—	158	2	no	—
260	C₂²×C₄	—	158	1	no	—
261	C₂²×C₄	—	158	1	no	—
262	C₂²×C₄	—	158	1	no	—
263	C₂²×C₄	—	158	1	no	—
264	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
266	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	316	8	no	—
277	C₂×C₈	—	316	1	no	—
278	C₂×C₈	—	316	1	no	—
279	C₂×C₈	—	316	1	no	—
280	C₂×C₈	—	316	1	no	—
281	C₂×C₈	—	158	2	no	—
282	C₂×C₈	—	158	2	no	—
283	C₂×C₈	—	158	2	no	—
284	C₂×C₈	—	158	2	no	—
285	C₂×C₈	—	158	2	no	—
286	C₄²	—	79	2	no	—
287	C₄²	—	79	2	no	—
288	C₄²	—	79	2	no	—
289	C₄²	—	79	2	no	—
290	C₄²	—	79	2	no	—
291	C₄²	—	79	2	no	—
292	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
293	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
294	C₄²	D₄×D₇₉	1	1	yes	characteristic
295	C₄²	D₄×D₇₉	1	1	yes	characteristic
296	C₂³	—	79	2	no	—
297	C₂³	—	79	4	no	—
298	C₂³	—	158	2	no	—
299	C₂³	—	158	1	no	—
300	C₂³	—	158	1	no	—
301	C₂³	C₄⋊C₄×D₇₉	1	1	yes	characteristic
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	79	2	no	—
304	C₂×C₄	—	79	2	no	—
305	C₂×C₄	—	79	2	no	—
306	C₂×C₄	—	158	4	no	—
307	C₂×C₄	—	158	4	no	—
308	C₂×C₄	—	158	2	no	—
309	C₂×C₄	—	158	1	no	—
310	C₂×C₄	—	158	1	no	—
311	C₂×C₄	—	158	2	no	—
312	C₂×C₄	—	158	2	no	—
313	C₂×C₄	—	158	2	no	—
314	C₂×C₄	—	158	4	no	—
315	C₂×C₄	—	158	2	no	—
316	C₂×C₄	—	158	2	no	—
317	C₂×C₄	—	158	2	no	—
318	C₂×C₄	—	158	2	no	—
319	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
320	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
321	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
322	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
323	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
324	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
325	C₂×C₄	#14579167	1	2	yes	—
326	C₂×C₄	#14579169	1	2	yes	—
327	C₂×C₄	#14579185	1	1	yes	characteristic
328	C₂×C₄	#14579188	1	1	yes	characteristic
329	C₂×C₄	#14579190	1	1	yes	characteristic
330	C₂×C₄	#14579193	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	316	2	no	—
335	C₈	—	316	2	no	—
336	C₈	—	316	2	no	—
337	C₂²	—	79	4	no	—
338	C₂²	—	158	4	no	—
339	C₂²	—	158	4	no	—
340	C₂²	—	158	2	no	—
341	C₂²	—	158	2	no	—
342	C₂²	#95×D₇₉	1	1	yes	characteristic
343	C₂²	#19098588	1	2	yes	—
344	C₂²	#19098602	1	1	yes	characteristic
345	C₂²	#19098604	1	1	yes	characteristic
346	C₂²	#108×D₇₉	1	2	yes	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	158	2	no	—
350	C₄	—	158	2	no	—
351	C₄	—	158	2	no	—
352	C₄	#19098517	1	1	yes	characteristic
353	C₄	#19098520	1	1	yes	characteristic
354	C₄	#19098528	1	2	yes	—
355	C₄	#19098561	1	1	yes	characteristic
356	C₄	#19098563	1	1	yes	characteristic
357	C₄	#19098566	1	1	yes	characteristic
358	C₄	#19098570	1	1	yes	characteristic
359	C₂	—	158	4	no	—
360	C₂	#23762182	1	2	yes	—
361	C₂	#23762184	1	2	yes	—
362	C₂	#23763095	1	1	yes	characteristic
363	C₂	#23763118	1	1	yes	characteristic
364	C₂	#341×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1277	C₂
socle	C₂³×C₇₉	C₂×C₄⋊C₄
prime radical	C₂³×D₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1277	D₇₉
hypocenter	C₇₉	#7879
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
4. #23761235	C₂	semidirect
5. #23761234	C₂	semidirect
6. #23761234	C₂	semidirect
7. C₇₉×#1277	C₂	semidirect
189. #1277	D₇₉	semidirect
190. C₇₉	#7879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098619	2
C₂×#19098614	2
C₂×C₄⋊C₈×D₇₉	2
#23761235	2
#23761234	2
C₇₉×#1277	2
#7879	79