Subgroups — #34257740

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772479	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23762220	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762219	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762218	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11929614	C₂	1	2	yes	maximal, complemented
7	#23762204	C₂	1	2	yes	maximal, complemented
8	#23762145	C₂	1	1	yes	maximal, characteristic
9	#23761793	C₂	1	2	yes	maximal
10	#23761685	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₇₉	C₂²	1	2	yes	—
14	Q₈×C₇₉⋊C₈	C₂²	1	2	yes	—
15	C₈×Dic₁₅₈	—	2	2	no	—
16	C₈×Dic₁₅₈	C₂²	1	2	yes	—
17	#19098687	—	2	2	no	—
18	#19098687	C₂²	1	2	yes	—
19	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₇₉	C₄	1	1	yes	characteristic
21	#19098657	C₄	1	1	yes	characteristic
22	C₄×#11929614	C₂²	1	2	yes	—
23	#19098655	C₄	1	2	yes	—
24	#19098615	—	2	2	no	—
25	#19098615	C₂²	1	2	yes	—
26	#19098614	—	2	2	no	—
27	#19098614	C₂²	1	1	yes	characteristic
28	#19098614	C₂²	1	1	yes	characteristic
29	#107×D₇₉	C₄	1	1	yes	characteristic
30	#103×D₇₉	C₄	1	2	yes	—
31	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
32	#19098315	C₂²	1	2	yes	—
33	#19098314	C₂²	1	1	yes	characteristic
34	#19098295	C₂²	1	2	yes	—
35	Q₈⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
36	#19098293	C₂²	1	2	yes	—
37	#19098284	C₂²	1	1	yes	characteristic
38	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₇₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
42	#334×C₇₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₉	C₂²	1	2	yes	—
44	#322×C₇₉	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₇₉	C₂×C₄	1	2	yes	—
46	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
47	C₂×#11929614	C₂×C₄	1	2	yes	—
48	Q₁₆×D₇₉	C₈	1	8	yes	complemented
49	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
51	Q₈×Dic₇₉	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₅₈	—	2	2	no	—
53	C₄×Dic₁₅₈	C₂³	1	2	yes	—
54	#14579126	—	2	2	no	—
55	#14579126	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×D₇₉	—	2	2	no	—
57	C₄⋊C₄×D₇₉	C₂×C₄	1	2	yes	—
58	C₄²×D₇₉	—	2	2	no	—
59	C₄²×D₇₉	C₂³	1	1	yes	characteristic
60	#14579095	C₂×C₄	1	1	yes	characteristic
61	#14579094	C₂×C₄	1	2	yes	—
62	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
63	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
64	Dic₇₉⋊C₈	—	2	2	no	—
65	Dic₇₉⋊C₈	—	2	2	no	—
66	Dic₇₉⋊C₈	—	2	1	no	—
67	Dic₇₉⋊C₈	D₄	1	2	yes	—
68	#14579086	C₂×C₄	1	2	yes	—
69	#14579085	C₂×C₄	1	2	yes	—
70	#14579084	C₂×C₄	1	2	yes	—
71	(C₂×D₇₉)⋊C₈	—	2	2	no	—
72	(C₂×D₇₉)⋊C₈	—	2	1	no	—
73	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
74	C₄×C₇₉⋊C₈	—	2	2	no	—
75	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×Dic₇₉	—	2	2	no	—
79	C₈×Dic₇₉	—	2	2	no	—
80	C₈×Dic₇₉	—	2	2	no	—
81	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
83	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₇₉	C₂³	1	2	yes	—
85	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₇₉	—	2	2	no	—
87	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
88	#103×C₇₉	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₇₉	—	2	2	no	—
90	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
93	C₂×Dic₁₅₈	—	2	2	no	—
94	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
95	Q₈×D₇₉	—	2	4	no	—
96	Q₈×D₇₉	C₂×C₈	1	8	yes	—
97	C₂×C₄×D₇₉	—	2	2	no	—
98	C₂×C₄×D₇₉	—	2	2	no	—
99	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₇₉	#41	1	1	yes	characteristic
102	Dic₃₁₆	C₂×C₈	1	4	yes	—
103	#11929614	C₂×C₈	1	8	yes	—
104	C₂×C₇₉⋊C₈	—	2	2	no	—
105	C₂×C₇₉⋊C₈	—	2	1	no	—
106	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
109	C₈×D₇₉	—	2	2	no	—
110	C₈×D₇₉	C₂×C₈	1	4	yes	—
111	C₄⋊Dic₇₉	—	2	2	no	—
112	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
113	Dic₇₉⋊C₄	—	2	2	no	—
114	Dic₇₉⋊C₄	—	2	2	no	—
115	Dic₇₉⋊C₄	—	2	2	no	—
116	C₄×Dic₇₉	—	2	2	no	—
117	C₄×Dic₇₉	—	2	2	no	—
118	C₄×Dic₇₉	—	2	2	no	—
119	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₇₉	#41	1	1	yes	characteristic
122	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
123	Q₁₆×C₇₉	C₂×C₈	1	4	yes	—
124	C₂×C₈×C₇₉	—	2	2	no	—
125	C₂×C₈×C₇₉	—	2	1	no	—
126	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
129	C₄⋊C₄×C₇₉	—	2	2	no	—
130	C₄⋊C₄×C₇₉	C₂²×C₄	1	2	yes	—
131	C₄²×C₇₉	—	2	2	no	—
132	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
133	C₂²×D₇₉	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₇₉	—	2	2	no	—
135	Q₈×C₇₉	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₇₉	—	2	2	no	—
137	C₂×C₄×C₇₉	—	2	2	no	—
138	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₇₉	—	2	2	no	—
142	C₂×Dic₇₉	—	2	2	no	—
143	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
146	C₄×D₇₉	—	2	8	no	—
147	C₄×D₇₉	—	2	2	no	—
148	C₄×D₇₉	—	4	1	no	—
149	C₄×D₇₉	—	8	2	no	—
150	C₄×D₇₉	C₂²×C₈	1	2	yes	—
151	C₄×D₇₉	#131	1	2	yes	—
152	Dic₁₅₈	—	2	2	no	—
153	Dic₁₅₈	—	2	8	no	—
154	Dic₁₅₈	C₂²×C₈	1	4	yes	—
155	C₈×C₇₉	—	2	1	no	—
156	C₈×C₇₉	—	2	2	no	—
157	C₈×C₇₉	—	4	1	no	—
158	C₈×C₇₉	—	4	2	no	—
159	C₈×C₇₉	C₂²×C₈	1	2	yes	—
160	C₇₉⋊C₈	—	2	1	no	—
161	C₇₉⋊C₈	—	2	2	no	—
162	C₇₉⋊C₈	—	4	1	no	—
163	C₇₉⋊C₈	—	4	2	no	—
164	C₇₉⋊C₈	C₂²×C₈	1	2	yes	—
165	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂×D₇₉	—	2	1	no	—
167	C₂×D₇₉	—	2	1	no	—
168	C₂×D₇₉	C₈×D₄	1	2	yes	—
169	C₄×C₇₉	—	2	4	no	—
170	C₄×C₇₉	—	2	1	no	—
171	C₄×C₇₉	—	4	2	no	—
172	C₄×C₇₉	C₂³×C₈	1	1	yes	characteristic
173	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
174	C₄×C₇₉	#578	1	2	yes	—
175	Dic₇₉	—	2	4	no	—
176	Dic₇₉	—	2	1	no	—
177	Dic₇₉	—	2	1	no	—
178	Dic₇₉	—	4	2	no	—
179	Dic₇₉	C₈×D₄	1	2	yes	—
180	#17856	—	79	1	no	maximal, Sylow
181	C₂×C₇₉	C₂×C₈×D₄	1	1	yes	characteristic
182	C₂×C₇₉	#2950	1	1	yes	characteristic
183	C₂×C₇₉	#2957	1	1	yes	characteristic
184	D₇₉	—	2	2	no	—
185	#2982	—	79	2	no	—
186	C₂×C₄×Q₁₆	—	79	1	no	—
187	#1579	—	79	1	no	—
188	#1579	—	79	1	no	—
189	#1579	—	79	1	no	—
190	#1579	D₇₉	1	1	yes	characteristic, complemented
191	C₈×Q₁₆	—	79	2	no	—
192	C₈×Q₁₆	—	79	2	no	—
193	#1564	—	79	1	no	—
194	#1479	—	79	2	no	—
195	#1450	—	79	1	no	—
196	C₇₉	#17856	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₁₆	—	79	1	no	—
198	C₂×C₄×Q₈	—	79	2	no	—
199	C₈×Q₈	—	79	2	no	—
200	C₈×Q₈	—	79	2	no	—
201	C₈×Q₈	—	79	2	no	—
202	C₈×Q₈	—	158	2	no	—
203	C₈×Q₈	—	158	2	no	—
204	C₈×Q₈	C₂×D₇₉	1	2	yes	—
205	C₄×Q₁₆	—	79	1	no	—
206	C₄×Q₁₆	—	79	2	no	—
207	C₄×Q₁₆	—	79	1	no	—
208	C₄×Q₁₆	—	79	2	no	—
209	C₄×Q₁₆	—	79	1	no	—
210	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
211	#431	—	79	1	no	—
212	#431	—	79	1	no	—
213	#431	—	79	2	no	—
214	#431	—	158	2	no	—
215	#431	—	158	2	no	—
216	C₂×#107	—	79	1	no	—
217	C₂×#103	—	79	2	no	—
218	C₂×C₄×C₈	—	79	1	no	—
219	#335	—	79	2	no	—
220	#334	—	79	1	no	—
221	#334	C₂×D₇₉	1	1	yes	characteristic
222	Q₈⋊C₈	—	79	2	no	—
223	Q₈⋊C₈	—	79	2	no	—
224	Q₈⋊C₈	—	79	2	no	—
225	Q₈⋊C₈	C₂×D₇₉	1	2	yes	—
226	#322	—	79	1	no	—
227	#322	C₂×D₇₉	1	1	yes	characteristic
228	C₈²	—	79	1	no	—
229	C₈²	—	79	1	no	—
230	C₂²×Q₈	—	79	2	no	—
231	C₂×Q₁₆	—	79	2	no	—
232	C₂×Q₁₆	—	79	8	no	—
233	C₂×Q₁₆	—	79	1	no	—
234	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
235	C₂²×C₈	—	79	1	no	—
236	C₂²×C₈	—	79	1	no	—
237	C₄×Q₈	—	79	2	no	—
238	C₄×Q₈	—	79	2	no	—
239	C₄×Q₈	—	79	2	no	—
240	C₄×Q₈	—	158	2	no	—
241	C₄×Q₈	—	158	2	no	—
242	C₄×Q₈	C₂²×D₇₉	1	2	yes	—
243	C₂×C₄⋊C₄	—	79	2	no	—
244	C₂×C₄⋊C₄	—	158	2	no	—
245	C₂×C₄²	—	79	1	no	—
246	C₂×C₄²	—	158	2	no	—
247	#107	—	79	1	no	—
248	#107	—	79	2	no	—
249	#107	C₄×D₇₉	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	79	2	no	—
252	C₄⋊C₈	—	79	2	no	—
253	C₄⋊C₈	—	158	2	no	—
254	C₄⋊C₈	—	158	1	no	—
255	C₄⋊C₈	—	158	2	no	—
256	C₄⋊C₈	—	158	2	no	—
257	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
258	#103	—	79	2	no	—
259	#103	—	79	2	no	—
260	#103	—	79	2	no	—
261	#103	C₄×D₇₉	1	2	yes	—
262	C₂²⋊C₈	—	79	2	no	—
263	C₂²⋊C₈	—	158	1	no	—
264	C₂²⋊C₈	—	158	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	79	1	no	—
267	C₄×C₈	—	79	1	no	—
268	C₄×C₈	—	79	1	no	—
269	C₄×C₈	—	79	1	no	—
270	C₄×C₈	—	79	1	no	—
271	C₄×C₈	—	158	2	no	—
272	C₄×C₈	—	158	2	no	—
273	C₄×C₈	—	158	2	no	—
274	C₄×C₈	—	158	2	no	—
275	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
278	C₂×Q₈	—	79	8	no	—
279	C₂×Q₈	—	79	2	no	—
280	C₂×Q₈	—	158	2	no	—
281	C₂×Q₈	—	158	4	no	—
282	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
283	C₂²×C₄	—	79	1	no	—
284	C₂²×C₄	—	79	1	no	—
285	C₂²×C₄	—	79	1	no	—
286	C₂²×C₄	—	158	2	no	—
287	C₂²×C₄	—	158	2	no	—
288	Q₁₆	—	79	8	no	—
289	Q₁₆	—	79	4	no	—
290	Q₁₆	C₈×D₇₉	1	4	yes	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	79	1	no	—
294	C₂×C₈	—	79	4	no	—
295	C₂×C₈	—	79	1	no	—
296	C₂×C₈	—	79	2	no	—
297	C₂×C₈	—	158	2	no	—
298	C₂×C₈	—	158	1	no	—
299	C₂×C₈	—	158	2	no	—
300	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
302	C₂×C₈	D₄×D₇₉	1	2	yes	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	79	2	no	—
305	C₄⋊C₄	—	158	2	no	—
306	C₄⋊C₄	—	158	2	no	—
307	C₄⋊C₄	—	158	2	no	—
308	C₄⋊C₄	—	158	2	no	—
309	C₄⋊C₄	C₂×C₄×D₇₉	1	2	yes	—
310	C₄²	—	2	2	no	—
311	C₄²	—	79	1	no	—
312	C₄²	—	79	1	no	—
313	C₄²	—	79	1	no	—
314	C₄²	—	158	2	no	—
315	C₄²	—	158	2	no	—
316	C₄²	—	158	2	no	—
317	C₄²	C₂³×D₇₉	1	1	yes	characteristic
318	C₂³	—	79	1	no	—
319	Q₈	—	2	2	no	—
320	Q₈	—	79	4	no	—
321	Q₈	—	158	8	no	—
322	Q₈	—	158	2	no	—
323	Q₈	C₂×C₈×D₇₉	1	4	yes	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	316	1	no	—
327	C₂×C₄	—	79	1	no	—
328	C₂×C₄	—	79	2	no	—
329	C₂×C₄	—	79	2	no	—
330	C₂×C₄	—	79	1	no	—
331	C₂×C₄	—	79	1	no	—
332	C₂×C₄	—	632	2	no	—
333	C₂×C₄	—	158	8	no	—
334	C₂×C₄	—	158	2	no	—
335	C₂×C₄	—	158	2	no	—
336	C₂×C₄	—	158	2	no	—
337	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
338	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
339	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	2	2	no	—
342	C₈	—	4	1	no	—
343	C₈	—	4	2	no	—
344	C₈	—	316	1	no	—
345	C₈	—	316	2	no	—
346	C₈	—	79	2	no	—
347	C₈	—	158	1	no	—
348	C₈	—	158	2	no	—
349	C₈	C₂×C₈×D₇₉	1	2	yes	—
350	C₂²	—	79	2	no	—
351	C₂²	—	158	1	no	—
352	C₂²	—	158	1	no	—
353	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
354	C₄	—	2	1	no	—
355	C₄	—	2	4	no	—
356	C₄	—	4	2	no	—
357	C₄	—	316	2	no	—
358	C₄	—	79	2	no	—
359	C₄	—	158	1	no	—
360	C₄	—	158	1	no	—
361	C₄	—	158	4	no	—
362	C₄	C₂²×C₈×D₇₉	1	1	yes	characteristic
363	C₄	#131×D₇₉	1	1	yes	characteristic
364	C₄	#19099379	1	2	yes	—
365	C₂	—	158	2	no	—
366	C₂	C₈×D₄×D₇₉	1	1	yes	characteristic
367	C₂	#23772265	1	1	yes	characteristic
368	C₂	#23772314	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₄×C₇₉	C₂³×C₈
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1579	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂²×D₇₉	C₄×D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1579	D₇₉
hypocenter	C₇₉	#17856
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #23762220	C₂	semidirect
4. #23762219	C₂	semidirect
5. #23762218	C₂	semidirect
6. C₈×#11929614	C₂	semidirect
7. #23762204	C₂	semidirect
11. C₇₉×#1579	C₂	semidirect
48. Q₁₆×D₇₉	C₈	semidirect
190. #1579	D₇₉	semidirect
196. C₇₉	#17856	semidirect

Maximal subgroups

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

subgroup	index
#23772479	2
C₄×Q₁₆×D₇₉	2
#23762220	2
#23762219	2
#23762218	2
C₈×#11929614	2
#23762204	2
#23762145	2
#23761793	2
#23761685	2
C₇₉×#1579	2
#17856	79