Subgroups — #34260300

Order 20224.

564 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773983	C₂	1	1	yes	maximal, characteristic
2	#23772498	C₂	1	2	yes	maximal
3	#23762308	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762248	C₂	1	2	yes	maximal, complemented
5	#23762247	C₂	1	2	yes	maximal, complemented
6	#23762220	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762219	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762145	C₂	1	1	yes	maximal, characteristic
9	#23761819	C₂	1	2	yes	maximal
10	#23761686	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099117	C₂²	1	2	yes	—
13	#19098967	C₄	1	1	yes	characteristic
14	#19098941	C₄	1	1	yes	characteristic
15	#19098912	C₄	1	2	yes	—
16	#19098805	C₄	1	1	yes	characteristic
17	#19098803	C₄	1	2	yes	—
18	#19098801	C₄	1	1	yes	characteristic
19	#19098691	—	2	2	no	—
20	#19098690	C₂²	1	2	yes	—
21	Q₈×C₇₉⋊C₈	C₂²	1	2	yes	—
22	C₈×Dic₁₅₈	C₂²	1	2	yes	—
23	#19098687	—	2	2	no	—
24	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
25	C₄×#11929614	C₂²	1	2	yes	—
26	#19098625	—	2	2	no	—
27	#19098623	—	2	2	no	—
28	#19098615	C₂²	1	2	yes	—
29	#19098614	C₂²	1	1	yes	characteristic
30	C₄×#11929607	C₂²	1	1	yes	characteristic
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	#19098285	C₂²	1	1	yes	characteristic
38	#19098284	C₂²	1	1	yes	characteristic
39	#19098284	C₂²	1	1	yes	characteristic
40	#446×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×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11929614	C₂×C₄	1	2	yes	—
47	C₂×#11929607	—	2	1	no	—
48	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
50	#14579160	C₂×C₄	1	2	yes	—
51	#14579159	C₂×C₄	1	2	yes	—
52	#14579151	—	2	2	no	—
53	#14579141	C₂×C₄	1	2	yes	—
54	#14579139	—	2	2	no	—
55	#14579137	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₅₈	—	2	2	no	—
57	C₄×Dic₁₅₈	C₂³	1	2	yes	—
58	#14579118	—	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	D₄	1	2	yes	—
67	#14579086	C₂×C₄	1	2	yes	—
68	#14579085	C₂×C₄	1	2	yes	—
69	#14579084	C₂×C₄	1	2	yes	—
70	(C₂×D₇₉)⋊C₈	—	2	2	no	—
71	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
72	#14579071	—	2	2	no	—
73	#14579071	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₇₉	C₂³	1	2	yes	—
84	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₇₉	—	2	2	no	—
86	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
87	#103×C₇₉	C₂×C₄	1	2	yes	—
88	#97×C₇₉	—	2	2	no	—
89	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₅₈	—	2	2	no	—
93	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₇₉	#41	1	1	yes	characteristic
97	Dic₃₁₆	—	2	2	no	—
98	#11929614	—	4	2	no	—
99	#11929607	—	4	2	no	—
100	C₂×C₇₉⋊C₈	—	2	2	no	—
101	C₂×C₇₉⋊C₈	—	2	1	no	—
102	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
105	C₈×D₇₉	—	2	2	no	—
106	C₈×D₇₉	—	4	1	no	—
107	C₄⋊Dic₇₉	—	2	2	no	—
108	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
109	Dic₇₉⋊C₄	—	2	2	no	—
110	Dic₇₉⋊C₄	—	2	2	no	—
111	Dic₇₉⋊C₄	—	2	2	no	—
112	Dic₇₉⋊C₄	—	2	2	no	—
113	Dic₇₉⋊C₄	—	2	2	no	—
114	(C₂×D₇₉)⋊C₄	—	2	2	no	—
115	(C₂×D₇₉)⋊C₄	—	2	2	no	—
116	C₄×Dic₇₉	—	2	2	no	—
117	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
118	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
119	C₄×Dic₇₉	#41	1	1	yes	characteristic
120	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
121	Q₁₆×C₇₉	—	2	2	no	—
122	C₂×C₈×C₇₉	—	2	2	no	—
123	C₂×C₈×C₇₉	—	2	1	no	—
124	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
127	C₄⋊C₄×C₇₉	—	2	2	no	—
128	C₄⋊C₄×C₇₉	C₂²×C₄	1	2	yes	—
129	C₄²×C₇₉	—	2	2	no	—
130	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
131	C₂²×D₇₉	C₄×D₄	1	1	yes	characteristic
132	Q₈×C₇₉	—	2	2	no	—
133	Q₈×C₇₉	—	2	2	no	—
134	C₂×C₄×C₇₉	—	2	2	no	—
135	C₂×C₄×C₇₉	—	2	2	no	—
136	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₇₉	—	2	2	no	—
140	C₂×Dic₇₉	—	2	2	no	—
141	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
143	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
144	C₄×D₇₉	—	2	1	no	—
145	C₄×D₇₉	—	2	1	no	—
146	C₄×D₇₉	—	2	2	no	—
147	C₄×D₇₉	—	2	2	no	—
148	Dic₁₅₈	—	2	2	no	—
149	Dic₁₅₈	—	2	2	no	—
150	Dic₁₅₈	—	4	4	no	—
151	C₈×C₇₉	—	2	1	no	—
152	C₈×C₇₉	—	2	2	no	—
153	C₈×C₇₉	—	4	1	no	—
154	C₈×C₇₉	—	4	2	no	—
155	C₈×C₇₉	#131	1	2	yes	—
156	C₇₉⋊C₈	—	2	1	no	—
157	C₇₉⋊C₈	—	2	1	no	—
158	C₇₉⋊C₈	—	2	2	no	—
159	C₇₉⋊C₈	—	4	1	no	—
160	C₇₉⋊C₈	—	4	2	no	—
161	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×D₇₉	—	2	1	no	—
163	C₂×D₇₉	—	2	1	no	—
164	C₂×D₇₉	—	2	1	no	—
165	C₄×C₇₉	—	2	1	no	—
166	C₄×C₇₉	—	4	2	no	—
167	C₄×C₇₉	—	4	2	no	—
168	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
169	C₄×C₇₉	#568	1	1	yes	characteristic
170	C₄×C₇₉	#578	1	2	yes	—
171	Dic₇₉	—	2	1	no	—
172	Dic₇₉	—	2	1	no	—
173	Dic₇₉	—	4	2	no	—
174	Dic₇₉	—	4	2	no	—
175	Dic₇₉	#444	1	2	yes	—
176	#18090	—	79	1	no	maximal, Sylow
177	C₂×C₇₉	#2935	1	1	yes	characteristic
178	C₂×C₇₉	#2950	1	1	yes	characteristic
179	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
180	D₇₉	—	4	1	no	—
181	#3170	—	79	1	no	—
182	#2986	—	79	2	no	—
183	#1593	—	79	1	no	—
184	#1593	D₇₉	1	1	yes	characteristic, complemented
185	#1584	—	79	2	no	—
186	#1584	—	79	2	no	—
187	#1579	—	79	1	no	—
188	#1579	—	79	1	no	—
189	#1564	—	79	1	no	—
190	#1485	—	79	2	no	—
191	#1451	—	79	1	no	—
192	C₇₉	#18090	1	1	yes	characteristic, complemented, Sylow
193	#533	—	79	2	no	—
194	#500	—	79	1	no	—
195	#494	—	79	1	no	—
196	#487	—	79	2	no	—
197	#467	—	79	1	no	—
198	#467	—	79	1	no	—
199	#467	—	79	2	no	—
200	#446	—	79	2	no	—
201	#446	—	158	2	no	—
202	#446	C₂×D₇₉	1	2	yes	—
203	C₈×Q₈	—	79	2	no	—
204	C₈×Q₈	—	79	2	no	—
205	C₈×Q₈	—	158	2	no	—
206	C₄×Q₁₆	—	79	2	no	—
207	C₄×Q₁₆	—	79	1	no	—
208	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
209	#433	—	158	2	no	—
210	#433	—	158	2	no	—
211	#431	—	79	1	no	—
212	#431	—	79	2	no	—
213	C₄×M₁₆	—	79	1	no	—
214	C₂×C₄×C₈	—	79	1	no	—
215	#335	—	79	2	no	—
216	#334	—	79	1	no	—
217	#334	C₂×D₇₉	1	1	yes	characteristic
218	Q₈⋊C₈	—	79	2	no	—
219	Q₈⋊C₈	—	79	2	no	—
220	Q₈⋊C₈	—	79	2	no	—
221	Q₈⋊C₈	C₂×D₇₉	1	2	yes	—
222	#322	—	79	1	no	—
223	#322	—	79	1	no	—
224	#322	—	79	1	no	—
225	#322	C₂×D₇₉	1	1	yes	characteristic
226	C₂×Q₁₆	—	79	2	no	—
227	C₂×Q₁₆	—	79	1	no	—
228	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
229	C₂×M₁₆	—	158	1	no	—
230	C₂²×C₈	—	79	1	no	—
231	C₂²×C₈	—	79	1	no	—
232	C₄⋊Q₈	—	79	2	no	—
233	C₄⋊Q₈	—	79	2	no	—
234	#125	—	158	2	no	—
235	C₂²⋊Q₈	—	79	2	no	—
236	C₂²⋊Q₈	—	79	2	no	—
237	C₂²⋊Q₈	—	158	2	no	—
238	C₄×Q₈	—	79	2	no	—
239	C₄×Q₈	—	158	2	no	—
240	C₄×Q₈	C₂²×D₇₉	1	2	yes	—
241	#117	—	158	2	no	—
242	C₂×C₄²	—	79	1	no	—
243	#107	—	79	1	no	—
244	#107	—	79	2	no	—
245	#107	C₄×D₇₉	1	1	yes	characteristic
246	C₄⋊C₈	—	2	2	no	—
247	C₄⋊C₈	—	79	2	no	—
248	C₄⋊C₈	—	79	2	no	—
249	C₄⋊C₈	—	158	2	no	—
250	C₄⋊C₈	—	158	2	no	—
251	C₄⋊C₈	—	158	2	no	—
252	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
253	#103	—	79	2	no	—
254	#103	—	79	2	no	—
255	#103	—	79	2	no	—
256	#103	C₄×D₇₉	1	2	yes	—
257	C₂²⋊C₈	—	79	2	no	—
258	C₂²⋊C₈	—	158	2	no	—
259	#97	—	2	2	no	—
260	#97	—	158	1	no	—
261	#97	—	158	2	no	—
262	C₄×C₈	—	79	1	no	—
263	C₄×C₈	—	79	1	no	—
264	C₄×C₈	—	79	1	no	—
265	C₄×C₈	—	79	1	no	—
266	C₄×C₈	—	79	1	no	—
267	C₄×C₈	—	158	2	no	—
268	C₄×C₈	—	158	2	no	—
269	C₄×C₈	—	158	2	no	—
270	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
273	C₂×Q₈	—	79	2	no	—
274	C₂×Q₈	—	158	2	no	—
275	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
276	C₂²×C₄	—	79	1	no	—
277	C₂²×C₄	—	79	1	no	—
278	C₂²×C₄	—	79	1	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	316	2	no	—
281	Q₁₆	—	158	2	no	—
282	M₁₆	—	316	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	316	1	no	—
286	C₂×C₈	—	79	1	no	—
287	C₂×C₈	—	79	1	no	—
288	C₂×C₈	—	79	2	no	—
289	C₂×C₈	—	158	1	no	—
290	C₂×C₈	—	158	2	no	—
291	C₂×C₈	—	158	2	no	—
292	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
294	C₂×C₈	D₄×D₇₉	1	2	yes	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	79	2	no	—
297	C₄⋊C₄	—	158	2	no	—
298	C₄⋊C₄	—	158	2	no	—
299	C₄⋊C₄	—	158	2	no	—
300	C₄⋊C₄	—	158	2	no	—
301	C₄⋊C₄	—	158	2	no	—
302	C₄⋊C₄	—	158	2	no	—
303	C₄⋊C₄	C₂×C₄×D₇₉	1	2	yes	—
304	C₂²⋊C₄	—	158	2	no	—
305	C₂²⋊C₄	—	158	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	79	1	no	—
308	C₄²	—	79	1	no	—
309	C₄²	—	79	1	no	—
310	C₄²	—	158	2	no	—
311	C₄²	C₂³×D₇₉	1	1	yes	characteristic
312	C₂³	—	79	1	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	2	2	no	—
315	Q₈	—	316	4	no	—
316	Q₈	—	158	2	no	—
317	Q₈	—	158	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	79	1	no	—
321	C₂×C₄	—	79	1	no	—
322	C₂×C₄	—	79	1	no	—
323	C₂×C₄	—	158	2	no	—
324	C₂×C₄	—	158	2	no	—
325	C₂×C₄	—	158	2	no	—
326	C₂×C₄	—	158	1	no	—
327	C₂×C₄	—	158	1	no	—
328	C₂×C₄	—	158	2	no	—
329	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
331	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	2	no	—
336	C₈	—	316	1	no	—
337	C₈	—	316	2	no	—
338	C₈	—	158	1	no	—
339	C₈	—	158	1	no	—
340	C₈	—	158	2	no	—
341	C₈	#14579169	1	2	yes	—
342	C₂²	—	158	1	no	—
343	C₂²	—	158	1	no	—
344	C₂²	—	158	1	no	—
345	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	316	2	no	—
350	C₄	—	316	2	no	—
351	C₄	—	79	2	no	—
352	C₄	—	158	1	no	—
353	C₄	—	158	1	no	—
354	C₄	C₂×#14579169	1	1	yes	characteristic
355	C₄	#19099328	1	1	yes	characteristic
356	C₄	#19099379	1	2	yes	—
357	C₂	—	316	1	no	—
358	C₂	#23772145	1	1	yes	characteristic
359	C₂	#23772265	1	1	yes	characteristic
360	C₂	#444×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1593	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂²×D₇₉	C₄×D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1593	D₇₉
hypocenter	C₇₉	#18090
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #23762308	C₂	semidirect
4. #23762248	C₂	semidirect
5. #23762247	C₂	semidirect
6. #23762220	C₂	semidirect
7. #23762219	C₂	semidirect
11. C₇₉×#1593	C₂	semidirect
184. #1593	D₇₉	semidirect
192. C₇₉	#18090	semidirect

Maximal subgroups

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

subgroup	index
#23773983	2
#23772498	2
#23762308	2
#23762248	2
#23762247	2
#23762220	2
#23762219	2
#23762145	2
#23761819	2
#23761686	2
C₇₉×#1593	2
#18090	79