Subgroups — #34260319

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773973	C₂	1	1	yes	maximal, characteristic
2	#23772559	C₂	1	2	yes	maximal, complemented
3	#23762315	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762309	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762243	C₂	1	2	yes	maximal, complemented
6	#23762239	C₂	1	2	yes	maximal, complemented
7	#23762223	C₂	1	1	yes	maximal, characteristic, complemented
8	#334×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#23761812	C₂	1	2	yes	maximal, complemented
10	#23761686	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099100	C₂²	1	2	yes	—
13	#19098964	C₄	1	1	yes	characteristic
14	#19098948	C₄	1	1	yes	characteristic
15	#19098919	C₄	1	2	yes	—
16	#19098801	C₄	1	1	yes	characteristic
17	#19098800	C₄	1	1	yes	characteristic
18	#19098795	C₄	1	2	yes	—
19	#19098694	C₂²	1	2	yes	—
20	C₄×#11929611	C₂²	1	2	yes	—
21	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
22	#19098642	C₂²	1	2	yes	—
23	#19098633	—	2	2	no	—
24	C₈×D₃₁₆	—	2	2	no	—
25	#19098628	C₂²	1	2	yes	—
26	#19098622	—	2	2	no	—
27	#19098614	C₂²	1	1	yes	characteristic
28	#19098588	—	2	2	no	—
29	C₄⋊C₈×D₇₉	C₂²	1	2	yes	—
30	C₄×#11929607	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
32	#19098314	C₂²	1	1	yes	characteristic
33	#19098313	C₂²	1	2	yes	—
34	Q₈⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
35	#19098292	C₂²	1	2	yes	—
36	#19098290	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	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×#11929611	C₂×C₄	1	2	yes	—
46	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
47	C₂×#11929607	—	2	1	no	—
48	C₂×#11929607	—	2	2	no	—
49	C₂×C₈×D₇₉	—	2	2	no	—
50	C₂×C₈×D₇₉	—	2	2	no	—
51	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
53	#14579156	—	2	2	no	—
54	#14579152	C₂×C₄	1	2	yes	—
55	#14579150	C₂×C₄	1	2	yes	—
56	#14579146	—	2	2	no	—
57	#14579137	C₂×C₄	1	2	yes	—
58	#14579136	C₂×C₄	1	2	yes	—
59	C₄×D₃₁₆	—	2	2	no	—
60	C₄×D₃₁₆	C₂³	1	2	yes	—
61	#14579118	—	2	2	no	—
62	C₄²×D₇₉	C₂³	1	1	yes	characteristic
63	#14579095	C₂×C₄	1	1	yes	characteristic
64	#14579092	C₂×C₄	1	2	yes	—
65	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
66	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
67	Dic₇₉⋊C₈	—	2	2	no	—
68	Dic₇₉⋊C₈	D₄	1	2	yes	—
69	#14579085	C₂×C₄	1	2	yes	—
70	#14579083	C₂×C₄	1	2	yes	—
71	#14579081	C₂×C₄	1	2	yes	—
72	(C₂×D₇₉)⋊C₈	—	2	2	no	—
73	(C₂×D₇₉)⋊C₈	—	2	2	no	—
74	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
75	#14579072	—	2	2	no	—
76	#14579071	—	2	1	no	—
77	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
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	C₄×C₈×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×D₃₁₆	—	2	2	no	—
93	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₇₉	—	2	2	no	—
95	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₇₉	#41	1	1	yes	characteristic
98	#11929611	—	4	2	no	—
99	D₆₃₂	—	2	2	no	—
100	#11929607	—	4	2	no	—
101	#11929607	—	8	2	no	—
102	C₂×C₇₉⋊C₈	—	2	2	no	—
103	C₂×C₇₉⋊C₈	—	2	1	no	—
104	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
107	C₈×D₇₉	—	2	2	no	—
108	C₈×D₇₉	—	4	1	no	—
109	C₈×D₇₉	—	8	2	no	—
110	C₈×D₇₉	—	8	2	no	—
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	(C₂×D₇₉)⋊C₄	—	2	2	no	—
116	(C₂×D₇₉)⋊C₄	—	2	2	no	—
117	(C₂×D₇₉)⋊C₄	—	2	2	no	—
118	(C₂×D₇₉)⋊C₄	—	2	2	no	—
119	(C₂×D₇₉)⋊C₄	—	2	2	no	—
120	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₇₉	#41	1	1	yes	characteristic
123	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
124	Q₁₆×C₇₉	—	2	2	no	—
125	C₂×C₈×C₇₉	—	2	2	no	—
126	C₂×C₈×C₇₉	—	2	1	no	—
127	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₇₉	—	2	2	no	—
131	C₄⋊C₄×C₇₉	C₂²×C₄	1	2	yes	—
132	C₄²×C₇₉	—	2	2	no	—
133	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
134	C₂²×D₇₉	—	2	2	no	—
135	C₂²×D₇₉	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₇₉	—	2	2	no	—
137	Q₈×C₇₉	—	2	2	no	—
138	C₂×C₄×C₇₉	—	2	2	no	—
139	C₂×C₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
147	D₃₁₆	—	2	2	no	—
148	D₃₁₆	—	2	2	no	—
149	D₃₁₆	—	4	4	no	—
150	C₄×D₇₉	—	2	2	no	—
151	C₄×D₇₉	—	2	1	no	—
152	C₄×D₇₉	—	2	1	no	—
153	C₄×D₇₉	—	2	2	no	—
154	C₄×D₇₉	—	4	2	no	—
155	C₄×D₇₉	—	4	2	no	—
156	C₈×C₇₉	—	2	1	no	—
157	C₈×C₇₉	—	2	2	no	—
158	C₈×C₇₉	—	4	1	no	—
159	C₈×C₇₉	—	4	2	no	—
160	C₈×C₇₉	#131	1	2	yes	—
161	C₇₉⋊C₈	—	2	1	no	—
162	C₇₉⋊C₈	—	2	1	no	—
163	C₇₉⋊C₈	—	2	2	no	—
164	C₇₉⋊C₈	—	4	1	no	—
165	C₇₉⋊C₈	—	4	2	no	—
166	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×D₇₉	—	2	1	no	—
168	C₂×D₇₉	—	2	1	no	—
169	C₂×D₇₉	—	4	2	no	—
170	C₂×D₇₉	—	4	2	no	—
171	C₂×D₇₉	—	4	2	no	—
172	C₂×D₇₉	#444	1	2	yes	—
173	C₄×C₇₉	—	2	1	no	—
174	C₄×C₇₉	—	4	2	no	—
175	C₄×C₇₉	—	4	2	no	—
176	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
177	C₄×C₇₉	#568	1	1	yes	characteristic
178	C₄×C₇₉	#577	1	2	yes	—
179	Dic₇₉	—	2	1	no	—
180	Dic₇₉	—	2	1	no	—
181	Dic₇₉	—	2	1	no	—
182	Dic₇₉	—	4	2	no	—
183	#18093	—	79	1	no	maximal, Sylow
184	C₂×C₇₉	#2935	1	1	yes	characteristic
185	C₂×C₇₉	#2952	1	1	yes	characteristic
186	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
187	D₇₉	—	2	2	no	—
188	D₇₉	—	8	2	no	—
189	#3168	—	79	1	no	—
190	#2994	—	79	2	no	—
191	#1594	—	79	1	no	—
192	#1593	—	79	1	no	—
193	#1583	—	79	2	no	—
194	#1582	—	79	2	no	—
195	#1580	—	79	1	no	—
196	#1579	D₇₉	1	1	yes	characteristic, complemented
197	C₂×#334	—	79	1	no	—
198	#1483	—	79	2	no	—
199	#1451	—	79	1	no	—
200	C₇₉	#18093	1	1	yes	characteristic, complemented, Sylow
201	#529	—	79	2	no	—
202	#499	—	79	1	no	—
203	#495	—	79	1	no	—
204	#488	—	79	2	no	—
205	#467	—	79	1	no	—
206	#466	—	79	1	no	—
207	#465	—	79	2	no	—
208	#446	—	79	2	no	—
209	C₈×Q₈	C₂×D₇₉	1	2	yes	—
210	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
211	C₄×SD₁₆	—	79	2	no	—
212	C₄×D₈	—	79	1	no	—
213	#436	—	79	2	no	—
214	#435	—	158	2	no	—
215	C₈×D₄	—	79	2	no	—
216	C₈×D₄	—	158	2	no	—
217	#433	—	158	2	no	—
218	#431	—	79	1	no	—
219	#424	—	158	2	no	—
220	C₂×C₄⋊C₈	—	79	2	no	—
221	C₄×M₁₆	—	79	1	no	—
222	C₂×C₄×C₈	—	79	1	no	—
223	#334	—	79	2	no	—
224	#334	—	79	1	no	—
225	#334	C₂×D₇₉	1	1	yes	characteristic
226	Q₈⋊C₈	—	79	2	no	—
227	Q₈⋊C₈	C₂×D₇₉	1	2	yes	—
228	D₄⋊C₈	—	79	2	no	—
229	D₄⋊C₈	—	79	2	no	—
230	#322	—	79	1	no	—
231	#322	—	79	1	no	—
232	#322	—	79	1	no	—
233	#322	C₂×D₇₉	1	1	yes	characteristic
234	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
235	C₂×SD₁₆	—	79	2	no	—
236	C₂×D₈	—	79	1	no	—
237	C₂×M₁₆	—	158	1	no	—
238	C₂×M₁₆	—	158	2	no	—
239	C₂²×C₈	—	79	1	no	—
240	C₂²×C₈	—	79	1	no	—
241	C₂²×C₈	—	158	2	no	—
242	C₂²×C₈	—	158	2	no	—
243	#126	—	158	2	no	—
244	#125	—	79	2	no	—
245	#124	—	79	2	no	—
246	#123	—	158	2	no	—
247	C₂²⋊Q₈	—	79	2	no	—
248	#121	—	79	2	no	—
249	C₄×Q₈	C₂²×D₇₉	1	2	yes	—
250	C₄×D₄	—	79	2	no	—
251	C₄×D₄	—	158	2	no	—
252	#117	—	158	2	no	—
253	C₂×C₄²	—	79	1	no	—
254	#107	—	79	1	no	—
255	#107	C₄×D₇₉	1	1	yes	characteristic
256	#106	—	79	2	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	79	2	no	—
259	C₄⋊C₈	—	79	2	no	—
260	C₄⋊C₈	—	158	2	no	—
261	C₄⋊C₈	—	158	2	no	—
262	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
263	#103	—	79	2	no	—
264	#103	C₄×D₇₉	1	2	yes	—
265	#102	—	79	2	no	—
266	#102	—	79	2	no	—
267	C₂²⋊C₈	—	79	2	no	—
268	C₂²⋊C₈	—	158	2	no	—
269	C₂²⋊C₈	—	158	2	no	—
270	#97	—	158	2	no	—
271	#97	—	158	1	no	—
272	C₄×C₈	—	2	2	no	—
273	C₄×C₈	—	79	1	no	—
274	C₄×C₈	—	79	1	no	—
275	C₄×C₈	—	79	1	no	—
276	C₄×C₈	—	79	1	no	—
277	C₄×C₈	—	79	1	no	—
278	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
281	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
282	C₂×D₄	—	79	2	no	—
283	C₂×D₄	—	158	2	no	—
284	C₂²×C₄	—	79	1	no	—
285	C₂²×C₄	—	79	1	no	—
286	C₂²×C₄	—	79	1	no	—
287	C₂²×C₄	—	158	2	no	—
288	Q₁₆	—	2	2	no	—
289	SD₁₆	—	316	2	no	—
290	D₈	—	158	2	no	—
291	M₁₆	—	316	2	no	—
292	M₁₆	—	632	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	316	1	no	—
296	C₂×C₈	—	79	1	no	—
297	C₂×C₈	—	79	1	no	—
298	C₂×C₈	—	79	2	no	—
299	C₂×C₈	—	632	2	no	—
300	C₂×C₈	—	632	2	no	—
301	C₂×C₈	—	158	1	no	—
302	C₂×C₈	—	158	2	no	—
303	C₂×C₈	—	158	2	no	—
304	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
306	C₂×C₈	D₄×D₇₉	1	2	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	79	2	no	—
309	C₄⋊C₄	—	158	2	no	—
310	C₄⋊C₄	—	158	2	no	—
311	C₄⋊C₄	—	158	2	no	—
312	C₄⋊C₄	C₂×C₄×D₇₉	1	2	yes	—
313	C₂²⋊C₄	—	158	2	no	—
314	C₂²⋊C₄	—	158	2	no	—
315	C₂²⋊C₄	—	158	2	no	—
316	C₂²⋊C₄	—	158	2	no	—
317	C₂²⋊C₄	—	158	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	79	1	no	—
320	C₄²	—	79	1	no	—
321	C₄²	—	79	1	no	—
322	C₄²	C₂³×D₇₉	1	1	yes	characteristic
323	C₂³	—	79	1	no	—
324	C₂³	—	158	2	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	D₄	—	316	4	no	—
328	D₄	—	158	2	no	—
329	D₄	—	158	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	316	2	no	—
333	C₂×C₄	—	316	2	no	—
334	C₂×C₄	—	79	1	no	—
335	C₂×C₄	—	79	1	no	—
336	C₂×C₄	—	79	1	no	—
337	C₂×C₄	—	158	2	no	—
338	C₂×C₄	—	158	2	no	—
339	C₂×C₄	—	158	1	no	—
340	C₂×C₄	—	158	1	no	—
341	C₂×C₄	—	158	2	no	—
342	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	316	1	no	—
350	C₈	—	316	2	no	—
351	C₈	—	158	1	no	—
352	C₈	—	158	1	no	—
353	C₈	—	158	2	no	—
354	C₈	#14579167	1	2	yes	—
355	C₂²	—	316	2	no	—
356	C₂²	—	316	2	no	—
357	C₂²	—	316	2	no	—
358	C₂²	—	79	2	no	—
359	C₂²	—	158	1	no	—
360	C₂²	—	158	1	no	—
361	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	2	no	—
365	C₄	—	316	2	no	—
366	C₄	—	158	1	no	—
367	C₄	—	158	1	no	—
368	C₄	—	158	1	no	—
369	C₄	C₂×#14579167	1	1	yes	characteristic
370	C₄	#19099328	1	1	yes	characteristic
371	C₄	#19099372	1	2	yes	—
372	C₂	—	632	2	no	—
373	C₂	—	158	2	no	—
374	C₂	#23772139	1	1	yes	characteristic
375	C₂	#23772280	1	1	yes	characteristic
376	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×#1579	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#19098800	C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1579	D₇₉
hypocenter	C₇₉	#18093
LCS	G ▷ C₂×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
2. #23772559	C₂	semidirect
3. #23762315	C₂	semidirect
4. #23762309	C₂	semidirect
5. #23762243	C₂	semidirect
6. #23762239	C₂	semidirect
7. #23762223	C₂	semidirect
8. #334×D₇₉	C₂	semidirect
9. #23761812	C₂	semidirect
10. #23761686	C₂	semidirect
11. C₇₉×#1579	C₂	semidirect
196. #1579	D₇₉	semidirect
200. C₇₉	#18093	semidirect

Maximal subgroups

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

subgroup	index
#23773973	2
#23772559	2
#23762315	2
#23762309	2
#23762243	2
#23762239	2
#23762223	2
#334×D₇₉	2
#23761812	2
#23761686	2
C₇₉×#1579	2
#18093	79