Subgroups — #34270908

Order 20224.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773911	C₂	1	1	yes	maximal, characteristic
2	#23773438	C₂	1	2	yes	maximal
3	#446×D₇₉	C₂	1	1	yes	maximal, characteristic
4	#23762929	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762928	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762915	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762905	C₂	1	2	yes	maximal, complemented
8	#23762904	C₂	1	2	yes	maximal, complemented
9	#23762145	C₂	1	1	yes	maximal, characteristic
10	#23762056	C₂	1	2	yes	maximal
11	C₇₉×#1694	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099118	C₂²	1	2	yes	—
13	C₄×Q₈×D₇₉	C₂²	1	1	yes	characteristic
14	#19098968	C₂²	1	1	yes	characteristic
15	#19098964	—	2	1	no	—
16	#19098964	—	2	1	no	—
17	#19098959	C₂²	1	2	yes	—
18	#19098897	—	2	2	no	—
19	#19098859	C₂²	1	2	yes	—
20	#19098858	C₂²	1	2	yes	—
21	#19098770	C₂²	1	2	yes	—
22	#19098717	—	2	2	no	—
23	#19098694	C₂²	1	1	yes	characteristic
24	#19098693	C₂²	1	1	yes	characteristic
25	#19098692	—	2	1	no	—
26	#19098691	—	2	1	no	—
27	#19098690	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₇₉	—	2	2	no	—
29	#19098655	—	2	2	no	—
30	#19098615	C₂²	1	2	yes	—
31	#19098600	—	2	1	no	—
32	#19098600	—	2	1	no	—
33	C₄⋊C₈×D₇₉	—	2	1	no	—
34	C₄⋊C₈×D₇₉	C₂²	1	1	yes	characteristic
35	#97×D₇₉	—	2	1	no	—
36	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
37	#19098315	C₂²	1	2	yes	—
38	#19098314	C₂²	1	1	yes	characteristic
39	#19098311	C₂²	1	2	yes	—
40	#19098311	C₂²	1	2	yes	—
41	#19098309	C₂²	1	2	yes	—
42	#500×C₇₉	C₂²	1	1	yes	characteristic
43	#462×C₇₉	C₂²	1	2	yes	—
44	#446×C₇₉	C₂²	1	1	yes	characteristic
45	#334×C₇₉	C₂²	1	1	yes	characteristic
46	#332×C₇₉	C₂²	1	2	yes	—
47	C₂×Q₈×D₇₉	D₄	1	1	yes	characteristic
48	C₂×#11929614	—	2	2	no	—
49	C₂×C₈×D₇₉	—	2	1	no	—
50	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
52	#14579161	C₂³	1	2	yes	—
53	#14579152	—	2	2	no	—
54	#14579141	—	2	2	no	—
55	#14579137	—	2	2	no	—
56	Q₈×Dic₇₉	—	2	2	no	—
57	Q₈×Dic₇₉	D₄	1	1	yes	characteristic
58	C₄×Dic₁₅₈	—	2	1	no	—
59	C₄×Dic₁₅₈	C₂³	1	1	yes	characteristic
60	#14579126	—	2	1	no	—
61	#14579126	—	2	2	no	—
62	#14579126	D₄	1	1	yes	characteristic
63	#14579119	—	2	2	no	—
64	C₄⋊C₄×D₇₉	—	2	1	no	—
65	C₄⋊C₄×D₇₉	D₄	1	1	yes	characteristic
66	C₄²×D₇₉	—	2	1	no	—
67	C₄²×D₇₉	C₂³	1	1	yes	characteristic
68	#14579095	—	2	1	no	—
69	#14579095	—	2	1	no	—
70	#14579095	—	2	2	no	—
71	#14579094	—	2	2	no	—
72	#14579092	—	2	2	no	—
73	#14579092	—	2	2	no	—
74	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
75	C₄⋊(C₇₉⋊C₈)	C₂³	1	1	yes	characteristic
76	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
77	Dic₇₉⋊C₈	—	2	1	no	—
78	Dic₇₉⋊C₈	—	2	1	no	—
79	Dic₇₉⋊C₈	—	2	1	no	—
80	#14579086	—	2	2	no	—
81	#14579085	—	2	2	no	—
82	#14579085	—	2	2	no	—
83	(C₂×D₇₉)⋊C₈	—	2	2	no	—
84	#14579072	—	2	1	no	—
85	#14579071	—	2	1	no	—
86	#14579071	—	2	1	no	—
87	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
88	C₈×Dic₇₉	—	2	2	no	—
89	C₈×Dic₇₉	D₄	1	1	yes	characteristic
90	C₈×Dic₇₉	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₇₉	—	2	2	no	—
92	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
93	C₄×Q₈×C₇₉	C₂³	1	1	yes	characteristic
94	#107×C₇₉	—	2	1	no	—
95	#107×C₇₉	—	2	1	no	—
96	C₄⋊C₈×C₇₉	—	2	1	no	—
97	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
99	#103×C₇₉	—	2	2	no	—
100	#97×C₇₉	—	2	1	no	—
101	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₅₈	—	2	1	no	—
103	C₂×Dic₁₅₈	—	2	2	no	—
104	C₂×Dic₁₅₈	C₂×D₄	1	1	yes	characteristic
105	Q₈×D₇₉	—	4	2	no	—
106	C₂×C₄×D₇₉	—	2	1	no	—
107	C₂×C₄×D₇₉	—	2	1	no	—
108	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₇₉	#41	1	1	yes	characteristic
111	#11929614	—	4	4	no	—
112	C₂×C₇₉⋊C₈	—	2	2	no	—
113	C₂×C₇₉⋊C₈	—	2	1	no	—
114	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
116	C₈×D₇₉	—	2	2	no	—
117	C₈×D₇₉	—	4	2	no	—
118	C₈×D₇₉	Q₁₆	1	4	yes	complemented
119	C₄⋊Dic₇₉	—	2	1	no	—
120	C₄⋊Dic₇₉	—	2	2	no	—
121	C₄⋊Dic₇₉	—	2	2	no	—
122	C₄⋊Dic₇₉	—	4	2	no	—
123	C₄⋊Dic₇₉	C₂×D₄	1	1	yes	characteristic
124	Dic₇₉⋊C₄	—	2	1	no	—
125	Dic₇₉⋊C₄	—	2	1	no	—
126	Dic₇₉⋊C₄	—	2	1	no	—
127	Dic₇₉⋊C₄	—	4	2	no	—
128	(C₂×D₇₉)⋊C₄	—	4	2	no	—
129	C₄×Dic₇₉	—	2	1	no	—
130	C₄×Dic₇₉	—	2	1	no	—
131	C₄×Dic₇₉	—	2	1	no	—
132	C₄×Dic₇₉	—	4	2	no	—
133	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₇₉	#41	1	1	yes	characteristic
136	C₂×Q₈×C₇₉	—	2	2	no	—
137	C₂×Q₈×C₇₉	C₂×D₄	1	1	yes	characteristic
138	Q₁₆×C₇₉	—	4	4	no	—
139	C₂×C₈×C₇₉	—	2	1	no	—
140	C₂×C₈×C₇₉	—	2	2	no	—
141	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₇₉	—	2	1	no	—
144	C₄⋊C₄×C₇₉	—	2	2	no	—
145	C₄⋊C₄×C₇₉	—	2	2	no	—
146	C₄⋊C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₇₉	—	2	1	no	—
148	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
149	C₂²×D₇₉	#121	1	1	yes	characteristic
150	Q₈×C₇₉	—	2	2	no	—
151	Q₈×C₇₉	—	4	2	no	—
152	Q₈×C₇₉	—	4	2	no	—
153	C₂×C₄×C₇₉	—	2	1	no	—
154	C₂×C₄×C₇₉	—	2	1	no	—
155	C₂×C₄×C₇₉	—	4	2	no	—
156	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₇₉	—	2	1	no	—
160	C₂×Dic₇₉	—	2	1	no	—
161	C₂×Dic₇₉	—	4	2	no	—
162	C₂×Dic₇₉	#121	1	1	yes	characteristic
163	C₂×Dic₇₉	#121	1	1	yes	characteristic
164	C₂×Dic₇₉	#121	1	1	yes	characteristic
165	C₄×D₇₉	—	2	2	no	—
166	C₄×D₇₉	—	4	1	no	—
167	C₄×D₇₉	—	4	2	no	—
168	C₄×D₇₉	—	4	2	no	—
169	C₄×D₇₉	C₂×Q₁₆	1	2	yes	—
170	C₄×D₇₉	#137	1	2	yes	—
171	Dic₁₅₈	—	2	2	no	—
172	Dic₁₅₈	—	4	2	no	—
173	Dic₁₅₈	—	8	2	no	—
174	C₈×C₇₉	—	2	1	no	—
175	C₈×C₇₉	—	4	1	no	—
176	C₈×C₇₉	—	4	2	no	—
177	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
178	C₇₉⋊C₈	—	2	1	no	—
179	C₇₉⋊C₈	—	4	1	no	—
180	C₇₉⋊C₈	—	4	2	no	—
181	C₇₉⋊C₈	C₂×Q₁₆	1	2	yes	—
182	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
183	C₂×D₇₉	—	2	1	no	—
184	C₂×D₇₉	—	2	1	no	—
185	C₂×D₇₉	#462	1	2	yes	—
186	C₄×C₇₉	—	2	1	no	—
187	C₄×C₇₉	—	4	1	no	—
188	C₄×C₇₉	—	4	1	no	—
189	C₄×C₇₉	—	8	2	no	—
190	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
191	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
192	C₄×C₇₉	#575	1	2	yes	—
193	Dic₇₉	—	2	1	no	—
194	Dic₇₉	—	2	1	no	—
195	Dic₇₉	—	4	1	no	—
196	Dic₇₉	—	4	1	no	—
197	Dic₇₉	—	8	2	no	—
198	Dic₇₉	#462	1	2	yes	—
199	#19150	—	79	1	no	maximal, Sylow
200	C₂×C₇₉	C₂×#462	1	1	yes	characteristic
201	C₂×C₇₉	#3054	1	1	yes	characteristic
202	C₂×C₇₉	#3067	1	1	yes	characteristic
203	D₇₉	—	2	2	no	—
204	#3160	—	79	1	no	—
205	#3104	—	79	2	no	—
206	C₂×#446	—	79	1	no	—
207	#1696	—	79	1	no	—
208	#1696	—	79	1	no	—
209	#1694	—	79	1	no	—
210	#1694	D₇₉	1	1	yes	characteristic, complemented
211	#1692	—	79	2	no	—
212	#1692	—	79	2	no	—
213	#1564	—	79	1	no	—
214	#1542	—	79	2	no	—
215	C₇₉	#19150	1	1	yes	characteristic, complemented, Sylow
216	#533	—	79	2	no	—
217	C₂×C₄×Q₈	—	79	1	no	—
218	#500	—	79	1	no	—
219	#500	C₂×D₇₉	1	1	yes	characteristic
220	#499	—	158	1	no	—
221	#499	—	158	1	no	—
222	#498	—	79	2	no	—
223	#484	—	158	2	no	—
224	#477	—	79	2	no	—
225	#477	—	79	2	no	—
226	#462	—	79	2	no	—
227	#462	C₂×D₇₉	1	2	yes	—
228	#451	—	158	2	no	—
229	#446	—	79	1	no	—
230	#446	—	79	1	no	—
231	#446	—	79	1	no	—
232	#446	—	158	1	no	—
233	#446	—	158	1	no	—
234	#446	C₂×D₇₉	1	1	yes	characteristic
235	C₄×Q₁₆	—	158	2	no	—
236	C₄×Q₁₆	—	158	2	no	—
237	#431	—	79	2	no	—
238	#427	—	158	1	no	—
239	#427	—	158	1	no	—
240	C₂×C₄⋊C₈	—	79	1	no	—
241	C₂×C₄⋊C₈	—	158	1	no	—
242	C₂×#97	—	158	1	no	—
243	C₂×C₄×C₈	—	79	1	no	—
244	#335	—	79	2	no	—
245	#334	—	79	1	no	—
246	#334	C₂×D₇₉	1	1	yes	characteristic
247	#333	—	79	2	no	—
248	#333	—	79	2	no	—
249	#332	—	79	2	no	—
250	#332	C₂×D₇₉	1	2	yes	—
251	C₂²×Q₈	—	79	1	no	—
252	C₂×Q₁₆	—	2	2	no	—
253	C₂×Q₁₆	—	158	2	no	—
254	C₂²×C₈	—	79	1	no	—
255	C₂²×C₈	—	79	1	no	—
256	C₂²×C₈	—	158	1	no	—
257	C₄⋊Q₈	—	79	2	no	—
258	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
259	#125	—	158	2	no	—
260	C₂²⋊Q₈	—	158	2	no	—
261	C₂²⋊Q₈	—	158	2	no	—
262	C₄×Q₈	—	79	1	no	—
263	C₄×Q₈	—	79	1	no	—
264	C₄×Q₈	—	79	1	no	—
265	C₄×Q₈	—	158	1	no	—
266	C₄×Q₈	—	158	2	no	—
267	C₄×Q₈	—	158	2	no	—
268	C₄×Q₈	—	158	1	no	—
269	C₄×Q₈	C₂²×D₇₉	1	1	yes	characteristic
270	#117	—	158	2	no	—
271	C₂×C₄⋊C₄	—	79	1	no	—
272	C₂×C₄⋊C₄	—	158	1	no	—
273	C₂×C₄²	—	79	1	no	—
274	C₂×C₄²	—	158	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	1	no	—
277	#107	—	158	2	no	—
278	#107	—	158	1	no	—
279	#107	—	158	1	no	—
280	#107	—	158	2	no	—
281	#106	—	158	2	no	—
282	#106	—	158	2	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	79	1	no	—
285	C₄⋊C₈	—	79	2	no	—
286	C₄⋊C₈	—	158	1	no	—
287	C₄⋊C₈	—	158	1	no	—
288	C₄⋊C₈	—	158	1	no	—
289	C₄⋊C₈	—	158	1	no	—
290	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
291	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
292	#103	—	2	2	no	—
293	#103	—	158	2	no	—
294	#103	—	158	2	no	—
295	#103	—	158	2	no	—
296	C₂²⋊C₈	—	158	2	no	—
297	#97	—	2	1	no	—
298	#97	—	158	1	no	—
299	#97	—	158	1	no	—
300	#97	—	158	1	no	—
301	C₄×C₈	—	79	1	no	—
302	C₄×C₈	—	79	1	no	—
303	C₄×C₈	—	79	1	no	—
304	C₄×C₈	—	158	2	no	—
305	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	316	2	no	—
308	C₂×Q₈	—	79	1	no	—
309	C₂×Q₈	—	158	1	no	—
310	C₂×Q₈	—	158	2	no	—
311	C₂×Q₈	D₄×D₇₉	1	1	yes	characteristic
312	C₂²×C₄	—	79	1	no	—
313	C₂²×C₄	—	79	1	no	—
314	C₂²×C₄	—	79	1	no	—
315	C₂²×C₄	—	158	1	no	—
316	C₂²×C₄	—	158	1	no	—
317	Q₁₆	—	4	4	no	—
318	Q₁₆	—	316	4	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	316	2	no	—
322	C₂×C₈	—	79	1	no	—
323	C₂×C₈	—	79	4	no	—
324	C₂×C₈	—	79	1	no	—
325	C₂×C₈	—	158	2	no	—
326	C₂×C₈	—	158	2	no	—
327	C₂×C₈	—	158	1	no	—
328	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
329	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	316	2	no	—
334	C₄⋊C₄	—	316	2	no	—
335	C₄⋊C₄	—	79	1	no	—
336	C₄⋊C₄	—	158	1	no	—
337	C₄⋊C₄	—	158	1	no	—
338	C₄⋊C₄	—	158	1	no	—
339	C₄⋊C₄	—	158	2	no	—
340	C₄⋊C₄	—	158	2	no	—
341	C₄⋊C₄	—	158	1	no	—
342	C₄⋊C₄	D₄×D₇₉	1	1	yes	characteristic
343	C₂²⋊C₄	—	316	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	316	2	no	—
346	C₄²	—	79	1	no	—
347	C₄²	—	79	1	no	—
348	C₄²	—	79	1	no	—
349	C₄²	—	158	1	no	—
350	C₄²	—	158	1	no	—
351	C₄²	—	158	1	no	—
352	C₄²	C₂³×D₇₉	1	1	yes	characteristic
353	C₂³	—	79	1	no	—
354	Q₈	—	2	2	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	316	2	no	—
358	Q₈	—	632	2	no	—
359	Q₈	—	158	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	316	1	no	—
364	C₂×C₄	—	316	2	no	—
365	C₂×C₄	—	316	2	no	—
366	C₂×C₄	—	316	2	no	—
367	C₂×C₄	—	79	1	no	—
368	C₂×C₄	—	79	2	no	—
369	C₂×C₄	—	79	2	no	—
370	C₂×C₄	—	79	1	no	—
371	C₂×C₄	—	79	1	no	—
372	C₂×C₄	—	158	1	no	—
373	C₂×C₄	—	158	1	no	—
374	C₂×C₄	—	158	2	no	—
375	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
377	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	4	2	no	—
381	C₈	—	316	1	no	—
382	C₈	—	316	2	no	—
383	C₈	—	79	2	no	—
384	C₈	—	158	1	no	—
385	C₈	Q₁₆×D₇₉	1	2	yes	—
386	C₂²	—	79	2	no	—
387	C₂²	—	158	1	no	—
388	C₂²	—	158	1	no	—
389	C₂²	#121×D₇₉	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	316	1	no	—
395	C₄	—	316	1	no	—
396	C₄	—	79	2	no	—
397	C₄	—	632	2	no	—
398	C₄	—	158	1	no	—
399	C₄	—	158	1	no	—
400	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
401	C₄	#137×D₇₉	1	1	yes	characteristic
402	C₄	#19099359	1	2	yes	—
403	C₂	—	158	2	no	—
404	C₂	#462×D₇₉	1	1	yes	characteristic
405	C₂	#23773007	1	1	yes	characteristic
406	C₂	#23773122	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₇₉
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1694	C₂
socle	C₂²×C₇₉	C₂×#121
prime radical	C₂²×D₇₉	#121
hypercenter	#1694	D₇₉
hypocenter	C₇₉	#19150
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1694

Complements

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

N	H ≅ G/N	type
4. #23762929	C₂	semidirect
5. #23762928	C₂	semidirect
6. #23762915	C₂	semidirect
7. #23762905	C₂	semidirect
8. #23762904	C₂	semidirect
11. C₇₉×#1694	C₂	semidirect
118. C₈×D₇₉	Q₁₆	semidirect
210. #1694	D₇₉	semidirect
215. C₇₉	#19150	semidirect

Maximal subgroups

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

subgroup	index
#23773911	2
#23773438	2
#446×D₇₉	2
#23762929	2
#23762928	2
#23762915	2
#23762905	2
#23762904	2
#23762145	2
#23762056	2
C₇₉×#1694	2
#19150	79