Subgroups — #34305874

Order 20224.

692 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772060	C₂	1	2	yes	maximal
2	C₂×#19098601	C₂	1	1	yes	maximal, characteristic
3	#23764151	C₂	1	2	yes	maximal, complemented
4	#23763998	C₂	1	4	yes	maximal, complemented
5	#23763497	C₂	1	2	yes	maximal
6	#23763287	C₂	1	2	yes	maximal, complemented
7	#23763176	C₂	1	1	yes	maximal, characteristic
8	C₇₉×#1810	C₂	1	1	yes	maximal, characteristic
9	C₂×#14579120	C₂²	1	2	yes	—
10	#19098621	—	2	4	no	—
11	#19098617	—	2	2	no	—
12	#19098613	—	2	2	no	—
13	#19098601	C₄	1	1	yes	characteristic, complemented
14	#19098601	C₄	1	1	yes	characteristic, complemented
15	#19098601	C₄	1	2	yes	complemented
16	#19098601	C₂²	1	2	yes	—
17	#19098601	C₂²	1	1	yes	characteristic
18	#19098601	C₂²	1	1	yes	characteristic
19	C₂×#14579095	C₄	1	1	yes	characteristic, complemented
20	C₂×#14579095	C₄	1	1	yes	characteristic, complemented
21	C₂×#14579093	C₄	1	2	yes	complemented
22	#19098591	C₂²	1	4	yes	—
23	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
24	#19098521	C₂²	1	4	yes	—
25	C₂×#14579071	C₂²	1	2	yes	—
26	C₄×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
27	#19098324	C₂²	1	4	yes	—
28	#19098324	C₂²	1	4	yes	—
29	#19098317	C₂²	1	2	yes	—
30	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×#108×C₇₉	C₂²	1	2	yes	—
32	#407×C₇₉	C₂²	1	2	yes	—
33	#341×C₇₉	C₂²	1	2	yes	—
34	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
35	#14579120	—	2	2	no	—
36	#14579120	—	2	2	no	—
37	#14579120	C₂×C₄	1	2	yes	—
38	#14579120	C₂×C₄	1	2	yes	—
39	#14579120	C₂³	1	2	yes	—
40	#14579120	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
44	C₂×C₂²⋊Dic₇₉	—	2	1	no	—
45	C₂×C₄×Dic₇₉	—	2	2	no	—
46	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
47	#14579095	C₂×C₄	1	2	yes	—
48	#14579095	C₂×C₄	1	2	yes	—
49	#14579095	C₂×C₄	1	1	yes	characteristic
50	#14579095	C₂×C₄	1	1	yes	characteristic
51	#14579095	C₂×C₄	1	1	yes	characteristic
52	#14579095	C₂×C₄	1	1	yes	characteristic
53	#14579093	C₂×C₄	1	2	yes	—
54	#14579093	C₂×C₄	1	2	yes	—
55	#14579093	C₂×C₄	1	2	yes	—
56	#14579093	C₂×C₄	1	2	yes	—
57	Dic₇₉⋊C₈	—	2	4	no	—
58	Dic₇₉⋊C₈	—	2	4	no	—
59	C₂²⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
60	C₂²⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
61	#14579071	—	2	4	no	—
62	#14579071	—	2	4	no	—
63	C₈×Dic₇₉	—	2	4	no	—
64	C₇₉⋊#95	—	2	1	no	—
65	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₇₉	C₂³	1	4	yes	—
67	C₂²×C₈×C₇₉	—	2	4	no	—
68	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
71	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	#108×C₇₉	—	2	4	no	—
75	C₂²⋊C₈×C₇₉	—	2	2	no	—
76	C₂²⋊C₈×C₇₉	—	2	2	no	—
77	C₂²×Dic₇₉	—	2	2	no	—
78	C₂²×Dic₇₉	—	2	2	no	—
79	C₂²×Dic₇₉	—	2	2	no	—
80	C₂×C₇₉⋊C₈	—	2	2	no	—
81	C₂×C₇₉⋊C₈	—	2	2	no	—
82	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
83	C₄⋊Dic₇₉	—	2	2	no	—
84	C₄⋊Dic₇₉	—	2	2	no	—
85	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
86	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
87	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
88	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
89	C₂²⋊Dic₇₉	—	2	2	no	—
90	C₂²⋊Dic₇₉	—	2	2	no	—
91	C₂²⋊Dic₇₉	—	2	2	no	—
92	C₂²⋊Dic₇₉	—	2	2	no	—
93	C₂²⋊Dic₇₉	—	4	2	no	—
94	C₄×Dic₇₉	—	2	2	no	—
95	C₄×Dic₇₉	—	2	2	no	—
96	C₄×Dic₇₉	—	2	2	no	—
97	C₄×Dic₇₉	—	2	2	no	—
98	C₄×Dic₇₉	—	2	2	no	—
99	C₄×Dic₇₉	—	2	2	no	—
100	C₂⁴×C₇₉	#41	1	1	yes	characteristic
101	C₂²×C₄×C₇₉	—	2	2	no	—
102	C₂²×C₄×C₇₉	—	2	2	no	—
103	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
108	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
109	M₁₆×C₇₉	—	2	4	no	—
110	M₁₆×C₇₉	—	2	4	no	—
111	C₂×C₈×C₇₉	—	2	2	no	—
112	C₂×C₈×C₇₉	—	2	2	no	—
113	C₂×C₈×C₇₉	—	2	2	no	—
114	C₂×C₈×C₇₉	—	2	2	no	—
115	C₂×C₈×C₇₉	—	2	4	no	—
116	C₂×C₈×C₇₉	—	2	2	no	—
117	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₇₉	#41	1	4	yes	—
122	C₂³×C₇₉	—	2	2	no	—
123	C₂³×C₇₉	—	2	2	no	—
124	C₂³×C₇₉	—	4	1	no	—
125	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₇₉	—	2	2	no	—
129	C₂×C₄×C₇₉	—	2	1	no	—
130	C₂×C₄×C₇₉	—	2	1	no	—
131	C₂×C₄×C₇₉	—	2	1	no	—
132	C₂×C₄×C₇₉	—	2	1	no	—
133	C₂×C₄×C₇₉	—	2	2	no	—
134	C₂×C₄×C₇₉	—	2	2	no	—
135	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
140	C₂×C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
141	C₂×Dic₇₉	—	2	2	no	—
142	C₂×Dic₇₉	—	2	2	no	—
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	2	2	no	—
145	C₂×Dic₇₉	—	4	4	no	—
146	C₂×Dic₇₉	—	4	2	no	—
147	C₂×Dic₇₉	—	4	2	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	—	8	2	no	—
150	C₈×C₇₉	—	2	1	no	—
151	C₈×C₇₉	—	2	1	no	—
152	C₈×C₇₉	—	4	4	no	—
153	C₈×C₇₉	#117	1	2	yes	—
154	C₈×C₇₉	#117	1	2	yes	—
155	C₇₉⋊C₈	—	2	4	no	—
156	C₂²×C₇₉	—	2	1	no	—
157	C₂²×C₇₉	—	2	1	no	—
158	C₂²×C₇₉	—	2	1	no	—
159	C₂²×C₇₉	—	2	1	no	—
160	C₂²×C₇₉	—	2	1	no	—
161	C₂²×C₇₉	—	2	1	no	—
162	C₂²×C₇₉	—	2	1	no	—
163	C₂²×C₇₉	—	4	4	no	—
164	C₂²×C₇₉	#444	1	2	yes	—
165	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₇₉	#552	1	2	yes	—
167	C₄×C₇₉	—	2	1	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
170	C₄×C₇₉	#519	1	1	yes	characteristic
171	C₄×C₇₉	#547	1	2	yes	—
172	Dic₇₉	—	4	2	no	—
173	Dic₇₉	—	4	2	no	—
174	Dic₇₉	—	4	2	no	—
175	Dic₇₉	—	4	2	no	—
176	Dic₇₉	—	8	2	no	—
177	#22607	—	79	1	no	maximal, Sylow
178	C₂×C₇₉	—	2	2	no	—
179	C₂×C₇₉	—	2	2	no	—
180	C₂×C₇₉	—	4	1	no	—
181	C₂×C₇₉	#2344	1	1	yes	characteristic
182	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
183	C₂×C₇₉	#2957	1	1	yes	characteristic
184	#2924	—	79	2	no	—
185	C₂×#427	—	79	1	no	—
186	#1946	—	79	2	no	—
187	#1922	—	79	4	no	—
188	#1826	—	79	2	no	—
189	#1810	D₇₉	1	1	yes	characteristic
190	#1773	—	79	2	no	—
191	#1752	—	79	1	no	—
192	C₇₉	#22607	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
194	C₂×#117	—	79	2	no	—
195	#433	—	158	4	no	—
196	#432	—	158	2	no	—
197	#431	—	158	2	no	—
198	C₂×#108	C₂×D₇₉	1	2	yes	—
199	#427	—	79	2	no	—
200	#427	—	79	1	no	—
201	#427	—	79	1	no	—
202	#427	—	79	1	no	—
203	#427	—	79	1	no	—
204	#427	—	79	2	no	—
205	C₂×#107	—	79	1	no	—
206	C₂×#107	—	79	1	no	—
207	C₂×#106	—	79	2	no	—
208	#424	—	79	4	no	—
209	#407	C₂×D₇₉	1	2	yes	—
210	C₂×C₂²⋊C₈	—	79	1	no	—
211	#405	—	79	4	no	—
212	C₂×#97	—	79	2	no	—
213	C₄×C₂²⋊C₄	—	79	1	no	—
214	#341	C₂×D₇₉	1	2	yes	—
215	#339	—	79	4	no	—
216	#339	—	79	4	no	—
217	#336	—	79	2	no	—
218	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
219	C₂×M₁₆	C₂²×D₇₉	1	4	yes	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	79	2	no	—
222	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
223	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
225	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
228	#117	—	79	2	no	—
229	#117	—	79	2	no	—
230	#117	—	79	2	no	—
231	#117	—	79	2	no	—
232	#117	—	158	2	no	—
233	#117	—	158	2	no	—
234	C₂×C₄⋊C₄	—	79	2	no	—
235	C₂×C₄⋊C₄	—	79	2	no	—
236	C₂×C₂²⋊C₄	—	158	1	no	—
237	C₂×C₂²⋊C₄	—	158	2	no	—
238	C₂×C₄²	—	79	2	no	—
239	C₂×C₄²	—	158	2	no	—
240	#108	—	2	4	no	—
241	#107	—	79	2	no	—
242	#107	—	79	2	no	—
243	#107	—	79	1	no	—
244	#107	—	79	1	no	—
245	#107	—	79	1	no	—
246	#107	—	79	1	no	—
247	#106	—	79	2	no	—
248	#106	—	79	2	no	—
249	#106	—	79	2	no	—
250	#106	—	79	2	no	—
251	C₄⋊C₈	—	158	4	no	—
252	C₄⋊C₈	—	158	4	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₂²⋊C₈	—	79	2	no	—
256	C₂²⋊C₈	—	79	2	no	—
257	#97	—	158	4	no	—
258	#97	—	158	4	no	—
259	C₄×C₈	—	158	4	no	—
260	#95	—	158	1	no	—
261	C₂⁴	#11929625	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	158	2	no	—
265	C₂²×C₄	—	158	2	no	—
266	C₂²×C₄	—	158	2	no	—
267	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
271	C₂²×C₄	#11929625	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
273	M₁₆	—	2	4	no	—
274	M₁₆	—	2	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	79	4	no	—
282	C₂×C₈	—	158	2	no	—
283	C₂×C₈	—	158	2	no	—
284	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
286	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
288	C₂×C₈	#11929623	1	4	yes	—
289	C₄⋊C₄	—	79	2	no	—
290	C₄⋊C₄	—	79	2	no	—
291	C₄⋊C₄	—	79	2	no	—
292	C₄⋊C₄	—	79	2	no	—
293	C₄⋊C₄	—	158	2	no	—
294	C₄⋊C₄	—	158	2	no	—
295	C₂²⋊C₄	—	316	2	no	—
296	C₂²⋊C₄	—	158	2	no	—
297	C₂²⋊C₄	—	158	2	no	—
298	C₂²⋊C₄	—	158	2	no	—
299	C₂²⋊C₄	—	158	2	no	—
300	C₄²	—	158	2	no	—
301	C₄²	—	158	2	no	—
302	C₄²	—	158	2	no	—
303	C₄²	—	158	2	no	—
304	C₄²	—	158	2	no	—
305	C₄²	—	158	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	#14579123	1	1	yes	characteristic
310	C₂³	#14579123	1	1	yes	characteristic
311	C₂³	C₂×#11929625	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	316	4	no	—
320	C₂×C₄	—	316	2	no	—
321	C₂×C₄	—	316	2	no	—
322	C₂×C₄	—	316	2	no	—
323	C₂×C₄	—	632	2	no	—
324	C₂×C₄	—	158	2	no	—
325	C₂×C₄	—	158	2	no	—
326	C₂×C₄	—	158	2	no	—
327	C₂×C₄	—	158	2	no	—
328	C₂×C₄	#14579123	1	1	yes	characteristic
329	C₂×C₄	#14579123	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
331	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
332	C₂×C₄	C₂×#11929623	1	2	yes	—
333	C₂×C₄	#14579208	1	2	yes	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	4	4	no	—
337	C₈	—	158	4	no	—
338	C₈	#14579117	1	2	yes	—
339	C₈	#14579117	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	#19098684	1	2	yes	—
349	C₂²	C₂×#14579123	1	1	yes	characteristic
350	C₂²	#19099255	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	316	2	no	—
354	C₄	—	316	2	no	—
355	C₄	—	316	2	no	—
356	C₄	—	316	2	no	—
357	C₄	—	632	2	no	—
358	C₄	C₂×#14579117	1	1	yes	characteristic
359	C₄	#19099040	1	1	yes	characteristic
360	C₄	#19099213	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	#23766271	1	1	yes	characteristic
365	C₂	C₂×#19098684	1	1	yes	characteristic
366	C₂	#23772316	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1810	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂⁴×C₇₉	#41
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1810	D₇₉
hypocenter	C₇₉	#22607
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1810

Complements

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

N	H ≅ G/N	type
3. #23764151	C₂	semidirect
4. #23763998	C₂	semidirect
6. #23763287	C₂	semidirect
13. #19098601	C₄	semidirect
14. #19098601	C₄	semidirect
15. #19098601	C₄	semidirect
19. C₂×#14579095	C₄	semidirect
20. C₂×#14579095	C₄	semidirect
21. C₂×#14579093	C₄	semidirect
192. C₇₉	#22607	semidirect

Maximal subgroups

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

subgroup	index
#23772060	2
C₂×#19098601	2
#23764151	2
#23763998	2
#23763497	2
#23763287	2
#23763176	2
C₇₉×#1810	2
#22607	79