Subgroups — #34288642

Order 20224.

608 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771825	C₂	1	1	yes	maximal, characteristic
2	#23763546	C₂	1	2	yes	maximal, complemented
3	#23763525	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763523	C₂	1	1	yes	maximal, characteristic, complemented
5	#23763503	C₂	1	2	yes	maximal
6	#23763447	C₂	1	1	yes	maximal, characteristic
7	#23763443	C₂	1	1	yes	maximal, characteristic
8	#23763284	C₂	1	1	yes	maximal, characteristic, complemented
9	#23763201	C₂	1	2	yes	maximal
10	#23763155	C₂	1	2	yes	maximal, complemented
11	C₇₉×#1773	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579167	C₂²	1	1	yes	characteristic
13	C₂×#14579167	C₂²	1	1	yes	characteristic
14	C₄×#11929624	C₂²	1	1	yes	characteristic
15	C₂×#14579097	C₂²	1	1	yes	characteristic
16	#19098601	C₂²	1	1	yes	characteristic
17	#19098545	C₂²	1	2	yes	—
18	#19098536	C₂²	1	2	yes	—
19	#19098520	C₄	1	2	yes	complemented
20	#19098520	C₄	1	2	yes	complemented
21	#19098518	C₂²	1	2	yes	—
22	#19098518	C₂²	1	1	yes	characteristic
23	#19098518	C₂²	1	1	yes	characteristic
24	C₄×#11929608	C₂²	1	1	yes	characteristic
25	#19098515	C₄	1	2	yes	complemented
26	#97×D₇₉	C₄	1	2	yes	complemented
27	#19098335	C₂²	1	2	yes	—
28	#19098333	C₂²	1	2	yes	—
29	#19098323	C₂²	1	2	yes	—
30	C₇₉⋊#338	C₂²	1	2	yes	—
31	#19098320	C₂²	1	2	yes	—
32	#19098319	C₂²	1	2	yes	—
33	#19098319	C₂²	1	2	yes	—
34	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
35	#427×C₇₉	C₂²	1	1	yes	characteristic
36	#405×C₇₉	C₂²	1	2	yes	—
37	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
38	#339×C₇₉	C₂²	1	2	yes	—
39	C₂×#11929624	C₂³	1	1	yes	characteristic
40	#14579167	—	2	2	no	—
41	#14579167	—	2	1	no	—
42	#14579167	—	2	1	no	—
43	#14579167	—	2	1	no	—
44	#14579167	—	2	1	no	—
45	#14579167	—	2	2	no	—
46	C₂×#11929608	C₂³	1	2	yes	—
47	C₂×#11929608	C₂³	1	1	yes	characteristic
48	C₂×#11929608	C₂³	1	1	yes	characteristic
49	C₂×#11929607	C₂×C₄	1	2	yes	—
50	C₂×#11929607	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₅₈	—	2	1	no	—
54	C₄×D₃₁₆	—	2	1	no	—
55	C₄×C₂²⋊D₇₉	—	2	2	no	—
56	#14579120	C₂³	1	2	yes	—
57	#14579120	C₂³	1	1	yes	characteristic
58	#14579118	C₂×C₄	1	2	yes	—
59	C₄²×D₇₉	C₂×C₄	1	2	yes	—
60	#14579097	—	2	1	no	—
61	#14579097	—	2	1	no	—
62	#14579095	—	2	1	no	—
63	#14579093	—	2	1	no	—
64	#14579079	—	2	2	no	—
65	#14579077	—	2	2	no	—
66	#14579075	—	2	4	no	—
67	#14579072	C₂×C₄	1	2	yes	—
68	#14579071	—	2	2	no	—
69	#14579071	C₂×C₄	1	2	yes	—
70	#14579071	C₂×C₄	1	2	yes	—
71	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
72	C₈×Dic₇₉	—	2	2	no	—
73	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
74	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
75	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
76	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	#117×C₇₉	C₂³	1	2	yes	—
79	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
80	#108×C₇₉	—	2	1	no	—
81	#108×C₇₉	—	2	1	no	—
82	#107×C₇₉	—	2	1	no	—
83	#106×C₇₉	—	2	1	no	—
84	#97×C₇₉	—	2	2	no	—
85	#97×C₇₉	C₂×C₄	1	1	yes	characteristic
86	#97×C₇₉	C₂×C₄	1	1	yes	characteristic
87	#97×C₇₉	C₂×C₄	1	1	yes	characteristic
88	#97×C₇₉	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₇₉	—	2	2	no	—
90	#11929624	—	2	1	no	—
91	#11929624	—	2	1	no	—
92	#11929624	—	2	1	no	—
93	#11929624	—	2	1	no	—
94	C₂×Dic₁₅₈	#41	1	1	yes	characteristic
95	C₂×D₃₁₆	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₇₉	#41	1	2	yes	—
97	C₂×C₄×D₇₉	—	2	2	no	—
98	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
99	#11929608	—	2	1	no	—
100	#11929608	—	2	1	no	—
101	#11929608	—	2	1	no	—
102	#11929608	—	2	1	no	—
103	#11929608	—	2	2	no	—
104	#11929608	—	2	2	no	—
105	#11929607	—	2	2	no	—
106	#11929607	—	2	2	no	—
107	#11929607	—	2	4	no	—
108	C₂×C₇₉⋊C₈	—	2	2	no	—
109	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
111	C₈×D₇₉	—	2	2	no	—
112	C₈×D₇₉	—	2	2	no	—
113	C₈×D₇₉	—	2	4	no	—
114	C₄⋊Dic₇₉	—	2	1	no	—
115	C₄⋊Dic₇₉	—	2	2	no	—
116	Dic₇₉⋊C₄	—	2	2	no	—
117	C₂²⋊Dic₇₉	—	2	1	no	—
118	C₂²⋊Dic₇₉	—	2	2	no	—
119	(C₂×D₇₉)⋊C₄	—	2	2	no	—
120	C₄×Dic₇₉	—	2	2	no	—
121	C₄×Dic₇₉	—	2	2	no	—
122	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₇₉	—	2	1	no	—
124	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
125	M₁₆×C₇₉	—	2	2	no	—
126	M₁₆×C₇₉	—	2	2	no	—
127	C₂×C₈×C₇₉	—	2	2	no	—
128	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₇₉	#41	1	2	yes	—
133	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
134	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
135	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
136	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
137	C₂×C₈×C₇₉	#41	1	2	yes	—
138	C₄⋊C₄×C₇₉	—	2	2	no	—
139	C₂²⋊C₄×C₇₉	—	2	2	no	—
140	C₄²×C₇₉	—	2	1	no	—
141	C₄²×C₇₉	—	2	2	no	—
142	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
144	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
145	C₂²×D₇₉	#117	1	2	yes	—
146	C₂×C₄×C₇₉	—	2	1	no	—
147	C₂×C₄×C₇₉	—	2	1	no	—
148	C₂×C₄×C₇₉	—	4	2	no	—
149	C₂×C₄×C₇₉	—	8	1	no	—
150	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
156	C₂²⋊D₇₉	—	2	2	no	—
157	C₂²⋊D₇₉	—	2	2	no	—
158	C₂×Dic₇₉	—	2	2	no	—
159	C₂×Dic₇₉	—	4	2	no	—
160	C₂×Dic₇₉	#117	1	2	yes	—
161	D₃₁₆	—	2	1	no	—
162	D₃₁₆	—	2	1	no	—
163	C₄×D₇₉	—	2	2	no	—
164	C₄×D₇₉	—	2	2	no	—
165	C₄×D₇₉	—	8	2	no	—
166	Dic₁₅₈	—	2	1	no	—
167	Dic₁₅₈	—	2	1	no	—
168	C₈×C₇₉	—	4	2	no	—
169	C₈×C₇₉	#117	1	1	yes	characteristic
170	C₈×C₇₉	#117	1	1	yes	characteristic
171	C₈×C₇₉	#117	1	1	yes	characteristic
172	C₈×C₇₉	#117	1	1	yes	characteristic
173	C₈×C₇₉	#117	1	2	yes	—
174	C₈×C₇₉	#117	1	2	yes	—
175	C₇₉⋊C₈	—	2	2	no	—
176	C₇₉⋊C₈	—	2	2	no	—
177	C₇₉⋊C₈	—	4	2	no	—
178	C₂²×C₇₉	—	4	1	no	—
179	C₂²×C₇₉	C₂×#117	1	1	yes	characteristic
180	C₂²×C₇₉	#529	1	1	yes	characteristic
181	C₂²×C₇₉	#529	1	1	yes	characteristic
182	C₂×D₇₉	—	2	2	no	—
183	C₂×D₇₉	—	4	2	no	—
184	C₄×C₇₉	—	4	1	no	—
185	C₄×C₇₉	—	4	1	no	—
186	C₄×C₇₉	—	8	2	no	—
187	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
188	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
189	C₄×C₇₉	#529	1	1	yes	characteristic
190	C₄×C₇₉	#529	1	1	yes	characteristic
191	Dic₇₉	—	2	2	no	—
192	Dic₇₉	—	4	2	no	—
193	Dic₇₉	—	8	2	no	—
194	#20788	—	79	1	no	maximal, Sylow
195	C₂×C₇₉	—	2	1	no	—
196	C₂×C₇₉	—	2	1	no	—
197	C₂×C₇₉	—	2	1	no	—
198	C₂×C₇₉	#2283	1	1	yes	characteristic
199	D₇₉	—	4	2	no	—
200	#2875	—	79	1	no	—
201	#1841	—	79	2	no	—
202	#1833	—	79	1	no	—
203	#1833	—	79	1	no	—
204	#1827	—	79	2	no	—
205	#1813	—	79	1	no	—
206	#1812	—	79	1	no	—
207	#1773	—	79	1	no	—
208	#1773	D₇₉	1	1	yes	characteristic, complemented
209	#1757	—	79	2	no	—
210	#1747	—	79	2	no	—
211	C₇₉	#20788	1	1	yes	characteristic, complemented, Sylow
212	C₂×#131	—	79	1	no	—
213	C₂×#131	—	79	1	no	—
214	C₄×#41	—	79	1	no	—
215	C₂×#108	—	79	1	no	—
216	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
217	#427	—	79	1	no	—
218	#427	C₂×D₇₉	1	1	yes	characteristic
219	#413	—	79	2	no	—
220	#410	—	79	2	no	—
221	#405	—	79	1	no	—
222	#405	—	79	1	no	—
223	#405	—	79	2	no	—
224	#405	—	79	2	no	—
225	#405	—	79	2	no	—
226	#405	C₂×D₇₉	1	2	yes	—
227	C₄×M₁₆	—	79	1	no	—
228	C₄×M₁₆	—	79	2	no	—
229	C₂×#97	—	79	2	no	—
230	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
231	#344	—	79	2	no	—
232	#344	—	79	2	no	—
233	#339	—	79	2	no	—
234	#339	C₂×D₇₉	1	2	yes	—
235	#338	—	79	2	no	—
236	H(3,ℤ/4)	—	79	2	no	—
237	H(3,ℤ/4)	—	79	2	no	—
238	H(3,ℤ/4)	—	79	2	no	—
239	C₂×#41	—	79	1	no	—
240	#131	—	158	2	no	—
241	#131	—	158	1	no	—
242	#131	—	158	1	no	—
243	#131	—	158	1	no	—
244	#131	—	158	1	no	—
245	#131	—	158	2	no	—
246	C₂×M₁₆	—	79	1	no	—
247	C₂×M₁₆	—	79	1	no	—
248	C₂×M₁₆	—	79	2	no	—
249	C₂×M₁₆	—	79	2	no	—
250	C₂×M₁₆	—	79	2	no	—
251	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
252	C₂²×C₈	—	79	2	no	—
253	C₂²×C₈	—	79	2	no	—
254	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
255	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
256	C₄×Q₈	—	158	1	no	—
257	C₄×D₄	—	158	1	no	—
258	C₄×D₄	—	158	2	no	—
259	#117	—	79	1	no	—
260	#117	—	79	2	no	—
261	#117	—	79	2	no	—
262	#117	C₂²×D₇₉	1	2	yes	—
263	C₂×C₄²	—	79	2	no	—
264	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
265	#108	—	2	1	no	—
266	#108	—	2	1	no	—
267	#108	—	158	1	no	—
268	#108	—	158	1	no	—
269	#107	—	2	1	no	—
270	#107	—	158	1	no	—
271	#106	—	2	1	no	—
272	#106	—	158	1	no	—
273	#101	—	158	2	no	—
274	#100	—	158	2	no	—
275	#99	—	158	4	no	—
276	#97	—	2	2	no	—
277	#97	—	79	2	no	—
278	#97	—	79	2	no	—
279	#97	—	79	2	no	—
280	#97	—	158	2	no	—
281	#97	C₄×D₇₉	1	1	yes	characteristic
282	#97	C₄×D₇₉	1	1	yes	characteristic
283	#97	C₄×D₇₉	1	1	yes	characteristic
284	#97	C₄×D₇₉	1	1	yes	characteristic
285	C₄×C₈	—	2	2	no	—
286	C₄×C₈	—	79	2	no	—
287	C₄×C₈	—	79	2	no	—
288	C₄×C₈	—	79	2	no	—
289	C₄×C₈	—	158	2	no	—
290	#41	—	158	1	no	—
291	#41	—	158	1	no	—
292	#41	—	158	1	no	—
293	#41	—	158	1	no	—
294	C₂×Q₈	—	79	1	no	—
295	C₂×D₄	—	79	2	no	—
296	C₂×D₄	—	79	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	79	2	no	—
299	C₂²×C₄	—	158	2	no	—
300	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	2	no	—
303	M₁₆	—	158	2	no	—
304	M₁₆	—	158	2	no	—
305	M₁₆	—	158	1	no	—
306	M₁₆	—	158	1	no	—
307	M₁₆	—	158	1	no	—
308	M₁₆	—	158	1	no	—
309	M₁₆	—	158	2	no	—
310	M₁₆	—	158	2	no	—
311	M₁₆	—	158	4	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	79	2	no	—
314	C₂×C₈	—	79	2	no	—
315	C₂×C₈	—	158	2	no	—
316	C₂×C₈	—	158	2	no	—
317	C₂×C₈	—	158	2	no	—
318	C₂×C₈	—	158	4	no	—
319	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
320	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
321	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
323	C₂×C₈	#11929623	1	2	yes	—
324	C₂×C₈	#11929624	1	1	yes	characteristic
325	C₂×C₈	#11929624	1	1	yes	characteristic
326	C₂×C₈	#11929624	1	1	yes	characteristic
327	C₂×C₈	#11929624	1	1	yes	characteristic
328	C₂×C₈	#11929625	1	2	yes	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	158	2	no	—
331	C₄⋊C₄	—	158	2	no	—
332	C₄⋊C₄	—	158	1	no	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	158	2	no	—
335	C₂²⋊C₄	—	158	2	no	—
336	C₂²⋊C₄	—	158	1	no	—
337	C₄²	—	2	1	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	79	2	no	—
340	C₄²	—	158	2	no	—
341	C₄²	—	158	2	no	—
342	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
343	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
344	C₂³	—	79	2	no	—
345	C₂³	#41×D₇₉	1	1	yes	characteristic
346	Q₈	—	158	1	no	—
347	Q₈	—	158	1	no	—
348	D₄	—	158	2	no	—
349	D₄	—	158	2	no	—
350	D₄	—	158	1	no	—
351	D₄	—	158	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	8	1	no	—
356	C₂×C₄	—	316	2	no	—
357	C₂×C₄	—	79	2	no	—
358	C₂×C₄	—	632	2	no	—
359	C₂×C₄	—	158	2	no	—
360	C₂×C₄	—	158	2	no	—
361	C₂×C₄	—	158	2	no	—
362	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
363	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
364	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
365	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
366	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
367	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
368	C₈	—	4	2	no	—
369	C₈	—	316	2	no	—
370	C₈	—	158	2	no	—
371	C₈	—	158	2	no	—
372	C₈	#14579118	1	1	yes	characteristic
373	C₈	#14579118	1	1	yes	characteristic
374	C₈	#14579118	1	1	yes	characteristic
375	C₈	#14579118	1	1	yes	characteristic
376	C₈	#14579119	1	2	yes	—
377	C₈	#14579119	1	2	yes	—
378	C₂²	—	4	1	no	—
379	C₂²	—	316	2	no	—
380	C₂²	—	158	2	no	—
381	C₂²	#117×D₇₉	1	1	yes	characteristic
382	C₂²	#19099095	1	1	yes	characteristic
383	C₂²	#19099100	1	1	yes	characteristic
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	316	2	no	—
388	C₄	—	632	2	no	—
389	C₄	—	158	2	no	—
390	C₄	C₂×#14579118	1	1	yes	characteristic
391	C₄	C₂×#14579119	1	1	yes	characteristic
392	C₄	#19099095	1	1	yes	characteristic
393	C₄	#19099100	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	316	2	no	—
398	C₂	#23765796	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14579119
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1773	C₂
socle	C₂×C₇₉	#2283
prime radical	C₂×#11929624	C₂³
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1773	D₇₉
hypocenter	C₇₉	#20788
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1773

Complements

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

N	H ≅ G/N	type
2. #23763546	C₂	semidirect
3. #23763525	C₂	semidirect
4. #23763523	C₂	semidirect
8. #23763284	C₂	semidirect
10. #23763155	C₂	semidirect
11. C₇₉×#1773	C₂	semidirect
19. #19098520	C₄	semidirect
20. #19098520	C₄	semidirect
25. #19098515	C₄	semidirect
26. #97×D₇₉	C₄	semidirect
208. #1773	D₇₉	semidirect
211. C₇₉	#20788	semidirect

Maximal subgroups

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

subgroup	index
#23771825	2
#23763546	2
#23763525	2
#23763523	2
#23763503	2
#23763447	2
#23763443	2
#23763284	2
#23763201	2
#23763155	2
C₇₉×#1773	2
#20788	79