Subgroups — #34288640

Order 20224.

608 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771826	C₂	1	1	yes	maximal, characteristic
2	#23763548	C₂	1	2	yes	maximal, complemented
3	#23763521	C₂	1	2	yes	maximal, complemented
4	#23763505	C₂	1	1	yes	maximal, characteristic
5	#23763501	C₂	1	1	yes	maximal, characteristic
6	#23763448	C₂	1	1	yes	maximal, characteristic
7	#23763444	C₂	1	1	yes	maximal, characteristic
8	#23763285	C₂	1	2	yes	maximal, complemented
9	#23763200	C₂	1	1	yes	maximal, characteristic
10	#23763155	C₂	1	2	yes	maximal, complemented
11	C₇₉×#1757	C₂	1	1	yes	maximal, characteristic
12	C₂×#14579168	C₂²	1	1	yes	characteristic
13	#41×Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×#14579096	C₂²	1	2	yes	—
15	#19098599	C₂²	1	2	yes	—
16	#19098547	C₂²	1	1	yes	characteristic
17	#19098537	C₂²	1	1	yes	characteristic
18	#19098521	C₄	1	2	yes	complemented
19	#19098521	C₂²	1	1	yes	characteristic
20	#19098518	C₄	1	2	yes	complemented
21	#19098518	C₂²	1	2	yes	—
22	#19098518	C₂²	1	1	yes	characteristic
23	M₁₆×Dic₇₉	C₄	1	2	yes	complemented
24	M₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
25	C₂×#14579071	C₄	1	2	yes	complemented
26	C₂×#14579071	C₂²	1	1	yes	characteristic
27	#19098334	C₂²	1	2	yes	—
28	#19098324	C₂²	1	2	yes	—
29	#19098322	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×#131×C₇₉	C₂²	1	1	yes	characteristic
35	#413×C₇₉	C₂²	1	1	yes	characteristic
36	#410×C₇₉	C₂²	1	1	yes	characteristic
37	#405×C₇₉	C₂²	1	2	yes	—
38	#344×C₇₉	C₂²	1	2	yes	—
39	#14579168	—	2	2	no	—
40	#14579168	—	2	2	no	—
41	C₂×#11929608	C₂×C₄	1	2	yes	—
42	C₂×#11929608	C₂³	1	2	yes	—
43	C₂×#11929608	C₂³	1	1	yes	characteristic
44	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
46	Q₈×Dic₇₉	—	2	1	no	—
47	D₄×Dic₇₉	—	2	2	no	—
48	D₄×Dic₇₉	—	2	1	no	—
49	#14579120	C₂×C₄	1	2	yes	—
50	#14579120	C₂³	1	2	yes	—
51	#14579120	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
54	#14579096	—	2	2	no	—
55	#14579096	—	2	2	no	—
56	#14579094	—	2	2	no	—
57	#14579092	—	2	2	no	—
58	#14579080	—	2	1	no	—
59	#14579078	—	2	1	no	—
60	#14579076	—	2	2	no	—
61	#14579071	—	2	2	no	—
62	#14579071	C₂×C₄	1	1	yes	characteristic
63	#14579071	C₂×C₄	1	2	yes	—
64	#14579071	C₂×C₄	1	2	yes	—
65	#14579071	C₂×C₄	1	2	yes	—
66	#14579071	C₂×C₄	1	1	yes	characteristic
67	#14579071	C₂×C₄	1	1	yes	characteristic
68	#14579071	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₇₉	—	2	2	no	—
70	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
71	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
72	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
73	C₂×#41×C₇₉	C₂³	1	1	yes	characteristic
74	#131×C₇₉	—	2	2	no	—
75	#131×C₇₉	—	2	2	no	—
76	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
77	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
78	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
80	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
81	#117×C₇₉	C₂³	1	2	yes	—
82	#101×C₇₉	—	2	1	no	—
83	#100×C₇₉	—	2	1	no	—
84	#99×C₇₉	—	2	2	no	—
85	#97×C₇₉	—	2	2	no	—
86	C₄×C₈×C₇₉	—	2	2	no	—
87	C₂²×Dic₇₉	—	2	2	no	—
88	C₂²×Dic₇₉	—	2	1	no	—
89	#11929608	—	2	4	no	—
90	#11929608	—	2	2	no	—
91	#11929608	—	2	2	no	—
92	#11929608	—	2	2	no	—
93	C₂×C₇₉⋊C₈	—	2	2	no	—
94	C₂×C₇₉⋊C₈	—	2	2	no	—
95	C₂×C₇₉⋊C₈	—	2	2	no	—
96	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₇₉⋊C₈	#41	1	2	yes	—
100	C₂×C₇₉⋊C₈	#41	1	2	yes	—
101	C₄⋊Dic₇₉	—	2	2	no	—
102	C₄⋊Dic₇₉	—	2	1	no	—
103	C₄⋊Dic₇₉	—	2	2	no	—
104	C₂²⋊Dic₇₉	—	2	2	no	—
105	C₂²⋊Dic₇₉	—	2	1	no	—
106	C₂²⋊Dic₇₉	—	2	2	no	—
107	C₄×Dic₇₉	—	2	2	no	—
108	C₄×Dic₇₉	—	2	1	no	—
109	C₄×Dic₇₉	—	2	2	no	—
110	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
111	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
113	#41×C₇₉	—	2	2	no	—
114	#41×C₇₉	—	2	2	no	—
115	C₂×Q₈×C₇₉	#41	1	1	yes	characteristic
116	C₂×D₄×C₇₉	#41	1	2	yes	—
117	C₂×D₄×C₇₉	#41	1	1	yes	characteristic
118	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
120	M₁₆×C₇₉	—	2	4	no	—
121	M₁₆×C₇₉	—	2	2	no	—
122	M₁₆×C₇₉	—	2	2	no	—
123	M₁₆×C₇₉	—	2	2	no	—
124	C₂×C₈×C₇₉	—	2	2	no	—
125	C₂×C₈×C₇₉	—	2	2	no	—
126	C₂×C₈×C₇₉	—	2	2	no	—
127	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₇₉	#41	1	2	yes	—
131	C₂×C₈×C₇₉	#41	1	2	yes	—
132	C₄⋊C₄×C₇₉	—	2	2	no	—
133	C₂²⋊C₄×C₇₉	—	2	2	no	—
134	C₄²×C₇₉	—	2	2	no	—
135	C₂³×C₇₉	#117	1	2	yes	—
136	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
137	Q₈×C₇₉	—	2	2	no	—
138	D₄×C₇₉	—	2	4	no	—
139	D₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	—	2	2	no	—
141	C₂×C₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	#117	1	2	yes	—
144	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
146	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
147	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₇₉	—	2	1	no	—
149	C₂×Dic₇₉	—	2	2	no	—
150	C₂×Dic₇₉	—	2	1	no	—
151	C₂×Dic₇₉	—	4	2	no	—
152	C₂×Dic₇₉	—	8	1	no	—
153	C₂×Dic₇₉	—	8	2	no	—
154	C₈×C₇₉	—	2	2	no	—
155	C₈×C₇₉	—	4	2	no	—
156	C₈×C₇₉	#117	1	1	yes	characteristic
157	C₈×C₇₉	#117	1	1	yes	characteristic
158	C₈×C₇₉	#117	1	2	yes	—
159	C₇₉⋊C₈	—	2	2	no	—
160	C₇₉⋊C₈	—	4	2	no	—
161	C₇₉⋊C₈	#117	1	1	yes	characteristic
162	C₇₉⋊C₈	#117	1	1	yes	characteristic
163	C₇₉⋊C₈	#117	1	2	yes	—
164	C₂²×C₇₉	—	2	2	no	—
165	C₂²×C₇₉	—	4	1	no	—
166	C₂²×C₇₉	—	4	2	no	—
167	C₂²×C₇₉	C₂×#117	1	1	yes	characteristic
168	C₂²×C₇₉	#529	1	2	yes	—
169	C₄×C₇₉	—	2	2	no	—
170	C₄×C₇₉	—	8	2	no	—
171	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
172	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
173	C₄×C₇₉	#529	1	2	yes	—
174	Dic₇₉	—	4	2	no	—
175	Dic₇₉	—	4	1	no	—
176	Dic₇₉	—	4	1	no	—
177	Dic₇₉	—	8	2	no	—
178	#20788	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	2	1	no	—
180	C₂×C₇₉	—	2	2	no	—
181	C₂×C₇₉	—	4	2	no	—
182	C₂×C₇₉	#2283	1	1	yes	characteristic
183	#2875	—	79	1	no	—
184	#1841	—	79	2	no	—
185	#1833	—	79	2	no	—
186	#1827	—	79	1	no	—
187	#1827	—	79	1	no	—
188	#1813	—	79	1	no	—
189	#1812	—	79	1	no	—
190	#1773	—	79	2	no	—
191	#1757	—	79	1	no	—
192	#1757	D₇₉	1	1	yes	characteristic
193	#1747	—	79	2	no	—
194	C₇₉	#20788	1	1	yes	characteristic, complemented, Sylow
195	C₂×#131	—	79	1	no	—
196	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
197	C₄×#41	—	79	1	no	—
198	C₂×#108	—	79	2	no	—
199	#427	—	79	2	no	—
200	#413	—	79	1	no	—
201	#413	C₂×D₇₉	1	1	yes	characteristic
202	#410	—	79	1	no	—
203	#410	C₂×D₇₉	1	1	yes	characteristic
204	#405	—	79	1	no	—
205	#405	—	79	2	no	—
206	#405	—	79	1	no	—
207	#405	—	79	2	no	—
208	#405	—	79	2	no	—
209	#405	C₂×D₇₉	1	2	yes	—
210	C₄×M₁₆	—	79	1	no	—
211	C₄×M₁₆	—	79	2	no	—
212	C₂×#97	—	79	1	no	—
213	C₂×#97	—	79	2	no	—
214	#344	—	79	2	no	—
215	#344	C₂×D₇₉	1	2	yes	—
216	#339	—	79	2	no	—
217	#339	—	79	2	no	—
218	#338	—	79	2	no	—
219	H(3,ℤ/4)	—	79	2	no	—
220	H(3,ℤ/4)	—	79	2	no	—
221	H(3,ℤ/4)	—	79	2	no	—
222	C₂×#41	C₂²×D₇₉	1	1	yes	characteristic
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	158	2	no	—
226	#131	—	158	2	no	—
227	C₂×M₁₆	—	79	2	no	—
228	C₂×M₁₆	—	79	1	no	—
229	C₂×M₁₆	—	79	2	no	—
230	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
231	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
232	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
233	C₂²×C₈	—	79	1	no	—
234	C₂²×C₈	—	79	2	no	—
235	C₂²×C₈	C₄×D₇₉	1	2	yes	—
236	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
237	C₄×Q₈	—	158	1	no	—
238	C₄×D₄	—	158	1	no	—
239	C₄×D₄	—	158	2	no	—
240	#117	—	79	2	no	—
241	#117	—	79	1	no	—
242	#117	—	79	2	no	—
243	#117	C₂²×D₇₉	1	2	yes	—
244	C₂×C₄²	—	79	2	no	—
245	C₂×C₄²	—	79	1	no	—
246	#108	—	158	2	no	—
247	#108	—	158	2	no	—
248	#107	—	158	2	no	—
249	#106	—	158	2	no	—
250	#101	—	2	1	no	—
251	#101	—	158	1	no	—
252	#100	—	2	1	no	—
253	#100	—	158	1	no	—
254	#99	—	2	2	no	—
255	#99	—	158	2	no	—
256	#97	—	2	2	no	—
257	#97	—	79	1	no	—
258	#97	—	79	1	no	—
259	#97	—	79	2	no	—
260	#97	—	79	1	no	—
261	#97	—	79	2	no	—
262	#97	—	79	1	no	—
263	#97	—	79	2	no	—
264	#97	—	158	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	79	2	no	—
267	C₄×C₈	—	79	2	no	—
268	C₄×C₈	—	79	2	no	—
269	C₄×C₈	—	158	2	no	—
270	#41	—	2	2	no	—
271	#41	—	2	2	no	—
272	C₂×Q₈	#11929624	1	1	yes	characteristic
273	C₂×D₄	#11929624	1	1	yes	characteristic
274	C₂×D₄	#11929624	1	2	yes	—
275	C₂²×C₄	—	158	1	no	—
276	C₂²×C₄	—	158	2	no	—
277	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
278	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	158	2	no	—
284	M₁₆	—	158	2	no	—
285	M₁₆	—	158	4	no	—
286	M₁₆	—	158	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	79	2	no	—
291	C₂×C₈	—	79	2	no	—
292	C₂×C₈	—	79	1	no	—
293	C₂×C₈	—	79	1	no	—
294	C₂×C₈	—	79	2	no	—
295	C₂×C₈	—	158	2	no	—
296	C₂×C₈	—	158	2	no	—
297	C₂×C₈	—	158	2	no	—
298	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
299	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
301	C₂×C₈	#11929623	1	2	yes	—
302	C₂×C₈	#11929623	1	2	yes	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	158	2	no	—
305	C₄⋊C₄	—	158	1	no	—
306	C₄⋊C₄	—	158	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	158	2	no	—
309	C₂²⋊C₄	—	158	1	no	—
310	C₂²⋊C₄	—	158	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	79	2	no	—
313	C₄²	—	79	1	no	—
314	C₄²	—	79	1	no	—
315	C₄²	—	158	2	no	—
316	C₄²	—	158	1	no	—
317	C₄²	—	158	2	no	—
318	C₂³	#14579118	1	2	yes	—
319	C₂³	C₂×#11929624	1	1	yes	characteristic
320	Q₈	—	2	2	no	—
321	D₄	—	2	4	no	—
322	D₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	316	2	no	—
327	C₂×C₄	—	632	1	no	—
328	C₂×C₄	—	632	2	no	—
329	C₂×C₄	—	158	1	no	—
330	C₂×C₄	—	158	1	no	—
331	C₂×C₄	—	158	2	no	—
332	C₂×C₄	#14579118	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
334	C₂×C₄	#41×D₇₉	1	2	yes	—
335	C₂×C₄	C₂×#11929623	1	2	yes	—
336	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	316	2	no	—
340	C₈	—	79	1	no	—
341	C₈	—	79	1	no	—
342	C₈	—	79	2	no	—
343	C₈	—	158	2	no	—
344	C₈	#14579117	1	1	yes	characteristic
345	C₈	#14579117	1	1	yes	characteristic
346	C₈	#14579117	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	C₂×#14579118	1	1	yes	characteristic
351	C₂²	#19099093	1	2	yes	—
352	C₄	—	2	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	316	1	no	—
355	C₄	—	316	1	no	—
356	C₄	—	316	2	no	—
357	C₄	—	632	2	no	—
358	C₄	#117×D₇₉	1	1	yes	characteristic
359	C₄	C₂×#14579117	1	1	yes	characteristic
360	C₄	#19099093	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	1	no	—
363	C₂	—	4	2	no	—
364	C₂	#23765794	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14579117
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1757	C₂
socle	C₂×C₇₉	#2283
prime radical	C₂×#41×C₇₉	C₂³
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1757	D₇₉
hypocenter	C₇₉	#20788
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1757

Complements

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

N	H ≅ G/N	type
2. #23763548	C₂	semidirect
3. #23763521	C₂	semidirect
8. #23763285	C₂	semidirect
10. #23763155	C₂	semidirect
18. #19098521	C₄	semidirect
20. #19098518	C₄	semidirect
23. M₁₆×Dic₇₉	C₄	semidirect
25. C₂×#14579071	C₄	semidirect
194. C₇₉	#20788	semidirect

Maximal subgroups

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

subgroup	index
#23771826	2
#23763548	2
#23763521	2
#23763505	2
#23763501	2
#23763448	2
#23763444	2
#23763285	2
#23763200	2
#23763155	2
C₇₉×#1757	2
#20788	79