Subgroups — #34182139

Order 20224.

454 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772440	C₂	1	1	yes	maximal, characteristic
2	#23764193	C₂	1	1	yes	maximal, characteristic
3	C₄×#14579077	C₂	1	1	yes	maximal, characteristic
4	#23762150	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761238	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761236	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
9	#19098642	—	2	1	no	—
10	#19098642	—	2	1	no	—
11	#19098640	—	2	1	no	—
12	#19098640	—	2	1	no	—
13	#19098616	C₂²	1	1	yes	characteristic
14	C₂×#14579077	C₄	1	1	yes	characteristic
15	C₂×#14579077	C₄	1	1	yes	characteristic
16	#19098525	—	2	2	no	—
17	C₂×(C₂×D₇₉)⋊C₈	—	2	2	no	—
18	C₄×#11929608	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	#19098331	C₄	1	1	yes	characteristic
21	#19098331	C₄	1	1	yes	characteristic
22	#19098315	—	2	1	no	—
23	#19098313	—	2	1	no	—
24	#431×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
27	#335×C₇₉	—	2	1	no	—
28	#334×C₇₉	—	2	1	no	—
29	C₂²×D₃₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₇₉	D₄	1	2	yes	—
31	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
32	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
33	C₂×#11929607	—	2	2	no	—
34	C₂×#11929607	—	2	2	no	—
35	C₂²×C₇₉⋊C₈	—	2	1	no	—
36	C₄×D₃₁₆	—	2	1	no	—
37	C₄×D₃₁₆	—	2	1	no	—
38	C₄×D₃₁₆	—	2	1	no	—
39	C₄×D₃₁₆	—	2	1	no	—
40	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₇₉)⋊C₄	Q₈	1	2	yes	—
42	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
43	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
44	#14579077	—	2	2	no	—
45	#14579077	—	2	2	no	—
46	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
47	(C₂×D₇₉)⋊C₈	—	2	2	no	—
48	(C₂×D₇₉)⋊C₈	—	2	2	no	—
49	(C₂×D₇₉)⋊C₈	—	4	4	no	—
50	#14579072	—	2	1	no	—
51	C₄×C₇₉⋊C₈	—	2	1	no	—
52	C₄×C₇₉⋊C₈	—	2	1	no	—
53	C₄×C₇₉⋊C₈	—	2	1	no	—
54	C₄×C₇₉⋊C₈	—	2	1	no	—
55	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₉	—	2	1	no	—
58	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₉	—	2	1	no	—
60	C₄⋊C₈×C₇₉	—	2	1	no	—
61	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
62	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₇₉	—	2	1	no	—
64	#97×C₇₉	—	2	1	no	—
65	C₄×C₈×C₇₉	—	2	1	no	—
66	C₄×C₈×C₇₉	—	2	1	no	—
67	C₂³×D₇₉	C₄⋊C₄	1	2	yes	—
68	C₂×D₃₁₆	—	2	1	no	—
69	C₂×D₃₁₆	—	2	1	no	—
70	C₂×D₃₁₆	—	2	1	no	—
71	C₂×D₃₁₆	—	2	1	no	—
72	C₂×D₃₁₆	—	2	1	no	—
73	C₂×D₃₁₆	—	2	1	no	—
74	C₂²×Dic₇₉	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×D₇₉	—	2	2	no	—
76	C₂×C₄×D₇₉	—	2	2	no	—
77	C₂×C₄×D₇₉	—	4	2	no	—
78	C₂×C₄×D₇₉	—	4	2	no	—
79	#11929608	—	2	1	no	—
80	#11929608	—	2	1	no	—
81	#11929608	—	2	1	no	—
82	#11929608	M₁₆	1	2	yes	complemented
83	#11929607	—	8	2	no	—
84	#11929607	—	8	2	no	—
85	C₂×C₇₉⋊C₈	—	2	2	no	—
86	C₂×C₇₉⋊C₈	—	2	1	no	—
87	C₂×C₇₉⋊C₈	—	2	1	no	—
88	C₂×C₇₉⋊C₈	—	2	1	no	—
89	C₂×C₇₉⋊C₈	—	2	1	no	—
90	C₂×C₇₉⋊C₈	—	4	1	no	—
91	C₂×C₇₉⋊C₈	—	4	1	no	—
92	C₄⋊Dic₇₉	—	2	1	no	—
93	C₄⋊Dic₇₉	—	2	1	no	—
94	(C₂×D₇₉)⋊C₄	—	2	2	no	—
95	(C₂×D₇₉)⋊C₄	—	2	2	no	—
96	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
99	M₁₆×C₇₉	—	2	1	no	—
100	M₁₆×C₇₉	—	2	1	no	—
101	M₁₆×C₇₉	—	2	1	no	—
102	M₁₆×C₇₉	M₁₆	1	2	yes	complemented
103	C₂×C₈×C₇₉	—	2	1	no	—
104	C₂×C₈×C₇₉	—	2	1	no	—
105	C₂×C₈×C₇₉	—	2	1	no	—
106	C₂×C₈×C₇₉	—	2	1	no	—
107	C₂×C₈×C₇₉	—	4	2	no	—
108	C₂×C₈×C₇₉	—	4	1	no	—
109	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇₉	#41	1	1	yes	characteristic
112	C₄²×C₇₉	#41	1	1	yes	characteristic
113	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₇₉	—	2	2	no	—
115	C₂²×D₇₉	—	2	2	no	—
116	C₂²×D₇₉	—	2	2	no	—
117	C₂²×D₇₉	—	4	2	no	—
118	C₂²×D₇₉	—	4	2	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	4	1	no	—
122	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
132	C₂×Dic₇₉	—	2	2	no	—
133	C₂×Dic₇₉	—	4	2	no	—
134	D₃₁₆	—	4	1	no	—
135	D₃₁₆	—	4	1	no	—
136	D₃₁₆	—	4	1	no	—
137	D₃₁₆	—	4	1	no	—
138	C₄×D₇₉	—	4	2	no	—
139	C₄×D₇₉	—	4	2	no	—
140	C₄×D₇₉	—	8	2	no	—
141	C₈×C₇₉	—	2	2	no	—
142	C₈×C₇₉	—	4	1	no	—
143	C₈×C₇₉	—	4	1	no	—
144	C₈×C₇₉	—	4	1	no	—
145	C₈×C₇₉	—	8	1	no	—
146	C₇₉⋊C₈	—	2	2	no	—
147	C₇₉⋊C₈	—	4	2	no	—
148	C₇₉⋊C₈	—	4	1	no	—
149	C₇₉⋊C₈	—	4	1	no	—
150	C₇₉⋊C₈	—	4	1	no	—
151	C₂²×C₇₉	—	2	1	no	—
152	C₂²×C₇₉	—	2	1	no	—
153	C₂²×C₇₉	#385	1	1	yes	characteristic
154	C₂²×C₇₉	#423	1	1	yes	characteristic
155	C₂²×C₇₉	#424	1	1	yes	characteristic
156	C₂×D₇₉	—	4	4	no	—
157	C₂×D₇₉	—	4	2	no	—
158	C₂×D₇₉	—	4	2	no	—
159	C₂×D₇₉	—	4	2	no	—
160	C₂×D₇₉	—	8	2	no	—
161	C₄×C₇₉	—	2	1	no	—
162	C₄×C₇₉	—	2	1	no	—
163	C₄×C₇₉	—	2	1	no	—
164	C₄×C₇₉	#435	1	1	yes	characteristic
165	C₄×C₇₉	#435	1	1	yes	characteristic
166	C₄×C₇₉	#436	1	1	yes	characteristic
167	C₄×C₇₉	#436	1	1	yes	characteristic
168	C₄×C₇₉	#453	1	1	yes	characteristic
169	C₄×C₇₉	#455	1	1	yes	characteristic
170	Dic₇₉	—	4	2	no	—
171	#8837	—	79	1	no	maximal, Sylow
172	C₂×C₇₉	—	2	1	no	—
173	C₂×C₇₉	—	2	1	no	—
174	C₂×C₇₉	#1851	1	1	yes	characteristic
175	C₂×C₇₉	#1856	1	1	yes	characteristic
176	C₂×C₇₉	#1858	1	1	yes	characteristic
177	D₇₉	—	8	2	no	—
178	#2975	—	79	1	no	—
179	#1954	—	79	1	no	—
180	C₄×#100	—	79	1	no	—
181	#1565	—	79	1	no	—
182	#1565	D₇₉	1	1	yes	characteristic, complemented
183	#1278	—	79	1	no	—
184	#1278	—	79	1	no	—
185	C₇₉	#8837	1	1	yes	characteristic, complemented, Sylow
186	C₂×C₄×D₄	—	79	1	no	—
187	#436	—	158	1	no	—
188	#436	—	158	1	no	—
189	#436	—	158	1	no	—
190	#436	—	158	1	no	—
191	#431	—	79	1	no	—
192	#431	C₂×D₇₉	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
194	C₂×#100	—	79	1	no	—
195	C₂×#100	—	79	1	no	—
196	#407	—	158	2	no	—
197	C₂×C₂²⋊C₈	—	158	2	no	—
198	C₄×M₁₆	—	79	1	no	—
199	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
200	C₂×C₄×C₈	—	79	1	no	—
201	#343	—	79	1	no	—
202	#343	—	79	1	no	—
203	#335	—	2	1	no	—
204	#335	—	158	1	no	—
205	#334	—	2	1	no	—
206	#334	—	158	1	no	—
207	C₂²×D₄	—	79	1	no	—
208	C₂³×C₄	—	79	2	no	—
209	C₂×M₁₆	—	79	1	no	—
210	C₂×M₁₆	—	79	1	no	—
211	C₂×M₁₆	—	158	2	no	—
212	C₂×M₁₆	—	158	2	no	—
213	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
214	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	158	1	no	—
217	C₄×D₄	—	158	1	no	—
218	C₄×D₄	—	158	1	no	—
219	C₄×D₄	—	158	1	no	—
220	C₄×D₄	—	158	1	no	—
221	C₂×C₄⋊C₄	—	79	1	no	—
222	C₂×C₂²⋊C₄	—	79	2	no	—
223	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	79	1	no	—
227	C₄⋊C₈	—	79	1	no	—
228	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
229	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
230	#100	—	158	2	no	—
231	#100	—	158	2	no	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₂²⋊C₈	—	316	4	no	—
234	C₂²⋊C₈	—	158	2	no	—
235	C₂²⋊C₈	—	158	2	no	—
236	C₂²⋊C₈	—	158	1	no	—
237	#97	—	2	1	no	—
238	#97	—	158	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	158	1	no	—
242	C₄×C₈	—	158	1	no	—
243	C₄×C₈	—	158	1	no	—
244	C₄×C₈	—	158	1	no	—
245	C₂⁴	—	79	2	no	—
246	C₂×D₄	—	158	1	no	—
247	C₂×D₄	—	158	1	no	—
248	C₂×D₄	—	158	1	no	—
249	C₂×D₄	—	158	1	no	—
250	C₂×D₄	—	158	1	no	—
251	C₂×D₄	—	158	1	no	—
252	C₂²×C₄	—	316	2	no	—
253	C₂²×C₄	—	316	2	no	—
254	C₂²×C₄	—	79	2	no	—
255	C₂²×C₄	—	158	2	no	—
256	C₂²×C₄	—	158	2	no	—
257	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
260	M₁₆	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	79	2	no	—
264	M₁₆	—	632	2	no	—
265	M₁₆	—	632	2	no	—
266	M₁₆	—	158	1	no	—
267	M₁₆	—	158	1	no	—
268	M₁₆	—	158	1	no	—
269	M₁₆	#11929607	1	2	yes	complemented
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	316	1	no	—
277	C₂×C₈	—	316	1	no	—
278	C₂×C₈	—	158	2	no	—
279	C₂×C₈	—	158	1	no	—
280	C₂×C₈	—	158	1	no	—
281	C₂×C₈	—	158	1	no	—
282	C₂×C₈	—	158	1	no	—
283	C₄⋊C₄	—	158	1	no	—
284	C₄⋊C₄	—	158	1	no	—
285	C₂²⋊C₄	—	158	2	no	—
286	C₂²⋊C₄	—	158	2	no	—
287	C₄²	D₄×D₇₉	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
289	C₄²	#11929624	1	1	yes	characteristic
290	C₄²	#11929625	1	1	yes	characteristic
291	C₂³	—	316	2	no	—
292	C₂³	—	316	2	no	—
293	C₂³	—	158	2	no	—
294	C₂³	—	158	2	no	—
295	C₂³	—	158	2	no	—
296	C₂³	C₄⋊C₄×D₇₉	1	1	yes	characteristic
297	D₄	—	316	1	no	—
298	D₄	—	316	1	no	—
299	D₄	—	316	1	no	—
300	D₄	—	316	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	316	2	no	—
305	C₂×C₄	—	316	2	no	—
306	C₂×C₄	—	316	2	no	—
307	C₂×C₄	—	632	2	no	—
308	C₂×C₄	—	158	2	no	—
309	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
310	C₂×C₄	#14579123	1	1	yes	characteristic
311	C₂×C₄	#14579129	1	1	yes	characteristic
312	C₂×C₄	#14579137	1	1	yes	characteristic
313	C₂×C₄	#14579139	1	1	yes	characteristic
314	C₂×C₄	#14579146	1	1	yes	characteristic
315	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
316	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
317	C₂×C₄	#14579167	1	1	yes	characteristic
318	C₂×C₄	#14579169	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₈	—	4	1	no	—
323	C₈	—	8	1	no	—
324	C₈	—	316	2	no	—
325	C₈	—	316	1	no	—
326	C₈	—	316	1	no	—
327	C₈	—	316	1	no	—
328	C₈	—	158	2	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	316	4	no	—
332	C₂²	—	316	2	no	—
333	C₂²	—	316	2	no	—
334	C₂²	—	316	2	no	—
335	C₂²	—	632	2	no	—
336	C₂²	#19098434	1	1	yes	characteristic
337	C₂²	#19098585	1	1	yes	characteristic
338	C₂²	#19098588	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	316	2	no	—
343	C₄	#19098633	1	1	yes	characteristic
344	C₄	#19098634	1	1	yes	characteristic
345	C₄	#19098639	1	1	yes	characteristic
346	C₄	#19098640	1	1	yes	characteristic
347	C₄	#19098729	1	1	yes	characteristic
348	C₄	#19098739	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	632	2	no	—
352	C₂	#23763581	1	1	yes	characteristic
353	C₂	#23763608	1	1	yes	characteristic
354	C₂	#23763618	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1565	C₂
socle	C₂²×C₇₉	#385
prime radical	C₂²×D₃₁₆	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1565	D₇₉
hypocenter	C₇₉	#8837
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #23762150	C₂	semidirect
5. #23761238	C₂	semidirect
6. #23761236	C₂	semidirect
7. C₇₉×#1565	C₂	semidirect
82. #11929608	M₁₆	semidirect
102. M₁₆×C₇₉	M₁₆	semidirect
182. #1565	D₇₉	semidirect
185. C₇₉	#8837	semidirect
269. M₁₆	#11929607	semidirect

Maximal subgroups

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

subgroup	index
#23772440	2
#23764193	2
C₄×#14579077	2
#23762150	2
#23761238	2
#23761236	2
C₇₉×#1565	2
#8837	79