Subgroups — #34181422

Order 20224.

434 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098640	C₂	1	1	yes	maximal, characteristic
2	#23763537	C₂	1	1	yes	maximal, characteristic, complemented
3	#23762146	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761826	C₂	1	1	yes	maximal, characteristic
5	#23761824	C₂	1	1	yes	maximal, characteristic
6	#23761297	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11929607	—	2	1	no	—
9	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
10	#19098640	—	2	1	no	—
11	#19098640	—	2	1	no	—
12	#19098640	C₄	1	2	yes	—
13	#19098640	C₄	1	2	yes	—
14	#19098616	C₂²	1	1	yes	characteristic
15	#19098587	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₇₉)⋊C₈	—	2	1	no	—
17	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
18	#19098327	—	2	1	no	—
19	#19098327	—	2	1	no	—
20	#19098315	—	2	1	no	—
21	#19098313	—	2	1	no	—
22	#19098292	—	2	2	no	—
23	#19098290	—	2	2	no	—
24	#431×C₇₉	C₂²	1	1	yes	characteristic
25	#423×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²×D₃₁₆	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₇₉	—	2	1	no	—
29	C₂×#11929608	—	2	1	no	—
30	C₂×#11929607	—	2	2	no	—
31	C₂×#11929607	—	2	2	no	—
32	C₂×#11929607	—	4	2	no	—
33	C₂×#11929607	—	4	2	no	—
34	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
35	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
36	C₄×D₃₁₆	—	2	1	no	—
37	C₄×D₃₁₆	—	2	1	no	—
38	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
39	C₄×D₃₁₆	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₇₉	Q₈	1	1	yes	characteristic
41	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
42	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
43	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
44	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
45	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
46	(C₂×D₇₉)⋊C₈	—	2	2	no	—
47	(C₂×D₇₉)⋊C₈	—	2	2	no	—
48	C₄×C₇₉⋊C₈	—	2	1	no	—
49	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
52	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₇₉	—	2	1	no	—
54	C₂²×C₈×C₇₉	—	2	1	no	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₉	—	2	1	no	—
57	C₄⋊C₈×C₇₉	—	2	1	no	—
58	C₄⋊C₈×C₇₉	—	2	1	no	—
59	C₄⋊C₈×C₇₉	—	2	1	no	—
60	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₇₉	—	2	1	no	—
62	C₄×C₈×C₇₉	—	2	1	no	—
63	C₂³×D₇₉	—	2	1	no	—
64	C₂×D₃₁₆	—	2	1	no	—
65	C₂×D₃₁₆	—	4	2	no	—
66	C₂×D₃₁₆	C₄⋊C₄	1	2	yes	—
67	C₂²×Dic₇₉	—	2	1	no	—
68	C₂×C₄×D₇₉	—	2	2	no	—
69	C₂×C₄×D₇₉	—	2	2	no	—
70	C₂×C₄×D₇₉	—	4	1	no	—
71	C₂×C₄×D₇₉	—	4	1	no	—
72	C₂×C₄×D₇₉	—	4	1	no	—
73	C₂×C₄×D₇₉	—	4	1	no	—
74	#11929608	—	4	1	no	—
75	#11929608	—	4	1	no	—
76	#11929607	—	4	4	no	—
77	#11929607	—	4	4	no	—
78	C₂×C₇₉⋊C₈	—	2	1	no	—
79	C₂×C₇₉⋊C₈	—	2	1	no	—
80	C₂×C₇₉⋊C₈	—	2	1	no	—
81	C₂×C₇₉⋊C₈	—	2	1	no	—
82	C₂×C₇₉⋊C₈	—	4	1	no	—
83	C₂×C₇₉⋊C₈	—	4	1	no	—
84	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₄⋊Dic₇₉	—	2	1	no	—
89	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
90	(C₂×D₇₉)⋊C₄	—	2	2	no	—
91	(C₂×D₇₉)⋊C₄	—	2	2	no	—
92	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
95	M₁₆×C₇₉	—	4	1	no	—
96	M₁₆×C₇₉	—	4	1	no	—
97	C₂×C₈×C₇₉	—	2	2	no	—
98	C₂×C₈×C₇₉	—	4	1	no	—
99	C₂×C₈×C₇₉	—	4	1	no	—
100	C₂×C₈×C₇₉	—	4	1	no	—
101	C₂×C₈×C₇₉	—	4	1	no	—
102	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇₉	#41	1	1	yes	characteristic
106	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂²×D₇₉	—	2	2	no	—
108	C₂²×D₇₉	—	4	2	no	—
109	C₂²×D₇₉	—	4	2	no	—
110	C₂²×D₇₉	—	4	1	no	—
111	C₂²×D₇₉	—	4	1	no	—
112	C₂×C₄×C₇₉	—	2	2	no	—
113	C₂×C₄×C₇₉	—	4	1	no	—
114	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
121	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
122	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×Dic₇₉	—	2	2	no	—
125	C₂×Dic₇₉	—	4	1	no	—
126	C₂×Dic₇₉	—	4	1	no	—
127	D₃₁₆	—	2	4	no	—
128	D₃₁₆	—	4	2	no	—
129	C₄×D₇₉	—	4	2	no	—
130	C₄×D₇₉	—	4	2	no	—
131	C₄×D₇₉	—	4	2	no	—
132	C₄×D₇₉	—	4	2	no	—
133	C₈×C₇₉	—	4	2	no	—
134	C₈×C₇₉	—	8	1	no	—
135	C₈×C₇₉	—	8	1	no	—
136	C₇₉⋊C₈	—	2	1	no	—
137	C₇₉⋊C₈	—	2	1	no	—
138	C₇₉⋊C₈	—	2	1	no	—
139	C₇₉⋊C₈	—	2	1	no	—
140	C₇₉⋊C₈	—	8	1	no	—
141	C₇₉⋊C₈	—	8	1	no	—
142	C₂²×C₇₉	—	2	1	no	—
143	C₂²×C₇₉	—	2	1	no	—
144	C₂²×C₇₉	#380	1	1	yes	characteristic
145	C₂²×C₇₉	#478	1	1	yes	characteristic
146	C₂²×C₇₉	#478	1	1	yes	characteristic
147	C₂×D₇₉	—	4	2	no	—
148	C₂×D₇₉	—	4	2	no	—
149	C₂×D₇₉	—	4	2	no	—
150	C₂×D₇₉	—	8	2	no	—
151	C₂×D₇₉	—	8	2	no	—
152	C₄×C₇₉	—	2	1	no	—
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	#416	1	1	yes	characteristic
156	C₄×C₇₉	#418	1	1	yes	characteristic
157	C₄×C₇₉	#430	1	2	yes	—
158	C₄×C₇₉	#463	1	1	yes	characteristic
159	C₄×C₇₉	#463	1	1	yes	characteristic
160	Dic₇₉	—	4	2	no	—
161	#8664	—	79	1	no	maximal, Sylow
162	C₂×C₇₉	—	2	1	no	—
163	C₂×C₇₉	—	2	1	no	—
164	C₂×C₇₉	#1804	1	1	yes	characteristic
165	C₂×C₇₉	#1811	1	1	yes	characteristic
166	C₂×C₇₉	#1814	1	1	yes	characteristic
167	D₇₉	—	8	2	no	—
168	C₂×#436	—	79	1	no	—
169	#1837	—	79	1	no	—
170	#1564	—	79	1	no	—
171	#1486	—	79	1	no	—
172	#1486	—	79	1	no	—
173	#1301	—	79	1	no	—
174	#1301	D₇₉	1	1	yes	characteristic, complemented
175	C₇₉	#8664	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	—	158	1	no	—
177	C₂×C₄×D₄	—	79	1	no	—
178	#436	—	79	2	no	—
179	#436	—	79	2	no	—
180	#436	—	158	1	no	—
181	#436	—	158	1	no	—
182	#431	—	79	1	no	—
183	#431	C₂×D₇₉	1	1	yes	characteristic
184	#423	—	79	1	no	—
185	#423	C₂×D₇₉	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
187	C₂×C₂²⋊C₈	—	158	1	no	—
188	C₂×C₄×C₈	—	79	1	no	—
189	#341	—	158	1	no	—
190	#341	—	158	1	no	—
191	#335	—	158	1	no	—
192	#334	—	158	1	no	—
193	D₄⋊C₈	—	158	2	no	—
194	D₄⋊C₈	—	158	2	no	—
195	C₂²×D₄	—	79	1	no	—
196	C₂³×C₄	—	158	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	316	2	no	—
199	C₂×M₁₆	—	316	2	no	—
200	C₂×M₁₆	—	158	2	no	—
201	C₂×M₁₆	—	158	2	no	—
202	C₂×M₁₆	—	158	1	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	—	79	1	no	—
205	C₂²×C₈	—	79	1	no	—
206	C₄×D₄	—	79	2	no	—
207	C₄×D₄	—	79	2	no	—
208	C₄×D₄	—	158	1	no	—
209	C₄×D₄	—	158	1	no	—
210	C₂×C₄⋊C₄	—	79	1	no	—
211	C₂×C₂²⋊C₄	—	158	1	no	—
212	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	158	1	no	—
218	C₄⋊C₈	—	158	1	no	—
219	C₄⋊C₈	—	158	1	no	—
220	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	158	2	no	—
223	C₂²⋊C₈	—	158	2	no	—
224	C₂²⋊C₈	—	158	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	79	1	no	—
227	C₄×C₈	—	79	1	no	—
228	C₄×C₈	—	79	1	no	—
229	C₄×C₈	—	79	1	no	—
230	C₄×C₈	—	158	1	no	—
231	C₂⁴	—	158	1	no	—
232	C₂×D₄	—	316	2	no	—
233	C₂×D₄	—	79	2	no	—
234	C₂×D₄	—	158	1	no	—
235	C₂²×C₄	—	316	1	no	—
236	C₂²×C₄	—	316	1	no	—
237	C₂²×C₄	—	316	1	no	—
238	C₂²×C₄	—	316	1	no	—
239	C₂²×C₄	—	158	1	no	—
240	C₂²×C₄	—	158	2	no	—
241	C₂²×C₄	—	158	2	no	—
242	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
243	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
244	C₂²×C₄	#11929624	1	1	yes	characteristic
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	316	4	no	—
248	M₁₆	—	316	4	no	—
249	M₁₆	—	316	1	no	—
250	M₁₆	—	316	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	316	1	no	—
257	C₂×C₈	—	316	1	no	—
258	C₂×C₈	—	79	1	no	—
259	C₂×C₈	—	79	1	no	—
260	C₂×C₈	—	79	1	no	—
261	C₂×C₈	—	79	1	no	—
262	C₂×C₈	—	158	1	no	—
263	C₂×C₈	—	158	1	no	—
264	C₂×C₈	—	158	1	no	—
265	C₂×C₈	—	158	1	no	—
266	C₄⋊C₄	—	79	2	no	—
267	C₄⋊C₄	—	158	1	no	—
268	C₂²⋊C₄	—	158	2	no	—
269	C₂²⋊C₄	—	158	2	no	—
270	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
271	C₄²	D₄×D₇₉	1	1	yes	characteristic
272	C₄²	D₄×D₇₉	1	1	yes	characteristic
273	C₄²	#11929624	1	1	yes	characteristic
274	C₂³	—	316	2	no	—
275	C₂³	—	316	2	no	—
276	C₂³	—	316	1	no	—
277	C₂³	—	316	1	no	—
278	C₂³	—	158	2	no	—
279	C₂³	#14579139	1	1	yes	characteristic
280	D₄	—	316	2	no	—
281	D₄	—	158	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	316	1	no	—
285	C₂×C₄	—	316	1	no	—
286	C₂×C₄	—	316	2	no	—
287	C₂×C₄	—	316	2	no	—
288	C₂×C₄	—	316	2	no	—
289	C₂×C₄	—	316	2	no	—
290	C₂×C₄	—	158	2	no	—
291	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
292	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
293	C₂×C₄	#14579118	1	1	yes	characteristic
294	C₂×C₄	#14579134	1	1	yes	characteristic
295	C₂×C₄	#14579134	1	1	yes	characteristic
296	C₂×C₄	#14579139	1	1	yes	characteristic
297	C₂×C₄	#14579182	1	1	yes	characteristic
298	C₂×C₄	#14579184	1	1	yes	characteristic
299	C₂×C₄	#14579185	1	1	yes	characteristic
300	C₂×C₄	#14579188	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	8	1	no	—
303	C₈	—	8	1	no	—
304	C₈	—	632	1	no	—
305	C₈	—	632	1	no	—
306	C₈	—	158	1	no	—
307	C₈	—	158	1	no	—
308	C₈	—	158	1	no	—
309	C₈	—	158	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	316	2	no	—
313	C₂²	—	316	2	no	—
314	C₂²	—	316	2	no	—
315	C₂²	—	632	2	no	—
316	C₂²	—	632	2	no	—
317	C₂²	#19098422	1	1	yes	characteristic
318	C₂²	#19098862	1	1	yes	characteristic
319	C₂²	#19098866	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	316	2	no	—
324	C₄	#19098558	1	1	yes	characteristic
325	C₄	#19098566	1	1	yes	characteristic
326	C₄	#19098610	1	2	yes	—
327	C₄	#19098777	1	1	yes	characteristic
328	C₄	#19098781	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	632	2	no	—
332	C₂	#23763409	1	1	yes	characteristic
333	C₂	#23763441	1	1	yes	characteristic
334	C₂	#23763453	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579134
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1301	C₂
socle	C₂²×C₇₉	#380
prime radical	C₂²×D₃₁₆	Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1301	D₇₉
hypocenter	C₇₉	#8664
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #23763537	C₂	semidirect
3. #23762146	C₂	semidirect
6. #23761297	C₂	semidirect
7. C₇₉×#1301	C₂	semidirect
174. #1301	D₇₉	semidirect
175. C₇₉	#8664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098640	2
#23763537	2
#23762146	2
#23761826	2
#23761824	2
#23761297	2
C₇₉×#1301	2
#8664	79