Subgroups — #34443637

Order 20224.

622 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#23773827	C₂	1	1	yes	maximal, characteristic
3	#23765635	C₂	1	4	yes	maximal, complemented
4	#23765629	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765628	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765623	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765109	C₂	1	4	yes	maximal, complemented
8	C₄×#14579100	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#2240	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579180	—	2	1	no	—
11	C₂×#14579180	C₂²	1	2	yes	—
12	#19099111	—	2	1	no	—
13	C₄⋊Q₈×D₇₉	—	2	1	no	—
14	C₂×#14579100	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098967	—	2	1	no	—
17	#19098967	C₂²	1	2	yes	—
18	#19098966	—	2	1	no	—
19	#19098946	—	2	1	no	—
20	#19098946	C₂²	1	2	yes	—
21	#19098943	C₂²	1	1	yes	characteristic
22	#19098935	—	2	1	no	—
23	#107×D₇₉	—	2	1	no	—
24	#107×D₇₉	C₂²	1	2	yes	—
25	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
26	#19098368	C₂²	1	4	yes	—
27	#19098366	C₂²	1	4	yes	—
28	#19098364	C₂²	1	4	yes	—
29	#19098338	C₂²	1	2	yes	—
30	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
31	#500×C₇₉	C₂²	1	1	yes	characteristic
32	#493×C₇₉	C₂²	1	1	yes	characteristic
33	#357×C₇₉	C₂²	1	4	yes	—
34	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
35	C₂×#11929623	—	2	2	no	—
36	C₂×Q₈×D₇₉	—	4	1	no	—
37	#14579180	—	2	8	no	—
38	#14579180	—	4	4	no	—
39	C₂×Dic₃₁₆	—	2	1	no	—
40	C₂×Dic₃₁₆	C₂³	1	2	yes	—
41	C₂×#11929612	—	2	2	no	—
42	C₂×#11929612	—	2	2	no	—
43	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
44	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
45	#14579161	—	2	1	no	—
46	#14579161	—	2	1	no	—
47	#14579160	—	2	2	no	—
48	#14579160	—	4	1	no	—
49	#14579159	—	4	1	no	—
50	#14579148	—	2	2	no	—
51	#14579132	—	4	1	no	—
52	#14579132	—	4	1	no	—
53	D₄×Dic₇₉	—	4	1	no	—
54	D₄×Dic₇₉	—	4	1	no	—
55	C₄⋊C₄×D₇₉	—	2	2	no	—
56	C₄⋊C₄×D₇₉	—	4	1	no	—
57	C₄²×D₇₉	D₄	1	1	yes	characteristic
58	#14579100	—	2	4	no	—
59	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
60	#14579095	—	2	1	no	—
61	#14579095	C₂³	1	2	yes	—
62	#14579094	—	2	2	no	—
63	#14579094	—	2	2	no	—
64	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₇₉	D₄	1	1	yes	characteristic
66	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₇₉	—	2	1	no	—
68	C₂×D₈×C₇₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₇₉	—	2	1	no	—
70	#127×C₇₉	—	2	1	no	—
71	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
72	#107×C₇₉	—	2	1	no	—
73	#107×C₇₉	C₂³	1	2	yes	—
74	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	#11929623	—	4	4	no	—
76	#11929623	—	4	4	no	—
77	C₂×Dic₁₅₈	—	2	2	no	—
78	C₂×Dic₁₅₈	—	4	2	no	—
79	C₂×Dic₁₅₈	—	4	1	no	—
80	Q₈×D₇₉	—	8	4	no	—
81	C₂×C₂²⋊D₇₉	—	4	2	no	—
82	C₂²×Dic₇₉	—	4	2	no	—
83	C₂×C₄×D₇₉	—	4	2	no	—
84	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
85	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
86	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
87	Dic₃₁₆	—	2	4	no	—
88	Dic₃₁₆	—	4	2	no	—
89	#11929612	—	4	4	no	—
90	#11929612	—	4	4	no	—
91	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₈×D₇₉	SD₁₆	1	4	yes	—
94	C₈×D₇₉	#41	1	4	yes	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	C₄⋊Dic₇₉	—	4	1	no	—
97	C₄⋊Dic₇₉	—	4	1	no	—
98	C₄⋊Dic₇₉	—	4	1	no	—
99	Dic₇₉⋊C₄	—	4	2	no	—
100	Dic₇₉⋊C₄	—	4	2	no	—
101	C₂²⋊Dic₇₉	—	4	2	no	—
102	C₂²⋊Dic₇₉	—	4	2	no	—
103	C₄×Dic₇₉	D₈	1	1	yes	characteristic
104	C₄×Dic₇₉	D₈	1	1	yes	characteristic
105	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
106	C₇₉⋊C₁₆	—	2	2	no	—
107	C₂×Q₈×C₇₉	—	4	1	no	—
108	C₂×D₄×C₇₉	—	2	2	no	—
109	C₂×D₄×C₇₉	—	4	1	no	—
110	C₂×D₄×C₇₉	—	4	1	no	—
111	D₈×C₇₉	—	2	4	no	—
112	D₈×C₇₉	—	4	2	no	—
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₇₉	—	2	2	no	—
116	C₄⋊C₄×C₇₉	—	4	1	no	—
117	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₉	—	2	2	no	—
119	C₂³×C₇₉	—	4	2	no	—
120	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₇₉	—	8	2	no	—
122	D₄×C₇₉	—	4	2	no	—
123	D₄×C₇₉	—	4	2	no	—
124	D₄×C₇₉	—	8	2	no	—
125	D₄×C₇₉	—	8	2	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₇₉	—	8	4	no	—
131	C₂×Dic₇₉	—	4	2	no	—
132	C₂×Dic₇₉	—	4	2	no	—
133	C₂×Dic₇₉	—	8	4	no	—
134	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
136	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
137	C₄×D₇₉	—	2	2	no	—
138	C₄×D₇₉	—	2	2	no	—
139	C₄×D₇₉	—	8	4	no	—
140	C₄×D₇₉	#124	1	2	yes	—
141	C₄×D₇₉	C₂×SD₁₆	1	2	yes	—
142	Dic₁₅₈	—	4	2	no	—
143	Dic₁₅₈	—	4	2	no	—
144	Dic₁₅₈	—	8	4	no	—
145	Dic₁₅₈	—	8	2	no	—
146	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
147	C₈×C₇₉	C₂×#41	1	2	yes	—
148	C₇₉⋊C₈	#124	1	2	yes	—
149	C₇₉⋊C₈	C₂×SD₁₆	1	2	yes	—
150	C₂²×C₇₉	—	8	2	no	—
151	C₂²×C₇₉	—	8	2	no	—
152	C₂²×C₇₉	—	8	2	no	—
153	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₇₉	—	2	1	no	—
155	C₂×D₇₉	—	2	1	no	—
156	C₂×D₇₉	#486	1	2	yes	—
157	C₄×C₇₉	—	8	2	no	—
158	C₄×C₇₉	#509	1	2	yes	—
159	C₄×C₇₉	#510	1	2	yes	—
160	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
161	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
162	Dic₇₉	—	2	1	no	—
163	Dic₇₉	—	2	1	no	—
164	Dic₇₉	—	8	2	no	—
165	Dic₇₉	—	8	2	no	—
166	Dic₇₉	#486	1	2	yes	—
167	#33707	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	—	16	2	no	—
169	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
170	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
171	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
172	D₇₉	—	2	2	no	—
173	C₂×#500	—	79	1	no	—
174	#3147	—	79	1	no	—
175	#2243	—	79	4	no	—
176	#2241	—	79	1	no	—
177	#2241	—	79	1	no	—
178	#2240	—	79	1	no	—
179	#2240	D₇₉	1	1	yes	characteristic, complemented
180	#2139	—	79	4	no	—
181	C₄×M₃₂	—	79	1	no	—
182	C₇₉	#33707	1	1	yes	characteristic, complemented, Sylow
183	C₂×#135	—	79	2	no	—
184	C₂×#135	—	158	1	no	—
185	#532	—	158	1	no	—
186	C₂×C₄⋊Q₈	—	158	1	no	—
187	C₂×M₃₂	—	79	2	no	—
188	#500	—	79	2	no	—
189	#500	—	79	1	no	—
190	#500	—	158	1	no	—
191	#500	—	158	1	no	—
192	#500	C₂×D₇₉	1	1	yes	characteristic
193	#495	—	79	2	no	—
194	#495	—	158	1	no	—
195	#494	—	79	1	no	—
196	#493	C₂×D₇₉	1	1	yes	characteristic
197	#492	—	158	1	no	—
198	C₂×#107	—	79	2	no	—
199	C₂×#107	—	158	1	no	—
200	C₂×C₄×C₈	—	79	1	no	—
201	#358	—	79	4	no	—
202	#358	—	79	4	no	—
203	#357	—	79	4	no	—
204	#357	C₂×D₇₉	1	4	yes	—
205	#346	—	79	2	no	—
206	C₄×C₁₆	—	79	1	no	—
207	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
208	C₂×#41	—	158	2	no	—
209	C₂²×Q₈	—	316	1	no	—
210	#135	—	316	4	no	—
211	#135	—	158	8	no	—
212	C₂×Q₁₆	—	79	2	no	—
213	C₂×Q₁₆	—	158	1	no	—
214	C₂×SD₁₆	—	158	2	no	—
215	C₂×SD₁₆	—	158	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	C₂²×D₇₉	1	2	yes	—
218	C₂²×C₈	—	79	1	no	—
219	C₂²×C₈	—	79	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	316	1	no	—
222	C₄⋊Q₈	—	316	1	no	—
223	C₄⋊Q₈	—	158	2	no	—
224	C₄⋊Q₈	—	158	1	no	—
225	C₄⋊Q₈	—	158	1	no	—
226	#127	—	2	1	no	—
227	#124	—	158	2	no	—
228	#121	—	316	1	no	—
229	#121	—	316	1	no	—
230	C₄×D₄	—	316	1	no	—
231	C₄×D₄	—	316	1	no	—
232	C₂×C₄⋊C₄	—	316	1	no	—
233	C₂×C₄⋊C₄	—	158	2	no	—
234	C₂×C₄²	—	79	1	no	—
235	M₃₂	—	158	4	no	—
236	C₂×C₁₆	—	79	2	no	—
237	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
238	#107	—	2	1	no	—
239	#107	—	79	2	no	—
240	#107	—	158	2	no	—
241	#107	—	158	2	no	—
242	#107	—	158	1	no	—
243	#107	C₂²×D₇₉	1	2	yes	—
244	C₄×C₈	—	79	1	no	—
245	C₄×C₈	—	79	1	no	—
246	C₄×C₈	—	79	1	no	—
247	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
248	#41	—	316	4	no	—
249	#41	—	316	4	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	316	2	no	—
252	C₂×Q₈	—	316	1	no	—
253	C₂×Q₈	—	632	4	no	—
254	C₂×Q₈	—	158	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	316	2	no	—
259	C₂²×C₄	—	316	2	no	—
260	C₂²×C₄	—	316	2	no	—
261	C₂²×C₄	—	79	1	no	—
262	C₂²×C₄	—	79	1	no	—
263	C₂²×C₄	—	79	1	no	—
264	Q₁₆	—	316	2	no	—
265	Q₁₆	—	158	4	no	—
266	SD₁₆	—	316	4	no	—
267	SD₁₆	—	316	4	no	—
268	D₈	—	2	4	no	—
269	D₈	—	4	2	no	—
270	C₂×C₈	—	79	1	no	—
271	C₂×C₈	—	79	1	no	—
272	C₂×C₈	—	79	4	no	—
273	C₂×C₈	—	79	4	no	—
274	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	316	2	no	—
279	C₄⋊C₄	—	316	2	no	—
280	C₄⋊C₄	—	316	1	no	—
281	C₄⋊C₄	—	316	1	no	—
282	C₄⋊C₄	—	316	1	no	—
283	C₄⋊C₄	—	158	2	no	—
284	C₂²⋊C₄	—	316	2	no	—
285	C₂²⋊C₄	—	316	2	no	—
286	C₄²	—	79	1	no	—
287	C₄²	—	79	1	no	—
288	C₄²	—	79	1	no	—
289	C₄²	D₄×D₇₉	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	158	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	79	1	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	316	2	no	—
296	Q₈	—	316	2	no	—
297	Q₈	—	632	4	no	—
298	Q₈	—	632	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	632	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	316	2	no	—
306	C₂×C₄	—	316	2	no	—
307	C₂×C₄	—	79	1	no	—
308	C₂×C₄	—	79	2	no	—
309	C₂×C₄	—	79	2	no	—
310	C₂×C₄	—	79	1	no	—
311	C₂×C₄	—	79	1	no	—
312	C₂×C₄	—	632	4	no	—
313	C₂×C₄	—	632	4	no	—
314	C₂×C₄	—	158	2	no	—
315	C₂×C₄	—	158	2	no	—
316	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
317	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
319	C₈	—	79	2	no	—
320	C₈	—	79	2	no	—
321	C₈	SD₁₆×D₇₉	1	2	yes	—
322	C₈	#41×D₇₉	1	2	yes	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	79	2	no	—
327	C₂²	—	158	1	no	—
328	C₂²	—	158	1	no	—
329	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
330	C₄	—	8	2	no	—
331	C₄	—	79	2	no	—
332	C₄	—	632	2	no	—
333	C₄	—	632	2	no	—
334	C₄	—	158	1	no	—
335	C₄	—	158	1	no	—
336	C₄	#19098997	1	2	yes	—
337	C₄	#19099002	1	2	yes	—
338	C₄	#124×D₇₉	1	1	yes	characteristic
339	C₄	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
340	C₂	—	16	2	no	—
341	C₂	—	158	2	no	—
342	C₂	#486×D₇₉	1	1	yes	characteristic
343	C₂	C₂×#19098997	1	1	yes	characteristic
344	C₂	C₂×#19099002	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2240	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23773827	C₂
hypercenter	#2240	D₇₉
hypocenter	C₇₉	#33707
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #500×D₇₉	C₂	semidirect
3. #23765635	C₂	semidirect
4. #23765629	C₂	semidirect
5. #23765628	C₂	semidirect
6. #23765623	C₂	semidirect
7. #23765109	C₂	semidirect
8. C₄×#14579100	C₂	semidirect
9. C₇₉×#2240	C₂	semidirect
179. #2240	D₇₉	semidirect
182. C₇₉	#33707	semidirect

Maximal subgroups

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

subgroup	index
#500×D₇₉	2
#23773827	2
#23765635	2
#23765629	2
#23765628	2
#23765623	2
#23765109	2
C₄×#14579100	2
C₇₉×#2240	2
#33707	79