Subgroups — #34443566

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773911	C₂	1	1	yes	maximal, characteristic, complemented
2	#23773827	C₂	1	1	yes	maximal, characteristic
3	#23765630	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	#23765104	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₇₉	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	#19099118	—	2	1	no	—
13	#19099111	—	2	1	no	—
14	C₂×C₁₆×D₇₉	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098964	—	2	1	no	—
17	#19098964	C₂²	1	2	yes	—
18	#19098959	—	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	#19098600	—	2	1	no	—
24	#19098600	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	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₇₉	C₂²	1	2	yes	—
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	#14579180	—	2	8	no	—
37	#14579180	—	4	4	no	—
38	C₂×Dic₃₁₆	—	2	1	no	—
39	C₂×Dic₃₁₆	C₂³	1	2	yes	—
40	C₂×#11929612	—	2	2	no	—
41	C₂×#11929612	—	2	2	no	—
42	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
43	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
44	#14579161	—	2	1	no	—
45	#14579161	—	2	1	no	—
46	#14579152	—	2	2	no	—
47	#14579148	—	2	2	no	—
48	#14579141	—	4	1	no	—
49	#14579137	—	4	1	no	—
50	#14579132	—	4	1	no	—
51	#14579132	—	4	1	no	—
52	Q₈×Dic₇₉	—	4	1	no	—
53	#14579126	—	4	1	no	—
54	D₄×Dic₇₉	—	4	1	no	—
55	D₄×Dic₇₉	—	4	1	no	—
56	#14579119	—	2	2	no	—
57	C₄²×D₇₉	D₄	1	1	yes	characteristic
58	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
59	C₁₆×D₇₉	—	2	4	no	—
60	#14579095	—	2	1	no	—
61	#14579095	C₂³	1	2	yes	—
62	#14579092	—	2	2	no	—
63	#14579092	—	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	1	no	—
79	C₂×C₂²⋊D₇₉	—	4	2	no	—
80	C₂²×Dic₇₉	—	4	2	no	—
81	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
82	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
83	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
84	Dic₃₁₆	—	2	4	no	—
85	Dic₃₁₆	—	4	2	no	—
86	#11929612	—	4	4	no	—
87	#11929612	—	4	4	no	—
88	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₈×D₇₉	SD₁₆	1	4	yes	—
91	C₈×D₇₉	#41	1	4	yes	—
92	C₄⋊Dic₇₉	—	2	2	no	—
93	C₄⋊Dic₇₉	—	4	2	no	—
94	C₄⋊Dic₇₉	—	4	1	no	—
95	C₄⋊Dic₇₉	—	4	1	no	—
96	C₄⋊Dic₇₉	—	4	1	no	—
97	Dic₇₉⋊C₄	—	4	2	no	—
98	C₂²⋊Dic₇₉	—	4	2	no	—
99	C₂²⋊Dic₇₉	—	4	2	no	—
100	(C₂×D₇₉)⋊C₄	—	4	2	no	—
101	C₄×Dic₇₉	—	4	2	no	—
102	C₄×Dic₇₉	D₈	1	1	yes	characteristic
103	C₄×Dic₇₉	D₈	1	1	yes	characteristic
104	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
105	C₇₉⋊C₁₆	—	2	2	no	—
106	C₂×Q₈×C₇₉	—	4	1	no	—
107	C₂×D₄×C₇₉	—	2	2	no	—
108	C₂×D₄×C₇₉	—	4	1	no	—
109	C₂×D₄×C₇₉	—	4	1	no	—
110	D₈×C₇₉	—	2	4	no	—
111	D₈×C₇₉	—	4	2	no	—
112	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₉	—	2	2	no	—
115	C₄⋊C₄×C₇₉	—	4	1	no	—
116	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₉	—	2	2	no	—
118	C₂³×C₇₉	—	4	2	no	—
119	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₇₉	—	8	2	no	—
121	D₄×C₇₉	—	4	2	no	—
122	D₄×C₇₉	—	4	2	no	—
123	D₄×C₇₉	—	8	2	no	—
124	D₄×C₇₉	—	8	2	no	—
125	C₂×C₄×C₇₉	—	4	2	no	—
126	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₇₉	—	8	4	no	—
130	C₂×Dic₇₉	—	4	2	no	—
131	C₂×Dic₇₉	—	4	2	no	—
132	C₂×Dic₇₉	—	8	4	no	—
133	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
136	C₄×D₇₉	—	2	2	no	—
137	C₄×D₇₉	—	2	2	no	—
138	C₄×D₇₉	#124	1	2	yes	—
139	C₄×D₇₉	C₂×SD₁₆	1	2	yes	—
140	Dic₁₅₈	—	4	2	no	—
141	Dic₁₅₈	—	4	2	no	—
142	Dic₁₅₈	—	8	2	no	—
143	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
144	C₈×C₇₉	C₂×#41	1	2	yes	—
145	C₇₉⋊C₈	#124	1	2	yes	—
146	C₇₉⋊C₈	C₂×SD₁₆	1	2	yes	—
147	C₂²×C₇₉	—	8	2	no	—
148	C₂²×C₇₉	—	8	2	no	—
149	C₂²×C₇₉	—	8	2	no	—
150	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₇₉	—	2	1	no	—
152	C₂×D₇₉	—	2	1	no	—
153	C₂×D₇₉	#486	1	2	yes	—
154	C₄×C₇₉	—	8	2	no	—
155	C₄×C₇₉	#508	1	2	yes	—
156	C₄×C₇₉	#508	1	2	yes	—
157	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
158	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₇₉	—	2	1	no	—
160	Dic₇₉	—	2	1	no	—
161	Dic₇₉	—	8	2	no	—
162	Dic₇₉	—	8	2	no	—
163	Dic₇₉	#486	1	2	yes	—
164	#33697	—	79	1	no	maximal, Sylow
165	C₂×C₇₉	—	16	2	no	—
166	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
167	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
168	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
169	D₇₉	—	2	2	no	—
170	#3160	—	79	1	no	—
171	#3147	—	79	1	no	—
172	#2242	—	79	4	no	—
173	#2241	—	79	1	no	—
174	#2241	—	79	1	no	—
175	#2240	—	79	1	no	—
176	#2240	D₇₉	1	1	yes	characteristic, complemented
177	#2138	—	79	4	no	—
178	C₂×C₄×C₁₆	—	79	1	no	—
179	C₇₉	#33697	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	—	79	2	no	—
181	C₂×#135	—	158	1	no	—
182	#533	—	158	1	no	—
183	#532	—	158	1	no	—
184	C₂²×C₁₆	—	79	2	no	—
185	#500	—	79	1	no	—
186	#500	C₂×D₇₉	1	1	yes	characteristic
187	#499	—	79	2	no	—
188	#499	—	158	1	no	—
189	#498	—	158	1	no	—
190	#495	—	79	2	no	—
191	#495	—	158	1	no	—
192	#494	—	79	1	no	—
193	#493	C₂×D₇₉	1	1	yes	characteristic
194	#492	—	158	1	no	—
195	#427	—	79	2	no	—
196	#427	—	158	1	no	—
197	C₂×C₄×C₈	—	79	1	no	—
198	#358	—	79	4	no	—
199	#358	—	79	4	no	—
200	#357	—	79	4	no	—
201	#357	C₂×D₇₉	1	4	yes	—
202	C₄×C₁₆	—	79	2	no	—
203	C₄×C₁₆	—	79	1	no	—
204	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
205	C₂×#41	—	158	2	no	—
206	#135	—	316	4	no	—
207	#135	—	158	8	no	—
208	C₂×Q₁₆	—	79	2	no	—
209	C₂×Q₁₆	—	158	1	no	—
210	C₂×SD₁₆	—	158	2	no	—
211	C₂×SD₁₆	—	158	2	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	C₂²×D₇₉	1	2	yes	—
214	C₂²×C₈	—	79	1	no	—
215	C₂²×C₈	—	79	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	158	1	no	—
218	C₄⋊Q₈	—	158	1	no	—
219	#127	—	2	1	no	—
220	#125	—	158	2	no	—
221	#124	—	158	2	no	—
222	C₂²⋊Q₈	—	316	1	no	—
223	C₂²⋊Q₈	—	316	1	no	—
224	#121	—	316	1	no	—
225	#121	—	316	1	no	—
226	C₄×Q₈	—	316	1	no	—
227	C₄×Q₈	—	316	1	no	—
228	C₄×D₄	—	316	1	no	—
229	C₄×D₄	—	316	1	no	—
230	#117	—	158	2	no	—
231	C₂×C₄²	—	79	1	no	—
232	C₂×C₁₆	—	79	2	no	—
233	C₂×C₁₆	—	158	4	no	—
234	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
235	#107	—	2	1	no	—
236	#107	—	79	2	no	—
237	#107	—	158	1	no	—
238	#107	C₂²×D₇₉	1	2	yes	—
239	#106	—	158	2	no	—
240	#106	—	158	2	no	—
241	C₄×C₈	—	79	1	no	—
242	C₄×C₈	—	79	1	no	—
243	C₄×C₈	—	79	1	no	—
244	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
245	#41	—	316	4	no	—
246	#41	—	316	4	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	316	1	no	—
249	C₂×Q₈	—	158	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	316	2	no	—
254	C₂²×C₄	—	316	2	no	—
255	C₂²×C₄	—	79	1	no	—
256	C₂²×C₄	—	79	1	no	—
257	C₂²×C₄	—	79	1	no	—
258	Q₁₆	—	316	2	no	—
259	Q₁₆	—	158	4	no	—
260	SD₁₆	—	316	4	no	—
261	SD₁₆	—	316	4	no	—
262	D₈	—	2	4	no	—
263	D₈	—	4	2	no	—
264	C₂×C₈	—	79	1	no	—
265	C₂×C₈	—	79	1	no	—
266	C₂×C₈	—	79	4	no	—
267	C₂×C₈	—	79	4	no	—
268	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	316	2	no	—
273	C₄⋊C₄	—	316	2	no	—
274	C₄⋊C₄	—	316	1	no	—
275	C₄⋊C₄	—	316	1	no	—
276	C₄⋊C₄	—	316	1	no	—
277	C₄⋊C₄	—	158	2	no	—
278	C₂²⋊C₄	—	316	2	no	—
279	C₂²⋊C₄	—	316	2	no	—
280	C₂²⋊C₄	—	316	2	no	—
281	C₄²	—	316	2	no	—
282	C₄²	—	79	1	no	—
283	C₄²	—	79	1	no	—
284	C₄²	—	79	1	no	—
285	C₄²	D₄×D₇₉	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	158	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	79	1	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	316	2	no	—
292	Q₈	—	316	2	no	—
293	Q₈	—	632	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	632	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	316	2	no	—
301	C₂×C₄	—	316	2	no	—
302	C₂×C₄	—	79	1	no	—
303	C₂×C₄	—	79	2	no	—
304	C₂×C₄	—	79	2	no	—
305	C₂×C₄	—	79	1	no	—
306	C₂×C₄	—	79	1	no	—
307	C₂×C₄	—	632	4	no	—
308	C₂×C₄	—	158	2	no	—
309	C₂×C₄	—	158	2	no	—
310	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
311	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
313	C₈	—	79	2	no	—
314	C₈	—	79	2	no	—
315	C₈	SD₁₆×D₇₉	1	2	yes	—
316	C₈	#41×D₇₉	1	2	yes	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	79	2	no	—
321	C₂²	—	158	1	no	—
322	C₂²	—	158	1	no	—
323	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	79	2	no	—
326	C₄	—	632	2	no	—
327	C₄	—	632	2	no	—
328	C₄	—	158	1	no	—
329	C₄	—	158	1	no	—
330	C₄	#19098992	1	2	yes	—
331	C₄	#19098994	1	2	yes	—
332	C₄	#124×D₇₉	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
334	C₂	—	16	2	no	—
335	C₂	—	158	2	no	—
336	C₂	#486×D₇₉	1	1	yes	characteristic
337	C₂	C₂×#19098992	1	1	yes	characteristic
338	C₂	C₂×#19098994	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	#33697
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. #23773911	C₂	semidirect
3. #23765630	C₂	semidirect
4. #23765629	C₂	semidirect
5. #23765628	C₂	semidirect
6. #23765623	C₂	semidirect
7. #23765104	C₂	semidirect
8. C₄×C₁₆×D₇₉	C₂	semidirect
9. C₇₉×#2240	C₂	semidirect
176. #2240	D₇₉	semidirect
179. C₇₉	#33697	semidirect

Maximal subgroups

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

subgroup	index
#23773911	2
#23773827	2
#23765630	2
#23765629	2
#23765628	2
#23765623	2
#23765104	2
C₄×C₁₆×D₇₉	2
C₇₉×#2240	2
#33697	79