Subgroups — #34445926

Order 20224.

486 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776386	C₂	1	1	yes	maximal, characteristic, complemented
2	#23776265	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765630	C₂	1	2	yes	maximal, complemented
4	#23765628	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765563	C₂	1	2	yes	maximal, complemented
6	#23765547	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765453	C₂	1	2	yes	maximal
8	#23765445	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₆×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765255	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
12	#19099283	—	2	1	no	—
13	#19099270	—	2	1	no	—
14	C₂×#14579105	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098964	C₂²	1	2	yes	—
17	#19098946	C₂²	1	2	yes	—
18	#19098943	C₂²	1	1	yes	characteristic
19	#19098861	—	2	2	no	—
20	#19098853	—	2	1	no	—
21	#19098784	—	2	2	no	—
22	#19098768	—	2	1	no	—
23	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
24	#19098657	C₂²	1	2	yes	—
25	D₈×Dic₇₉	C₂²	1	2	yes	—
26	#19098390	C₂²	1	2	yes	—
27	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
28	Dic₇₉⋊C₁₆	C₂²	1	2	yes	—
29	#19098368	C₂²	1	2	yes	—
30	#19098368	C₂²	1	2	yes	—
31	#19098366	C₂²	1	2	yes	—
32	#19098364	C₂²	1	2	yes	—
33	C₁₆×Dic₇₉	C₂²	1	2	yes	—
34	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
37	#366×C₇₉	C₂²	1	1	yes	characteristic
38	#357×C₇₉	C₂²	1	1	yes	characteristic
39	#357×C₇₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₁₆	—	2	1	no	—
42	C₂×Dic₃₁₆	C₂³	1	2	yes	—
43	C₂×#11929612	—	2	2	no	—
44	#14579161	—	2	1	no	—
45	#14579161	—	2	1	no	—
46	#14579154	—	4	2	no	—
47	#14579152	—	2	2	no	—
48	#14579148	—	2	2	no	—
49	#14579140	—	4	1	no	—
50	#14579140	—	4	1	no	—
51	#14579138	—	4	2	no	—
52	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
53	#14579126	—	2	2	no	—
54	D₄×Dic₇₉	—	2	2	no	—
55	#14579120	—	4	1	no	—
56	C₂×C₄⋊Dic₇₉	—	4	1	no	—
57	#14579105	—	2	4	no	—
58	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
59	#14579095	—	2	1	no	—
60	#14579095	C₂³	1	2	yes	—
61	#14579092	—	2	2	no	—
62	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
63	Dic₇₉⋊C₈	D₄	1	2	yes	—
64	#14579084	—	2	2	no	—
65	#14579082	—	2	2	no	—
66	C₈×Dic₇₉	C₂³	1	2	yes	—
67	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₇₉	—	2	1	no	—
70	C₄×D₄×C₇₉	—	2	1	no	—
71	D₁₆×C₇₉	—	2	2	no	—
72	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	#107×C₇₉	C₂³	1	1	yes	characteristic
75	#107×C₇₉	C₂³	1	1	yes	characteristic
76	#102×C₇₉	—	2	1	no	—
77	#102×C₇₉	—	2	1	no	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₅₈	—	2	2	no	—
80	C₂×Dic₁₅₈	—	4	1	no	—
81	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
82	C₂²×Dic₇₉	—	4	2	no	—
83	Dic₃₁₆	—	2	2	no	—
84	Dic₃₁₆	—	2	2	no	—
85	Dic₃₁₆	—	8	1	no	—
86	#11929612	—	8	2	no	—
87	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
88	C₄⋊Dic₇₉	—	2	2	no	—
89	C₄⋊Dic₇₉	—	4	2	no	—
90	C₄⋊Dic₇₉	—	4	1	no	—
91	C₄⋊Dic₇₉	—	4	1	no	—
92	C₄⋊Dic₇₉	—	4	1	no	—
93	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
94	Dic₇₉⋊C₄	—	4	2	no	—
95	Dic₇₉⋊C₄	—	4	2	no	—
96	Dic₇₉⋊C₄	D₈	1	2	yes	—
97	C₂²⋊Dic₇₉	—	4	2	no	—
98	C₂²⋊Dic₇₉	—	4	2	no	—
99	C₂²⋊Dic₇₉	—	4	2	no	—
100	C₄×Dic₇₉	—	4	2	no	—
101	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
102	C₇₉⋊C₁₆	—	2	2	no	—
103	C₂×D₄×C₇₉	—	2	1	no	—
104	C₂×D₄×C₇₉	—	2	1	no	—
105	C₂²×C₄×C₇₉	—	4	1	no	—
106	C₂²×C₄×C₇₉	—	4	1	no	—
107	D₈×C₇₉	—	2	1	no	—
108	D₈×C₇₉	—	2	1	no	—
109	D₈×C₇₉	—	2	1	no	—
110	D₈×C₇₉	#41	1	2	yes	—
111	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₉	—	2	1	no	—
114	C₄⋊C₄×C₇₉	—	2	1	no	—
115	C₂²⋊C₄×C₇₉	—	4	1	no	—
116	C₂²⋊C₄×C₇₉	—	4	1	no	—
117	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₉	—	2	1	no	—
119	C₁₆×C₇₉	—	2	1	no	—
120	C₂³×C₇₉	—	4	1	no	—
121	C₂³×C₇₉	—	4	1	no	—
122	D₄×C₇₉	—	2	2	no	—
123	D₄×C₇₉	—	4	1	no	—
124	D₄×C₇₉	—	4	1	no	—
125	D₄×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	1	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	8	2	no	—
129	C₂×C₄×C₇₉	—	8	2	no	—
130	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₇₉	—	4	2	no	—
134	C₂×Dic₇₉	—	4	2	no	—
135	C₂×Dic₇₉	—	16	2	no	—
136	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
137	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
138	Dic₁₅₈	—	2	2	no	—
139	Dic₁₅₈	—	4	2	no	—
140	Dic₁₅₈	—	4	2	no	—
141	Dic₁₅₈	—	16	1	no	—
142	C₈×C₇₉	—	2	1	no	—
143	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₇₉⋊C₈	—	2	2	no	—
146	C₂²×C₇₉	—	4	2	no	—
147	C₂²×C₇₉	—	8	1	no	—
148	C₂²×C₇₉	—	8	1	no	—
149	C₂²×C₇₉	—	8	1	no	—
150	C₂²×C₇₉	—	8	1	no	—
151	C₂²×C₇₉	—	8	1	no	—
152	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	8	1	no	—
155	C₄×C₇₉	—	8	1	no	—
156	C₄×C₇₉	#508	1	2	yes	—
157	C₄×C₇₉	#548	1	1	yes	characteristic
158	C₄×C₇₉	#577	1	1	yes	characteristic
159	Dic₇₉	—	2	2	no	—
160	Dic₇₉	—	2	2	no	—
161	Dic₇₉	—	8	2	no	—
162	Dic₇₉	—	8	2	no	—
163	#33927	—	79	1	no	maximal, Sylow
164	C₂×C₇₉	—	8	2	no	—
165	C₂×C₇₉	—	16	1	no	—
166	C₂×C₇₉	#3358	1	1	yes	characteristic
167	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
168	C₂×C₇₉	#3416	1	1	yes	characteristic
169	#3393	—	79	1	no	—
170	#3380	—	79	1	no	—
171	#2242	—	79	2	no	—
172	#2241	—	79	1	no	—
173	#2229	—	79	2	no	—
174	#2226	—	79	1	no	—
175	#2210	—	79	2	no	—
176	#2208	—	79	1	no	—
177	Q₈×C₁₆	—	79	1	no	—
178	C₄×SD₃₂	—	79	2	no	—
179	C₄×D₁₆	D₇₉	1	1	yes	characteristic
180	C₇₉	#33927	1	1	yes	characteristic, complemented, Sylow
181	#556	—	158	1	no	—
182	#554	—	158	1	no	—
183	C₂×SD₃₂	—	79	2	no	—
184	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
185	#500	—	79	1	no	—
186	#499	—	79	2	no	—
187	#495	—	79	2	no	—
188	#494	—	79	1	no	—
189	#478	—	158	2	no	—
190	#476	—	158	1	no	—
191	#464	—	158	2	no	—
192	#461	—	158	1	no	—
193	C₈×Q₈	—	79	1	no	—
194	C₄×Q₁₆	—	79	2	no	—
195	C₄×D₈	—	79	2	no	—
196	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
197	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
198	#367	—	79	2	no	—
199	#366	C₂×D₇₉	1	1	yes	characteristic
200	C₄⋊C₁₆	—	79	2	no	—
201	C₄⋊C₁₆	—	79	1	no	—
202	#358	—	79	2	no	—
203	#358	—	79	2	no	—
204	#358	—	79	2	no	—
205	#357	—	79	2	no	—
206	#357	C₂×D₇₉	1	1	yes	characteristic
207	#357	C₂×D₇₉	1	1	yes	characteristic
208	C₄×C₁₆	—	79	2	no	—
209	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
210	C₂×Q₁₆	—	79	2	no	—
211	C₂×Q₁₆	—	158	1	no	—
212	C₂×SD₁₆	—	158	2	no	—
213	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
214	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	158	1	no	—
216	C₄⋊Q₈	—	158	1	no	—
217	#126	—	316	2	no	—
218	#125	—	158	2	no	—
219	#124	—	158	2	no	—
220	C₂²⋊Q₈	—	316	2	no	—
221	C₂²⋊Q₈	—	316	1	no	—
222	C₂²⋊Q₈	—	316	1	no	—
223	C₄×Q₈	—	79	1	no	—
224	C₄×Q₈	—	158	2	no	—
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	—	158	2	no	—
228	#117	—	316	1	no	—
229	C₂×C₄⋊C₄	—	316	1	no	—
230	SD₃₂	—	158	4	no	—
231	D₁₆	—	2	2	no	—
232	C₂×C₁₆	—	79	2	no	—
233	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
235	#107	—	79	2	no	—
236	#107	—	158	1	no	—
237	#107	C₂²×D₇₉	1	1	yes	characteristic
238	#107	C₂²×D₇₉	1	1	yes	characteristic
239	#106	—	158	2	no	—
240	C₄⋊C₈	—	79	2	no	—
241	C₄⋊C₈	—	79	1	no	—
242	#103	—	158	2	no	—
243	#102	—	2	1	no	—
244	#102	—	2	1	no	—
245	#102	—	158	2	no	—
246	C₄×C₈	—	79	2	no	—
247	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
248	C₂×Q₈	—	316	1	no	—
249	C₂×Q₈	—	79	1	no	—
250	C₂×Q₈	—	158	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	316	2	no	—
256	Q₁₆	—	632	1	no	—
257	Q₁₆	—	158	2	no	—
258	Q₁₆	—	158	2	no	—
259	SD₁₆	—	632	2	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	D₈	—	2	1	no	—
263	D₈	#11929624	1	2	yes	—
264	C₂×C₈	—	79	2	no	—
265	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	316	2	no	—
270	C₄⋊C₄	—	316	2	no	—
271	C₄⋊C₄	—	316	2	no	—
272	C₄⋊C₄	—	316	1	no	—
273	C₄⋊C₄	—	316	1	no	—
274	C₄⋊C₄	—	316	1	no	—
275	C₄⋊C₄	—	79	2	no	—
276	C₄⋊C₄	—	79	1	no	—
277	C₄⋊C₄	—	158	2	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	316	2	no	—
281	C₂²⋊C₄	—	316	2	no	—
282	C₂²⋊C₄	—	316	2	no	—
283	C₄²	—	316	2	no	—
284	C₄²	—	79	2	no	—
285	C₄²	D₄×D₇₉	1	1	yes	characteristic
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	158	2	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	Q₈	—	316	2	no	—
292	Q₈	—	316	2	no	—
293	Q₈	—	158	2	no	—
294	Q₈	—	1264	1	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	316	2	no	—
304	C₂×C₄	—	316	2	no	—
305	C₂×C₄	—	79	2	no	—
306	C₂×C₄	—	79	2	no	—
307	C₂×C₄	—	1264	2	no	—
308	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
309	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	158	2	no	—
313	C₈	C₂×#11929624	1	1	yes	characteristic
314	C₈	#14579208	1	1	yes	characteristic
315	C₂²	—	4	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	632	2	no	—
326	C₄	—	632	2	no	—
327	C₄	—	158	2	no	—
328	C₄	—	158	2	no	—
329	C₄	#19098992	1	2	yes	—
330	C₄	#19099222	1	1	yes	characteristic
331	C₄	#19099370	1	1	yes	characteristic
332	C₂	—	8	2	no	—
333	C₂	—	16	1	no	—
334	C₂	#23776033	1	1	yes	characteristic
335	C₂	C₂×#19098992	1	1	yes	characteristic
336	C₂	#23776543	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₄×D₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₁₆×C₇₉	C₂²
hypercenter	C₄×D₁₆	D₇₉
hypocenter	C₇₉	#33927
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #23776386	C₂	semidirect
2. #23776265	C₂	semidirect
3. #23765630	C₂	semidirect
4. #23765628	C₂	semidirect
5. #23765563	C₂	semidirect
6. #23765547	C₂	semidirect
8. #23765445	C₂	semidirect
9. C₁₆×Dic₁₅₈	C₂	semidirect
10. #23765255	C₂	semidirect
180. C₇₉	#33927	semidirect

Maximal subgroups

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

subgroup	index
#23776386	2
#23776265	2
#23765630	2
#23765628	2
#23765563	2
#23765547	2
#23765453	2
#23765445	2
C₁₆×Dic₁₅₈	2
#23765255	2
C₄×D₁₆×C₇₉	2
#33927	79