Subgroups — #34204310

Order 20224.

510 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098364	C₂	1	2	yes	maximal
3	#23763560	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761491	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14579095	—	2	1	no	—
8	C₂×#14579095	C₂²	1	2	yes	—
9	#19098433	—	2	1	no	—
10	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
11	#19098364	—	2	4	no	—
12	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₇₉	—	2	1	no	—
14	C₄×D₈×C₇₉	—	2	2	no	—
15	C₄×D₈×C₇₉	C₄	1	4	yes	complemented
16	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#102×C₇₉	—	2	1	no	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇₉	—	2	2	no	—
20	C₂×C₄⋊Dic₇₉	—	4	1	no	—
21	C₂×C₄⋊Dic₇₉	—	4	1	no	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	C₂×C₇₉⋊C₁₆	—	2	2	no	—
25	#14579095	—	2	2	no	—
26	#14579095	—	2	2	no	—
27	#14579095	—	4	2	no	—
28	C₂²×D₄×C₇₉	—	2	1	no	—
29	C₂³×C₄×C₇₉	—	4	1	no	—
30	C₂×D₈×C₇₉	—	2	1	no	—
31	C₂×D₈×C₇₉	—	2	1	no	—
32	C₂×D₈×C₇₉	—	2	1	no	—
33	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×D₈×C₇₉	D₄	1	2	yes	—
35	C₂×D₈×C₇₉	Q₈	1	2	yes	—
36	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₇₉	—	2	4	no	—
39	C₄×D₄×C₇₉	—	4	2	no	—
40	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	#107×C₇₉	—	2	1	no	—
44	#107×C₇₉	C₂×C₄	1	2	yes	—
45	#102×C₇₉	—	2	4	no	—
46	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₇₉	D₄	1	2	yes	—
48	C₂²×Dic₇₉	—	4	2	no	—
49	C₄⋊Dic₇₉	—	4	2	no	—
50	C₄⋊Dic₇₉	—	4	2	no	—
51	C₄⋊Dic₇₉	—	8	2	no	—
52	C₄⋊Dic₇₉	—	8	2	no	—
53	C₇₉⋊C₁₆	—	4	2	no	—
54	C₂⁴×C₇₉	—	4	1	no	—
55	C₂×D₄×C₇₉	—	2	2	no	—
56	C₂×D₄×C₇₉	—	2	2	no	—
57	C₂×D₄×C₇₉	—	2	2	no	—
58	C₂×D₄×C₇₉	—	4	1	no	—
59	C₂×D₄×C₇₉	—	4	1	no	—
60	C₂×D₄×C₇₉	—	4	1	no	—
61	C₂²×C₄×C₇₉	—	4	1	no	—
62	C₂²×C₄×C₇₉	—	4	4	no	—
63	C₂²×C₄×C₇₉	—	8	2	no	—
64	C₂²×C₄×C₇₉	—	8	2	no	—
65	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	D₈×C₇₉	—	2	2	no	—
69	D₈×C₇₉	—	2	2	no	—
70	D₈×C₇₉	—	2	2	no	—
71	D₈×C₇₉	C₄⋊C₄	1	4	yes	complemented
72	C₂×C₈×C₇₉	—	2	1	no	—
73	C₂×C₈×C₇₉	—	2	1	no	—
74	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
81	C₄⋊C₄×C₇₉	—	2	2	no	—
82	C₄⋊C₄×C₇₉	—	4	1	no	—
83	C₂²⋊C₄×C₇₉	—	4	4	no	—
84	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₇₉	SD₁₆	1	2	yes	—
86	C₂³×C₇₉	—	4	2	no	—
87	C₂³×C₇₉	—	4	2	no	—
88	C₂³×C₇₉	—	4	2	no	—
89	C₂³×C₇₉	—	8	1	no	—
90	C₂³×C₇₉	—	8	1	no	—
91	C₂³×C₇₉	—	8	1	no	—
92	C₂³×C₇₉	—	8	1	no	—
93	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
94	D₄×C₇₉	—	2	4	no	—
95	D₄×C₇₉	—	4	2	no	—
96	D₄×C₇₉	—	4	2	no	—
97	D₄×C₇₉	—	4	2	no	—
98	C₂×C₄×C₇₉	—	2	1	no	—
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	4	2	no	—
101	C₂×C₄×C₇₉	—	8	1	no	—
102	C₂×C₄×C₇₉	—	8	1	no	—
103	C₂×C₄×C₇₉	—	8	8	no	—
104	C₂×C₄×C₇₉	#102	1	2	yes	—
105	C₂×C₄×C₇₉	#102	1	2	yes	—
106	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
107	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
108	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
114	C₂×Dic₇₉	—	8	2	no	—
115	C₂×Dic₇₉	—	8	2	no	—
116	C₂×Dic₇₉	—	8	2	no	—
117	C₈×C₇₉	—	2	2	no	—
118	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₇₉	#117	1	1	yes	characteristic
120	C₈×C₇₉	#121	1	1	yes	characteristic
121	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₇₉	—	4	4	no	—
123	C₂²×C₇₉	—	8	2	no	—
124	C₂²×C₇₉	—	8	2	no	—
125	C₂²×C₇₉	—	8	2	no	—
126	C₂²×C₇₉	—	8	2	no	—
127	C₂²×C₇₉	—	8	2	no	—
128	C₂²×C₇₉	—	8	2	no	—
129	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
130	C₂²×C₇₉	#459	1	1	yes	characteristic
131	C₂²×C₇₉	#474	1	1	yes	characteristic
132	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
133	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
134	C₂²×C₇₉	#509	1	1	yes	characteristic
135	C₂²×C₇₉	#510	1	1	yes	characteristic
136	C₄×C₇₉	—	2	2	no	—
137	C₄×C₇₉	—	8	2	no	—
138	C₄×C₇₉	#357	1	4	yes	complemented
139	C₄×C₇₉	#380	1	1	yes	characteristic
140	C₄×C₇₉	#417	1	1	yes	characteristic
141	C₄×C₇₉	#461	1	1	yes	characteristic
142	C₄×C₇₉	#476	1	1	yes	characteristic
143	Dic₇₉	—	16	2	no	—
144	#12811	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	—	8	4	no	—
146	C₂×C₇₉	#1802	1	1	yes	characteristic
147	C₂×C₇₉	C₂×#357	1	1	yes	characteristic
148	C₂×C₇₉	#2139	1	1	yes	characteristic
149	C₂×C₇₉	#2206	1	1	yes	characteristic
150	C₂×C₇₉	#2208	1	1	yes	characteristic
151	C₂×C₇₉	#2224	1	1	yes	characteristic
152	C₂×C₇₉	#2226	1	1	yes	characteristic
153	C₂×C₄×D₈	D₇₉	1	1	yes	characteristic
154	C₂×C₄⋊C₁₆	—	79	1	no	—
155	C₂×#357	—	79	2	no	—
156	#1845	—	79	1	no	—
157	#1380	—	79	2	no	—
158	C₇₉	#12811	1	1	yes	characteristic, complemented, Sylow
159	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
160	C₂×C₄×D₄	—	2	1	no	—
161	C₂²×C₁₆	—	79	2	no	—
162	C₄×D₈	—	2	2	no	—
163	C₄×D₈	Dic₇₉	1	4	yes	complemented
164	C₂×#107	—	79	2	no	—
165	C₂×#107	—	158	1	no	—
166	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
167	C₂×#102	—	2	1	no	—
168	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
169	#384	—	158	1	no	—
170	C₄⋊C₁₆	—	158	2	no	—
171	#357	—	158	4	no	—
172	C₂²×D₄	—	2	1	no	—
173	C₂³×C₄	—	4	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	Dic₁₅₈	1	2	yes	—
178	C₂×D₈	D₃₁₆	1	2	yes	—
179	C₂×D₈	C₂×Dic₇₉	1	2	yes	—
180	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	—	4	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	316	1	no	—
186	C₂×C₄⋊C₄	—	316	1	no	—
187	C₂×C₄⋊C₄	—	158	2	no	—
188	C₂×C₂²⋊C₄	—	4	1	no	—
189	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
190	C₂×C₁₆	—	158	2	no	—
191	C₂×C₁₆	—	158	2	no	—
192	C₂×C₁₆	—	158	2	no	—
193	#107	—	2	1	no	—
194	#107	—	316	2	no	—
195	#107	—	158	2	no	—
196	#107	—	158	2	no	—
197	#107	C₂×Dic₇₉	1	2	yes	—
198	#102	—	2	4	no	—
199	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
200	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
201	C₂⁴	—	4	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	8	2	no	—
211	C₂²×C₄	—	8	2	no	—
212	C₂²×C₄	—	316	2	no	—
213	C₂²×C₄	#11929609	1	1	yes	characteristic
214	C₂²×C₄	#11929609	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
216	D₈	—	2	2	no	—
217	D₈	—	2	2	no	—
218	D₈	—	2	2	no	—
219	D₈	C₄⋊Dic₇₉	1	4	yes	complemented
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
223	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
224	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
225	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
226	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
227	C₂×C₈	#11929624	1	1	yes	characteristic
228	C₂×C₈	#11929624	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	316	2	no	—
232	C₄⋊C₄	—	316	2	no	—
233	C₄⋊C₄	—	632	2	no	—
234	C₄⋊C₄	—	632	2	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
237	C₄²	#11929612	1	2	yes	—
238	C₁₆	—	316	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	—	8	1	no	—
246	C₂³	C₂×#11929609	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	632	2	no	—
258	C₂×C₄	—	632	2	no	—
259	C₂×C₄	—	632	2	no	—
260	C₂×C₄	#14579082	1	2	yes	—
261	C₂×C₄	#14579082	1	2	yes	—
262	C₂×C₄	#14579102	1	2	yes	—
263	C₂×C₄	#14579105	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
265	C₂×C₄	#14579135	1	1	yes	characteristic
266	C₂×C₄	#14579140	1	1	yes	characteristic
267	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
268	C₂×C₄	#14579186	1	1	yes	characteristic
269	C₂×C₄	#14579191	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
272	C₈	#14579120	1	1	yes	characteristic
273	C₈	#14579135	1	1	yes	characteristic
274	C₈	#14579140	1	1	yes	characteristic
275	C₂²	—	4	4	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₂×#14579082	1	1	yes	characteristic
283	C₂²	#19098758	1	1	yes	characteristic
284	C₂²	#19098843	1	1	yes	characteristic
285	C₂²	C₂×#14579102	1	1	yes	characteristic
286	C₂²	C₂×#14579105	1	1	yes	characteristic
287	C₂²	#19098998	1	1	yes	characteristic
288	C₂²	#19099003	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	1264	2	no	—
292	C₄	#19098364	1	4	yes	complemented
293	C₄	#19098423	1	1	yes	characteristic
294	C₄	#19098564	1	1	yes	characteristic
295	C₄	#19098768	1	1	yes	characteristic
296	C₄	#19098853	1	1	yes	characteristic
297	C₂	—	8	4	no	—
298	C₂	#23763400	1	1	yes	characteristic
299	C₂	C₂×#19098364	1	1	yes	characteristic
300	C₂	#23765112	1	1	yes	characteristic
301	C₂	#23765435	1	1	yes	characteristic
302	C₂	#23765445	1	1	yes	characteristic
303	C₂	#23765537	1	1	yes	characteristic
304	C₂	#23765547	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄×D₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₈×C₇₉	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×D₈	D₇₉
hypocenter	C₇₉	#12811
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
3. #23763560	C₂	semidirect
4. #23761491	C₂	semidirect
15. C₄×D₈×C₇₉	C₄	semidirect
71. D₈×C₇₉	C₄⋊C₄	semidirect
138. C₄×C₇₉	#357	semidirect
158. C₇₉	#12811	semidirect
163. C₄×D₈	Dic₇₉	semidirect
219. D₈	C₄⋊Dic₇₉	semidirect
292. C₄	#19098364	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₂×#19098364	2
#23763560	2
#23761491	2
C₂×C₄×D₈×C₇₉	2
#12811	79