Subgroups — #34181330

Order 20224.

526 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic
2	#23763397	C₂	1	2	yes	maximal
3	C₂×#19098314	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761283	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₉	—	2	1	no	—
7	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
8	C₈×D₃₁₆	—	2	2	no	—
9	C₈×D₃₁₆	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14579081	—	2	2	no	—
13	C₂×(C₂×D₇₉)⋊C₈	—	2	1	no	—
14	#19098325	—	2	2	no	—
15	#19098314	—	2	2	no	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#384×C₇₉	C₂²	1	2	yes	—
18	#340×C₇₉	—	2	1	no	—
19	#340×C₇₉	—	2	1	no	—
20	C₂²×D₃₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₇₉	—	2	1	no	—
22	C₂²×C₇₉⋊C₈	—	2	1	no	—
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	C₂×C₈×D₇₉	—	2	4	no	—
25	C₂×C₈×D₇₉	—	4	4	no	—
26	C₄×D₃₁₆	—	2	1	no	—
27	C₄×D₃₁₆	—	2	1	no	—
28	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
29	C₄×D₃₁₆	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
31	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
33	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
34	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
36	#14579081	—	4	4	no	—
37	(C₂×D₇₉)⋊C₈	—	2	4	no	—
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₉	D₄	1	2	yes	—
46	C₂³×D₇₉	—	2	1	no	—
47	C₂×D₃₁₆	—	2	1	no	—
48	C₂×D₃₁₆	—	2	2	no	—
49	C₂×D₃₁₆	C₄⋊C₄	1	2	yes	—
50	C₂×D₃₁₆	SD₁₆	1	4	yes	—
51	C₂²×Dic₇₉	—	2	1	no	—
52	C₂×C₄×D₇₉	—	2	2	no	—
53	C₂×C₄×D₇₉	—	2	2	no	—
54	C₂×C₄×D₇₉	—	4	2	no	—
55	C₂×C₄×D₇₉	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	2	2	no	—
57	C₂×C₇₉⋊C₈	—	4	2	no	—
58	C₂×C₇₉⋊C₈	—	4	2	no	—
59	C₂×C₇₉⋊C₈	—	4	2	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₈×D₇₉	—	4	8	no	—
62	C₄⋊Dic₇₉	—	2	1	no	—
63	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
64	(C₂×D₇₉)⋊C₄	—	2	2	no	—
65	(C₂×D₇₉)⋊C₄	—	2	2	no	—
66	C₂²×C₄×C₇₉	—	4	2	no	—
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
70	C₂×C₈×C₇₉	—	2	2	no	—
71	C₂×C₈×C₇₉	—	2	2	no	—
72	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
74	C₄⋊C₄×C₇₉	—	4	2	no	—
75	C₄⋊C₄×C₇₉	—	4	2	no	—
76	C₄⋊C₄×C₇₉	—	4	2	no	—
77	C₄⋊C₄×C₇₉	—	4	2	no	—
78	C₄⋊C₄×C₇₉	—	8	2	no	—
79	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₇₉	#41	1	1	yes	characteristic
83	C₂³×C₇₉	#121	1	1	yes	characteristic
84	C₂²×D₇₉	—	2	2	no	—
85	C₂²×D₇₉	—	2	4	no	—
86	C₂²×D₇₉	—	4	2	no	—
87	C₂²×D₇₉	—	4	1	no	—
88	C₂²×D₇₉	—	4	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	8	2	no	—
92	C₂×C₄×C₇₉	—	8	2	no	—
93	C₂×C₄×C₇₉	—	8	2	no	—
94	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
97	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄×C₇₉	#135	1	2	yes	—
102	C₂×C₄×C₇₉	#135	1	2	yes	—
103	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
104	C₂×Dic₇₉	—	2	2	no	—
105	C₂×Dic₇₉	—	4	2	no	—
106	D₃₁₆	—	2	2	no	—
107	D₃₁₆	—	2	4	no	—
108	D₃₁₆	#106	1	4	yes	—
109	C₄×D₇₉	—	4	2	no	—
110	C₄×D₇₉	—	4	2	no	—
111	C₄×D₇₉	—	4	4	no	—
112	C₈×C₇₉	—	2	2	no	—
113	C₈×C₇₉	—	2	2	no	—
114	C₇₉⋊C₈	—	4	2	no	—
115	C₇₉⋊C₈	—	8	2	no	—
116	C₂²×C₇₉	#380	1	1	yes	characteristic
117	C₂²×C₇₉	#463	1	2	yes	—
118	C₂²×C₇₉	#465	1	2	yes	—
119	C₂²×C₇₉	#472	1	1	yes	characteristic
120	C₂²×C₇₉	#473	1	1	yes	characteristic
121	C₂×D₇₉	—	2	4	no	—
122	C₂×D₇₉	—	4	2	no	—
123	C₂×D₇₉	—	4	2	no	—
124	C₂×D₇₉	—	4	4	no	—
125	C₂×D₇₉	—	4	4	no	—
126	C₄×C₇₉	—	2	1	no	—
127	C₄×C₇₉	—	2	1	no	—
128	C₄×C₇₉	—	16	2	no	—
129	C₄×C₇₉	#416	1	1	yes	characteristic
130	C₄×C₇₉	#418	1	1	yes	characteristic
131	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
132	C₄×C₇₉	#427	1	1	yes	characteristic
133	C₄×C₇₉	#465	1	2	yes	—
134	C₄×C₇₉	#478	1	2	yes	—
135	Dic₇₉	—	4	2	no	—
136	#8646	—	79	1	no	maximal, Sylow
137	C₂×C₇₉	#1677	1	2	yes	—
138	C₂×C₇₉	#1679	1	2	yes	—
139	C₂×C₇₉	#1804	1	1	yes	characteristic
140	C₂×C₇₉	#1807	1	1	yes	characteristic
141	C₂×C₇₉	#1816	1	1	yes	characteristic
142	D₇₉	—	4	4	no	—
143	C₂×C₈×D₄	—	79	1	no	—
144	#1834	D₇₉	1	1	yes	characteristic, complemented
145	#1802	—	79	2	no	—
146	C₂×#334	—	79	1	no	—
147	#1296	—	79	2	no	—
148	C₇₉	#8646	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₈	—	158	1	no	—
150	C₂×C₄×D₄	—	79	1	no	—
151	C₈×D₄	—	79	4	no	—
152	C₈×D₄	—	158	2	no	—
153	C₂×C₄⋊C₈	—	79	1	no	—
154	C₂×C₄⋊C₈	—	79	2	no	—
155	C₂×#102	—	158	2	no	—
156	C₂×C₂²⋊C₈	—	158	1	no	—
157	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
158	#384	C₂×D₇₉	1	2	yes	—
159	#340	—	2	1	no	—
160	#340	—	2	1	no	—
161	#340	—	158	2	no	—
162	#334	—	158	2	no	—
163	C₂²×D₄	—	79	1	no	—
164	C₂³×C₄	—	158	1	no	—
165	C₂²×C₈	—	316	4	no	—
166	C₂²×C₈	—	158	2	no	—
167	C₂²×C₈	—	158	1	no	—
168	C₂²×C₈	—	158	4	no	—
169	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
171	C₄×D₄	—	79	2	no	—
172	C₄×D₄	—	79	2	no	—
173	C₄×D₄	—	158	1	no	—
174	C₄×D₄	—	158	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	79	1	no	—
179	C₂×C₂²⋊C₄	—	158	1	no	—
180	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
181	C₄⋊C₈	—	79	2	no	—
182	C₄⋊C₈	—	158	2	no	—
183	C₄⋊C₈	—	158	1	no	—
184	C₄⋊C₈	—	158	2	no	—
185	#102	—	316	4	no	—
186	C₂²⋊C₈	—	158	4	no	—
187	C₄×C₈	C₄×D₇₉	1	2	yes	—
188	C₄×C₈	D₃₁₆	1	2	yes	—
189	C₂⁴	—	158	1	no	—
190	C₂×D₄	—	79	2	no	—
191	C₂×D₄	—	79	4	no	—
192	C₂×D₄	—	158	1	no	—
193	C₂×D₄	—	158	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	316	2	no	—
196	C₂²×C₄	—	316	2	no	—
197	C₂²×C₄	—	158	1	no	—
198	C₂²×C₄	—	158	2	no	—
199	C₂²×C₄	—	158	2	no	—
200	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂²×C₄	#11929623	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	316	8	no	—
206	C₂×C₈	—	316	2	no	—
207	C₂×C₈	—	316	2	no	—
208	C₂×C₈	—	316	2	no	—
209	C₂×C₈	—	316	2	no	—
210	C₂×C₈	—	158	2	no	—
211	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
212	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	8	2	no	—
218	C₄⋊C₄	—	79	2	no	—
219	C₄⋊C₄	—	158	1	no	—
220	C₂²⋊C₄	—	158	2	no	—
221	C₂²⋊C₄	—	158	2	no	—
222	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
223	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
224	C₄²	Q₈×D₇₉	1	1	yes	characteristic
225	C₄²	#11929625	1	1	yes	characteristic
226	C₂³	—	316	2	no	—
227	C₂³	—	316	1	no	—
228	C₂³	—	316	1	no	—
229	C₂³	—	158	2	no	—
230	C₂³	—	158	4	no	—
231	C₂³	#14579132	1	1	yes	characteristic
232	D₄	—	79	4	no	—
233	D₄	—	158	4	no	—
234	D₄	—	158	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	316	2	no	—
241	C₂×C₄	—	316	2	no	—
242	C₂×C₄	—	316	2	no	—
243	C₂×C₄	—	316	4	no	—
244	C₂×C₄	—	158	2	no	—
245	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
246	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
247	C₂×C₄	#14579119	1	1	yes	characteristic
248	C₂×C₄	#14579136	1	1	yes	characteristic
249	C₂×C₄	#14579137	1	1	yes	characteristic
250	C₂×C₄	#14579141	1	1	yes	characteristic
251	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
252	C₂×C₄	#14579181	1	2	yes	—
253	C₂×C₄	#14579182	1	2	yes	—
254	C₂×C₄	#14579186	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	316	2	no	—
258	C₈	—	632	2	no	—
259	C₂²	—	316	2	no	—
260	C₂²	—	316	4	no	—
261	C₂²	—	316	4	no	—
262	C₂²	—	316	2	no	—
263	C₂²	—	158	4	no	—
264	C₂²	#19098420	1	1	yes	characteristic
265	C₂²	#19098775	1	2	yes	—
266	C₂²	#19098789	1	2	yes	—
267	C₂²	#19098830	1	1	yes	characteristic
268	C₂²	#19098835	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	16	2	no	—
272	C₄	—	316	2	no	—
273	C₄	#19098557	1	1	yes	characteristic
274	C₄	#19098567	1	1	yes	characteristic
275	C₄	#106×D₇₉	1	1	yes	characteristic
276	C₄	#19098598	1	1	yes	characteristic
277	C₄	#19098793	1	2	yes	—
278	C₄	#19098860	1	2	yes	—
279	C₂	—	316	4	no	—
280	C₂	#23762806	1	2	yes	—
281	C₂	#23762820	1	2	yes	—
282	C₂	#23763407	1	1	yes	characteristic
283	C₂	#23763422	1	1	yes	characteristic
284	C₂	#23763465	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579132
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1834	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂²×D₃₁₆	D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1834	D₇₉
hypocenter	C₇₉	#8646
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
3. C₂×#19098314	C₂	semidirect
4. #23761283	C₂	semidirect
5. C₇₉×#1834	C₂	semidirect
9. C₈×D₃₁₆	C₄	semidirect
144. #1834	D₇₉	semidirect
148. C₇₉	#8646	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₁₆	2
#23763397	2
C₂×#19098314	2
#23761283	2
C₇₉×#1834	2
#8646	79