Subgroups — #34188288

Order 20224.

490 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579161	C₂	1	1	yes	maximal, characteristic
2	#23763517	C₂	1	1	yes	maximal, characteristic
3	#23761240	C₂	1	4	yes	maximal
4	C₇₉×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₅₈	—	2	1	no	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	#19098469	—	2	1	no	—
9	#19098461	—	2	1	no	—
10	#19098461	C₄	1	2	yes	—
11	#19098433	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₇₉	—	2	1	no	—
13	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
14	#19098318	—	2	1	no	—
15	#19098318	—	2	1	no	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
17	C₄³×C₇₉	C₂²	1	1	yes	characteristic
18	#336×C₇₉	—	2	1	no	—
19	#336×C₇₉	—	2	1	no	—
20	C₂²×Dic₁₅₈	—	2	1	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	#14579161	—	2	2	no	—
23	#14579161	C₈	1	4	yes	complemented
24	C₄×Dic₁₅₈	—	4	4	no	—
25	C₂×C₄⋊Dic₇₉	—	2	1	no	—
26	C₂×C₄⋊Dic₇₉	—	2	1	no	—
27	C₂×C₄⋊Dic₇₉	—	2	1	no	—
28	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
30	C₂×Dic₇₉⋊C₄	—	2	2	no	—
31	C₂×C₄×Dic₇₉	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
33	C₇₉⋊#95	—	2	2	no	—
34	C₇₉⋊#95	—	2	2	no	—
35	C₂²×C₈×C₇₉	—	2	2	no	—
36	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₇₉	—	2	2	no	—
44	C₄×C₈×C₇₉	Q₈	1	4	yes	complemented
45	C₂×Dic₁₅₈	—	2	4	no	—
46	C₂×Dic₁₅₈	—	4	2	no	—
47	C₂×Dic₁₅₈	—	4	2	no	—
48	C₂²×Dic₇₉	—	2	2	no	—
49	C₂²×Dic₇₉	—	2	2	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₂×C₇₉⋊C₈	—	4	2	no	—
52	C₂×C₇₉⋊C₈	—	4	2	no	—
53	C₄⋊Dic₇₉	—	2	4	no	—
54	C₄⋊Dic₇₉	—	2	2	no	—
55	C₄⋊Dic₇₉	—	2	4	no	—
56	C₄⋊Dic₇₉	—	4	2	no	—
57	C₄⋊Dic₇₉	—	4	2	no	—
58	C₄⋊Dic₇₉	C₂×C₈	1	4	yes	—
59	Dic₇₉⋊C₄	—	4	4	no	—
60	C₄×Dic₇₉	—	4	4	no	—
61	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₉	—	2	4	no	—
69	C₂×C₈×C₇₉	—	4	2	no	—
70	C₂×C₈×C₇₉	—	4	2	no	—
71	C₄²×C₇₉	—	2	2	no	—
72	C₄²×C₇₉	—	2	2	no	—
73	C₄²×C₇₉	—	2	2	no	—
74	C₄²×C₇₉	—	4	1	no	—
75	C₄²×C₇₉	—	4	1	no	—
76	C₄²×C₇₉	C₂×C₈	1	2	yes	—
77	C₄²×C₇₉	M₁₆	1	2	yes	—
78	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
79	C₄²×C₇₉	#41	1	2	yes	—
80	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	2	1	no	—
92	C₂×C₄×C₇₉	—	2	1	no	—
93	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#131	1	2	yes	—
102	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₇₉	#137	1	2	yes	—
104	C₂×Dic₇₉	—	2	4	no	—
105	C₂×Dic₇₉	—	4	2	no	—
106	C₂×Dic₇₉	—	4	2	no	—
107	C₂×Dic₇₉	—	4	2	no	—
108	C₂×Dic₇₉	—	4	2	no	—
109	C₂×Dic₇₉	—	4	2	no	—
110	Dic₁₅₈	—	4	8	no	—
111	C₈×C₇₉	—	4	4	no	—
112	C₇₉⋊C₈	—	8	2	no	—
113	C₂²×C₇₉	#391	1	1	yes	characteristic
114	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₇₉	#408	1	1	yes	characteristic
116	C₂²×C₇₉	#418	1	1	yes	characteristic
117	C₂²×C₇₉	#419	1	1	yes	characteristic
118	C₂²×C₇₉	#421	1	1	yes	characteristic
119	C₂²×C₇₉	#421	1	1	yes	characteristic
120	C₄×C₇₉	—	2	2	no	—
121	C₄×C₇₉	—	2	2	no	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	—	4	1	no	—
124	C₄×C₇₉	—	4	1	no	—
125	C₄×C₇₉	C₈×Q₈	1	2	yes	—
126	C₄×C₇₉	#446	1	2	yes	—
127	C₄×C₇₉	#489	1	2	yes	—
128	C₄×C₇₉	#501	1	2	yes	—
129	Dic₇₉	—	4	4	no	—
130	Dic₇₉	—	8	2	no	—
131	#9873	—	79	1	no	maximal, Sylow
132	C₂×C₇₉	#1478	1	1	yes	characteristic
133	C₂×C₇₉	#1479	1	1	yes	characteristic
134	C₂×C₇₉	#1496	1	1	yes	characteristic
135	C₂×C₇₉	#1497	1	1	yes	characteristic
136	C₂×C₇₉	#1984	1	1	yes	characteristic
137	C₂×C₇₉	#1993	1	1	yes	characteristic
138	C₂×C₇₉	#1995	1	1	yes	characteristic
139	C₄×C₄⋊Q₈	—	79	1	no	—
140	#1831	—	79	1	no	—
141	#1744	D₇₉	1	1	yes	characteristic
142	#1279	—	79	4	no	—
143	C₇₉	#9873	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	—	79	1	no	—
145	C₂×C₄×Q₈	—	158	1	no	—
146	C₂×C₄⋊C₈	—	79	2	no	—
147	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
148	#391	—	158	1	no	—
149	#389	—	79	2	no	—
150	#389	—	158	1	no	—
151	#384	—	79	1	no	—
152	C₄×C₄⋊C₄	—	79	2	no	—
153	C₄×C₄⋊C₄	—	158	1	no	—
154	C₄³	C₂×D₇₉	1	1	yes	characteristic
155	#336	—	2	1	no	—
156	#336	—	2	1	no	—
157	#336	—	158	1	no	—
158	#336	—	158	1	no	—
159	C₂²×Q₈	—	158	1	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	158	2	no	—
162	C₄⋊Q₈	—	79	4	no	—
163	C₄⋊Q₈	—	158	2	no	—
164	C₄×Q₈	—	316	4	no	—
165	C₂×C₄⋊C₄	—	79	2	no	—
166	C₂×C₄⋊C₄	—	79	2	no	—
167	C₂×C₄⋊C₄	—	158	1	no	—
168	C₂×C₄⋊C₄	—	158	2	no	—
169	C₂×C₄⋊C₄	—	158	1	no	—
170	C₂×C₄⋊C₄	—	158	1	no	—
171	C₂×C₄²	—	158	2	no	—
172	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
174	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
175	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
179	C₄⋊C₈	—	158	4	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	Dic₁₅₈	1	4	yes	complemented
182	#95	—	158	2	no	—
183	#95	—	158	2	no	—
184	C₂×Q₈	—	316	2	no	—
185	C₂×Q₈	—	316	2	no	—
186	C₂×Q₈	—	158	4	no	—
187	C₂²×C₄	—	158	2	no	—
188	C₂²×C₄	—	158	2	no	—
189	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	316	2	no	—
200	C₂×C₈	—	316	2	no	—
201	C₂×C₈	—	316	2	no	—
202	C₄⋊C₄	—	316	4	no	—
203	C₄⋊C₄	—	316	2	no	—
204	C₄⋊C₄	—	316	2	no	—
205	C₄⋊C₄	—	79	4	no	—
206	C₄⋊C₄	—	158	4	no	—
207	C₄⋊C₄	—	158	4	no	—
208	C₄⋊C₄	—	158	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	316	4	no	—
215	C₄²	C₈×D₇₉	1	2	yes	—
216	C₄²	#11929607	1	2	yes	—
217	C₄²	C₂×Dic₁₅₈	1	2	yes	—
218	C₄²	#11929624	1	2	yes	—
219	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
220	Q₈	—	316	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	316	2	no	—
234	C₂×C₄	—	316	2	no	—
235	C₂×C₄	—	316	2	no	—
236	C₂×C₄	—	316	2	no	—
237	C₂×C₄	—	316	2	no	—
238	C₂×C₄	—	158	4	no	—
239	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
240	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
241	C₂×C₄	C₄×Dic₁₅₈	1	2	yes	—
242	C₂×C₄	#14579140	1	2	yes	—
243	C₂×C₄	#14579149	1	1	yes	characteristic
244	C₂×C₄	#14579161	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
246	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
247	C₂×C₄	#14579167	1	2	yes	—
248	C₂×C₄	#14579189	1	2	yes	—
249	C₂×C₄	#14579194	1	2	yes	—
250	C₈	—	4	4	no	—
251	C₈	—	632	2	no	—
252	C₂²	#19098469	1	1	yes	characteristic
253	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
254	C₂²	#19098529	1	1	yes	characteristic
255	C₂²	#19098569	1	1	yes	characteristic
256	C₂²	#19098573	1	1	yes	characteristic
257	C₂²	#19098581	1	1	yes	characteristic
258	C₂²	#19098581	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	316	4	no	—
265	C₄	—	632	2	no	—
266	C₄	C₈×Dic₁₅₈	1	2	yes	—
267	C₄	#19098693	1	2	yes	—
268	C₄	#19098923	1	2	yes	—
269	C₄	#19098972	1	2	yes	—
270	C₂	#23761792	1	1	yes	characteristic
271	C₂	#23761796	1	1	yes	characteristic
272	C₂	#23761870	1	1	yes	characteristic
273	C₂	#23761874	1	1	yes	characteristic
274	C₂	#23764336	1	1	yes	characteristic
275	C₂	#23764373	1	1	yes	characteristic
276	C₂	#23764381	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1744	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#9873
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
23. #14579161	C₈	semidirect
44. C₄×C₈×C₇₉	Q₈	semidirect
143. C₇₉	#9873	semidirect
181. C₄×C₈	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579161	2
#23763517	2
#23761240	2
C₇₉×#1744	2
#9873	79