Subgroups — #34187648

Order 20224.

502 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763564	C₂	1	1	yes	maximal, characteristic, complemented
2	#23761960	C₂	1	4	yes	maximal, complemented
3	#23761683	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₉×#3147	C₂	1	1	yes	maximal, characteristic
5	#19098616	C₂²	1	2	yes	—
6	C₂×#14579095	—	2	1	no	—
7	C₂×#14579093	—	2	1	no	—
8	#19098429	C₂²	1	2	yes	—
9	#19098297	—	2	8	no	—
10	#19098284	—	2	1	no	—
11	C₈×C₇₉⋊C₈	—	2	1	no	—
12	C₂×#135×C₇₉	—	2	1	no	—
13	C₂×#135×C₇₉	—	2	1	no	—
14	#532×C₇₉	C₂²	1	2	yes	—
15	#495×C₇₉	—	2	1	no	—
16	#495×C₇₉	—	2	1	no	—
17	#494×C₇₉	C₄	1	1	yes	characteristic, complemented
18	#493×C₇₉	C₄	1	1	yes	characteristic, complemented
19	#492×C₇₉	C₄	1	2	yes	complemented
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇₉	—	2	2	no	—
22	#14579095	—	4	2	no	—
23	#14579093	—	4	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₇₉⋊#95	—	2	4	no	—
28	C₂×#41×C₇₉	—	2	2	no	—
29	#135×C₇₉	—	4	2	no	—
30	#135×C₇₉	—	4	2	no	—
31	#135×C₇₉	—	4	2	no	—
32	#135×C₇₉	—	4	2	no	—
33	C₂×Q₁₆×C₇₉	—	2	1	no	—
34	C₂×Q₁₆×C₇₉	—	2	1	no	—
35	C₂×SD₁₆×C₇₉	—	2	2	no	—
36	C₂×SD₁₆×C₇₉	—	2	2	no	—
37	C₂×D₈×C₇₉	—	2	1	no	—
38	C₂×D₈×C₇₉	—	2	1	no	—
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
42	#127×C₇₉	C₂×C₄	1	2	yes	—
43	#124×C₇₉	—	2	2	no	—
44	#121×C₇₉	—	2	4	no	—
45	C₄×D₄×C₇₉	—	2	4	no	—
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
51	C₂²×Dic₇₉	—	4	2	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₄⋊Dic₇₉	—	4	2	no	—
54	C₄⋊Dic₇₉	—	4	2	no	—
55	#41×C₇₉	—	4	2	no	—
56	#41×C₇₉	—	4	2	no	—
57	#41×C₇₉	—	4	2	no	—
58	#41×C₇₉	—	4	2	no	—
59	C₂×Q₈×C₇₉	—	2	2	no	—
60	C₂×D₄×C₇₉	—	2	2	no	—
61	C₂×D₄×C₇₉	—	2	4	no	—
62	C₂×D₄×C₇₉	—	4	2	no	—
63	C₂²×C₄×C₇₉	—	4	2	no	—
64	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₇₉	—	4	2	no	—
67	Q₁₆×C₇₉	—	4	2	no	—
68	SD₁₆×C₇₉	—	4	4	no	—
69	SD₁₆×C₇₉	—	4	4	no	—
70	D₈×C₇₉	—	4	2	no	—
71	D₈×C₇₉	—	4	2	no	—
72	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₇₉	D₈	1	2	yes	—
77	C₂×C₈×C₇₉	D₈	1	2	yes	—
78	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
79	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₇₉	—	2	4	no	—
81	C₂²⋊C₄×C₇₉	—	4	4	no	—
82	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇₉	#41	1	2	yes	—
85	C₂³×C₇₉	—	4	2	no	—
86	C₂³×C₇₉	#127	1	1	yes	characteristic
87	Q₈×C₇₉	—	4	2	no	—
88	Q₈×C₇₉	—	4	2	no	—
89	D₄×C₇₉	—	4	2	no	—
90	D₄×C₇₉	—	4	2	no	—
91	D₄×C₇₉	—	8	4	no	—
92	D₄×C₇₉	—	8	2	no	—
93	D₄×C₇₉	—	8	2	no	—
94	C₂×C₄×C₇₉	—	4	2	no	—
95	C₂×C₄×C₇₉	—	4	2	no	—
96	C₂×C₄×C₇₉	—	8	2	no	—
97	C₂×C₄×C₇₉	—	8	2	no	—
98	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₇₉	#121	1	2	yes	—
101	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×Dic₇₉	—	8	4	no	—
107	C₂×Dic₇₉	—	8	2	no	—
108	C₈×C₇₉	#102	1	2	yes	—
109	C₈×C₇₉	#102	1	2	yes	—
110	C₈×C₇₉	#102	1	2	yes	—
111	C₈×C₇₉	#102	1	2	yes	—
112	C₇₉⋊C₈	—	4	4	no	—
113	C₂²×C₇₉	—	2	2	no	—
114	C₂²×C₇₉	—	8	2	no	—
115	C₂²×C₇₉	—	8	4	no	—
116	C₂²×C₇₉	#390	1	1	yes	characteristic
117	C₂²×C₇₉	#492	1	1	yes	characteristic
118	C₂²×C₇₉	#493	1	1	yes	characteristic
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	—	2	2	no	—
121	C₄×C₇₉	—	8	2	no	—
122	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
123	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
124	C₄×C₇₉	#486	1	1	yes	characteristic
125	C₄×C₇₉	#486	1	1	yes	characteristic
126	Dic₇₉	—	16	2	no	—
127	#9756	—	79	1	no	maximal, Sylow
128	C₂×C₇₉	—	2	1	no	—
129	C₂×C₇₉	—	2	1	no	—
130	C₂×C₇₉	—	16	2	no	—
131	C₂×C₇₉	#1957	1	1	yes	characteristic
132	C₂×C₇₉	#1976	1	2	yes	—
133	#3147	D₇₉	1	1	yes	characteristic
134	#1846	—	79	1	no	—
135	#1517	—	79	4	no	—
136	#1450	—	79	1	no	—
137	C₇₉	#9756	1	1	yes	characteristic, complemented, Sylow
138	C₂×#135	—	2	1	no	—
139	C₂×#135	—	2	1	no	—
140	#532	C₂×D₇₉	1	2	yes	—
141	#495	—	2	1	no	—
142	#495	—	2	1	no	—
143	#494	Dic₇₉	1	1	yes	characteristic, complemented
144	#493	Dic₇₉	1	1	yes	characteristic, complemented
145	#492	Dic₇₉	1	2	yes	complemented
146	#431	—	79	2	no	—
147	C₂×#107	—	158	1	no	—
148	C₂×#106	—	158	1	no	—
149	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
150	#382	—	79	2	no	—
151	#327	—	158	8	no	—
152	#322	—	158	1	no	—
153	C₈²	—	158	1	no	—
154	C₂×#41	—	2	2	no	—
155	#135	—	4	2	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	#135	—	4	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
167	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
168	#127	C₂×Dic₇₉	1	2	yes	—
169	#124	—	2	2	no	—
170	#121	—	2	4	no	—
171	C₄×D₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	158	2	no	—
173	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
174	#107	—	316	2	no	—
175	#106	—	316	2	no	—
176	C₄⋊C₈	—	158	2	no	—
177	C₂²⋊C₈	—	79	4	no	—
178	C₄×C₈	—	158	2	no	—
179	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
180	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
183	#95	—	158	4	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	316	2	no	—
194	C₂²×C₄	D₄×D₇₉	1	2	yes	—
195	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
196	Q₁₆	—	4	2	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	2	no	—
202	C₂×C₈	—	158	4	no	—
203	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
207	C₂×C₈	#11929609	1	2	yes	—
208	C₂×C₈	#11929609	1	2	yes	—
209	C₂×C₈	#11929612	1	2	yes	—
210	C₂×C₈	#11929612	1	2	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	316	2	no	—
213	C₄⋊C₄	—	316	2	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
217	C₄²	#11929623	1	2	yes	—
218	C₂³	—	4	2	no	—
219	C₂³	#14579157	1	1	yes	characteristic
220	Q₈	—	4	2	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	4	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	632	4	no	—
232	C₂×C₄	—	632	2	no	—
233	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
235	C₂×C₄	#14579132	1	2	yes	—
236	C₂×C₄	#14579148	1	1	yes	characteristic
237	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
238	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
239	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
240	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
241	C₈	—	316	4	no	—
242	C₈	#14579082	1	2	yes	—
243	C₈	#14579082	1	2	yes	—
244	C₈	#14579082	1	2	yes	—
245	C₈	#14579082	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	2	no	—
249	C₂²	#19098466	1	1	yes	characteristic
250	C₂²	#19098935	1	1	yes	characteristic
251	C₂²	#19098939	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	1264	2	no	—
256	C₄	C₂×#14579082	1	1	yes	characteristic
257	C₄	C₂×#14579082	1	1	yes	characteristic
258	C₄	#19098909	1	1	yes	characteristic
259	C₄	#19098909	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	16	2	no	—
263	C₂	#23764207	1	1	yes	characteristic
264	C₂	#23764299	1	2	yes	—

	subgroup	G/H
center	C₂²	#19098466
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting, prime radical	C₇₉×#3147	C₂
socle	C₂²×C₇₉	#390
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3147	D₇₉
hypocenter	C₇₉	#9756
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #23763564	C₂	semidirect
2. #23761960	C₂	semidirect
3. #23761683	C₂	semidirect
17. #494×C₇₉	C₄	semidirect
18. #493×C₇₉	C₄	semidirect
19. #492×C₇₉	C₄	semidirect
137. C₇₉	#9756	semidirect
143. #494	Dic₇₉	semidirect
144. #493	Dic₇₉	semidirect
145. #492	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23763564	2
#23761960	2
#23761683	2
C₇₉×#3147	2
#9756	79