Subgroups — #34441315

Order 20224.

658 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098979	C₂	1	1	yes	maximal, characteristic
2	#23771997	C₂	1	2	yes	maximal
3	#23765189	C₂	1	2	yes	maximal, complemented
4	#23765185	C₂	1	4	yes	maximal, complemented
5	#23765183	C₂	1	2	yes	maximal, complemented
6	C₂×#19098391	C₂	1	1	yes	maximal, characteristic, complemented
7	#367×D₇₉	C₂	1	2	yes	maximal, complemented
8	C₇₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579167	C₂²	1	1	yes	characteristic
10	#19099043	—	2	2	no	—
11	#19098979	C₄	1	8	yes	complemented
12	C₂×#14579101	C₂²	1	1	yes	characteristic
13	C₂×#14579100	C₂²	1	2	yes	—
14	C₂×C₁₆×D₇₉	C₂²	1	2	yes	—
15	#19098606	—	2	2	no	—
16	#19098602	—	2	4	no	—
17	#19098600	C₂²	1	4	yes	—
18	C₂×#14579095	C₂²	1	2	yes	—
19	#107×D₇₉	C₂²	1	4	yes	—
20	#19098391	C₂²	1	2	yes	—
21	#19098391	C₂²	1	2	yes	—
22	#19098390	C₂²	1	4	yes	—
23	#19098388	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
25	#427×C₇₉	C₂²	1	2	yes	—
26	#367×C₇₉	C₂²	1	1	yes	characteristic
27	#367×C₇₉	C₂²	1	1	yes	characteristic
28	#366×C₇₉	C₂²	1	2	yes	—
29	C₂×#11929624	D₄	1	1	yes	characteristic
30	#14579167	C₂×C₄	1	4	yes	—
31	#14579167	Q₈	1	4	yes	—
32	C₂×#11929608	D₄	1	1	yes	characteristic
33	C₂×#11929607	D₄	1	2	yes	—
34	C₂×C₈×D₇₉	C₂³	1	2	yes	—
35	C₄×Dic₁₅₈	—	4	2	no	—
36	C₄×D₃₁₆	—	4	2	no	—
37	(C₂²⋊D₇₉)⋊C₄	—	4	4	no	—
38	#14579119	—	2	4	no	—
39	C₂×C₄⋊Dic₇₉	—	2	2	no	—
40	C₄⋊C₄×D₇₉	—	2	4	no	—
41	#14579101	C₂×C₄	1	4	yes	complemented
42	#14579100	C₂×C₄	1	8	yes	complemented
43	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
44	C₁₆×D₇₉	C₂×C₄	1	8	yes	complemented
45	#14579095	—	2	2	no	—
46	#14579095	C₂³	1	4	yes	—
47	#14579094	—	2	4	no	—
48	#14579092	—	2	4	no	—
49	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	#117×C₇₉	—	2	2	no	—
51	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
54	#107×C₇₉	C₂³	1	2	yes	—
55	#107×C₇₉	C₂³	1	2	yes	—
56	#106×C₇₉	—	2	2	no	—
57	#11929624	C₄⋊C₄	1	4	yes	—
58	#11929624	Q₁₆	1	4	yes	—
59	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
60	C₂×D₃₁₆	D₈	1	1	yes	characteristic
61	C₂×C₂²⋊D₇₉	D₈	1	2	yes	—
62	C₂²×Dic₇₉	—	4	2	no	—
63	C₂×C₄×D₇₉	—	4	4	no	—
64	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
65	#11929608	C₄⋊C₄	1	4	yes	—
66	#11929607	C₄⋊C₄	1	8	yes	—
67	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
68	C₈×D₇₉	C₂²×C₄	1	4	yes	—
69	C₈×D₇₉	C₂×Q₈	1	4	yes	—
70	C₄⋊Dic₇₉	—	2	4	no	—
71	C₄⋊Dic₇₉	—	4	2	no	—
72	Dic₇₉⋊C₄	—	4	4	no	—
73	(C₂×D₇₉)⋊C₄	—	4	4	no	—
74	C₄×Dic₇₉	—	4	4	no	—
75	C₇₉⋊C₁₆	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₇₉	—	2	2	no	—
82	C₄⋊C₄×C₇₉	—	2	2	no	—
83	C₂²⋊C₄×C₇₉	—	4	2	no	—
84	C₄²×C₇₉	—	4	2	no	—
85	C₁₆×C₇₉	C₂²×C₄	1	2	yes	—
86	C₁₆×C₇₉	C₂²×C₄	1	2	yes	—
87	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
88	C₂²×D₇₉	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₇₉	—	4	2	no	—
90	C₂×C₄×C₇₉	—	4	2	no	—
91	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
94	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
95	C₂²⋊D₇₉	#107	1	8	yes	—
96	C₂×Dic₇₉	—	4	4	no	—
97	C₂×Dic₇₉	—	16	2	no	—
98	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
99	D₃₁₆	#107	1	4	yes	—
100	C₄×D₇₉	—	16	4	no	—
101	C₄×D₇₉	C₂×C₄⋊C₄	1	4	yes	—
102	C₄×D₇₉	C₂×Q₁₆	1	4	yes	—
103	Dic₁₅₈	#107	1	4	yes	—
104	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
107	C₇₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
108	C₂²×C₇₉	—	2	2	no	—
109	C₂²×C₇₉	C₂×#107	1	2	yes	—
110	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
111	C₂×D₇₉	—	2	4	no	—
112	C₂×D₇₉	C₂×#107	1	4	yes	—
113	C₄×C₇₉	—	8	2	no	—
114	C₄×C₇₉	—	8	2	no	—
115	C₄×C₇₉	C₂×#107	1	2	yes	—
116	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
118	Dic₇₉	—	8	4	no	—
119	Dic₇₉	C₂×#107	1	4	yes	—
120	#33411	—	79	1	no	maximal, Sylow
121	C₂×C₇₉	—	2	2	no	—
122	C₂×C₇₉	C₂²×#107	1	1	yes	characteristic
123	C₂×C₇₉	#3416	1	2	yes	—
124	D₇₉	—	2	4	no	—
125	C₂×#504	—	79	1	no	—
126	#2913	—	79	2	no	—
127	#2159	—	79	2	no	—
128	#2159	—	79	4	no	—
129	#2158	—	79	2	no	—
130	#2158	D₇₉	1	1	yes	characteristic, complemented
131	C₂×#367	—	79	1	no	—
132	C₂×#367	—	79	2	no	—
133	C₇₉	#33411	1	1	yes	characteristic, complemented, Sylow
134	C₂×#131	—	79	1	no	—
135	#520	—	158	2	no	—
136	#504	—	79	8	no	—
137	C₂×M₃₂	—	79	1	no	—
138	C₂×M₃₂	—	79	2	no	—
139	C₂²×C₁₆	—	79	2	no	—
140	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
141	#428	—	158	4	no	—
142	#428	—	158	2	no	—
143	#427	—	79	4	no	—
144	#427	C₂×D₇₉	1	2	yes	—
145	C₂×#107	—	79	2	no	—
146	C₂×#107	—	79	4	no	—
147	#367	—	79	2	no	—
148	#367	—	79	2	no	—
149	#367	—	79	4	no	—
150	#367	C₂×D₇₉	1	1	yes	characteristic
151	#367	C₂×D₇₉	1	1	yes	characteristic
152	#366	—	79	4	no	—
153	#366	C₂×D₇₉	1	2	yes	—
154	C₂×#41	—	79	1	no	—
155	#131	—	79	4	no	—
156	#131	—	79	4	no	—
157	C₂×M₁₆	—	79	1	no	—
158	C₂×M₁₆	—	79	2	no	—
159	C₂²×C₈	—	79	2	no	—
160	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
161	C₄×Q₈	—	316	2	no	—
162	C₄×D₄	—	316	2	no	—
163	C₄×D₄	—	316	4	no	—
164	#117	—	2	2	no	—
165	#117	—	158	4	no	—
166	C₂×C₄⋊C₄	—	158	4	no	—
167	C₂×C₄⋊C₄	—	158	2	no	—
168	M₃₂	—	79	4	no	—
169	M₃₂	—	79	8	no	—
170	C₂×C₁₆	—	79	2	no	—
171	C₂×C₁₆	—	79	8	no	—
172	C₂×C₁₆	C₄×D₇₉	1	4	yes	complemented
173	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
174	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
175	#107	—	79	4	no	—
176	#107	—	158	4	no	—
177	#107	—	158	2	no	—
178	#107	C₂²×D₇₉	1	2	yes	—
179	#107	C₂²×D₇₉	1	2	yes	—
180	#106	—	2	2	no	—
181	#106	—	158	4	no	—
182	#41	—	79	4	no	—
183	#41	—	79	4	no	—
184	C₂×Q₈	—	79	1	no	—
185	C₂×D₄	—	79	2	no	—
186	C₂×D₄	—	79	1	no	—
187	C₂²×C₄	—	316	4	no	—
188	C₂²×C₄	—	316	2	no	—
189	C₂²×C₄	—	79	2	no	—
190	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
191	M₁₆	—	79	4	no	—
192	M₁₆	—	79	8	no	—
193	C₂×C₈	—	79	2	no	—
194	C₂×C₈	—	79	4	no	—
195	C₂×C₈	—	79	4	no	—
196	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
197	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
198	C₂×C₈	Q₈×D₇₉	1	2	yes	—
199	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	316	4	no	—
203	C₄⋊C₄	—	316	2	no	—
204	C₄⋊C₄	—	158	4	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	316	4	no	—
207	C₄²	—	4	2	no	—
208	C₄²	—	316	4	no	—
209	C₁₆	—	79	4	no	—
210	C₁₆	C₂×C₄×D₇₉	1	2	yes	—
211	C₁₆	C₂×C₄×D₇₉	1	2	yes	—
212	C₂³	—	79	2	no	—
213	C₂³	D₈×D₇₉	1	1	yes	characteristic
214	Q₈	—	79	4	no	—
215	D₄	—	79	8	no	—
216	D₄	—	79	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	316	4	no	—
220	C₂×C₄	—	79	2	no	—
221	C₂×C₄	—	79	4	no	—
222	C₂×C₄	—	79	4	no	—
223	C₂×C₄	—	1264	4	no	—
224	C₂×C₄	—	1264	2	no	—
225	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
226	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
227	C₂×C₄	Q₁₆×D₇₉	1	2	yes	—
228	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
229	C₈	—	79	4	no	—
230	C₈	C₄⋊C₄×D₇₉	1	2	yes	—
231	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
232	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	79	4	no	—
235	C₂²	—	158	4	no	—
236	C₂²	#107×D₇₉	1	2	yes	—
237	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
238	C₄	—	8	2	no	—
239	C₄	—	8	2	no	—
240	C₄	—	79	4	no	—
241	C₄	—	632	4	no	—
242	C₄	#107×D₇₉	1	2	yes	—
243	C₄	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
244	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	158	4	no	—
247	C₂	C₂×#107×D₇₉	1	1	yes	characteristic
248	C₂	#23776543	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2158	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#11929624	D₄
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	#2158	D₇₉
hypocenter	C₇₉	#33411
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #23765189	C₂	semidirect
4. #23765185	C₂	semidirect
5. #23765183	C₂	semidirect
6. C₂×#19098391	C₂	semidirect
7. #367×D₇₉	C₂	semidirect
8. C₇₉×#2158	C₂	semidirect
11. #19098979	C₄	semidirect
41. #14579101	C₂×C₄	semidirect
42. #14579100	C₂×C₄	semidirect
44. C₁₆×D₇₉	C₂×C₄	semidirect
51. C₂×C₁₆×C₇₉	C₂×C₄	semidirect
130. #2158	D₇₉	semidirect
133. C₇₉	#33411	semidirect
172. C₂×C₁₆	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098979	2
#23771997	2
#23765189	2
#23765185	2
#23765183	2
C₂×#19098391	2
#367×D₇₉	2
C₇₉×#2158	2
#33411	79