Subgroups — #34296612

Order 20224.

658 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23766705	C₂	1	1	yes	maximal, characteristic
2	#23764123	C₂	1	2	yes	maximal
3	#23763964	C₂	1	4	yes	maximal
4	#23763520	C₂	1	1	yes	maximal, characteristic
5	C₈×Dic₇₉⋊C₄	C₂	1	4	yes	maximal
6	C₈×C₄⋊Dic₇₉	C₂	1	2	yes	maximal
7	C₇₉×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×Dic₇₉⋊C₈	C₂²	1	4	yes	—
10	C₂×C₈×Dic₇₉	C₂²	1	4	yes	—
11	#19098505	C₄	1	4	yes	—
12	#19098497	C₄	1	4	yes	—
13	C₄×Dic₇₉⋊C₄	C₂²	1	4	yes	—
14	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
15	#19098318	C₂²	1	4	yes	—
16	#19098317	C₂²	1	4	yes	—
17	#19098317	C₂²	1	4	yes	—
18	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#97×C₇₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	#376×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×C₇₉⋊C₈	C₂³	1	4	yes	—
24	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
25	C₂×Dic₇₉⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×Dic₇₉⋊C₄	C₂×C₄	1	4	yes	—
27	C₂×C₄×Dic₇₉	C₂³	1	4	yes	—
28	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
29	Dic₇₉⋊C₈	—	2	8	no	—
30	C₈×Dic₇₉	—	2	8	no	—
31	C₇₉⋊#95	C₂×C₄	1	2	yes	—
32	C₇₉⋊#95	C₂×C₄	1	2	yes	—
33	C₇₉⋊#95	C₂×C₄	1	4	yes	—
34	C₇₉⋊#95	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
39	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	#97×C₇₉	—	2	4	no	—
41	#97×C₇₉	—	2	2	no	—
42	C₄×C₈×C₇₉	—	2	4	no	—
43	C₄×C₈×C₇₉	Q₈	1	4	yes	—
44	#95×C₇₉	C₂×C₄	1	4	yes	—
45	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
46	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
47	C₂×C₇₉⋊C₈	—	2	4	no	—
48	C₂×C₇₉⋊C₈	—	2	4	no	—
49	C₂×C₇₉⋊C₈	—	2	4	no	—
50	C₄⋊Dic₇₉	—	2	8	no	—
51	Dic₇₉⋊C₄	—	2	8	no	—
52	Dic₇₉⋊C₄	—	2	8	no	—
53	C₄×Dic₇₉	—	2	4	no	—
54	C₄×Dic₇₉	—	2	4	no	—
55	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₇₉	—	2	2	no	—
59	C₂×C₈×C₇₉	—	2	2	no	—
60	C₂×C₈×C₇₉	—	2	2	no	—
61	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₇₉	#41	1	4	yes	—
64	C₂×C₈×C₇₉	#41	1	4	yes	—
65	C₄²×C₇₉	—	2	2	no	—
66	C₄²×C₇₉	—	2	2	no	—
67	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
68	C₄²×C₇₉	#41	1	2	yes	—
69	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	—	2	4	no	—
71	C₂×C₄×C₇₉	—	2	4	no	—
72	C₂×C₄×C₇₉	—	2	4	no	—
73	C₂×C₄×C₇₉	—	2	4	no	—
74	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
75	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
77	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
79	C₂×Dic₇₉	—	2	4	no	—
80	C₂×Dic₇₉	—	2	4	no	—
81	C₂×Dic₇₉	—	2	4	no	—
82	C₂×Dic₇₉	—	2	4	no	—
83	C₂×Dic₇₉	—	2	4	no	—
84	C₂×Dic₇₉	#131	1	8	yes	—
85	C₈×C₇₉	—	2	4	no	—
86	C₈×C₇₉	—	2	2	no	—
87	C₈×C₇₉	#125	1	4	yes	—
88	C₇₉⋊C₈	—	4	4	no	—
89	C₂²×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
90	C₂²×C₇₉	C₄×#41	1	2	yes	—
91	C₂²×C₇₉	C₂×#131	1	4	yes	—
92	C₄×C₇₉	—	2	4	no	—
93	C₄×C₇₉	—	4	4	no	—
94	C₄×C₇₉	C₂×#125	1	1	yes	characteristic
95	C₄×C₇₉	#529	1	2	yes	—
96	C₄×C₇₉	#533	1	1	yes	characteristic
97	Dic₇₉	—	2	8	no	—
98	Dic₇₉	—	4	4	no	—
99	#21676	—	79	1	no	maximal, Sylow
100	C₂×C₇₉	C₄×#125	1	1	yes	characteristic
101	C₂×C₇₉	#2960	1	2	yes	—
102	C₂×C₇₉	#2966	1	4	yes	—
103	#2402	—	79	1	no	—
104	#1940	—	79	2	no	—
105	#1917	—	79	4	no	—
106	#1832	—	79	1	no	—
107	C₈×C₄⋊C₄	—	79	2	no	—
108	C₈×C₄⋊C₄	—	79	4	no	—
109	#1726	D₇₉	1	1	yes	characteristic
110	C₇₉	#21676	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊C₈	—	79	4	no	—
112	C₂×C₄⋊C₈	—	79	2	no	—
113	C₂×#97	C₂×D₇₉	1	2	yes	—
114	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
115	C₂×C₄×C₈	—	79	4	no	—
116	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
117	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
118	#400	—	79	4	no	—
119	#398	—	79	4	no	—
120	C₄×C₄⋊C₄	—	79	2	no	—
121	C₄×C₄⋊C₄	—	79	4	no	—
122	#376	C₂×D₇₉	1	1	yes	characteristic
123	#336	—	79	4	no	—
124	#336	—	79	4	no	—
125	#336	—	79	4	no	—
126	C₂²×C₈	—	79	4	no	—
127	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
129	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	79	4	no	—
131	C₂×C₄⋊C₄	—	79	4	no	—
132	C₂×C₄⋊C₄	—	79	4	no	—
133	C₂×C₄²	—	79	4	no	—
134	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
135	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
136	C₄⋊C₈	—	158	8	no	—
137	C₄⋊C₈	—	158	4	no	—
138	#97	—	2	4	no	—
139	#97	—	2	2	no	—
140	C₄×C₈	—	2	4	no	—
141	C₄×C₈	—	158	8	no	—
142	C₄×C₈	Dic₁₅₈	1	4	yes	—
143	#95	—	79	2	no	—
144	#95	—	79	4	no	—
145	#95	—	79	4	no	—
146	#95	—	79	2	no	—
147	#95	C₄×D₇₉	1	4	yes	—
148	C₂²×C₄	—	79	4	no	—
149	C₂²×C₄	—	79	4	no	—
150	C₂²×C₄	C₂×C₄×D₇₉	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
152	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	158	4	no	—
157	C₂×C₈	—	158	4	no	—
158	C₂×C₈	—	158	4	no	—
159	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
160	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
161	C₂×C₈	#11929624	1	4	yes	—
162	C₂×C₈	#11929624	1	4	yes	—
163	C₄⋊C₄	—	158	8	no	—
164	C₄⋊C₄	—	158	8	no	—
165	C₄⋊C₄	—	158	8	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	158	4	no	—
169	C₄²	—	158	4	no	—
170	C₄²	C₂×Dic₁₅₈	1	2	yes	—
171	C₄²	#11929624	1	2	yes	—
172	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	79	8	no	—
178	C₂×C₄	—	158	4	no	—
179	C₂×C₄	—	158	4	no	—
180	C₂×C₄	—	158	4	no	—
181	C₂×C₄	—	158	4	no	—
182	C₂×C₄	—	158	4	no	—
183	C₂×C₄	C₄×Dic₁₅₈	1	4	yes	—
184	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
185	C₂×C₄	C₂×#11929624	1	2	yes	—
186	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
187	C₂×C₄	C₂×#11929624	1	2	yes	—
188	C₈	—	2	4	no	—
189	C₈	—	2	2	no	—
190	C₈	—	316	4	no	—
191	C₈	#14579153	1	4	yes	—
192	C₂²	C₂×C₄×Dic₁₅₈	1	1	yes	characteristic
193	C₂²	C₄×#11929624	1	2	yes	—
194	C₂²	#131×D₇₉	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	4	no	—
197	C₄	—	316	4	no	—
198	C₄	—	158	8	no	—
199	C₄	C₂×#14579153	1	1	yes	characteristic
200	C₄	#19099097	1	2	yes	—
201	C₄	#19099120	1	1	yes	characteristic
202	C₂	C₄×#14579153	1	1	yes	characteristic
203	C₂	#23772332	1	2	yes	—
204	C₂	#23772371	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇₉
commutator, socle, prime radical	C₂³×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1726	C₂
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1726	D₇₉
hypocenter	C₇₉	#21676
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
110. C₇₉	#21676	semidirect

Maximal subgroups

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

subgroup	index
#23766705	2
#23764123	2
#23763964	2
#23763520	2
C₈×Dic₇₉⋊C₄	2
C₈×C₄⋊Dic₇₉	2
C₇₉×#1726	2
#21676	79