Subgroups — #346×Dic₇₉ #34203089

Order 20224.

510 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098338	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₇₉	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₉	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₉	C₄	1	1	yes	characteristic
13	#19098339	C₄	1	4	yes	complemented
14	#19098338	C₄	1	8	yes	complemented
15	C₁₆×Dic₇₉	C₄	1	8	yes	complemented
16	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#346×C₇₉	C₄	1	4	yes	complemented
19	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₉	C₈	1	4	yes	—
22	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
25	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
27	C₄×C₇₉⋊C₈	C₈	1	4	yes	—
28	C₈×Dic₇₉	C₈	1	4	yes	—
29	C₈×Dic₇₉	C₈	1	4	yes	—
30	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
31	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
36	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
40	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
41	C₂²×Dic₇₉	C₂×C₈	1	2	yes	—
42	C₂×C₇₉⋊C₈	C₄²	1	2	yes	—
43	C₂×C₇₉⋊C₈	C₄²	1	2	yes	—
44	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
45	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
46	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
47	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
48	C₄×Dic₇₉	—	2	8	no	—
49	C₄×Dic₇₉	C₄²	1	2	yes	—
50	C₄×Dic₇₉	C₄²	1	2	yes	—
51	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
52	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
53	C₇₉⋊C₁₆	C₄²	1	8	yes	complemented
54	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₄²	1	2	yes	—
57	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₄²	1	2	yes	—
64	C₄²×C₇₉	C₂×C₈	1	2	yes	—
65	C₁₆×C₇₉	C₄²	1	8	yes	complemented
66	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	—	2	4	no	—
68	C₂×C₄×C₇₉	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
75	C₂×Dic₇₉	—	2	4	no	—
76	C₂×Dic₇₉	C₄×C₈	1	4	yes	—
77	C₂×Dic₇₉	C₄×C₈	1	4	yes	—
78	C₂×Dic₇₉	M₃₂	1	8	yes	—
79	C₈×C₇₉	C₄×C₈	1	4	yes	—
80	C₈×C₇₉	C₂×C₄²	1	2	yes	—
81	C₈×C₇₉	C₂×C₄²	1	2	yes	—
82	C₇₉⋊C₈	C₄×C₈	1	4	yes	—
83	C₇₉⋊C₈	C₄×C₈	1	4	yes	—
84	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
85	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
86	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
87	C₂²×C₇₉	C₂×M₃₂	1	4	yes	—
88	C₄×C₇₉	—	2	4	no	—
89	C₄×C₇₉	C₄³	1	1	yes	characteristic
90	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
91	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
93	Dic₇₉	—	2	4	no	—
94	Dic₇₉	#346	1	8	yes	direct product factor
95	C₄×#346	—	79	1	no	maximal, Sylow
96	C₂×C₇₉	C₄²×C₈	1	1	yes	characteristic
97	C₂×C₇₉	C₂×#346	1	2	yes	—
98	C₂×C₇₉	C₄×M₃₂	1	2	yes	—
99	C₂×C₇₉	C₄×M₃₂	1	2	yes	—
100	C₂×#346	—	79	2	no	—
101	C₂×#346	—	79	1	no	—
102	C₂×#346	D₇₉	1	1	yes	characteristic
103	C₂×C₄×C₁₆	—	79	2	no	—
104	C₄²×C₈	—	79	1	no	—
105	C₇₉	C₄×#346	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₁₆	—	79	2	no	—
107	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
108	C₂×C₄×C₈	—	79	1	no	—
109	C₂×C₄×C₈	—	79	1	no	—
110	C₂×C₄×C₈	—	79	1	no	—
111	C₂×C₄×C₈	—	79	1	no	—
112	C₂×C₄×C₈	—	79	1	no	—
113	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
114	C₄³	—	79	1	no	—
115	#346	—	79	8	no	—
116	#346	—	79	4	no	—
117	#346	Dic₇₉	1	4	yes	direct product factor
118	C₄×C₁₆	—	79	8	no	—
119	C₂²×C₈	—	79	1	no	—
120	C₂²×C₈	—	79	1	no	—
121	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
123	C₂×C₄²	—	79	1	no	—
124	C₂×C₄²	—	79	1	no	—
125	C₂×C₄²	—	79	4	no	—
126	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
127	C₂×C₁₆	—	79	4	no	—
128	C₂×C₁₆	—	79	4	no	—
129	C₂×C₁₆	—	79	4	no	—
130	C₂×C₁₆	C₄×D₇₉	1	4	yes	—
131	C₂×C₁₆	C₄×D₇₉	1	4	yes	—
132	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
133	C₄×C₈	—	79	4	no	—
134	C₄×C₈	—	79	4	no	—
135	C₄×C₈	—	79	4	no	—
136	C₄×C₈	—	79	4	no	—
137	C₄×C₈	—	79	4	no	—
138	C₄×C₈	C₄×D₇₉	1	2	yes	—
139	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
140	C₂²×C₄	—	79	2	no	—
141	C₂²×C₄	—	79	2	no	—
142	C₂²×C₄	C₈×D₇₉	1	2	yes	—
143	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
144	C₂×C₈	—	79	2	no	—
145	C₂×C₈	—	79	2	no	—
146	C₂×C₈	—	79	2	no	—
147	C₂×C₈	—	79	2	no	—
148	C₂×C₈	—	79	2	no	—
149	C₂×C₈	—	79	2	no	—
150	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
151	C₂×C₈	C₈×D₇₉	1	2	yes	—
152	C₂×C₈	C₈×D₇₉	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
154	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
155	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
156	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
157	C₄²	—	79	2	no	—
158	C₄²	—	79	2	no	—
159	C₄²	—	79	2	no	—
160	C₄²	—	79	2	no	—
161	C₄²	—	158	8	no	—
162	C₄²	C₄×Dic₇₉	1	2	yes	—
163	C₄²	C₈×D₇₉	1	2	yes	—
164	C₁₆	—	79	8	no	—
165	C₁₆	C₄×Dic₇₉	1	8	yes	complemented
166	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	79	8	no	—
169	C₂×C₄	—	79	4	no	—
170	C₂×C₄	—	79	4	no	—
171	C₂×C₄	—	158	4	no	—
172	C₂×C₄	C₈×Dic₇₉	1	4	yes	—
173	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
174	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
176	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
179	C₈	—	79	4	no	—
180	C₈	—	79	4	no	—
181	C₈	C₈×Dic₇₉	1	4	yes	—
182	C₈	C₄²×D₇₉	1	2	yes	—
183	C₈	C₂×C₄×Dic₇₉	1	2	yes	—
184	C₂²	C₄×C₈×D₇₉	1	1	yes	characteristic
185	C₂²	C₄×C₈×D₇₉	1	1	yes	characteristic
186	C₂²	C₂×C₈×Dic₇₉	1	1	yes	characteristic
187	C₂²	M₃₂×D₇₉	1	4	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	79	8	no	—
190	C₄	—	158	4	no	—
191	C₄	C₄²×Dic₇₉	1	1	yes	characteristic
192	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
193	C₄	C₄×C₈×D₇₉	1	1	yes	characteristic
194	C₄	C₂×C₈×Dic₇₉	1	1	yes	characteristic
195	C₂	C₄×C₈×Dic₇₉	1	1	yes	characteristic
196	C₂	#346×D₇₉	1	2	yes	—
197	C₂	M₃₂×Dic₇₉	1	2	yes	—
198	C₂	#23764993	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₉
commutator	C₂×C₇₉	C₄²×C₈
Fitting	C₂×#346×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×C₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	C₄×#346
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. #19098339	C₄	semidirect
14. #19098338	C₄	semidirect
15. C₁₆×Dic₇₉	C₄	semidirect
18. #346×C₇₉	C₄	semidirect
53. C₇₉⋊C₁₆	C₄²	semidirect
65. C₁₆×C₇₉	C₄²	semidirect
94. Dic₇₉	#346	direct (N × H)
105. C₇₉	C₄×#346	semidirect
117. #346	Dic₇₉	direct (N × H)
165. C₁₆	C₄×Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098339	2
C₂×#19098338	2
C₂×C₁₆×Dic₇₉	2
C₄×C₈×Dic₇₉	2
C₂×#346×C₇₉	2
C₄×#346	79

Subgroups — #346×Dic79 #34203089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #346×Dic₇₉ #34203089