Subgroups — C₇₉×#2898 #23760520

Order 10112.

726 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#131×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#520×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#419×C₇₉	C₂	1	3	yes	maximal, complemented
5	#417×C₇₉	C₂	1	3	yes	maximal, complemented
6	#416×C₇₉	C₂	1	3	yes	maximal, complemented
7	C₂×#103×C₇₉	C₂	1	3	yes	maximal, complemented
8	2₋¹⁺⁴×C₇₉	—	2	4	no	—
9	2₋¹⁺⁴×C₇₉	C₄	1	8	yes	complemented
10	C₂×#41×C₇₉	—	2	3	no	—
11	C₂×#41×C₇₉	C₂²	1	3	yes	—
12	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₇₉	—	2	1	no	—
14	C₂²×Q₈×C₇₉	C₂²	1	3	yes	—
15	#131×C₇₉	—	2	4	no	—
16	C₂×M₁₆×C₇₉	C₂²	1	3	yes	—
17	C₂²×C₈×C₇₉	C₂²	1	3	yes	—
18	C₄×Q₈×C₇₉	—	2	1	no	—
19	C₄×D₄×C₇₉	—	2	3	no	—
20	#117×C₇₉	C₂²	1	3	yes	—
21	C₂×C₄⋊C₄×C₇₉	C₂²	1	3	yes	—
22	#103×C₇₉	C₂²	1	6	yes	—
23	#103×C₇₉	C₂²	1	3	yes	—
24	#103×C₇₉	C₂²	1	3	yes	—
25	#102×C₇₉	C₂²	1	3	yes	—
26	#102×C₇₉	C₂²	1	1	yes	characteristic
27	#41×C₇₉	—	2	24	no	—
28	#41×C₇₉	—	2	6	no	—
29	#41×C₇₉	C₂×C₄	1	4	yes	—
30	#41×C₇₉	C₂×C₄	1	12	yes	complemented
31	#41×C₇₉	D₄	1	4	yes	—
32	C₂×Q₈×C₇₉	—	2	8	no	—
33	C₂×Q₈×C₇₉	—	2	3	no	—
34	C₂×Q₈×C₇₉	—	2	6	no	—
35	C₂×Q₈×C₇₉	C₂×C₄	1	12	yes	complemented
36	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₇₉	C₂³	1	3	yes	—
38	C₂×D₄×C₇₉	—	2	3	no	—
39	C₂×D₄×C₇₉	D₄	1	3	yes	—
40	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	—	2	3	no	—
42	C₂²×C₄×C₇₉	—	2	3	no	—
43	C₂²×C₄×C₇₉	—	2	3	no	—
44	C₂²×C₄×C₇₉	C₂³	1	3	yes	—
45	M₁₆×C₇₉	—	2	6	no	—
46	C₂×C₈×C₇₉	—	2	6	no	—
47	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂³	1	3	yes	—
49	C₄⋊C₄×C₇₉	—	2	3	no	—
50	C₄⋊C₄×C₇₉	C₂³	1	3	yes	—
51	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
52	C₂²⋊C₄×C₇₉	—	2	3	no	—
53	C₄²×C₇₉	—	2	3	no	—
54	C₂³×C₇₉	—	2	1	no	—
55	C₂³×C₇₉	C₂×D₄	1	3	yes	—
56	Q₈×C₇₉	—	2	12	no	—
57	Q₈×C₇₉	—	2	3	no	—
58	Q₈×C₇₉	C₂²⋊C₄	1	4	yes	—
59	Q₈×C₇₉	C₂²×C₄	1	6	yes	—
60	D₄×C₇₉	—	2	12	no	—
61	D₄×C₇₉	—	2	1	no	—
62	D₄×C₇₉	C₂²⋊C₄	1	12	yes	—
63	D₄×C₇₉	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₇₉	—	2	12	no	—
65	C₂×C₄×C₇₉	—	2	3	no	—
66	C₂×C₄×C₇₉	—	2	12	no	—
67	C₂×C₄×C₇₉	—	2	1	no	—
68	C₂×C₄×C₇₉	—	2	3	no	—
69	C₂×C₄×C₇₉	—	8	3	no	—
70	C₂×C₄×C₇₉	C₂²×C₄	1	6	yes	—
71	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
72	C₂×C₄×C₇₉	C₂×D₄	1	6	yes	—
73	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
74	C₈×C₇₉	—	2	1	no	—
75	C₈×C₇₉	—	2	3	no	—
76	C₂²×C₇₉	—	2	3	no	—
77	C₂²×C₇₉	—	2	3	no	—
78	C₂²×C₇₉	—	2	2	no	—
79	C₂²×C₇₉	—	4	1	no	—
80	C₂²×C₇₉	—	4	1	no	—
81	C₂²×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
82	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₇₉	—	2	6	no	—
84	C₄×C₇₉	—	4	3	no	—
85	C₄×C₇₉	—	4	1	no	—
86	C₄×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
87	C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₇₉	—	2	6	no	—
90	C₂×C₇₉	—	4	2	no	—
91	C₂×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₇₉	#577	1	1	yes	characteristic
93	C₂×C₇₉	#578	1	1	yes	characteristic
94	#2898	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₇₉	#2898	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×2₋¹⁺⁴	C₂×C₇₉	1	1	yes	characteristic
97	C₂×#131	C₂×C₇₉	1	1	yes	characteristic, complemented
98	#520	C₂×C₇₉	1	1	yes	characteristic, complemented
99	#419	C₂×C₇₉	1	3	yes	complemented
100	#417	C₂×C₇₉	1	3	yes	complemented
101	#416	C₂×C₇₉	1	3	yes	complemented
102	C₂×#103	C₂×C₇₉	1	3	yes	complemented
103	2₋¹⁺⁴	—	2	4	no	—
104	2₋¹⁺⁴	C₄×C₇₉	1	8	yes	complemented
105	C₂×#41	—	2	3	no	—
106	C₂×#41	C₂²×C₇₉	1	3	yes	—
107	C₂×#41	C₂²×C₇₉	1	1	yes	characteristic
108	C₂²×Q₈	—	2	1	no	—
109	C₂²×Q₈	C₂²×C₇₉	1	3	yes	—
110	#131	—	2	4	no	—
111	C₂×M₁₆	C₂²×C₇₉	1	3	yes	—
112	C₂²×C₈	C₂²×C₇₉	1	3	yes	—
113	C₄×Q₈	—	2	1	no	—
114	C₄×D₄	—	2	3	no	—
115	#117	C₂²×C₇₉	1	3	yes	—
116	C₂×C₄⋊C₄	C₂²×C₇₉	1	3	yes	—
117	#103	C₂²×C₇₉	1	3	yes	—
118	#103	C₂²×C₇₉	1	6	yes	—
119	#103	C₂²×C₇₉	1	3	yes	—
120	#102	C₂²×C₇₉	1	1	yes	characteristic
121	#102	C₂²×C₇₉	1	3	yes	—
122	#41	—	2	24	no	—
123	#41	—	2	6	no	—
124	#41	C₂×C₄×C₇₉	1	4	yes	—
125	#41	C₂×C₄×C₇₉	1	12	yes	complemented
126	#41	D₄×C₇₉	1	4	yes	—
127	C₂×Q₈	—	2	8	no	—
128	C₂×Q₈	—	2	3	no	—
129	C₂×Q₈	—	2	6	no	—
130	C₂×Q₈	C₂×C₄×C₇₉	1	12	yes	complemented
131	C₂×Q₈	D₄×C₇₉	1	1	yes	characteristic
132	C₂×Q₈	C₂³×C₇₉	1	3	yes	—
133	C₂×D₄	—	2	3	no	—
134	C₂×D₄	D₄×C₇₉	1	3	yes	—
135	C₂×D₄	C₂³×C₇₉	1	1	yes	characteristic
136	C₂²×C₄	—	2	3	no	—
137	C₂²×C₄	—	2	3	no	—
138	C₂²×C₄	—	2	3	no	—
139	C₂²×C₄	C₂³×C₇₉	1	3	yes	—
140	M₁₆	—	2	6	no	—
141	C₂×C₈	—	2	6	no	—
142	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
143	C₂×C₈	C₂³×C₇₉	1	3	yes	—
144	C₄⋊C₄	—	2	3	no	—
145	C₄⋊C₄	C₂³×C₇₉	1	3	yes	—
146	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
147	C₂²⋊C₄	—	2	3	no	—
148	C₄²	—	2	3	no	—
149	C₂³	—	2	1	no	—
150	C₂³	C₂×D₄×C₇₉	1	3	yes	—
151	Q₈	—	2	12	no	—
152	Q₈	—	2	3	no	—
153	Q₈	C₂²⋊C₄×C₇₉	1	4	yes	—
154	Q₈	C₂²×C₄×C₇₉	1	6	yes	—
155	D₄	—	2	12	no	—
156	D₄	—	2	1	no	—
157	D₄	C₂²⋊C₄×C₇₉	1	12	yes	—
158	D₄	C₂²×C₄×C₇₉	1	2	yes	—
159	C₂×C₄	—	2	12	no	—
160	C₂×C₄	—	2	3	no	—
161	C₂×C₄	—	2	12	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	3	no	—
164	C₂×C₄	—	8	3	no	—
165	C₂×C₄	C₂²×C₄×C₇₉	1	6	yes	—
166	C₂×C₄	C₂×D₄×C₇₉	1	3	yes	—
167	C₂×C₄	C₂×D₄×C₇₉	1	6	yes	—
168	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	3	no	—
171	C₂²	—	2	3	no	—
172	C₂²	—	2	3	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	1	no	—
175	C₂²	—	4	1	no	—
176	C₂²	C₂×C₂²⋊C₄×C₇₉	1	6	yes	—
177	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
178	C₄	—	2	6	no	—
179	C₄	—	4	3	no	—
180	C₄	—	4	1	no	—
181	C₄	C₂×C₂²⋊C₄×C₇₉	1	6	yes	—
182	C₄	C₂³×C₄×C₇₉	1	1	yes	characteristic
183	C₄	C₂²×D₄×C₇₉	1	1	yes	characteristic
184	C₂	—	2	6	no	—
185	C₂	—	4	2	no	—
186	C₂	C₂²×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
187	C₂	#577×C₇₉	1	1	yes	characteristic
188	C₂	#578×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂²×D₄
commutator	C₄	C₂³×C₄×C₇₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₇₉
prime radical	C₂×2₋¹⁺⁴×C₇₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×#41×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#131×C₇₉	C₂	semidirect
3. #520×C₇₉	C₂	semidirect
4. #419×C₇₉	C₂	semidirect
5. #417×C₇₉	C₂	semidirect
6. #416×C₇₉	C₂	semidirect
7. C₂×#103×C₇₉	C₂	semidirect
9. 2₋¹⁺⁴×C₇₉	C₄	semidirect
30. #41×C₇₉	C₂×C₄	semidirect
35. C₂×Q₈×C₇₉	C₂×C₄	semidirect
94. #2898	C₇₉	direct (N × H)
95. C₇₉	#2898	direct (N × H)
97. C₂×#131	C₂×C₇₉	semidirect
98. #520	C₂×C₇₉	semidirect
99. #419	C₂×C₇₉	semidirect
100. #417	C₂×C₇₉	semidirect
101. #416	C₂×C₇₉	semidirect
102. C₂×#103	C₂×C₇₉	semidirect
104. 2₋¹⁺⁴	C₄×C₇₉	semidirect
125. #41	C₂×C₄×C₇₉	semidirect
130. C₂×Q₈	C₂×C₄×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₇₉	2
C₂×#131×C₇₉	2
#520×C₇₉	2
#419×C₇₉	2
#417×C₇₉	2
#416×C₇₉	2
C₂×#103×C₇₉	2
#2898	79

Subgroups — C79×#2898 #23760520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#2898 #23760520