Subgroups — C₇₉×#2546 #23760168

Order 10112.

540 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₇₉	C₂	1	2	yes	maximal
2	C₂×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#398×C₇₉	C₂	1	4	yes	maximal
5	#391×C₇₉	C₂	1	2	yes	maximal
6	#389×C₇₉	C₂	1	4	yes	maximal
7	C₄×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₇₉	C₂²	1	2	yes	—
9	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₇₉	—	2	8	no	—
11	C₄⋊Q₈×C₇₉	—	2	4	no	—
12	C₄×Q₈×C₇₉	—	2	4	no	—
13	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₇₉	C₂²	1	8	yes	—
17	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₇₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
22	#95×C₇₉	C₂²	1	2	yes	—
23	#95×C₇₉	C₂²	1	8	yes	—
24	C₂×Q₈×C₇₉	—	2	2	no	—
25	C₂×Q₈×C₇₉	—	2	4	no	—
26	C₂×Q₈×C₇₉	—	2	4	no	—
27	C₂×Q₈×C₇₉	—	2	2	no	—
28	C₂×Q₈×C₇₉	—	2	4	no	—
29	C₂×Q₈×C₇₉	D₄	1	4	yes	—
30	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₇₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₇₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₇₉	—	2	4	no	—
38	C₄⋊C₄×C₇₉	—	2	2	no	—
39	C₄⋊C₄×C₇₉	—	2	16	no	—
40	C₄⋊C₄×C₇₉	—	2	4	no	—
41	C₄⋊C₄×C₇₉	—	2	4	no	—
42	C₄⋊C₄×C₇₉	Q₈	1	8	yes	complemented
43	C₄²×C₇₉	—	2	4	no	—
44	C₄²×C₇₉	—	2	4	no	—
45	C₄²×C₇₉	D₄	1	4	yes	—
46	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
47	Q₈×C₇₉	—	2	8	no	—
48	Q₈×C₇₉	—	4	8	no	—
49	C₂×C₄×C₇₉	—	2	4	no	—
50	C₂×C₄×C₇₉	—	2	2	no	—
51	C₂×C₄×C₇₉	—	2	2	no	—
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	—	2	4	no	—
54	C₂×C₄×C₇₉	—	2	2	no	—
55	C₂×C₄×C₇₉	—	2	4	no	—
56	C₂×C₄×C₇₉	—	2	4	no	—
57	C₂×C₄×C₇₉	—	2	4	no	—
58	C₂×C₄×C₇₉	—	2	4	no	—
59	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₇₉	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₇₉	C₂×Q₈	1	8	yes	—
65	C₂×C₄×C₇₉	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₇₉	#41	1	2	yes	—
67	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₇₉	C₂²×Q₈	1	2	yes	—
70	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
71	C₂²×C₇₉	2₋¹⁺⁴	1	2	yes	—
72	C₄×C₇₉	—	2	4	no	—
73	C₄×C₇₉	—	2	2	no	—
74	C₄×C₇₉	—	2	8	no	—
75	C₄×C₇₉	—	4	4	no	—
76	C₄×C₇₉	—	4	2	no	—
77	C₄×C₇₉	#121	1	4	yes	—
78	C₄×C₇₉	C₄⋊Q₈	1	4	yes	complemented
79	C₂×C₇₉	C₂×#121	1	1	yes	characteristic
80	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₇₉	#536	1	1	yes	characteristic
82	C₂×C₇₉	D₄×Q₈	1	2	yes	—
83	C₂×C₇₉	#557	1	2	yes	—
84	#2546	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₇₉	#2546	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×C₄⋊Q₈	C₂×C₇₉	1	2	yes	—
87	C₂×C₄⋊Q₈	C₂×C₇₉	1	1	yes	characteristic
88	C₂×C₄×Q₈	C₂×C₇₉	1	1	yes	characteristic
89	#398	C₂×C₇₉	1	4	yes	—
90	#391	C₂×C₇₉	1	2	yes	—
91	#389	C₂×C₇₉	1	4	yes	—
92	C₄×C₄⋊C₄	C₂×C₇₉	1	1	yes	characteristic
93	C₂²×Q₈	C₂²×C₇₉	1	2	yes	—
94	C₂²×Q₈	C₂²×C₇₉	1	1	yes	characteristic
95	C₄⋊Q₈	—	2	8	no	—
96	C₄⋊Q₈	—	2	4	no	—
97	C₄×Q₈	—	2	4	no	—
98	C₂×C₄⋊C₄	C₂²×C₇₉	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂²×C₇₉	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₇₉	1	2	yes	—
101	C₂×C₄⋊C₄	C₂²×C₇₉	1	8	yes	—
102	C₂×C₄⋊C₄	C₂²×C₇₉	1	2	yes	—
103	C₂×C₄⋊C₄	C₂²×C₇₉	1	2	yes	—
104	C₂×C₄²	C₂²×C₇₉	1	2	yes	—
105	C₂×C₄²	C₂²×C₇₉	1	2	yes	—
106	C₂×C₄²	C₂²×C₇₉	1	1	yes	characteristic
107	#95	C₂²×C₇₉	1	2	yes	—
108	#95	C₂²×C₇₉	1	8	yes	—
109	C₂×Q₈	—	2	2	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	2	4	no	—
112	C₂×Q₈	—	2	2	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	D₄×C₇₉	1	4	yes	—
115	C₂²×C₄	C₂³×C₇₉	1	2	yes	—
116	C₂²×C₄	C₂³×C₇₉	1	4	yes	—
117	C₂²×C₄	C₂³×C₇₉	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₇₉	1	2	yes	—
119	C₂²×C₄	C₂³×C₇₉	1	4	yes	—
120	C₂²×C₄	C₂³×C₇₉	1	1	yes	characteristic
121	C₂²×C₄	C₂³×C₇₉	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	16	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	Q₈×C₇₉	1	8	yes	complemented
128	C₄²	—	2	4	no	—
129	C₄²	—	2	4	no	—
130	C₄²	D₄×C₇₉	1	4	yes	—
131	C₂³	C₂⁴×C₇₉	1	1	yes	characteristic
132	Q₈	—	2	8	no	—
133	Q₈	—	4	8	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
145	C₂×C₄	C₂×D₄×C₇₉	1	4	yes	—
146	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
147	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
148	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
149	C₂×C₄	C₂×Q₈×C₇₉	1	8	yes	—
150	C₂×C₄	C₂×Q₈×C₇₉	1	4	yes	—
151	C₂×C₄	#41×C₇₉	1	2	yes	—
152	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
153	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
154	C₂²	C₂²×Q₈×C₇₉	1	2	yes	—
155	C₂²	C₂×#41×C₇₉	1	1	yes	characteristic
156	C₂²	2₋¹⁺⁴×C₇₉	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	2	no	—
162	C₄	#121×C₇₉	1	4	yes	—
163	C₄	C₄⋊Q₈×C₇₉	1	4	yes	complemented
164	C₂	C₂×#121×C₇₉	1	1	yes	characteristic
165	C₂	C₂×C₄⋊Q₈×C₇₉	1	1	yes	characteristic
166	C₂	#536×C₇₉	1	1	yes	characteristic
167	C₂	D₄×Q₈×C₇₉	1	2	yes	—
168	C₂	#557×C₇₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
42. C₄⋊C₄×C₇₉	Q₈	semidirect
78. C₄×C₇₉	C₄⋊Q₈	semidirect
84. #2546	C₇₉	direct (N × H)
85. C₇₉	#2546	direct (N × H)
127. C₄⋊C₄	Q₈×C₇₉	semidirect
163. C₄	C₄⋊Q₈×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₇₉	2
C₂×C₄×Q₈×C₇₉	2
#398×C₇₉	2
#391×C₇₉	2
#389×C₇₉	2
C₄×C₄⋊C₄×C₇₉	2
#2546	79

Subgroups — C79×#2546 #23760168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#2546 #23760168