Subgroups — C₁₆×#41×C₇₉ #34145167

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₇₉	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₇₉	C₂	1	2	yes	maximal, complemented
3	D₄×C₁₆×C₇₉	C₂	1	6	yes	maximal, complemented
4	C₇₉×#2162	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₇₉	C₂	1	3	yes	maximal, complemented
6	C₄×#41×C₇₉	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₉	C₂²	1	6	yes	—
8	C₈×Q₈×C₇₉	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₈×D₄×C₇₉	C₄	1	3	yes	—
11	C₈×D₄×C₇₉	C₂²	1	3	yes	—
12	#431×C₇₉	C₄	1	3	yes	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	3	yes	—
14	C₄⋊C₁₆×C₇₉	C₂²	1	6	yes	—
15	C₂²⋊C₁₆×C₇₉	C₂²	1	6	yes	—
16	C₄×C₁₆×C₇₉	C₂²	1	3	yes	—
17	C₄×C₁₆×C₇₉	C₂²	1	6	yes	—
18	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#41×C₇₉	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₇₉	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₇₉	C₂³	1	3	yes	—
22	C₄×Q₈×C₇₉	C₈	1	1	yes	characteristic
23	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₄×C₇₉	C₈	1	3	yes	—
25	C₄×D₄×C₇₉	C₂×C₄	1	3	yes	—
26	#117×C₇₉	C₈	1	3	yes	—
27	C₂×C₄²×C₇₉	C₂×C₄	1	3	yes	—
28	C₂×C₁₆×C₇₉	—	2	12	no	—
29	C₂×C₁₆×C₇₉	C₂³	1	6	yes	—
30	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
31	C₄⋊C₈×C₇₉	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₇₉	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₇₉	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₇₉	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₇₉	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₇₉	C₂³	1	3	yes	—
37	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	#41×C₇₉	C₁₆	1	8	yes	direct product factor
39	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
40	C₂×D₄×C₇₉	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₇₉	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₇₉	—	2	6	no	—
45	C₂×C₈×C₇₉	—	2	6	no	—
46	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₇₉	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
50	C₄⋊C₄×C₇₉	C₂×C₈	1	3	yes	—
51	C₄⋊C₄×C₇₉	C₂×C₈	1	3	yes	—
52	C₂²⋊C₄×C₇₉	C₂×C₈	1	6	yes	—
53	C₄²×C₇₉	C₂×C₈	1	3	yes	—
54	C₄²×C₇₉	C₂×C₈	1	3	yes	—
55	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₇₉	C₂²×C₄	1	3	yes	—
57	C₁₆×C₇₉	—	2	6	no	—
58	C₁₆×C₇₉	#41	1	4	yes	direct product factor
59	C₂³×C₇₉	C₂²×C₈	1	3	yes	—
60	Q₈×C₇₉	C₂×C₁₆	1	4	yes	complemented
61	D₄×C₇₉	C₂×C₁₆	1	12	yes	complemented
62	C₂×C₄×C₇₉	—	2	3	no	—
63	C₂×C₄×C₇₉	—	2	3	no	—
64	C₂×C₄×C₇₉	—	2	6	no	—
65	C₂×C₄×C₇₉	C₂×C₁₆	1	12	yes	complemented
66	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₇₉	—	2	3	no	—
73	C₈×C₇₉	—	2	3	no	—
74	C₈×C₇₉	#131	1	2	yes	—
75	C₈×C₇₉	C₂×#41	1	2	yes	—
76	C₂²×C₇₉	—	2	3	no	—
77	C₂²×C₇₉	—	2	3	no	—
78	C₂²×C₇₉	C₂²×C₁₆	1	6	yes	—
79	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₇₉	—	2	3	no	—
81	C₄×C₇₉	—	2	3	no	—
82	C₄×C₇₉	C₂²×C₁₆	1	2	yes	—
83	C₄×C₇₉	C₂²×C₁₆	1	6	yes	—
84	C₄×C₇₉	#504	1	2	yes	—
85	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
86	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
87	C₁₆×#41	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₇₉	—	2	6	no	—
89	C₂×C₇₉	C₈×#41	1	1	yes	characteristic
90	C₂×C₇₉	C₂³×C₁₆	1	1	yes	characteristic
91	C₂×C₇₉	C₂×#504	1	1	yes	characteristic
92	C₈×#41	C₂×C₇₉	1	1	yes	characteristic
93	Q₈×C₁₆	C₂×C₇₉	1	2	yes	complemented
94	D₄×C₁₆	C₂×C₇₉	1	6	yes	complemented
95	#2162	C₂×C₇₉	1	3	yes	complemented
96	C₂×C₄×C₁₆	C₂×C₇₉	1	3	yes	complemented
97	C₇₉	C₁₆×#41	1	1	yes	characteristic, direct product factor, Sylow
98	C₄×#41	C₄×C₇₉	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₇₉	1	6	yes	—
100	C₈×Q₈	C₄×C₇₉	1	1	yes	characteristic
101	C₈×Q₈	C₂²×C₇₉	1	1	yes	characteristic
102	C₈×D₄	C₄×C₇₉	1	3	yes	—
103	C₈×D₄	C₂²×C₇₉	1	3	yes	—
104	#431	C₄×C₇₉	1	3	yes	—
105	C₂×C₄×C₈	C₂²×C₇₉	1	3	yes	—
106	C₄⋊C₁₆	C₂²×C₇₉	1	6	yes	—
107	C₂²⋊C₁₆	C₂²×C₇₉	1	6	yes	—
108	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
109	C₄×C₁₆	C₂²×C₇₉	1	3	yes	—
110	C₄×C₁₆	C₂²×C₇₉	1	6	yes	—
111	C₂×#41	C₈×C₇₉	1	1	yes	characteristic
112	C₂²×C₈	C₂×C₄×C₇₉	1	3	yes	—
113	C₂²×C₈	C₂³×C₇₉	1	3	yes	—
114	C₄×Q₈	C₈×C₇₉	1	1	yes	characteristic
115	C₄×Q₈	C₂×C₄×C₇₉	1	1	yes	characteristic
116	C₄×D₄	C₈×C₇₉	1	3	yes	—
117	C₄×D₄	C₂×C₄×C₇₉	1	3	yes	—
118	#117	C₈×C₇₉	1	3	yes	—
119	C₂×C₄²	C₂×C₄×C₇₉	1	3	yes	—
120	C₂×C₁₆	—	2	12	no	—
121	C₂×C₁₆	C₂³×C₇₉	1	6	yes	—
122	C₂×C₁₆	C₂³×C₇₉	1	2	yes	—
123	C₄⋊C₈	C₂×C₄×C₇₉	1	3	yes	—
124	C₄⋊C₈	C₂×C₄×C₇₉	1	3	yes	—
125	C₂²⋊C₈	C₂×C₄×C₇₉	1	6	yes	—
126	C₄×C₈	C₂×C₄×C₇₉	1	3	yes	—
127	C₄×C₈	C₂×C₄×C₇₉	1	3	yes	—
128	C₄×C₈	C₂³×C₇₉	1	1	yes	characteristic
129	C₄×C₈	C₂³×C₇₉	1	3	yes	—
130	#41	C₁₆×C₇₉	1	8	yes	direct product factor
131	C₂×Q₈	C₂×C₈×C₇₉	1	1	yes	characteristic
132	C₂×D₄	C₂×C₈×C₇₉	1	3	yes	—
133	C₂²×C₄	C₂×C₈×C₇₉	1	3	yes	—
134	C₂²×C₄	C₂×C₈×C₇₉	1	3	yes	—
135	C₂²×C₄	C₂²×C₄×C₇₉	1	3	yes	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	C₂²×C₄×C₇₉	1	3	yes	—
139	C₂×C₈	C₂²×C₄×C₇₉	1	3	yes	—
140	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
141	C₂×C₈	C₂⁴×C₇₉	1	1	yes	characteristic
142	C₄⋊C₄	C₂×C₈×C₇₉	1	3	yes	—
143	C₄⋊C₄	C₂×C₈×C₇₉	1	3	yes	—
144	C₂²⋊C₄	C₂×C₈×C₇₉	1	6	yes	—
145	C₄²	C₂×C₈×C₇₉	1	3	yes	—
146	C₄²	C₂×C₈×C₇₉	1	3	yes	—
147	C₄²	C₂²×C₄×C₇₉	1	1	yes	characteristic
148	C₄²	C₂²×C₄×C₇₉	1	3	yes	—
149	C₁₆	—	2	6	no	—
150	C₁₆	#41×C₇₉	1	4	yes	direct product factor
151	C₂³	C₂²×C₈×C₇₉	1	3	yes	—
152	Q₈	C₂×C₁₆×C₇₉	1	4	yes	complemented
153	D₄	C₂×C₁₆×C₇₉	1	12	yes	complemented
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	C₂×C₁₆×C₇₉	1	12	yes	complemented
158	C₂×C₄	C₂²×C₈×C₇₉	1	3	yes	—
159	C₂×C₄	C₂²×C₈×C₇₉	1	3	yes	—
160	C₂×C₄	C₂²×C₈×C₇₉	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₈×C₇₉	1	3	yes	—
162	C₂×C₄	C₂²×C₈×C₇₉	1	1	yes	characteristic
163	C₂×C₄	C₂³×C₄×C₇₉	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	2	3	no	—
166	C₈	#131×C₇₉	1	2	yes	—
167	C₈	C₂×#41×C₇₉	1	2	yes	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	C₂²×C₁₆×C₇₉	1	6	yes	—
171	C₂²	C₂³×C₈×C₇₉	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂²×C₁₆×C₇₉	1	2	yes	—
175	C₄	C₂²×C₁₆×C₇₉	1	6	yes	—
176	C₄	#504×C₇₉	1	2	yes	—
177	C₄	C₄×#41×C₇₉	1	1	yes	characteristic
178	C₄	C₂×#131×C₇₉	1	1	yes	characteristic
179	C₂	—	2	6	no	—
180	C₂	C₈×#41×C₇₉	1	1	yes	characteristic
181	C₂	C₂³×C₁₆×C₇₉	1	1	yes	characteristic
182	C₂	C₂×#504×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₆×C₇₉	C₂²
commutator	C₂	C₂³×C₁₆×C₇₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₇₉
socle	C₂²×C₇₉	C₂³×C₈
prime radical	C₂×#41×C₇₉	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×C₁₆×C₇₉	C₂	semidirect
3. D₄×C₁₆×C₇₉	C₂	semidirect
4. C₇₉×#2162	C₂	semidirect
5. C₂×C₄×C₁₆×C₇₉	C₂	semidirect
38. #41×C₇₉	C₁₆	direct (N × H)
58. C₁₆×C₇₉	#41	direct (N × H)
60. Q₈×C₇₉	C₂×C₁₆	semidirect
61. D₄×C₇₉	C₂×C₁₆	semidirect
65. C₂×C₄×C₇₉	C₂×C₁₆	semidirect
87. C₁₆×#41	C₇₉	direct (N × H)
93. Q₈×C₁₆	C₂×C₇₉	semidirect
94. D₄×C₁₆	C₂×C₇₉	semidirect
95. #2162	C₂×C₇₉	semidirect
96. C₂×C₄×C₁₆	C₂×C₇₉	semidirect
97. C₇₉	C₁₆×#41	direct (N × H)
130. #41	C₁₆×C₇₉	direct (N × H)
150. C₁₆	#41×C₇₉	direct (N × H)
152. Q₈	C₂×C₁₆×C₇₉	semidirect
153. D₄	C₂×C₁₆×C₇₉	semidirect
157. C₂×C₄	C₂×C₁₆×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₇₉	2
Q₈×C₁₆×C₇₉	2
D₄×C₁₆×C₇₉	2
C₇₉×#2162	2
C₂×C₄×C₁₆×C₇₉	2
C₁₆×#41	79

Subgroups — C16×#41×C79 #34145167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#41×C₇₉ #34145167