Subgroups — C₇₉×#12043 #34123732

Order 20224.

238 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×SD₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1371	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₉×#1331	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₃₂×C₇₉	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₇₉	—	2	1	no	—
10	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
11	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₇₉	—	2	1	no	—
13	#367×C₇₉	C₄	1	1	yes	characteristic
14	#358×C₇₉	C₄	1	1	yes	characteristic
15	#357×C₇₉	C₄	1	1	yes	characteristic
16	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
18	#335×C₇₉	C₂²	1	1	yes	characteristic
19	#335×C₇₉	C₂²	1	1	yes	characteristic
20	Q₈⋊C₈×C₇₉	—	2	1	no	—
21	D₄⋊C₈×C₇₉	—	2	1	no	—
22	C₈²×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×D₈×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	—	4	1	no	—
26	C₄×Q₈×C₇₉	—	2	1	no	—
27	C₄×D₄×C₇₉	—	2	1	no	—
28	SD₃₂×C₇₉	C₈	1	4	yes	complemented
29	C₂×C₁₆×C₇₉	—	2	1	no	—
30	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
32	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
33	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₇₉	—	2	1	no	—
35	C₄⋊C₈×C₇₉	—	2	1	no	—
36	C₄⋊C₈×C₇₉	—	4	1	no	—
37	#103×C₇₉	—	2	1	no	—
38	#102×C₇₉	—	2	1	no	—
39	C₂²⋊C₈×C₇₉	—	4	1	no	—
40	C₄×C₈×C₇₉	—	4	1	no	—
41	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₇₉	—	2	1	no	—
45	C₂×D₄×C₇₉	—	2	1	no	—
46	C₂²×C₄×C₇₉	—	4	1	no	—
47	Q₁₆×C₇₉	—	2	1	no	—
48	Q₁₆×C₇₉	C₂×C₈	1	2	yes	complemented
49	D₈×C₇₉	—	2	1	no	—
50	D₈×C₇₉	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₇₉	—	2	1	no	—
52	C₂×C₈×C₇₉	—	4	1	no	—
53	C₂×C₈×C₇₉	—	4	1	no	—
54	C₂×C₈×C₇₉	—	8	2	no	—
55	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
56	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
59	C₄⋊C₄×C₇₉	—	2	1	no	—
60	C₄⋊C₄×C₇₉	—	2	1	no	—
61	C₄⋊C₄×C₇₉	—	4	1	no	—
62	C₂²⋊C₄×C₇₉	—	4	1	no	—
63	C₄²×C₇₉	—	4	1	no	—
64	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₁₆×C₇₉	—	2	2	no	—
66	C₁₆×C₇₉	—	2	1	no	—
67	C₁₆×C₇₉	C₂×C₈	1	2	yes	complemented
68	C₂³×C₇₉	—	4	1	no	—
69	Q₈×C₇₉	—	2	2	no	—
70	Q₈×C₇₉	—	4	1	no	—
71	D₄×C₇₉	—	2	2	no	—
72	D₄×C₇₉	—	4	1	no	—
73	C₂×C₄×C₇₉	—	4	1	no	—
74	C₂×C₄×C₇₉	—	4	1	no	—
75	C₂×C₄×C₇₉	—	4	1	no	—
76	C₂×C₄×C₇₉	—	8	1	no	—
77	C₂×C₄×C₇₉	—	8	1	no	—
78	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
81	C₈×C₇₉	—	2	1	no	—
82	C₈×C₇₉	—	2	1	no	—
83	C₈×C₇₉	—	2	1	no	—
84	C₈×C₇₉	—	4	1	no	—
85	C₈×C₇₉	—	8	1	no	—
86	C₈×C₇₉	—	8	1	no	—
87	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
88	C₈×C₇₉	#131	1	1	yes	characteristic
89	C₂²×C₇₉	—	4	2	no	—
90	C₂²×C₇₉	—	8	1	no	—
91	C₂²×C₇₉	—	8	1	no	—
92	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
93	C₄×C₇₉	—	2	1	no	—
94	C₄×C₇₉	—	4	2	no	—
95	C₄×C₇₉	—	8	1	no	—
96	C₄×C₇₉	—	8	1	no	—
97	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
98	C₄×C₇₉	#443	1	1	yes	characteristic
99	C₄×C₇₉	#509	1	1	yes	characteristic
100	C₄×C₇₉	#510	1	1	yes	characteristic
101	#12043	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂×C₇₉	—	8	2	no	—
103	C₂×C₇₉	C₈×D₈	1	1	yes	characteristic
104	C₂×C₇₉	#2175	1	1	yes	characteristic
105	C₂×C₇₉	#2179	1	1	yes	characteristic
106	C₄×SD₃₂	C₂×C₇₉	1	1	yes	characteristic
107	C₈×Q₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
108	C₈×D₈	C₂×C₇₉	1	1	yes	characteristic
109	#1371	C₂×C₇₉	1	1	yes	characteristic, complemented
110	#1332	C₂×C₇₉	1	1	yes	characteristic, complemented
111	#1331	C₂×C₇₉	1	1	yes	characteristic
112	#1311	C₂×C₇₉	1	1	yes	characteristic, complemented
113	C₇₉	#12043	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×SD₃₂	C₄×C₇₉	1	1	yes	characteristic
115	C₈×Q₈	—	2	1	no	—
116	C₄×Q₁₆	C₂²×C₇₉	1	1	yes	characteristic
117	C₄×D₈	C₂²×C₇₉	1	1	yes	characteristic
118	C₈×D₄	—	2	1	no	—
119	#367	C₄×C₇₉	1	1	yes	characteristic
120	#358	C₄×C₇₉	1	1	yes	characteristic
121	#357	C₄×C₇₉	1	1	yes	characteristic
122	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
123	C₄×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
124	#335	C₂²×C₇₉	1	1	yes	characteristic
125	#335	C₂²×C₇₉	1	1	yes	characteristic
126	Q₈⋊C₈	—	2	1	no	—
127	D₄⋊C₈	—	2	1	no	—
128	C₈²	C₂²×C₇₉	1	1	yes	characteristic
129	C₂×Q₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
130	C₂×D₈	C₂×C₄×C₇₉	1	1	yes	characteristic
131	C₂²×C₈	—	4	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×D₄	—	2	1	no	—
134	SD₃₂	C₈×C₇₉	1	4	yes	complemented
135	C₂×C₁₆	—	2	1	no	—
136	C₂×C₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
137	C₂×C₁₆	C₂×C₄×C₇₉	1	1	yes	characteristic
138	#107	C₂×C₄×C₇₉	1	1	yes	characteristic
139	#107	C₂×C₄×C₇₉	1	1	yes	characteristic
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	4	1	no	—
143	#103	—	2	1	no	—
144	#102	—	2	1	no	—
145	C₂²⋊C₈	—	4	1	no	—
146	C₄×C₈	—	4	1	no	—
147	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
148	C₄×C₈	D₄×C₇₉	1	1	yes	characteristic
149	C₄×C₈	C₂³×C₇₉	1	1	yes	characteristic
150	C₂×Q₈	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂²×C₄	—	4	1	no	—
153	Q₁₆	—	2	1	no	—
154	Q₁₆	C₂×C₈×C₇₉	1	2	yes	complemented
155	D₈	—	2	1	no	—
156	D₈	C₂×C₈×C₇₉	1	2	yes	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	8	2	no	—
161	C₂×C₈	D₈×C₇₉	1	1	yes	characteristic
162	C₂×C₈	D₈×C₇₉	1	1	yes	characteristic
163	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
164	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₂²⋊C₄	—	4	1	no	—
169	C₄²	—	4	1	no	—
170	C₄²	C₂×D₄×C₇₉	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	1	no	—
173	C₁₆	C₂×C₈×C₇₉	1	2	yes	complemented
174	C₂³	—	4	1	no	—
175	Q₈	—	2	2	no	—
176	Q₈	—	4	1	no	—
177	D₄	—	2	2	no	—
178	D₄	—	4	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	8	1	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	C₄×D₄×C₇₉	1	1	yes	characteristic
185	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
186	C₂×C₄	#135×C₇₉	1	1	yes	characteristic
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	1	no	—
191	C₈	—	8	1	no	—
192	C₈	—	8	1	no	—
193	C₈	C₂²×C₈×C₇₉	1	1	yes	characteristic
194	C₈	#131×C₇₉	1	1	yes	characteristic
195	C₂²	—	4	2	no	—
196	C₂²	—	8	1	no	—
197	C₂²	—	8	1	no	—
198	C₂²	C₄×D₈×C₇₉	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	4	2	no	—
201	C₄	—	8	1	no	—
202	C₄	—	8	1	no	—
203	C₄	C₈×D₄×C₇₉	1	1	yes	characteristic
204	C₄	#443×C₇₉	1	1	yes	characteristic
205	C₄	#509×C₇₉	1	1	yes	characteristic
206	C₄	#510×C₇₉	1	1	yes	characteristic
207	C₂	—	8	2	no	—
208	C₂	C₈×D₈×C₇₉	1	1	yes	characteristic
209	C₂	C₇₉×#2175	1	1	yes	characteristic
210	C₂	C₇₉×#2179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇₉	C₂×D₈
commutator	C₈	C₂²×C₈×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	C₄×D₈
prime radical	C₂×D₈×C₇₉	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₈²×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×Q₁₆×C₇₉	C₂	semidirect
4. C₇₉×#1371	C₂	semidirect
5. C₇₉×#1332	C₂	semidirect
7. C₇₉×#1311	C₂	semidirect
28. SD₃₂×C₇₉	C₈	semidirect
48. Q₁₆×C₇₉	C₂×C₈	semidirect
67. C₁₆×C₇₉	C₂×C₈	semidirect
101. #12043	C₇₉	direct (N × H)
107. C₈×Q₁₆	C₂×C₇₉	semidirect
109. #1371	C₂×C₇₉	semidirect
110. #1332	C₂×C₇₉	semidirect
112. #1311	C₂×C₇₉	semidirect
113. C₇₉	#12043	direct (N × H)
134. SD₃₂	C₈×C₇₉	semidirect
154. Q₁₆	C₂×C₈×C₇₉	semidirect
173. C₁₆	C₂×C₈×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×SD₃₂×C₇₉	2
C₈×Q₁₆×C₇₉	2
C₈×D₈×C₇₉	2
C₇₉×#1371	2
C₇₉×#1332	2
C₇₉×#1331	2
C₇₉×#1311	2
#12043	79

Subgroups — C79×#12043 #34123732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#12043 #34123732