Subgroups — C₇₉×#17619 #34129308

Order 20224.

938 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₇₉×#2982	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#2351	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1518	C₂	1	6	yes	maximal
5	C₇₉×#1503	C₂	1	6	yes	maximal
6	Q₈²×C₇₉	C₄	1	4	yes	complemented
7	#557×C₇₉	—	2	4	no	—
8	#557×C₇₉	C₄	1	4	yes	complemented
9	#544×C₇₉	—	2	3	no	—
10	#533×C₇₉	C₂²	1	6	yes	—
11	#519×C₇₉	—	2	1	no	—
12	C₂×C₄×Q₈×C₇₉	—	2	1	no	—
13	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₇₉	—	2	4	no	—
15	#431×C₇₉	C₂²	1	6	yes	—
16	#389×C₇₉	—	2	3	no	—
17	#382×C₇₉	C₂²	1	3	yes	—
18	C₄×C₄⋊C₄×C₇₉	C₂²	1	3	yes	—
19	#328×C₇₉	C₂²	1	4	yes	—
20	#328×C₇₉	C₂²	1	12	yes	—
21	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₇₉	—	2	12	no	—
23	C₄⋊Q₈×C₇₉	C₂×C₄	1	6	yes	—
24	C₄⋊Q₈×C₇₉	C₂×C₄	1	6	yes	—
25	#125×C₇₉	—	2	12	no	—
26	#125×C₇₉	—	2	6	no	—
27	C₂²⋊Q₈×C₇₉	—	2	6	no	—
28	C₂²⋊Q₈×C₇₉	C₂³	1	4	yes	—
29	C₄×Q₈×C₇₉	—	2	4	no	—
30	C₄×Q₈×C₇₉	—	2	4	no	—
31	C₄×Q₈×C₇₉	—	2	6	no	—
32	C₄×Q₈×C₇₉	—	4	4	no	—
33	C₄×Q₈×C₇₉	C₂×C₄	1	12	yes	—
34	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
35	C₄×Q₈×C₇₉	D₄	1	4	yes	—
36	#117×C₇₉	—	2	6	no	—
37	C₂×C₄⋊C₄×C₇₉	—	2	3	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	3	no	—
39	C₂×C₄⋊C₄×C₇₉	D₄	1	3	yes	—
40	C₂×C₄²×C₇₉	—	2	3	no	—
41	C₂×C₄²×C₇₉	C₂³	1	3	yes	—
42	C₄⋊C₈×C₇₉	—	2	6	no	—
43	C₂²⋊C₈×C₇₉	C₂³	1	4	yes	—
44	C₄×C₈×C₇₉	—	2	6	no	—
45	#95×C₇₉	—	2	3	no	—
46	#95×C₇₉	C₂³	1	1	yes	characteristic
47	#95×C₇₉	C₂³	1	3	yes	—
48	C₂×Q₈×C₇₉	—	2	6	no	—
49	C₂×Q₈×C₇₉	C₂²⋊C₄	1	4	yes	—
50	C₂×Q₈×C₇₉	SD₁₆	1	4	yes	—
51	C₂×Q₈×C₇₉	Q₁₆	1	4	yes	—
52	C₂×Q₈×C₇₉	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₇₉	—	2	1	no	—
54	C₂²×C₄×C₇₉	—	2	3	no	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₇₉	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₇₉	—	2	4	no	—
59	C₄⋊C₄×C₇₉	—	2	6	no	—
60	C₄⋊C₄×C₇₉	—	2	6	no	—
61	C₄⋊C₄×C₇₉	—	2	6	no	—
62	C₄⋊C₄×C₇₉	—	2	12	no	—
63	C₄⋊C₄×C₇₉	—	2	4	no	—
64	C₄⋊C₄×C₇₉	—	4	6	no	—
65	C₄⋊C₄×C₇₉	—	4	6	no	—
66	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	12	yes	—
67	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
68	C₂²⋊C₄×C₇₉	—	2	4	no	—
69	C₄²×C₇₉	—	2	12	no	—
70	C₄²×C₇₉	—	4	6	no	—
71	C₄²×C₇₉	C₂²×C₄	1	6	yes	—
72	C₄²×C₇₉	C₂×D₄	1	6	yes	—
73	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
74	Q₈×C₇₉	—	4	4	no	—
75	Q₈×C₇₉	—	8	6	no	—
76	Q₈×C₇₉	#103	1	16	yes	—
77	C₂×C₄×C₇₉	—	2	2	no	—
78	C₂×C₄×C₇₉	—	2	4	no	—
79	C₂×C₄×C₇₉	—	2	4	no	—
80	C₂×C₄×C₇₉	—	4	3	no	—
81	C₂×C₄×C₇₉	—	4	1	no	—
82	C₂×C₄×C₇₉	—	4	1	no	—
83	C₂×C₄×C₇₉	—	4	1	no	—
84	C₂×C₄×C₇₉	—	4	3	no	—
85	C₂×C₄×C₇₉	—	4	3	no	—
86	C₂×C₄×C₇₉	—	4	3	no	—
87	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
88	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂×C₄×C₇₉	C₂×SD₁₆	1	6	yes	—
90	C₂×C₄×C₇₉	C₂×Q₁₆	1	6	yes	—
91	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
93	C₈×C₇₉	—	4	4	no	—
94	C₂²×C₇₉	—	2	1	no	—
95	C₂²×C₇₉	—	2	1	no	—
96	C₂²×C₇₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
98	C₂²×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
99	C₄×C₇₉	—	2	1	no	—
100	C₄×C₇₉	—	2	1	no	—
101	C₄×C₇₉	—	2	6	no	—
102	C₄×C₇₉	—	4	2	no	—
103	C₄×C₇₉	—	4	4	no	—
104	C₄×C₇₉	—	4	4	no	—
105	C₄×C₇₉	—	8	3	no	—
106	C₄×C₇₉	C₂×#103	1	12	yes	—
107	#17619	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂×C₇₉	—	2	1	no	—
109	C₂×C₇₉	—	2	1	no	—
110	C₂×C₇₉	#2884	1	1	yes	characteristic
111	C₂×C₇₉	C₂²×#103	1	1	yes	characteristic
112	C₂×C₇₉	#2901	1	1	yes	characteristic
113	#3502	C₂×C₇₉	1	1	yes	characteristic
114	#2982	C₂×C₇₉	1	1	yes	characteristic
115	#2351	C₂×C₇₉	1	1	yes	characteristic
116	#1518	C₂×C₇₉	1	6	yes	—
117	#1503	C₂×C₇₉	1	6	yes	—
118	C₇₉	#17619	1	1	yes	characteristic, direct product factor, Sylow
119	Q₈²	C₄×C₇₉	1	4	yes	complemented
120	#557	—	2	4	no	—
121	#557	C₄×C₇₉	1	4	yes	complemented
122	#544	—	2	3	no	—
123	#533	C₂²×C₇₉	1	6	yes	—
124	#519	—	2	1	no	—
125	C₂×C₄×Q₈	—	2	1	no	—
126	C₂×C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
127	C₈×Q₈	—	2	4	no	—
128	#431	C₂²×C₇₉	1	6	yes	—
129	#389	—	2	3	no	—
130	#382	C₂²×C₇₉	1	3	yes	—
131	C₄×C₄⋊C₄	C₂²×C₇₉	1	3	yes	—
132	#328	C₂²×C₇₉	1	4	yes	—
133	#328	C₂²×C₇₉	1	12	yes	—
134	C₂²×Q₈	D₄×C₇₉	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	12	no	—
136	C₄⋊Q₈	C₂×C₄×C₇₉	1	6	yes	—
137	C₄⋊Q₈	C₂×C₄×C₇₉	1	6	yes	—
138	#125	—	2	12	no	—
139	#125	—	2	6	no	—
140	C₂²⋊Q₈	—	2	6	no	—
141	C₂²⋊Q₈	C₂³×C₇₉	1	4	yes	—
142	C₄×Q₈	—	2	4	no	—
143	C₄×Q₈	—	2	4	no	—
144	C₄×Q₈	—	2	6	no	—
145	C₄×Q₈	—	4	4	no	—
146	C₄×Q₈	C₂×C₄×C₇₉	1	12	yes	—
147	C₄×Q₈	C₂×C₄×C₇₉	1	4	yes	—
148	C₄×Q₈	D₄×C₇₉	1	4	yes	—
149	#117	—	2	6	no	—
150	C₂×C₄⋊C₄	—	2	3	no	—
151	C₂×C₄⋊C₄	—	2	3	no	—
152	C₂×C₄⋊C₄	D₄×C₇₉	1	3	yes	—
153	C₂×C₄²	—	2	3	no	—
154	C₂×C₄²	C₂³×C₇₉	1	3	yes	—
155	C₄⋊C₈	—	2	6	no	—
156	C₂²⋊C₈	C₂³×C₇₉	1	4	yes	—
157	C₄×C₈	—	2	6	no	—
158	#95	—	2	3	no	—
159	#95	C₂³×C₇₉	1	1	yes	characteristic
160	#95	C₂³×C₇₉	1	3	yes	—
161	C₂×Q₈	—	2	6	no	—
162	C₂×Q₈	C₂²⋊C₄×C₇₉	1	4	yes	—
163	C₂×Q₈	SD₁₆×C₇₉	1	4	yes	—
164	C₂×Q₈	Q₁₆×C₇₉	1	4	yes	—
165	C₂×Q₈	C₂²×C₄×C₇₉	1	4	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	C₂×D₄×C₇₉	1	3	yes	—
169	C₂²×C₄	C₂×D₄×C₇₉	1	3	yes	—
170	C₂²×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
171	C₂×C₈	—	2	4	no	—
172	C₄⋊C₄	—	2	6	no	—
173	C₄⋊C₄	—	2	6	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	—	2	12	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	6	no	—
178	C₄⋊C₄	—	4	6	no	—
179	C₄⋊C₄	C₂²⋊C₄×C₇₉	1	12	yes	—
180	C₄⋊C₄	C₂²×C₄×C₇₉	1	4	yes	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	—	2	12	no	—
183	C₄²	—	4	6	no	—
184	C₄²	C₂²×C₄×C₇₉	1	6	yes	—
185	C₄²	C₂×D₄×C₇₉	1	6	yes	—
186	C₂³	C₂²×D₄×C₇₉	1	1	yes	characteristic
187	Q₈	—	4	4	no	—
188	Q₈	—	8	6	no	—
189	Q₈	#103×C₇₉	1	16	yes	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	3	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	3	no	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	—	4	3	no	—
200	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	6	yes	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	6	yes	—
202	C₂×C₄	C₂×SD₁₆×C₇₉	1	6	yes	—
203	C₂×C₄	C₂×Q₁₆×C₇₉	1	6	yes	—
204	C₂×C₄	C₂³×C₄×C₇₉	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₇₉	1	1	yes	characteristic
206	C₈	—	4	4	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	C₂²×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
210	C₂²	C₂²×SD₁₆×C₇₉	1	1	yes	characteristic
211	C₂²	C₂²×Q₁₆×C₇₉	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	6	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	3	no	—
219	C₄	C₂×#103×C₇₉	1	12	yes	—
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₇₉×#2884	1	1	yes	characteristic
223	C₂	C₂²×#103×C₇₉	1	1	yes	characteristic
224	C₂	C₇₉×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₇₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₉
prime radical	C₂³×C₇₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄×Q₈×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. Q₈²×C₇₉	C₄	semidirect
8. #557×C₇₉	C₄	semidirect
107. #17619	C₇₉	direct (N × H)
118. C₇₉	#17619	direct (N × H)
119. Q₈²	C₄×C₇₉	semidirect
121. #557	C₄×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₇₉×#3502	2
C₇₉×#2982	2
C₇₉×#2351	2
C₇₉×#1518	2
C₇₉×#1503	2
#17619	79

Subgroups — C79×#17619 #34129308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#17619 #34129308