Subgroups — C₇₉×#3365 #23760987

Order 10112.

348 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#559×C₇₉	C₂	1	1	yes	maximal, characteristic
2	#557×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#488×C₇₉	C₂	1	2	yes	maximal, complemented
4	#486×C₇₉	C₂	1	1	yes	maximal, characteristic
5	#478×C₇₉	C₂	1	2	yes	maximal, complemented
6	#475×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#464×C₇₉	C₂	1	2	yes	maximal, complemented
8	#460×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×SD₁₆×C₇₉	C₂	1	2	yes	maximal, complemented
12	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₇₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₇₉	—	2	1	no	—
15	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
16	#127×C₇₉	C₂²	1	1	yes	characteristic
17	#125×C₇₉	—	2	2	no	—
18	#125×C₇₉	C₂²	1	2	yes	—
19	#124×C₇₉	—	2	1	no	—
20	#124×C₇₉	C₂²	1	2	yes	—
21	#123×C₇₉	—	2	2	no	—
22	C₄×Q₈×C₇₉	—	2	1	no	—
23	C₄×Q₈×C₇₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₇₉	—	2	1	no	—
28	C₄×D₄×C₇₉	C₂²	1	2	yes	—
29	#107×C₇₉	C₂²	1	1	yes	characteristic
30	#106×C₇₉	C₂²	1	2	yes	—
31	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
33	#103×C₇₉	C₂²	1	1	yes	characteristic
34	#103×C₇₉	C₂²	1	1	yes	characteristic
35	#103×C₇₉	C₂²	1	2	yes	—
36	#102×C₇₉	C₂²	1	2	yes	—
37	#102×C₇₉	C₂²	1	2	yes	—
38	#102×C₇₉	C₂²	1	2	yes	—
39	C₄×C₈×C₇₉	C₂²	1	2	yes	—
40	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₇₉	—	2	1	no	—
42	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₇₉	—	2	1	no	—
46	C₂×D₄×C₇₉	—	2	1	no	—
47	C₂×D₄×C₇₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₇₉	—	2	2	no	—
49	Q₁₆×C₇₉	—	2	2	no	—
50	SD₁₆×C₇₉	—	2	4	no	—
51	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂³	1	2	yes	—
54	C₄⋊C₄×C₇₉	—	2	2	no	—
55	C₄⋊C₄×C₇₉	—	2	1	no	—
56	C₄⋊C₄×C₇₉	—	2	1	no	—
57	C₄⋊C₄×C₇₉	—	2	2	no	—
58	C₄⋊C₄×C₇₉	—	2	1	no	—
59	C₄⋊C₄×C₇₉	—	2	2	no	—
60	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
62	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
63	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₇₉	—	2	2	no	—
66	C₂²⋊C₄×C₇₉	—	2	2	no	—
67	C₂²⋊C₄×C₇₉	—	2	2	no	—
68	C₄²×C₇₉	—	2	1	no	—
69	C₄²×C₇₉	—	2	2	no	—
70	C₄²×C₇₉	—	2	1	no	—
71	C₄²×C₇₉	C₂³	1	2	yes	—
72	C₄²×C₇₉	C₂³	1	1	yes	characteristic
73	C₂³×C₇₉	—	2	2	no	—
74	Q₈×C₇₉	—	2	2	no	—
75	Q₈×C₇₉	—	2	1	no	—
76	Q₈×C₇₉	—	2	1	no	—
77	Q₈×C₇₉	—	2	1	no	—
78	Q₈×C₇₉	—	4	2	no	—
79	Q₈×C₇₉	#41	1	2	yes	—
80	D₄×C₇₉	—	2	2	no	—
81	D₄×C₇₉	—	2	2	no	—
82	D₄×C₇₉	—	4	2	no	—
83	D₄×C₇₉	—	8	1	no	—
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	2	no	—
89	C₂×C₄×C₇₉	—	8	2	no	—
90	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
95	C₈×C₇₉	—	2	2	no	—
96	C₈×C₇₉	—	2	1	no	—
97	C₈×C₇₉	—	2	1	no	—
98	C₂²×C₇₉	—	4	2	no	—
99	C₂²×C₇₉	—	4	2	no	—
100	C₂²×C₇₉	—	4	2	no	—
101	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₇₉	—	2	2	no	—
103	C₄×C₇₉	—	2	2	no	—
104	C₄×C₇₉	—	2	2	no	—
105	C₄×C₇₉	—	2	1	no	—
106	C₄×C₇₉	—	4	1	no	—
107	C₄×C₇₉	—	4	2	no	—
108	C₄×C₇₉	—	4	1	no	—
109	C₄×C₇₉	—	4	1	no	—
110	C₄×C₇₉	#135	1	2	yes	—
111	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
112	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂×C₇₉	—	8	2	no	—
114	C₂×C₇₉	#548	1	1	yes	characteristic
115	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
116	C₂×C₇₉	#577	1	1	yes	characteristic
117	#3365	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₇₉	#3365	1	1	yes	characteristic, direct product factor, Sylow
119	#559	C₂×C₇₉	1	1	yes	characteristic
120	#557	C₂×C₇₉	1	1	yes	characteristic, complemented
121	#488	C₂×C₇₉	1	2	yes	complemented
122	#486	C₂×C₇₉	1	1	yes	characteristic
123	#478	C₂×C₇₉	1	2	yes	complemented
124	#475	C₂×C₇₉	1	1	yes	characteristic, complemented
125	#464	C₂×C₇₉	1	2	yes	complemented
126	#460	C₂×C₇₉	1	1	yes	characteristic
127	C₈×Q₈	C₂×C₇₉	1	1	yes	characteristic, complemented
128	C₄×Q₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
129	C₄×SD₁₆	C₂×C₇₉	1	2	yes	complemented
130	C₂×Q₁₆	C₂²×C₇₉	1	1	yes	characteristic
131	C₂×SD₁₆	C₂²×C₇₉	1	2	yes	—
132	C₄⋊Q₈	—	2	1	no	—
133	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
134	#127	C₂²×C₇₉	1	1	yes	characteristic
135	#125	—	2	2	no	—
136	#125	C₂²×C₇₉	1	2	yes	—
137	#124	—	2	1	no	—
138	#124	C₂²×C₇₉	1	2	yes	—
139	#123	—	2	2	no	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	C₂²×C₇₉	1	2	yes	—
142	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
143	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
144	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	C₂²×C₇₉	1	2	yes	—
147	#107	C₂²×C₇₉	1	1	yes	characteristic
148	#106	C₂²×C₇₉	1	2	yes	—
149	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₇₉	1	2	yes	—
151	#103	C₂²×C₇₉	1	1	yes	characteristic
152	#103	C₂²×C₇₉	1	1	yes	characteristic
153	#103	C₂²×C₇₉	1	2	yes	—
154	#102	C₂²×C₇₉	1	2	yes	—
155	#102	C₂²×C₇₉	1	2	yes	—
156	#102	C₂²×C₇₉	1	2	yes	—
157	C₄×C₈	C₂²×C₇₉	1	2	yes	—
158	C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	D₄×C₇₉	1	1	yes	characteristic
161	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
162	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	C₂³×C₇₉	1	2	yes	—
166	C₂²×C₄	—	2	2	no	—
167	Q₁₆	—	2	2	no	—
168	SD₁₆	—	2	4	no	—
169	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
170	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
171	C₂×C₈	C₂³×C₇₉	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	D₄×C₇₉	1	1	yes	characteristic
179	C₄⋊C₄	D₄×C₇₉	1	2	yes	—
180	C₄⋊C₄	C₂³×C₇₉	1	2	yes	—
181	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	1	no	—
189	C₄²	C₂³×C₇₉	1	2	yes	—
190	C₄²	C₂³×C₇₉	1	1	yes	characteristic
191	C₂³	—	2	2	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	1	no	—
194	Q₈	—	2	1	no	—
195	Q₈	—	2	1	no	—
196	Q₈	—	4	2	no	—
197	Q₈	#41×C₇₉	1	2	yes	—
198	D₄	—	2	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	8	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
210	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
212	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	1	no	—
228	C₄	#135×C₇₉	1	2	yes	—
229	C₄	C₂×#41×C₇₉	1	1	yes	characteristic
230	C₄	2₋¹⁺⁴×C₇₉	1	1	yes	characteristic
231	C₂	—	8	2	no	—
232	C₂	#548×C₇₉	1	1	yes	characteristic
233	C₂	C₂×#135×C₇₉	1	1	yes	characteristic
234	C₂	#577×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₉
prime radical	#127×C₇₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄×Q₈×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #557×C₇₉	C₂	semidirect
3. #488×C₇₉	C₂	semidirect
5. #478×C₇₉	C₂	semidirect
6. #475×C₇₉	C₂	semidirect
7. #464×C₇₉	C₂	semidirect
9. C₈×Q₈×C₇₉	C₂	semidirect
10. C₄×Q₁₆×C₇₉	C₂	semidirect
11. C₄×SD₁₆×C₇₉	C₂	semidirect
117. #3365	C₇₉	direct (N × H)
118. C₇₉	#3365	direct (N × H)
120. #557	C₂×C₇₉	semidirect
121. #488	C₂×C₇₉	semidirect
123. #478	C₂×C₇₉	semidirect
124. #475	C₂×C₇₉	semidirect
125. #464	C₂×C₇₉	semidirect
127. C₈×Q₈	C₂×C₇₉	semidirect
128. C₄×Q₁₆	C₂×C₇₉	semidirect
129. C₄×SD₁₆	C₂×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
#559×C₇₉	2
#557×C₇₉	2
#488×C₇₉	2
#486×C₇₉	2
#478×C₇₉	2
#475×C₇₉	2
#464×C₇₉	2
#460×C₇₉	2
C₈×Q₈×C₇₉	2
C₄×Q₁₆×C₇₉	2
C₄×SD₁₆×C₇₉	2
#3365	79

Subgroups — C79×#3365 #23760987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#3365 #23760987