Subgroups — C₇₉×#3226 #23760848

Order 10112.

404 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#532×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#500×C₇₉	C₂	1	2	yes	maximal, complemented
4	#486×C₇₉	C₂	1	2	yes	maximal, complemented
5	#480×C₇₉	C₂	1	4	yes	maximal, complemented
6	#474×C₇₉	C₂	1	4	yes	maximal, complemented
7	#431×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₇₉	—	2	1	no	—
9	C₂²×Q₈×C₇₉	—	2	1	no	—
10	C₄⋊Q₈×C₇₉	—	2	1	no	—
11	C₄⋊Q₈×C₇₉	—	2	1	no	—
12	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
15	#127×C₇₉	C₂²	1	1	yes	characteristic
16	#124×C₇₉	—	2	1	no	—
17	#121×C₇₉	—	2	1	no	—
18	#121×C₇₉	C₂²	1	2	yes	—
19	C₄×D₄×C₇₉	—	2	1	no	—
20	C₄×D₄×C₇₉	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
22	C₂×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
23	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
24	#107×C₇₉	C₂²	1	8	yes	—
25	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
26	#102×C₇₉	C₂²	1	8	yes	—
27	C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
28	C₄×C₈×C₇₉	C₂²	1	2	yes	—
29	#41×C₇₉	—	4	4	no	—
30	C₂×Q₈×C₇₉	—	2	1	no	—
31	C₂×Q₈×C₇₉	—	2	2	no	—
32	C₂×Q₈×C₇₉	—	2	2	no	—
33	C₂×Q₈×C₇₉	—	4	4	no	—
34	C₂×D₄×C₇₉	—	2	2	no	—
35	C₂×D₄×C₇₉	—	2	1	no	—
36	C₂×D₄×C₇₉	—	2	1	no	—
37	C₂×D₄×C₇₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₇₉	—	2	2	no	—
39	C₂²×C₄×C₇₉	—	2	2	no	—
40	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂³	1	4	yes	—
44	C₄⋊C₄×C₇₉	—	2	2	no	—
45	C₄⋊C₄×C₇₉	—	2	2	no	—
46	C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₄⋊C₄×C₇₉	—	2	1	no	—
48	C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
49	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
50	C₂²⋊C₄×C₇₉	—	2	2	no	—
51	C₂²⋊C₄×C₇₉	—	2	2	no	—
52	C₄²×C₇₉	D₄	1	1	yes	characteristic
53	C₄²×C₇₉	D₄	1	1	yes	characteristic
54	C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	C₂³×C₇₉	—	2	2	no	—
57	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
58	Q₈×C₇₉	—	4	4	no	—
59	Q₈×C₇₉	—	4	4	no	—
60	Q₈×C₇₉	—	4	2	no	—
61	D₄×C₇₉	—	2	4	no	—
62	D₄×C₇₉	—	4	4	no	—
63	D₄×C₇₉	—	4	2	no	—
64	D₄×C₇₉	—	4	2	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	4	no	—
69	C₂×C₄×C₇₉	—	4	4	no	—
70	C₂×C₄×C₇₉	—	4	4	no	—
71	C₂×C₄×C₇₉	D₈	1	4	yes	—
72	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
78	C₈×C₇₉	—	2	4	no	—
79	C₂²×C₇₉	—	2	1	no	—
80	C₂²×C₇₉	—	2	1	no	—
81	C₂²×C₇₉	—	4	2	no	—
82	C₂²×C₇₉	—	4	2	no	—
83	C₂²×C₇₉	—	4	2	no	—
84	C₂²×C₇₉	C₂×D₈	1	2	yes	—
85	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₇₉	—	2	1	no	—
87	C₄×C₇₉	—	2	1	no	—
88	C₄×C₇₉	—	4	2	no	—
89	C₄×C₇₉	—	4	4	no	—
90	C₄×C₇₉	C₂×D₈	1	2	yes	—
91	C₄×C₇₉	C₂×D₈	1	2	yes	—
92	C₄×C₇₉	#137	1	2	yes	—
93	C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
94	C₂×C₇₉	—	2	2	no	—
95	C₂×C₇₉	—	8	2	no	—
96	C₂×C₇₉	#536	1	1	yes	characteristic
97	C₂×C₇₉	C₂²×D₈	1	1	yes	characteristic
98	C₂×C₇₉	C₂×#137	1	1	yes	characteristic
99	#3226	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₇₉	#3226	1	1	yes	characteristic, direct product factor, Sylow
101	#532	C₂×C₇₉	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	C₂×C₇₉	1	1	yes	characteristic, complemented
103	#500	C₂×C₇₉	1	2	yes	complemented
104	#486	C₂×C₇₉	1	2	yes	complemented
105	#480	C₂×C₇₉	1	4	yes	complemented
106	#474	C₂×C₇₉	1	4	yes	complemented
107	#431	C₂×C₇₉	1	1	yes	characteristic, complemented
108	C₂×#41	—	2	1	no	—
109	C₂²×Q₈	—	2	1	no	—
110	C₄⋊Q₈	—	2	1	no	—
111	C₄⋊Q₈	—	2	1	no	—
112	C₄⋊Q₈	C₂²×C₇₉	1	2	yes	—
113	C₄⋊Q₈	C₂²×C₇₉	1	2	yes	—
114	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
115	#127	C₂²×C₇₉	1	1	yes	characteristic
116	#124	—	2	1	no	—
117	#121	—	2	1	no	—
118	#121	C₂²×C₇₉	1	2	yes	—
119	C₄×D₄	—	2	1	no	—
120	C₄×D₄	C₂²×C₇₉	1	2	yes	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	C₂²×C₇₉	1	2	yes	—
123	C₂×C₄²	C₂²×C₇₉	1	1	yes	characteristic
124	#107	C₂²×C₇₉	1	8	yes	—
125	C₄⋊C₈	C₂²×C₇₉	1	2	yes	—
126	#102	C₂²×C₇₉	1	8	yes	—
127	C₂²⋊C₈	C₂²×C₇₉	1	2	yes	—
128	C₄×C₈	C₂²×C₇₉	1	2	yes	—
129	#41	—	4	4	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	—	2	1	no	—
136	C₂×D₄	—	2	1	no	—
137	C₂×D₄	C₂³×C₇₉	1	2	yes	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	D₄×C₇₉	1	1	yes	characteristic
141	C₂²×C₄	D₄×C₇₉	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₇₉	1	1	yes	characteristic
143	C₂×C₈	C₂³×C₇₉	1	4	yes	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	C₂³×C₇₉	1	4	yes	—
149	C₄⋊C₄	C₂³×C₇₉	1	2	yes	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₄²	D₄×C₇₉	1	1	yes	characteristic
153	C₄²	D₄×C₇₉	1	1	yes	characteristic
154	C₄²	C₂³×C₇₉	1	1	yes	characteristic
155	C₄²	C₂³×C₇₉	1	1	yes	characteristic
156	C₂³	—	2	2	no	—
157	C₂³	C₂×D₄×C₇₉	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	4	4	no	—
160	Q₈	—	4	2	no	—
161	D₄	—	2	4	no	—
162	D₄	—	4	4	no	—
163	D₄	—	4	2	no	—
164	D₄	—	4	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	D₈×C₇₉	1	4	yes	—
172	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
173	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
174	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
175	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
176	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
177	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
178	C₈	—	2	4	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×D₈×C₇₉	1	2	yes	—
185	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	4	no	—
190	C₄	C₂×D₈×C₇₉	1	2	yes	—
191	C₄	C₂×D₈×C₇₉	1	2	yes	—
192	C₄	#137×C₇₉	1	2	yes	—
193	C₄	2₋¹⁺⁴×C₇₉	1	2	yes	—
194	C₂	—	2	2	no	—
195	C₂	—	8	2	no	—
196	C₂	#536×C₇₉	1	1	yes	characteristic
197	C₂	C₂²×D₈×C₇₉	1	1	yes	characteristic
198	C₂	C₂×#137×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₉
prime radical	#532×C₇₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Q₈×C₇₉	C₂	semidirect
3. #500×C₇₉	C₂	semidirect
4. #486×C₇₉	C₂	semidirect
5. #480×C₇₉	C₂	semidirect
6. #474×C₇₉	C₂	semidirect
7. #431×C₇₉	C₂	semidirect
99. #3226	C₇₉	direct (N × H)
100. C₇₉	#3226	direct (N × H)
102. C₂×C₄⋊Q₈	C₂×C₇₉	semidirect
103. #500	C₂×C₇₉	semidirect
104. #486	C₂×C₇₉	semidirect
105. #480	C₂×C₇₉	semidirect
106. #474	C₂×C₇₉	semidirect
107. #431	C₂×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
#532×C₇₉	2
C₂×C₄⋊Q₈×C₇₉	2
#500×C₇₉	2
#486×C₇₉	2
#480×C₇₉	2
#474×C₇₉	2
#431×C₇₉	2
#3226	79

Subgroups — C79×#3226 #23760848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#3226 #23760848