Subgroups — #34175459

Order 20224.

246 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#23763564	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
5	C₂×#14579095	C₄	1	2	yes	—
6	C₂×#14579093	—	2	1	no	—
7	#19098429	—	2	1	no	—
8	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#334×C₇₉	—	2	1	no	—
13	#334×C₇₉	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₇₉	—	2	2	no	—
15	C₂×C₇₉⋊C₁₆	—	2	2	no	—
16	C₂×C₇₉⋊C₁₆	—	4	1	no	—
17	C₂×C₇₉⋊C₁₆	—	4	1	no	—
18	#14579095	—	4	2	no	—
19	#14579093	—	4	1	no	—
20	#14579093	—	4	1	no	—
21	C₇₉⋊#95	—	4	1	no	—
22	C₇₉⋊#95	—	4	1	no	—
23	C₂²×C₈×C₇₉	—	4	1	no	—
24	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₉	—	2	2	no	—
28	C₄⋊C₈×C₇₉	—	4	1	no	—
29	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
33	C₂²×Dic₇₉	—	4	2	no	—
34	C₄⋊Dic₇₉	—	4	2	no	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₇₉⋊C₁₆	—	4	2	no	—
37	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
40	C₂×C₈×C₇₉	—	2	1	no	—
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₂×C₈×C₇₉	—	4	2	no	—
43	C₂×C₈×C₇₉	—	8	2	no	—
44	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
49	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
52	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
55	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
56	C₂³×C₇₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	2	2	no	—
59	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
61	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
65	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
68	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
69	C₂×Dic₇₉	—	8	2	no	—
70	C₂×Dic₇₉	—	8	2	no	—
71	C₂×Dic₇₉	—	8	2	no	—
72	C₈×C₇₉	—	2	1	no	—
73	C₈×C₇₉	—	2	1	no	—
74	C₈×C₇₉	—	8	2	no	—
75	C₈×C₇₉	#102	1	1	yes	characteristic
76	C₈×C₇₉	#103	1	1	yes	characteristic
77	C₈×C₇₉	#106	1	1	yes	characteristic
78	C₈×C₇₉	#107	1	1	yes	characteristic
79	C₂²×C₇₉	#331	1	1	yes	characteristic
80	C₂²×C₇₉	#332	1	1	yes	characteristic
81	C₂²×C₇₉	#340	1	1	yes	characteristic
82	C₂²×C₇₉	#359	1	2	yes	—
83	C₂²×C₇₉	#360	1	2	yes	—
84	C₄×C₇₉	—	2	1	no	—
85	C₄×C₇₉	—	2	1	no	—
86	C₄×C₇₉	—	2	1	no	—
87	C₄×C₇₉	—	2	1	no	—
88	C₄×C₇₉	#332	1	1	yes	characteristic
89	C₄×C₇₉	#333	1	1	yes	characteristic
90	C₄×C₇₉	#340	1	1	yes	characteristic
91	C₄×C₇₉	#343	1	1	yes	characteristic
92	Dic₇₉	—	16	2	no	—
93	#7275	—	79	1	no	maximal, Sylow
94	C₂×C₇₉	#1296	1	1	yes	characteristic
95	C₂×C₇₉	#1360	1	2	yes	—
96	C₂×C₇₉	#1363	1	2	yes	—
97	C₂×C₇₉	#1382	1	1	yes	characteristic
98	C₂×C₇₉	#1386	1	1	yes	characteristic
99	C₂×C₄⋊C₁₆	—	79	1	no	—
100	#1846	—	79	1	no	—
101	C₂×#334	D₇₉	1	1	yes	characteristic
102	C₇₉	#7275	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₁₆	—	158	1	no	—
104	C₂×#107	—	79	2	no	—
105	C₂×#106	—	158	1	no	—
106	C₂×C₄⋊C₈	—	2	1	no	—
107	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
108	#382	—	158	1	no	—
109	C₄⋊C₁₆	—	79	2	no	—
110	C₄⋊C₁₆	—	158	1	no	—
111	#334	—	2	1	no	—
112	#334	Dic₇₉	1	2	yes	complemented
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
115	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	158	2	no	—
117	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
118	C₂×C₁₆	—	316	1	no	—
119	C₂×C₁₆	—	316	1	no	—
120	C₂×C₁₆	—	158	2	no	—
121	#107	—	316	2	no	—
122	#106	—	316	1	no	—
123	#106	—	316	1	no	—
124	C₄⋊C₈	—	2	2	no	—
125	C₄⋊C₈	—	4	1	no	—
126	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
127	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
130	#95	—	316	1	no	—
131	#95	—	316	1	no	—
132	C₂²×C₄	—	316	2	no	—
133	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
134	C₂²×C₄	#11929609	1	1	yes	characteristic
135	C₂²×C₄	#11929611	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
141	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
142	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
144	C₂×C₈	D₆₃₂	1	1	yes	characteristic
145	C₂×C₈	#11929613	1	1	yes	characteristic
146	C₂×C₈	#11929613	1	1	yes	characteristic
147	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
148	C₄⋊C₄	—	316	2	no	—
149	C₄⋊C₄	—	316	2	no	—
150	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
151	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
152	C₄²	#11929612	1	1	yes	characteristic
153	C₄²	#11929614	1	1	yes	characteristic
154	C₁₆	—	316	2	no	—
155	C₂³	#14579081	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	632	2	no	—
159	C₂×C₄	—	632	2	no	—
160	C₂×C₄	—	632	2	no	—
161	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
162	C₂×C₄	#14579077	1	1	yes	characteristic
163	C₂×C₄	#14579079	1	1	yes	characteristic
164	C₂×C₄	#14579082	1	1	yes	characteristic
165	C₂×C₄	#14579083	1	1	yes	characteristic
166	C₂×C₄	#14579084	1	1	yes	characteristic
167	C₂×C₄	#14579085	1	1	yes	characteristic
168	C₂×C₄	#14579086	1	1	yes	characteristic
169	C₂×C₄	#14579092	1	1	yes	characteristic
170	C₂×C₄	#14579094	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	8	2	no	—
174	C₈	#14579083	1	1	yes	characteristic
175	C₈	#14579086	1	1	yes	characteristic
176	C₈	#14579093	1	1	yes	characteristic
177	C₈	#14579095	1	1	yes	characteristic
178	C₂²	#19098306	1	1	yes	characteristic
179	C₂²	#19098310	1	1	yes	characteristic
180	C₂²	#19098325	1	1	yes	characteristic
181	C₂²	#19098369	1	2	yes	—
182	C₂²	#19098372	1	2	yes	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	1264	2	no	—
188	C₄	#19098308	1	1	yes	characteristic
189	C₄	#19098311	1	1	yes	characteristic
190	C₄	#19098326	1	1	yes	characteristic
191	C₄	#19098331	1	1	yes	characteristic
192	C₂	#23761284	1	1	yes	characteristic
193	C₂	#23761442	1	2	yes	—
194	C₂	#23761453	1	2	yes	—
195	C₂	#23761497	1	1	yes	characteristic
196	C₂	#23761507	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579081
commutator	C₂×C₈×C₇₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×#334×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#102
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×#334	D₇₉
hypocenter	C₇₉	#7275
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₇₉⋊C₁₆)	C₄	semidirect
13. #334×C₇₉	C₄	semidirect
102. C₇₉	#7275	semidirect
112. #334	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₉⋊C₁₆)	2
#23763564	2
C₂×#334×C₇₉	2
#7275	79