Subgroups — #34175172

Order 20224.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 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	C₈×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₇₉	—	2	1	no	—
7	C₄×C₄⋊Dic₇₉	C₄	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	complemented
10	#19098317	—	2	1	no	—
11	C₂²×C₁₆×C₇₉	—	2	1	no	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₉	—	2	1	no	—
14	C₄×C₁₆×C₇₉	C₄	1	2	yes	complemented
15	C₂²×C₇₉⋊C₈	—	2	1	no	—
16	C₂×C₄⋊Dic₇₉	C₈	1	2	yes	—
17	C₂×C₄×Dic₇₉	—	2	1	no	—
18	C₂×C₇₉⋊C₁₆	—	2	2	no	—
19	C₂×C₇₉⋊C₁₆	—	4	1	no	—
20	C₂×C₇₉⋊C₁₆	—	4	1	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
22	C₄⋊(C₇₉⋊C₈)	C₈	1	2	yes	—
23	C₈×Dic₇₉	—	4	1	no	—
24	C₈×Dic₇₉	—	4	1	no	—
25	C₇₉⋊#95	—	2	1	no	—
26	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₉	—	2	2	no	—
30	C₂×C₁₆×C₇₉	—	2	2	no	—
31	C₂×C₁₆×C₇₉	—	2	2	no	—
32	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
36	C₂²×Dic₇₉	—	2	2	no	—
37	C₂×C₇₉⋊C₈	—	2	2	no	—
38	C₂×C₇₉⋊C₈	—	4	1	no	—
39	C₂×C₇₉⋊C₈	—	4	1	no	—
40	C₄⋊Dic₇₉	—	2	2	no	—
41	C₄⋊Dic₇₉	C₁₆	1	4	yes	complemented
42	C₄×Dic₇₉	—	4	1	no	—
43	C₄×Dic₇₉	—	4	1	no	—
44	C₇₉⋊C₁₆	—	4	2	no	—
45	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
48	C₂×C₈×C₇₉	—	2	1	no	—
49	C₂×C₈×C₇₉	—	2	1	no	—
50	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
55	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
58	C₄²×C₇₉	C₄²	1	1	yes	characteristic
59	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
62	C₁₆×C₇₉	—	2	4	no	—
63	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₇₉	M₃₂	1	2	yes	—
77	C₂×Dic₇₉	—	2	4	no	—
78	C₂×Dic₇₉	—	4	2	no	—
79	C₂×Dic₇₉	—	4	2	no	—
80	C₈×C₇₉	—	2	1	no	—
81	C₈×C₇₉	—	2	1	no	—
82	C₈×C₇₉	#102	1	1	yes	characteristic
83	C₈×C₇₉	#103	1	1	yes	characteristic
84	C₈×C₇₉	#106	1	1	yes	characteristic
85	C₈×C₇₉	#107	1	1	yes	characteristic
86	C₇₉⋊C₈	—	4	2	no	—
87	C₂²×C₇₉	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₉	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₇₉	#336	1	1	yes	characteristic
90	C₂²×C₇₉	C₂²⋊C₁₆	1	2	yes	—
91	C₂²×C₇₉	#350	1	2	yes	—
92	C₄×C₇₉	—	2	1	no	—
93	C₄×C₇₉	—	2	1	no	—
94	C₄×C₇₉	D₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₉	Q₈⋊C₈	1	1	yes	characteristic
96	C₄×C₇₉	#339	1	1	yes	characteristic
97	C₄×C₇₉	#340	1	1	yes	characteristic
98	C₄×C₇₉	C₄×C₁₆	1	1	yes	characteristic
99	C₄×C₇₉	#346	1	1	yes	characteristic
100	C₄×C₇₉	C₄⋊C₁₆	1	2	yes	—
101	Dic₇₉	—	4	4	no	—
102	#7167	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	#1279	1	1	yes	characteristic
104	C₂×C₇₉	D₄⋊C₁₆	1	2	yes	—
105	C₂×C₇₉	Q₈⋊C₁₆	1	2	yes	—
106	C₂×C₇₉	#1374	1	1	yes	characteristic
107	C₂×C₇₉	#1379	1	1	yes	characteristic
108	C₂×C₄⋊C₁₆	—	79	1	no	—
109	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
110	C₈×C₄⋊C₄	—	79	1	no	—
111	C₇₉	#7167	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₁₆	—	2	1	no	—
113	C₂²×C₁₆	—	158	1	no	—
114	C₂×C₄⋊C₈	—	79	1	no	—
115	C₂×C₄×C₈	—	158	1	no	—
116	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	79	1	no	—
118	C₄⋊C₁₆	—	79	2	no	—
119	C₄⋊C₁₆	—	158	1	no	—
120	C₄×C₁₆	—	2	1	no	—
121	C₄×C₁₆	Dic₇₉	1	2	yes	complemented
122	#336	—	158	1	no	—
123	C₂²×C₈	—	158	1	no	—
124	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
126	C₂×C₄⋊C₄	—	79	2	no	—
127	C₂×C₄²	—	158	1	no	—
128	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	316	1	no	—
133	C₂×C₁₆	—	316	1	no	—
134	C₂×C₁₆	—	158	2	no	—
135	C₄⋊C₈	—	79	2	no	—
136	C₄⋊C₈	—	158	1	no	—
137	C₄×C₈	—	316	1	no	—
138	C₄×C₈	—	316	1	no	—
139	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
140	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
141	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
143	#95	—	158	1	no	—
144	C₂²×C₄	—	158	2	no	—
145	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
147	C₂²×C₄	#11929607	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	316	1	no	—
151	C₂×C₈	—	316	1	no	—
152	C₂×C₈	—	158	2	no	—
153	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
154	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
155	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
156	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
157	C₂×C₈	D₆₃₂	1	1	yes	characteristic
158	C₂×C₈	#11929613	1	1	yes	characteristic
159	C₂×C₈	#11929613	1	1	yes	characteristic
160	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
161	C₄⋊C₄	—	79	4	no	—
162	C₄⋊C₄	—	158	2	no	—
163	C₄²	—	316	1	no	—
164	C₄²	—	316	1	no	—
165	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
166	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
167	C₄²	C₈×D₇₉	1	1	yes	characteristic
168	C₄²	#11929607	1	1	yes	characteristic
169	C₁₆	—	2	4	no	—
170	C₁₆	—	316	2	no	—
171	C₂³	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	316	2	no	—
174	C₂×C₄	—	316	2	no	—
175	C₂×C₄	—	158	4	no	—
176	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
177	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
178	C₂×C₄	#14579071	1	1	yes	characteristic
179	C₂×C₄	(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
180	C₂×C₄	#14579083	1	1	yes	characteristic
181	C₂×C₄	#14579086	1	1	yes	characteristic
182	C₂×C₄	#14579089	1	1	yes	characteristic
183	C₂×C₄	#14579089	1	1	yes	characteristic
184	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
185	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
186	C₂×C₄	C₁₆×D₇₉	1	2	yes	—
187	C₂×C₄	#14579100	1	2	yes	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	316	2	no	—
191	C₈	#14579083	1	1	yes	characteristic
192	C₈	#14579086	1	1	yes	characteristic
193	C₈	#14579093	1	1	yes	characteristic
194	C₈	#14579095	1	1	yes	characteristic
195	C₂²	#19098292	1	1	yes	characteristic
196	C₂²	#19098295	1	1	yes	characteristic
197	C₂²	#19098317	1	1	yes	characteristic
198	C₂²	(C₂×D₇₉)⋊C₁₆	1	2	yes	—
199	C₂²	#19098346	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	316	4	no	—
203	C₄	#19098292	1	1	yes	characteristic
204	C₄	#19098295	1	1	yes	characteristic
205	C₄	#19098323	1	1	yes	characteristic
206	C₄	#19098326	1	1	yes	characteristic
207	C₄	C₁₆×Dic₇₉	1	1	yes	characteristic
208	C₄	#19098338	1	1	yes	characteristic
209	C₄	Dic₇₉⋊C₁₆	1	2	yes	—
210	C₂	#23761240	1	1	yes	characteristic
211	C₂	#23761356	1	2	yes	—
212	C₂	#23761380	1	2	yes	—
213	C₂	#23761477	1	1	yes	characteristic
214	C₂	#23761488	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₃₁₆
commutator	C₄×C₇₉	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#7167
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₇₉⋊C₁₆)	C₄	semidirect
14. C₄×C₁₆×C₇₉	C₄	semidirect
41. C₄⋊Dic₇₉	C₁₆	semidirect
111. C₇₉	#7167	semidirect
121. C₄×C₁₆	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
C₈×C₄⋊Dic₇₉	2
C₂×C₄×C₁₆×C₇₉	2
#7167	79