Subgroups — #34177662

Order 20224.

802 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11929608	C₂	1	2	yes	maximal
2	C₇₉⋊#1273	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929608	C₄	1	4	yes	—
5	C₄×#11929608	—	2	4	no	—
6	C₄×#11929608	—	2	4	no	—
7	C₂×#14579072	C₄	1	4	yes	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
9	C₂²×M₁₆×C₇₉	C₄	1	2	yes	—
10	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₇₉	—	2	4	no	—
12	C₂×#97×C₇₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
14	C₂×#11929608	—	2	8	no	—
15	C₂×#11929608	—	2	16	no	—
16	C₂²×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
17	#14579072	—	2	8	no	—
18	C₄×C₇₉⋊C₈	—	2	4	no	—
19	C₄×C₇₉⋊C₈	—	2	4	no	—
20	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₇₉	—	2	4	no	—
23	C₂×M₁₆×C₇₉	—	2	8	no	—
24	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	D₄	1	4	yes	—
28	C₂×C₄²×C₇₉	D₄	1	2	yes	—
29	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
30	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
31	#97×C₇₉	—	2	4	no	—
32	C₄×C₈×C₇₉	—	2	4	no	—
33	#11929608	—	2	32	no	—
34	C₂×C₇₉⋊C₈	—	2	8	no	—
35	C₂×C₇₉⋊C₈	—	2	8	no	—
36	C₂×C₇₉⋊C₈	M₁₆	1	16	yes	complemented
37	C₂⁴×C₇₉	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	—	2	4	no	—
39	C₂²×C₄×C₇₉	—	2	8	no	—
40	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₇₉	—	2	16	no	—
49	C₂×C₈×C₇₉	—	2	4	no	—
50	C₂×C₈×C₇₉	—	2	4	no	—
51	C₂×C₈×C₇₉	M₁₆	1	8	yes	complemented
52	C₄²×C₇₉	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₇₉	C₄⋊C₄	1	4	yes	—
55	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
56	C₄²×C₇₉	C₂×D₄	1	2	yes	—
57	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₇₉	—	2	4	no	—
60	C₂³×C₇₉	—	2	2	no	—
61	C₂³×C₇₉	#95	1	2	yes	—
62	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	4	no	—
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	2	8	no	—
66	C₂×C₄×C₇₉	—	2	4	no	—
67	C₂×C₄×C₇₉	—	2	8	no	—
68	C₂×C₄×C₇₉	—	2	4	no	—
69	C₂×C₄×C₇₉	#95	1	2	yes	—
70	C₂×C₄×C₇₉	#95	1	4	yes	—
71	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
72	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×M₁₆	1	8	yes	—
76	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
77	C₈×C₇₉	—	2	8	no	—
78	C₇₉⋊C₈	—	2	16	no	—
79	C₂²×C₇₉	—	2	4	no	—
80	C₂²×C₇₉	—	2	2	no	—
81	C₂²×C₇₉	—	4	4	no	—
82	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
83	C₂²×C₇₉	C₄×M₁₆	1	4	yes	—
84	C₂²×C₇₉	C₄×M₁₆	1	2	yes	—
85	C₄×C₇₉	—	2	4	no	—
86	C₄×C₇₉	—	2	2	no	—
87	C₄×C₇₉	#407	1	4	yes	—
88	C₄×C₇₉	#407	1	2	yes	—
89	C₄×C₇₉	#423	1	4	yes	—
90	C₄×C₇₉	#423	1	2	yes	—
91	#7852	—	79	1	no	maximal, Sylow
92	C₂×C₇₉	—	4	2	no	—
93	C₂×C₇₉	#1455	1	4	yes	—
94	C₂×C₇₉	#1728	1	2	yes	—
95	C₂×C₇₉	#1728	1	1	yes	characteristic
96	C₂×C₄×M₁₆	—	79	2	no	—
97	C₂×C₄×M₁₆	D₇₉	1	1	yes	characteristic
98	#1273	—	79	4	no	—
99	C₇₉	#7852	1	1	yes	characteristic, complemented, Sylow
100	C₂²×M₁₆	—	79	4	no	—
101	C₂²×M₁₆	Dic₇₉	1	2	yes	—
102	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
103	C₄×M₁₆	—	2	4	no	—
104	C₄×M₁₆	—	158	4	no	—
105	C₄×M₁₆	—	158	4	no	—
106	C₂×#97	—	79	4	no	—
107	C₂×#97	Dic₇₉	1	2	yes	—
108	C₂×C₄×C₈	—	79	4	no	—
109	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
110	C₂³×C₄	C₄×D₇₉	1	2	yes	—
111	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
112	C₂×M₁₆	—	2	4	no	—
113	C₂×M₁₆	—	2	8	no	—
114	C₂×M₁₆	—	158	16	no	—
115	C₂×M₁₆	—	158	8	no	—
116	C₂²×C₈	—	79	8	no	—
117	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
118	C₂×C₄²	Dic₁₅₈	1	2	yes	—
119	C₂×C₄²	C₄×D₇₉	1	2	yes	—
120	C₂×C₄²	D₃₁₆	1	2	yes	—
121	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₇₉	1	4	yes	—
123	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
124	#97	—	2	4	no	—
125	#97	—	158	8	no	—
126	C₄×C₈	—	2	4	no	—
127	C₄×C₈	—	158	4	no	—
128	C₄×C₈	—	158	4	no	—
129	C₂⁴	C₄×Dic₇₉	1	1	yes	characteristic
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
133	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
134	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	4	yes	—
135	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
136	C₂²×C₄	Dic₇₉⋊C₄	1	4	yes	—
137	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
138	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
140	M₁₆	—	2	16	no	—
141	M₁₆	—	158	32	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	79	16	no	—
145	C₂×C₈	—	158	8	no	—
146	C₂×C₈	—	158	8	no	—
147	C₂×C₈	#11929608	1	8	yes	complemented
148	C₄²	(C₂×D₇₉)⋊C₄	1	4	yes	—
149	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
150	C₄²	Dic₇₉⋊C₄	1	4	yes	—
151	C₄²	C₄⋊Dic₇₉	1	2	yes	—
152	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
153	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
154	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
155	C₂³	—	2	4	no	—
156	C₂³	—	2	2	no	—
157	C₂³	C₇₉⋊#95	1	2	yes	—
158	C₂³	C₂×C₄×Dic₇₉	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	C₇₉⋊#95	1	2	yes	—
166	C₂×C₄	C₇₉⋊#95	1	4	yes	—
167	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
171	C₂×C₄	M₁₆×D₇₉	1	8	yes	—
172	C₂×C₄	C₂×#11929608	1	4	yes	—
173	C₈	—	2	8	no	—
174	C₈	—	158	16	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	4	4	no	—
178	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
179	C₂²	M₁₆×Dic₇₉	1	4	yes	—
180	C₂²	C₄×#11929608	1	2	yes	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	#19098526	1	4	yes	—
184	C₄	#19098527	1	2	yes	—
185	C₄	#19098586	1	4	yes	—
186	C₄	#19098587	1	2	yes	—
187	C₂	—	4	2	no	—
188	C₂	#23761703	1	4	yes	—
189	C₂	#23763093	1	2	yes	—
190	C₂	#23763094	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator, socle	C₂³×C₇₉	C₂×C₄²
Fitting	C₂×C₄×M₁₆×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₇₉
hypocenter	C₇₉	#7852
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₇₉⋊#1273	C₂	semidirect
36. C₂×C₇₉⋊C₈	M₁₆	semidirect
51. C₂×C₈×C₇₉	M₁₆	semidirect
99. C₇₉	#7852	semidirect
147. C₂×C₈	#11929608	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11929608	2
C₇₉⋊#1273	2
C₂×C₄×M₁₆×C₇₉	2
#7852	79