Subgroups — #34182072

Order 20224.

354 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763571	C₂	1	2	yes	maximal
2	#23762159	C₂	1	1	yes	maximal, characteristic
3	#23762158	C₂	1	2	yes	maximal
4	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579096	—	2	4	no	—
7	#19098587	C₂²	1	2	yes	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	#19098316	—	2	2	no	—
11	#19098315	—	2	4	no	—
12	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₇₉⋊C₈	C₄	1	1	yes	characteristic
16	#423×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#335×C₇₉	—	2	2	no	—
19	C₂×#11929608	—	2	2	no	—
20	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
21	C₂²×C₇₉⋊C₈	Q₈	1	2	yes	—
22	#14579096	—	4	8	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
30	C₄×C₇₉⋊C₈	Q₈	1	2	yes	—
31	C₂×M₁₆×C₇₉	—	2	2	no	—
32	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
33	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₉	—	2	2	no	—
35	C₄⋊C₈×C₇₉	—	2	2	no	—
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₉	D₄	1	2	yes	—
39	#11929608	—	4	4	no	—
40	C₂×C₇₉⋊C₈	—	2	4	no	—
41	C₂×C₇₉⋊C₈	—	4	2	no	—
42	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₇₉⋊C₈	D₈	1	4	yes	—
47	C₂×C₇₉⋊C₈	Q₁₆	1	4	yes	—
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
51	M₁₆×C₇₉	—	4	4	no	—
52	C₂×C₈×C₇₉	—	2	4	no	—
53	C₂×C₈×C₇₉	—	4	2	no	—
54	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
56	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
60	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
71	C₈×C₇₉	—	2	2	no	—
72	C₈×C₇₉	—	8	2	no	—
73	C₈×C₇₉	#108	1	4	yes	complemented
74	C₇₉⋊C₈	—	2	2	no	—
75	C₇₉⋊C₈	—	8	2	no	—
76	C₇₉⋊C₈	#107	1	4	yes	—
77	C₇₉⋊C₈	#107	1	4	yes	—
78	C₇₉⋊C₈	#108	1	4	yes	complemented
79	C₂²×C₇₉	—	2	1	no	—
80	C₂²×C₇₉	—	2	1	no	—
81	C₂²×C₇₉	#384	1	1	yes	characteristic
82	C₂²×C₇₉	#500	1	2	yes	—
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	—	2	1	no	—
85	C₄×C₇₉	C₂×#107	1	2	yes	—
86	C₄×C₇₉	C₂×#108	1	2	yes	—
87	C₄×C₇₉	C₂×#108	1	2	yes	—
88	C₄×C₇₉	#493	1	1	yes	characteristic
89	C₄×C₇₉	#494	1	1	yes	characteristic
90	#8823	—	79	1	no	maximal, Sylow
91	C₂×C₇₉	—	2	2	no	—
92	C₂×C₇₉	#1845	1	1	yes	characteristic
93	C₂×C₇₉	#1848	1	1	yes	characteristic
94	C₂×C₇₉	#1848	1	1	yes	characteristic
95	#1848	—	79	2	no	—
96	#1567	—	79	2	no	—
97	#1567	—	79	1	no	—
98	#1567	D₇₉	1	1	yes	characteristic
99	C₂×C₈²	—	79	1	no	—
100	C₇₉	#8823	1	1	yes	characteristic, complemented, Sylow
101	C₂×#108	—	158	4	no	—
102	#423	—	79	2	no	—
103	#423	C₂×D₇₉	1	2	yes	—
104	C₂×C₄×C₈	—	79	1	no	—
105	C₂×C₄×C₈	—	79	1	no	—
106	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
107	#335	—	2	2	no	—
108	#335	—	158	4	no	—
109	#335	—	158	2	no	—
110	C₈²	—	79	1	no	—
111	C₈²	—	79	1	no	—
112	C₈²	—	79	1	no	—
113	C₈²	—	79	1	no	—
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	158	2	no	—
116	C₂²×C₈	—	79	2	no	—
117	C₂²×C₈	—	79	2	no	—
118	C₂²×C₈	Dic₁₅₈	1	2	yes	—
119	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
120	#108	—	316	8	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	158	2	no	—
124	C₄⋊C₈	—	158	2	no	—
125	C₄×C₈	—	79	1	no	—
126	C₄×C₈	—	79	1	no	—
127	C₄×C₈	—	79	1	no	—
128	C₄×C₈	—	79	1	no	—
129	C₄×C₈	—	79	2	no	—
130	C₄×C₈	—	79	2	no	—
131	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
132	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
134	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
137	M₁₆	—	4	4	no	—
138	M₁₆	—	316	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	316	2	no	—
142	C₂×C₈	—	79	2	no	—
143	C₂×C₈	—	79	2	no	—
144	C₂×C₈	—	79	2	no	—
145	C₂×C₈	—	79	4	no	—
146	C₂×C₈	—	79	4	no	—
147	C₂×C₈	—	79	2	no	—
148	C₂×C₈	—	158	4	no	—
149	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
150	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
151	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
152	C₄²	D₄×D₇₉	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
154	C₄²	Q₈×D₇₉	1	1	yes	characteristic
155	C₂³	#14579160	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
161	C₂×C₄	#14579157	1	1	yes	characteristic
162	C₂×C₄	#14579159	1	1	yes	characteristic
163	C₂×C₄	#14579160	1	1	yes	characteristic
164	C₂×C₄	D₈×D₇₉	1	2	yes	—
165	C₂×C₄	Q₁₆×D₇₉	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	8	2	no	—
168	C₈	—	79	4	no	—
169	C₈	—	79	4	no	—
170	C₈	—	79	4	no	—
171	C₈	—	632	2	no	—
172	C₈	—	158	2	no	—
173	C₈	#14579096	1	4	yes	complemented
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	#19098432	1	1	yes	characteristic
177	C₂²	#19098967	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#107×D₇₉	1	2	yes	—
181	C₄	#108×D₇₉	1	2	yes	—
182	C₄	C₂×#14579096	1	2	yes	—
183	C₄	#19098938	1	1	yes	characteristic
184	C₄	#19098941	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#23763558	1	1	yes	characteristic
187	C₂	#23763570	1	1	yes	characteristic
188	C₂	#23763571	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1567	C₂
socle	C₂²×C₇₉	#384
prime radical	C₂³×C₇₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1567	D₇₉
hypocenter	C₇₉	#8823
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
73. C₈×C₇₉	#108	semidirect
78. C₇₉⋊C₈	#108	semidirect
100. C₇₉	#8823	semidirect
173. C₈	#14579096	semidirect

Maximal subgroups

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

subgroup	index
#23763571	2
#23762159	2
#23762158	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#1567	2
#8823	79