Subgroups — #34204705

Order 20224.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098390	C₂	1	4	yes	maximal
2	C₂×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#23763560	C₂	1	1	yes	maximal, characteristic
4	C₇₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14579095	—	2	1	no	—
7	C₂×#14579095	C₂²	1	2	yes	—
8	#19098433	—	2	1	no	—
9	#19098390	—	2	8	no	—
10	C₄×C₇₉⋊C₁₆	C₄	1	4	yes	complemented
11	C₂×#107×C₇₉	—	2	1	no	—
12	C₂×#107×C₇₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#384×C₇₉	—	2	1	no	—
15	C₂×C₄⋊Dic₇₉	—	2	2	no	—
16	C₂×C₄⋊Dic₇₉	—	4	2	no	—
17	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	—
19	C₂×C₇₉⋊C₁₆	Q₈	1	4	yes	—
20	#14579095	—	2	4	no	—
21	#14579095	—	4	2	no	—
22	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
26	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
27	#107×C₇₉	—	2	4	no	—
28	#107×C₇₉	—	4	2	no	—
29	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₇₉	Q₈	1	2	yes	—
31	C₂²×Dic₇₉	—	4	2	no	—
32	C₄⋊Dic₇₉	—	4	2	no	—
33	C₄⋊Dic₇₉	—	4	2	no	—
34	C₄⋊Dic₇₉	—	8	4	no	—
35	C₇₉⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
36	C₂²×C₄×C₇₉	—	4	2	no	—
37	C₂²×C₄×C₇₉	D₈	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₇₉	D₈	1	2	yes	—
41	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
42	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
46	C₄⋊C₄×C₇₉	—	4	2	no	—
47	C₄⋊C₄×C₇₉	—	4	2	no	—
48	C₄⋊C₄×C₇₉	—	8	4	no	—
49	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
50	C₄²×C₇₉	Q₁₆	1	2	yes	—
51	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	8	2	no	—
53	C₂×C₄×C₇₉	—	8	4	no	—
54	C₂×C₄×C₇₉	#107	1	4	yes	—
55	C₂×C₄×C₇₉	SD₃₂	1	4	yes	—
56	C₂×C₄×C₇₉	SD₃₂	1	4	yes	—
57	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₇₉	—	8	2	no	—
64	C₂×Dic₇₉	—	8	4	no	—
65	C₈×C₇₉	#107	1	4	yes	—
66	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
67	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
68	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
69	C₂²×C₇₉	#493	1	1	yes	characteristic
70	C₂²×C₇₉	#500	1	1	yes	characteristic
71	C₂²×C₇₉	C₂×SD₃₂	1	2	yes	—
72	C₂²×C₇₉	C₂×SD₃₂	1	2	yes	—
73	C₄×C₇₉	—	16	2	no	—
74	C₄×C₇₉	#367	1	8	yes	complemented
75	C₄×C₇₉	#384	1	1	yes	characteristic
76	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
77	C₄×C₇₉	#494	1	1	yes	characteristic
78	C₄×C₇₉	#500	1	1	yes	characteristic
79	Dic₇₉	—	16	2	no	—
80	#12906	—	79	1	no	maximal, Sylow
81	C₂×C₇₉	#1845	1	1	yes	characteristic
82	C₂×C₇₉	C₂×#367	1	2	yes	—
83	C₂×C₇₉	#2248	1	1	yes	characteristic
84	C₂×C₇₉	#2248	1	1	yes	characteristic
85	C₂×C₇₉	#2254	1	2	yes	—
86	C₂×#367	—	79	4	no	—
87	C₂×C₄×C₁₆	—	79	1	no	—
88	#1845	—	79	1	no	—
89	#1845	D₇₉	1	1	yes	characteristic
90	C₇₉	#12906	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₁₆	—	79	2	no	—
92	C₂×#107	—	2	1	no	—
93	C₂×#107	—	79	2	no	—
94	C₂×#107	—	158	1	no	—
95	C₂×#107	C₂×D₇₉	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
97	#384	—	2	1	no	—
98	#384	—	158	1	no	—
99	#367	—	158	8	no	—
100	C₄×C₁₆	—	79	4	no	—
101	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	4	2	no	—
105	C₂×C₄⋊C₄	—	316	2	no	—
106	C₂×C₄⋊C₄	—	158	2	no	—
107	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
108	C₂×C₁₆	—	79	4	no	—
109	C₂×C₁₆	—	79	4	no	—
110	C₂×C₁₆	—	79	4	no	—
111	#107	—	2	4	no	—
112	#107	—	4	2	no	—
113	#107	—	316	2	no	—
114	#107	—	158	4	no	—
115	C₄×C₈	Dic₁₅₈	1	2	yes	—
116	C₄×C₈	C₄×D₇₉	1	2	yes	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	316	2	no	—
119	C₂²×C₄	#11929609	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
121	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
122	C₂×C₈	#11929609	1	2	yes	—
123	C₂×C₈	#11929614	1	2	yes	—
124	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
125	C₂×C₈	D₄×D₇₉	1	2	yes	—
126	C₂×C₈	Q₈×D₇₉	1	2	yes	—
127	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄⋊C₄	—	316	2	no	—
132	C₄⋊C₄	—	316	2	no	—
133	C₄⋊C₄	—	632	4	no	—
134	C₄²	Dic₇₉⋊C₄	1	2	yes	—
135	C₄²	#11929614	1	2	yes	—
136	C₁₆	—	79	8	no	—
137	C₂³	C₂×#11929609	1	1	yes	characteristic
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	632	2	no	—
141	C₂×C₄	—	632	4	no	—
142	C₂×C₄	#14579094	1	4	yes	—
143	C₂×C₄	#14579104	1	4	yes	—
144	C₂×C₄	#14579105	1	4	yes	—
145	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
146	C₂×C₄	#14579157	1	1	yes	characteristic
147	C₂×C₄	#14579159	1	1	yes	characteristic
148	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
149	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
150	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
151	C₈	#14579094	1	4	yes	—
152	C₈	C₄⋊C₄×D₇₉	1	2	yes	—
153	C₈	#14579160	1	2	yes	—
154	C₂²	C₂×#14579094	1	1	yes	characteristic
155	C₂²	#19098939	1	1	yes	characteristic
156	C₂²	#19098966	1	1	yes	characteristic
157	C₂²	C₂×#14579104	1	2	yes	—
158	C₂²	C₂×#14579105	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	1264	2	no	—
161	C₄	#19098390	1	8	yes	complemented
162	C₄	#19098432	1	1	yes	characteristic
163	C₄	C₂×#14579094	1	1	yes	characteristic
164	C₄	#19098942	1	1	yes	characteristic
165	C₄	#19098966	1	1	yes	characteristic
166	C₂	#23763559	1	1	yes	characteristic
167	C₂	C₂×#19098390	1	2	yes	—
168	C₂	#23765655	1	1	yes	characteristic
169	C₂	#23765657	1	1	yes	characteristic
170	C₂	#23765680	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1845	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1845	D₇₉
hypocenter	C₇₉	#12906
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₄×C₇₉⋊C₁₆	C₄	semidirect
35. C₇₉⋊C₁₆	C₄⋊C₄	semidirect
74. C₄×C₇₉	#367	semidirect
90. C₇₉	#12906	semidirect
161. C₄	#19098390	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098390	2
C₂×C₄×C₇₉⋊C₁₆	2
#23763560	2
C₇₉×#1845	2
#12906	79