Subgroups — #34175205

Order 20224.

274 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579095	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579095	C₄	1	2	yes	—
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
6	C₄×C₄⋊Dic₇₉	—	2	1	no	—
7	#19098326	—	2	1	no	—
8	#19098314	—	2	1	no	—
9	#19098314	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₇₉	—	2	1	no	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	#346×C₇₉	—	2	1	no	—
13	#346×C₇₉	C₄	1	2	yes	complemented
14	C₂²×C₇₉⋊C₈	—	4	1	no	—
15	C₂×C₄⋊Dic₇₉	—	2	2	no	—
16	C₂×C₄×Dic₇₉	—	4	1	no	—
17	#14579095	—	2	2	no	—
18	#14579095	C₈	1	4	yes	complemented
19	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
20	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
21	C₇₉⋊#95	—	4	1	no	—
22	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₉	—	2	4	no	—
26	C₂×C₁₆×C₇₉	—	2	2	no	—
27	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
31	C₂²×Dic₇₉	—	4	2	no	—
32	C₂×C₇₉⋊C₈	—	4	2	no	—
33	C₂×C₇₉⋊C₈	—	8	2	no	—
34	C₄⋊Dic₇₉	—	2	4	no	—
35	C₄⋊Dic₇₉	—	4	2	no	—
36	C₄×Dic₇₉	—	8	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	2	no	—
41	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
47	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
50	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
51	C₁₆×C₇₉	—	2	4	no	—
52	C₂³×C₇₉	#102	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	2	2	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	2	1	no	—
56	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
57	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
59	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
61	C₂×C₄×C₇₉	#104	1	2	yes	—
62	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
64	C₂×Dic₇₉	—	4	4	no	—
65	C₂×Dic₇₉	—	8	2	no	—
66	C₂×Dic₇₉	—	8	2	no	—
67	C₈×C₇₉	—	2	1	no	—
68	C₈×C₇₉	—	2	1	no	—
69	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
70	C₈×C₇₉	#97	1	1	yes	characteristic
71	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
72	C₇₉⋊C₈	—	8	2	no	—
73	C₂²×C₇₉	D₄⋊C₈	1	2	yes	—
74	C₂²×C₇₉	#340	1	1	yes	characteristic
75	C₂²×C₇₉	#360	1	2	yes	—
76	C₂²×C₇₉	#361	1	1	yes	characteristic
77	C₂²×C₇₉	#362	1	1	yes	characteristic
78	C₄×C₇₉	—	2	1	no	—
79	C₄×C₇₉	—	2	1	no	—
80	C₄×C₇₉	—	2	1	no	—
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	Q₈⋊C₈	1	2	yes	—
83	C₄×C₇₉	#336	1	1	yes	characteristic
84	C₄×C₇₉	#339	1	1	yes	characteristic
85	Dic₇₉	—	8	4	no	—
86	#7178	—	79	1	no	maximal, Sylow
87	C₂×C₇₉	#1279	1	1	yes	characteristic
88	C₂×C₇₉	#1333	1	2	yes	—
89	C₂×C₇₉	#1334	1	2	yes	—
90	C₂×C₇₉	#1386	1	1	yes	characteristic
91	C₂×C₇₉	#1387	1	1	yes	characteristic
92	C₂×#346	D₇₉	1	1	yes	characteristic
93	C₄×#107	—	79	1	no	—
94	C₂×#334	—	79	1	no	—
95	C₇₉	#7178	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	2	1	no	—
97	C₂×#107	—	79	2	no	—
98	C₂×C₄⋊C₈	—	158	1	no	—
99	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
100	C₄×C₄⋊C₄	—	158	1	no	—
101	#346	—	2	1	no	—
102	#346	Dic₇₉	1	2	yes	complemented
103	#340	—	158	1	no	—
104	#334	—	79	2	no	—
105	#334	—	158	1	no	—
106	C₂²×C₈	—	316	1	no	—
107	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	158	2	no	—
110	C₂×C₄²	—	316	1	no	—
111	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	—	2	2	no	—
114	#107	—	79	4	no	—
115	#107	—	158	2	no	—
116	C₄⋊C₈	—	316	1	no	—
117	C₄⋊C₈	—	158	2	no	—
118	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
122	#95	—	316	1	no	—
123	C₂²×C₄	—	316	2	no	—
124	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
125	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
126	C₂²×C₄	#11929613	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	316	2	no	—
129	C₂×C₈	—	632	2	no	—
130	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
131	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
132	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
133	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
134	C₂×C₈	C₈×D₇₉	1	2	yes	—
135	C₂×C₈	#11929607	1	2	yes	—
136	C₄⋊C₄	—	316	2	no	—
137	C₄⋊C₄	—	158	4	no	—
138	C₄²	—	632	2	no	—
139	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
140	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
141	C₄²	#11929613	1	1	yes	characteristic
142	C₄²	Dic₃₁₆	1	1	yes	characteristic
143	C₁₆	—	2	4	no	—
144	C₂³	#14579083	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	316	4	no	—
149	C₂×C₄	—	632	2	no	—
150	C₂×C₄	—	632	2	no	—
151	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
152	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
153	C₂×C₄	#14579083	1	1	yes	characteristic
154	C₂×C₄	#14579086	1	1	yes	characteristic
155	C₂×C₄	#14579086	1	1	yes	characteristic
156	C₂×C₄	#14579089	1	2	yes	—
157	C₂×C₄	#14579093	1	1	yes	characteristic
158	C₂×C₄	#14579095	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	632	2	no	—
162	C₈	C₈×Dic₇₉	1	1	yes	characteristic
163	C₈	#14579071	1	1	yes	characteristic
164	C₈	Dic₇₉⋊C₈	1	2	yes	—
165	C₂²	#19098292	1	2	yes	—
166	C₂²	#19098326	1	1	yes	characteristic
167	C₂²	#19098374	1	2	yes	—
168	C₂²	#19098377	1	1	yes	characteristic
169	C₂²	#19098380	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	632	4	no	—
175	C₄	#19098295	1	2	yes	—
176	C₄	#19098317	1	1	yes	characteristic
177	C₄	#19098323	1	1	yes	characteristic
178	C₂	#23761240	1	1	yes	characteristic
179	C₂	#23761368	1	2	yes	—
180	C₂	#23761371	1	2	yes	—
181	C₂	#23761506	1	1	yes	characteristic
182	C₂	#23761509	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579083
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#102
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#7178
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
9. #19098314	C₄	semidirect
13. #346×C₇₉	C₄	semidirect
18. #14579095	C₈	semidirect
95. C₇₉	#7178	semidirect
102. #346	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579095	2
C₂×#19098314	2
C₂×#346×C₇₉	2
#7178	79