Subgroups — #34175206

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	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
6	#19098339	—	2	1	no	—
7	#19098339	C₄	1	2	yes	complemented
8	#19098314	—	2	1	no	—
9	#19098314	C₄	1	2	yes	complemented
10	C₂×#107×C₇₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
13	#340×C₇₉	—	2	1	no	—
14	C₂²×C₇₉⋊C₈	—	4	1	no	—
15	C₂×C₇₉⋊C₁₆	—	2	4	no	—
16	C₂×C₇₉⋊C₁₆	—	2	2	no	—
17	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
19	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
22	C₂×C₄²×C₇₉	—	4	1	no	—
23	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
24	#107×C₇₉	—	2	2	no	—
25	#107×C₇₉	C₈	1	4	yes	complemented
26	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
30	#95×C₇₉	—	4	1	no	—
31	C₂×C₇₉⋊C₈	—	4	2	no	—
32	C₂×C₇₉⋊C₈	—	8	2	no	—
33	C₇₉⋊C₁₆	—	2	4	no	—
34	C₂²×C₄×C₇₉	—	4	2	no	—
35	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
38	C₂×C₈×C₇₉	—	2	2	no	—
39	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
45	C₄⋊C₄×C₇₉	—	2	4	no	—
46	C₄⋊C₄×C₇₉	—	4	2	no	—
47	C₄²×C₇₉	—	8	2	no	—
48	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
51	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
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	—	4	4	no	—
57	C₂×C₄×C₇₉	—	8	2	no	—
58	C₂×C₄×C₇₉	—	8	2	no	—
59	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#104	1	2	yes	—
65	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
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	—	8	4	no	—
83	C₄×C₇₉	Q₈⋊C₈	1	2	yes	—
84	C₄×C₇₉	#336	1	1	yes	characteristic
85	C₄×C₇₉	#339	1	1	yes	characteristic
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	—	79	1	no	—
93	C₄×#107	D₇₉	1	1	yes	characteristic
94	C₂×#334	—	79	1	no	—
95	C₇₉	#7178	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	158	1	no	—
97	C₂×#107	Dic₇₉	1	2	yes	—
98	C₂×C₄⋊C₈	—	158	1	no	—
99	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
100	C₄×C₄⋊C₄	—	2	1	no	—
101	#346	—	79	2	no	—
102	#346	—	158	1	no	—
103	#340	—	2	1	no	—
104	#334	—	79	2	no	—
105	#334	—	158	1	no	—
106	C₂²×C₈	—	316	1	no	—
107	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	—	4	1	no	—
111	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
112	C₂×C₁₆	—	158	4	no	—
113	C₂×C₁₆	—	158	2	no	—
114	#107	—	2	2	no	—
115	#107	C₇₉⋊C₈	1	4	yes	complemented
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	C₄×D₇₉	1	1	yes	characteristic
121	C₄×C₈	D₃₁₆	1	1	yes	characteristic
122	#95	—	4	1	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
125	C₂²×C₄	#11929609	1	1	yes	characteristic
126	C₂²×C₄	#11929612	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	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
132	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
134	C₂×C₈	C₂×C₇₉⋊C₈	1	2	yes	—
135	C₂×C₈	#11929608	1	2	yes	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄²	—	8	2	no	—
139	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
140	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
141	C₄²	#11929611	1	1	yes	characteristic
142	C₄²	#11929614	1	1	yes	characteristic
143	C₁₆	—	158	4	no	—
144	C₂³	#14579082	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	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
152	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
153	C₂×C₄	#14579081	1	1	yes	characteristic
154	C₂×C₄	#14579084	1	1	yes	characteristic
155	C₂×C₄	#14579085	1	1	yes	characteristic
156	C₂×C₄	#14579088	1	2	yes	—
157	C₂×C₄	#14579092	1	1	yes	characteristic
158	C₂×C₄	#14579094	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	632	2	no	—
162	C₈	C₄×C₇₉⋊C₈	1	1	yes	characteristic
163	C₈	#14579072	1	1	yes	characteristic
164	C₈	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
165	C₂²	D₄⋊(C₇₉⋊C₈)	1	2	yes	—
166	C₂²	#19098325	1	1	yes	characteristic
167	C₂²	#19098373	1	2	yes	—
168	C₂²	#19098376	1	1	yes	characteristic
169	C₂²	#19098379	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	—	8	4	no	—
175	C₄	Q₈⋊(C₇₉⋊C₈)	1	2	yes	—
176	C₄	#19098318	1	1	yes	characteristic
177	C₄	#19098322	1	1	yes	characteristic
178	C₂	#23761241	1	1	yes	characteristic
179	C₂	#23761367	1	2	yes	—
180	C₂	#23761370	1	2	yes	—
181	C₂	#23761505	1	1	yes	characteristic
182	C₂	#23761508	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579082
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₄×#107×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#102
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₄×#107	D₇₉
hypocenter	C₇₉	#7178
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. #19098339	C₄	semidirect
9. #19098314	C₄	semidirect
25. #107×C₇₉	C₈	semidirect
95. C₇₉	#7178	semidirect
115. #107	C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098339	2
C₂×#19098314	2
C₄×#107×C₇₉	2
#7178	79