Subgroups — #34175185

Order 20224.

282 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098316	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	C₄×C₇₉⋊C₁₆	—	2	1	no	—
7	C₄×C₇₉⋊C₁₆	C₄	1	2	yes	complemented
8	#19098316	—	2	1	no	—
9	#19098316	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	1	no	—
33	C₂×C₇₉⋊C₈	—	8	1	no	—
34	C₇₉⋊C₁₆	—	2	4	no	—
35	C₂²×C₄×C₇₉	—	4	2	no	—
36	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
39	C₂×C₈×C₇₉	—	2	2	no	—
40	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
46	C₄⋊C₄×C₇₉	—	2	4	no	—
47	C₄⋊C₄×C₇₉	—	4	2	no	—
48	C₄²×C₇₉	—	8	1	no	—
49	C₄²×C₇₉	—	8	1	no	—
50	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
53	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
54	C₂³×C₇₉	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₉	—	2	2	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	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
69	C₂×C₄×C₇₉	SD₃₂	1	1	yes	characteristic
70	C₂×C₄×C₇₉	Q₃₂	1	1	yes	characteristic
71	C₈×C₇₉	—	2	1	no	—
72	C₈×C₇₉	—	2	1	no	—
73	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
74	C₈×C₇₉	#97	1	1	yes	characteristic
75	C₈×C₇₉	C₄⋊C₈	1	2	yes	—
76	C₇₉⋊C₈	—	8	2	no	—
77	C₂²×C₇₉	D₄⋊C₈	1	2	yes	—
78	C₂²×C₇₉	#340	1	1	yes	characteristic
79	C₂²×C₇₉	#357	1	1	yes	characteristic
80	C₂²×C₇₉	#358	1	1	yes	characteristic
81	C₂²×C₇₉	#359	1	2	yes	—
82	C₄×C₇₉	—	2	1	no	—
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	—	8	4	no	—
85	C₄×C₇₉	Q₈⋊C₈	1	2	yes	—
86	C₄×C₇₉	#336	1	1	yes	characteristic
87	C₄×C₇₉	#339	1	1	yes	characteristic
88	C₄×C₇₉	#357	1	1	yes	characteristic
89	C₄×C₇₉	#358	1	1	yes	characteristic
90	C₄×C₇₉	#366	1	1	yes	characteristic
91	C₄×C₇₉	#367	1	1	yes	characteristic
92	#7171	—	79	1	no	maximal, Sylow
93	C₂×C₇₉	#1279	1	1	yes	characteristic
94	C₂×C₇₉	#1331	1	2	yes	—
95	C₂×C₇₉	#1332	1	2	yes	—
96	C₂×C₇₉	#1380	1	1	yes	characteristic
97	C₂×C₇₉	#1382	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	79	1	no	—
99	C₄×#107	D₇₉	1	1	yes	characteristic
100	C₂×#335	—	79	1	no	—
101	C₇₉	#7171	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	158	1	no	—
103	C₂×#107	Dic₇₉	1	2	yes	—
104	C₂×C₄⋊C₈	—	158	1	no	—
105	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	1	no	—
107	C₄×C₁₆	—	79	2	no	—
108	C₄×C₁₆	—	158	1	no	—
109	#340	—	2	1	no	—
110	#335	—	79	2	no	—
111	#335	—	158	1	no	—
112	C₂²×C₈	—	316	1	no	—
113	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
118	C₂×C₁₆	—	158	4	no	—
119	C₂×C₁₆	—	158	2	no	—
120	#107	—	2	2	no	—
121	#107	C₇₉⋊C₈	1	4	yes	complemented
122	C₄⋊C₈	—	316	1	no	—
123	C₄⋊C₈	—	158	2	no	—
124	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
125	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
126	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
127	C₄×C₈	D₃₁₆	1	1	yes	characteristic
128	#95	—	4	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
131	C₂²×C₄	#11929609	1	1	yes	characteristic
132	C₂²×C₄	#11929612	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	316	2	no	—
135	C₂×C₈	—	632	1	no	—
136	C₂×C₈	—	632	1	no	—
137	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
138	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
139	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
141	C₂×C₈	C₂×C₇₉⋊C₈	1	2	yes	—
142	C₂×C₈	#11929608	1	2	yes	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	8	1	no	—
146	C₄²	—	8	1	no	—
147	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
148	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
149	C₄²	#11929611	1	1	yes	characteristic
150	C₄²	#11929614	1	1	yes	characteristic
151	C₁₆	—	158	4	no	—
152	C₂³	#14579082	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
159	C₂×C₄	#14579081	1	1	yes	characteristic
160	C₂×C₄	#14579084	1	1	yes	characteristic
161	C₂×C₄	#14579085	1	1	yes	characteristic
162	C₂×C₄	#14579088	1	2	yes	—
163	C₂×C₄	#14579092	1	1	yes	characteristic
164	C₂×C₄	#14579094	1	1	yes	characteristic
165	C₂×C₄	#14579102	1	1	yes	characteristic
166	C₂×C₄	#14579104	1	1	yes	characteristic
167	C₂×C₄	#14579105	1	1	yes	characteristic
168	C₂×C₄	#14579107	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	632	2	no	—
172	C₈	C₄×C₇₉⋊C₈	1	1	yes	characteristic
173	C₈	#14579072	1	1	yes	characteristic
174	C₈	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
175	C₂²	D₄⋊(C₇₉⋊C₈)	1	2	yes	—
176	C₂²	#19098325	1	1	yes	characteristic
177	C₂²	#19098364	1	1	yes	characteristic
178	C₂²	#19098367	1	1	yes	characteristic
179	C₂²	#19098370	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	Q₈⋊(C₇₉⋊C₈)	1	2	yes	—
184	C₄	#19098318	1	1	yes	characteristic
185	C₄	#19098322	1	1	yes	characteristic
186	C₄	#19098363	1	1	yes	characteristic
187	C₄	#19098366	1	1	yes	characteristic
188	C₄	#19098388	1	1	yes	characteristic
189	C₄	#19098390	1	1	yes	characteristic
190	C₂	#23761241	1	1	yes	characteristic
191	C₂	#23761361	1	2	yes	—
192	C₂	#23761364	1	2	yes	—
193	C₂	#23761491	1	1	yes	characteristic
194	C₂	#23761495	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11929609
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	#7171
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. C₄×C₇₉⋊C₁₆	C₄	semidirect
9. #19098316	C₄	semidirect
25. #107×C₇₉	C₈	semidirect
101. C₇₉	#7171	semidirect
121. #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×C₄×C₇₉⋊C₁₆	2
C₂×#19098316	2
C₄×#107×C₇₉	2
#7171	79