Subgroups — #34175209

Order 20224.

274 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579101	—	2	1	no	—
5	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
6	#19098384	—	2	1	no	—
7	#19098384	C₄	1	2	yes	complemented
8	C₈×C₇₉⋊C₈	—	2	1	no	—
9	C₈×C₇₉⋊C₈	C₄	1	2	yes	complemented
10	C₂×#108×C₇₉	C₄	1	1	yes	characteristic
11	#427×C₇₉	C₄	1	1	yes	characteristic
12	#405×C₇₉	—	2	1	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#339×C₇₉	—	2	1	no	—
15	C₂²×C₇₉⋊C₈	—	2	2	no	—
16	#14579101	—	2	2	no	—
17	#14579101	—	4	1	no	—
18	C₂×C₇₉⋊C₁₆	—	4	1	no	—
19	C₄×C₇₉⋊C₈	—	2	2	no	—
20	C₄×C₇₉⋊C₈	—	2	2	no	—
21	C₂×M₁₆×C₇₉	—	2	1	no	—
22	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
24	#117×C₇₉	—	2	1	no	—
25	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
26	#108×C₇₉	—	2	1	no	—
27	#108×C₇₉	C₈	1	2	yes	complemented
28	#107×C₇₉	C₈	1	2	yes	complemented
29	#106×C₇₉	—	2	1	no	—
30	#97×C₇₉	—	4	1	no	—
31	C₄×C₈×C₇₉	—	4	1	no	—
32	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
36	C₂×C₇₉⋊C₈	—	2	4	no	—
37	C₂×C₇₉⋊C₈	—	2	8	no	—
38	C₇₉⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₇₉	—	2	1	no	—
40	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	M₁₆×C₇₉	—	2	2	no	—
42	M₁₆×C₇₉	—	4	1	no	—
43	C₂×C₈×C₇₉	—	4	1	no	—
44	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
53	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
56	C₄⋊C₄×C₇₉	—	2	2	no	—
57	C₂²⋊C₄×C₇₉	—	4	1	no	—
58	C₄²×C₇₉	—	2	1	no	—
59	C₄²×C₇₉	—	4	1	no	—
60	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂³×C₇₉	#104	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	4	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	2	1	no	—
66	C₂×C₄×C₇₉	—	4	2	no	—
67	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
73	C₈×C₇₉	—	4	2	no	—
74	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
75	C₈×C₇₉	#97	1	1	yes	characteristic
76	C₈×C₇₉	#102	1	1	yes	characteristic
77	C₈×C₇₉	#103	1	1	yes	characteristic
78	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
80	C₈×C₇₉	#106	1	1	yes	characteristic
81	C₈×C₇₉	#107	1	1	yes	characteristic
82	C₇₉⋊C₈	—	2	8	no	—
83	C₂²×C₇₉	—	2	2	no	—
84	C₂²×C₇₉	D₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₇₉	Q₈⋊C₈	1	1	yes	characteristic
86	C₂²×C₇₉	#339	1	1	yes	characteristic
87	C₄×C₇₉	—	2	2	no	—
88	C₄×C₇₉	—	2	2	no	—
89	C₄×C₇₉	—	8	2	no	—
90	C₄×C₇₉	D₄⋊C₈	1	1	yes	characteristic
91	C₄×C₇₉	Q₈⋊C₈	1	1	yes	characteristic
92	C₄×C₇₉	#336	1	1	yes	characteristic
93	C₄×C₇₉	#340	1	1	yes	characteristic
94	#7179	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	—	2	1	no	—
96	C₂×C₇₉	—	2	1	no	—
97	C₂×C₇₉	#1279	1	1	yes	characteristic
98	C₂×C₇₉	#1335	1	2	yes	—
99	C₂×#364	—	79	1	no	—
100	#1772	D₇₉	1	1	yes	characteristic
101	C₂×C₈²	—	79	1	no	—
102	C₇₉	#7179	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₃₂	—	158	1	no	—
104	C₂×#108	Dic₇₉	1	1	yes	characteristic
105	#427	Dic₇₉	1	1	yes	characteristic
106	#405	—	2	1	no	—
107	C₂×C₄×C₈	—	158	1	no	—
108	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
109	#364	—	79	2	no	—
110	#364	—	158	1	no	—
111	#339	—	2	1	no	—
112	C₈²	—	79	2	no	—
113	C₈²	—	158	1	no	—
114	C₂×M₁₆	—	2	1	no	—
115	C₂²×C₈	—	158	2	no	—
116	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
117	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
118	#117	—	2	1	no	—
119	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
120	M₃₂	—	316	1	no	—
121	M₃₂	—	158	2	no	—
122	C₂×C₁₆	—	316	1	no	—
123	#108	—	2	1	no	—
124	#108	C₇₉⋊C₈	1	2	yes	complemented
125	#107	C₇₉⋊C₈	1	2	yes	complemented
126	#106	—	2	1	no	—
127	#97	—	4	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	—	158	2	no	—
130	C₄×C₈	—	158	2	no	—
131	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
132	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
133	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
134	C₄×C₈	D₃₁₆	1	1	yes	characteristic
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
137	M₁₆	—	2	2	no	—
138	M₁₆	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	158	4	no	—
141	C₂×C₈	—	158	8	no	—
142	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
143	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
144	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
145	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
146	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
147	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
148	C₂×C₈	#11929608	1	1	yes	characteristic
149	C₂×C₈	#11929608	1	1	yes	characteristic
150	C₂×C₈	#11929609	1	1	yes	characteristic
151	C₂×C₈	#11929611	1	1	yes	characteristic
152	C₂×C₈	#11929612	1	1	yes	characteristic
153	C₂×C₈	#11929614	1	1	yes	characteristic
154	C₄⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	4	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
159	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
160	C₁₆	—	316	2	no	—
161	C₂³	#14579088	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
168	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
169	C₂×C₄	#14579082	1	1	yes	characteristic
170	C₂×C₄	#14579085	1	1	yes	characteristic
171	C₂×C₄	#14579088	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	158	8	no	—
174	C₈	C₄×C₇₉⋊C₈	1	1	yes	characteristic
175	C₈	#14579072	1	1	yes	characteristic
176	C₈	#14579081	1	1	yes	characteristic
177	C₈	#14579084	1	1	yes	characteristic
178	C₈	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
179	C₈	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
180	C₈	#14579092	1	1	yes	characteristic
181	C₈	#14579094	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	D₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
184	C₂²	Q₈⋊(C₇₉⋊C₈)	1	1	yes	characteristic
185	C₂²	#19098322	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	8	2	no	—
189	C₄	D₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
190	C₄	Q₈⋊(C₇₉⋊C₈)	1	1	yes	characteristic
191	C₄	#19098318	1	1	yes	characteristic
192	C₄	#19098325	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#23761241	1	1	yes	characteristic
196	C₂	#23761373	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#11929609
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₇₉×#1772	C₂
socle	C₂²×C₇₉	D₄⋊C₈
prime radical	C₂³×C₇₉	#104
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#1772	D₇₉
hypocenter	C₇₉	#7179
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
7. #19098384	C₄	semidirect
9. C₈×C₇₉⋊C₈	C₄	semidirect
27. #108×C₇₉	C₈	semidirect
28. #107×C₇₉	C₈	semidirect
102. C₇₉	#7179	semidirect
124. #108	C₇₉⋊C₈	semidirect
125. #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×#19098384	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#1772	2
#7179	79