Subgroups — #34175179

Order 20224.

266 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765156	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579101	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579101	—	2	1	no	—
5	C₂×#14579101	—	2	1	no	—
6	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	complemented
8	#19098339	C₄	1	2	yes	complemented
9	C₄×C₇₉⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₈×C₇₉	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₉	—	2	1	no	—
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₉	C₄	1	1	yes	characteristic
14	#336×C₇₉	—	2	1	no	—
15	#14579101	—	2	2	no	—
16	#14579101	—	2	2	no	—
17	#14579101	—	4	1	no	—
18	#14579101	—	4	1	no	—
19	C₂×C₇₉⋊C₁₆	—	2	2	no	—
20	C₂×C₇₉⋊C₁₆	—	2	2	no	—
21	C₂²×C₈×C₇₉	—	2	1	no	—
22	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
25	C₂×C₄²×C₇₉	—	2	1	no	—
26	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₉	—	2	1	no	—
28	C₄⋊C₈×C₇₉	C₈	1	2	yes	—
29	C₄×C₈×C₇₉	—	4	1	no	—
30	C₄×C₈×C₇₉	—	4	1	no	—
31	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
35	#95×C₇₉	C₈	1	2	yes	—
36	C₇₉⋊C₁₆	—	2	4	no	—
37	C₇₉⋊C₁₆	—	4	2	no	—
38	C₂²×C₄×C₇₉	—	2	2	no	—
39	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₇₉	—	2	2	no	—
43	C₂×C₈×C₇₉	—	2	1	no	—
44	C₂×C₈×C₇₉	—	2	1	no	—
45	C₂×C₈×C₇₉	—	4	1	no	—
46	C₂×C₈×C₇₉	—	4	1	no	—
47	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
52	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
55	C₄⋊C₄×C₇₉	—	2	4	no	—
56	C₄²×C₇₉	—	4	1	no	—
57	C₄²×C₇₉	—	4	1	no	—
58	C₄²×C₇₉	C₄²	1	1	yes	characteristic
59	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
61	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
62	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	2	4	no	—
65	C₂×C₄×C₇₉	—	4	1	no	—
66	C₂×C₄×C₇₉	—	4	2	no	—
67	C₂×C₄×C₇₉	—	4	2	no	—
68	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₇₉	—	2	1	no	—
79	C₈×C₇₉	—	2	1	no	—
80	C₈×C₇₉	—	4	2	no	—
81	C₈×C₇₉	#102	1	1	yes	characteristic
82	C₈×C₇₉	#103	1	1	yes	characteristic
83	C₈×C₇₉	#106	1	1	yes	characteristic
84	C₈×C₇₉	#107	1	1	yes	characteristic
85	C₂²×C₇₉	—	2	1	no	—
86	C₂²×C₇₉	—	2	1	no	—
87	C₂²×C₇₉	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₉	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₇₉	#336	1	1	yes	characteristic
90	C₄×C₇₉	—	2	1	no	—
91	C₄×C₇₉	—	2	1	no	—
92	C₄×C₇₉	—	2	1	no	—
93	C₄×C₇₉	—	4	4	no	—
94	C₄×C₇₉	D₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₉	Q₈⋊C₈	1	1	yes	characteristic
96	C₄×C₇₉	#339	1	1	yes	characteristic
97	C₄×C₇₉	#340	1	1	yes	characteristic
98	C₄×C₇₉	#364	1	2	yes	—
99	#7169	—	79	1	no	maximal, Sylow
100	C₂×C₇₉	—	2	1	no	—
101	C₂×C₇₉	—	2	1	no	—
102	C₂×C₇₉	#1279	1	1	yes	characteristic
103	C₂×C₇₉	#1376	1	1	yes	characteristic
104	C₂×C₇₉	#1378	1	1	yes	characteristic
105	#2150	—	79	1	no	—
106	C₄×M₃₂	—	79	1	no	—
107	C₈×C₄⋊C₄	D₇₉	1	1	yes	characteristic
108	C₇₉	#7169	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₃₂	—	158	1	no	—
110	C₂×M₃₂	—	158	1	no	—
111	C₂×C₄⋊C₈	Dic₇₉	1	1	yes	characteristic
112	C₂×C₄×C₈	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
114	C₄×C₄⋊C₄	Dic₇₉	1	1	yes	characteristic
115	C₄⋊C₁₆	—	79	2	no	—
116	C₄⋊C₁₆	—	158	1	no	—
117	#346	—	79	2	no	—
118	C₄×C₁₆	—	158	1	no	—
119	#336	—	2	1	no	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
126	M₃₂	—	316	1	no	—
127	M₃₂	—	316	1	no	—
128	M₃₂	—	158	2	no	—
129	M₃₂	—	158	2	no	—
130	C₂×C₁₆	—	158	2	no	—
131	C₂×C₁₆	—	158	2	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	C₇₉⋊C₈	1	2	yes	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
137	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
138	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
139	C₄×C₈	D₃₁₆	1	1	yes	characteristic
140	#95	C₇₉⋊C₈	1	2	yes	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
144	C₂²×C₄	#11929608	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
151	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
152	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
153	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
154	C₂×C₈	#11929609	1	1	yes	characteristic
155	C₂×C₈	#11929611	1	1	yes	characteristic
156	C₂×C₈	#11929612	1	1	yes	characteristic
157	C₂×C₈	#11929614	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
162	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
163	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
164	C₄²	#11929608	1	1	yes	characteristic
165	C₁₆	—	316	2	no	—
166	C₁₆	—	158	4	no	—
167	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
174	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
175	C₂×C₄	#14579072	1	1	yes	characteristic
176	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
177	C₂×C₄	#14579082	1	1	yes	characteristic
178	C₂×C₄	#14579085	1	1	yes	characteristic
179	C₂×C₄	#14579088	1	1	yes	characteristic
180	C₂×C₄	#14579088	1	1	yes	characteristic
181	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
182	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	2	no	—
186	C₈	#14579081	1	1	yes	characteristic
187	C₈	#14579084	1	1	yes	characteristic
188	C₈	#14579092	1	1	yes	characteristic
189	C₈	#14579094	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	D₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
193	C₂²	Q₈⋊(C₇₉⋊C₈)	1	1	yes	characteristic
194	C₂²	#19098318	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	4	no	—
199	C₄	D₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
200	C₄	Q₈⋊(C₇₉⋊C₈)	1	1	yes	characteristic
201	C₄	#19098322	1	1	yes	characteristic
202	C₄	#19098325	1	1	yes	characteristic
203	C₄	#19098384	1	2	yes	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#23761241	1	1	yes	characteristic
207	C₂	#23761482	1	1	yes	characteristic
208	C₂	#23761486	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₈×C₄⋊C₄×C₇₉	C₂
socle	C₂²×C₇₉	#336
prime radical	C₂³×C₇₉	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₈×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#7169
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₄⋊(C₇₉⋊C₁₆)	C₄	semidirect
8. #19098339	C₄	semidirect
108. C₇₉	#7169	semidirect

Maximal subgroups

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

subgroup	index
#23765156	2
C₄×#14579101	2
C₈×C₄⋊C₄×C₇₉	2
#7169	79