Subgroups — #34175136

Order 20224.

218 conjugacy classes of subgroups.

The subgroup lattice has 190 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	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579101	—	2	1	no	—
5	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
6	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
9	C₄⋊(C₇₉⋊C₁₆)	C₄	1	2	yes	—
10	#431×C₇₉	C₄	1	1	yes	characteristic
11	#431×C₇₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	#322×C₇₉	—	2	1	no	—
14	C₈²×C₇₉	—	2	1	no	—
15	#14579101	—	4	1	no	—
16	#14579101	—	4	1	no	—
17	C₂×C₇₉⋊C₁₆	—	2	2	no	—
18	C₂×C₇₉⋊C₁₆	—	2	2	no	—
19	C₂×C₇₉⋊C₁₆	—	4	1	no	—
20	C₂×C₇₉⋊C₁₆	—	4	1	no	—
21	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₈×C₇₉	—	2	1	no	—
25	C₄⋊C₈×C₇₉	C₈	1	2	yes	—
26	C₂²⋊C₈×C₇₉	—	2	1	no	—
27	C₂²⋊C₈×C₇₉	C₈	1	2	yes	—
28	C₄×C₈×C₇₉	—	2	1	no	—
29	C₄×C₈×C₇₉	—	2	1	no	—
30	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₇₉⋊C₁₆	—	4	2	no	—
35	C₇₉⋊C₁₆	—	4	2	no	—
36	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₇₉	—	2	2	no	—
40	C₂×C₈×C₇₉	—	2	2	no	—
41	C₂×C₈×C₇₉	—	2	1	no	—
42	C₂×C₈×C₇₉	—	2	1	no	—
43	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
51	C₄²×C₇₉	C₄²	1	1	yes	characteristic
52	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
55	C₂³×C₇₉	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	4	1	no	—
58	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
65	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#108	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#108	1	1	yes	characteristic
68	C₈×C₇₉	—	2	1	no	—
69	C₈×C₇₉	—	2	1	no	—
70	C₈×C₇₉	—	4	2	no	—
71	C₈×C₇₉	—	4	2	no	—
72	C₈×C₇₉	#102	1	1	yes	characteristic
73	C₈×C₇₉	#102	1	1	yes	characteristic
74	C₈×C₇₉	#103	1	1	yes	characteristic
75	C₈×C₇₉	#103	1	1	yes	characteristic
76	C₂²×C₇₉	—	2	1	no	—
77	C₂²×C₇₉	—	2	1	no	—
78	C₂²×C₇₉	#334	1	1	yes	characteristic
79	C₂²×C₇₉	#335	1	1	yes	characteristic
80	C₂²×C₇₉	#336	1	1	yes	characteristic
81	C₄×C₇₉	—	2	1	no	—
82	C₄×C₇₉	—	2	1	no	—
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	#334	1	1	yes	characteristic
85	C₄×C₇₉	#335	1	1	yes	characteristic
86	C₄×C₇₉	#340	1	1	yes	characteristic
87	C₄×C₇₉	#341	1	1	yes	characteristic
88	C₄×C₇₉	#350	1	2	yes	—
89	#7153	—	79	1	no	maximal, Sylow
90	C₂×C₇₉	—	2	1	no	—
91	C₂×C₇₉	—	2	1	no	—
92	C₂×C₇₉	#1277	1	1	yes	characteristic
93	C₂×C₇₉	#1378	1	1	yes	characteristic
94	C₂×C₇₉	#1379	1	1	yes	characteristic
95	#2150	—	79	1	no	—
96	C₂×C₄⋊C₁₆	—	79	1	no	—
97	#1450	D₇₉	1	1	yes	characteristic
98	C₇₉	#7153	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	—	158	1	no	—
100	C₂²×C₁₆	—	158	1	no	—
101	#431	Dic₇₉	1	1	yes	characteristic
102	#431	Dic₇₉	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
104	C₄⋊C₁₆	—	79	2	no	—
105	C₄⋊C₁₆	—	79	2	no	—
106	C₄⋊C₁₆	—	158	1	no	—
107	C₄⋊C₁₆	—	158	1	no	—
108	#322	—	2	1	no	—
109	C₈²	—	2	1	no	—
110	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
111	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
113	M₃₂	—	316	1	no	—
114	M₃₂	—	316	1	no	—
115	C₂×C₁₆	—	316	1	no	—
116	C₂×C₁₆	—	316	1	no	—
117	C₂×C₁₆	—	158	2	no	—
118	C₂×C₁₆	—	158	2	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	C₇₉⋊C₈	1	2	yes	—
121	C₂²⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	C₇₉⋊C₈	1	2	yes	—
123	C₄×C₈	—	2	1	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
126	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
131	C₂²×C₄	#11929608	1	1	yes	characteristic
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
137	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
138	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
139	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
140	C₂×C₈	D₆₃₂	1	1	yes	characteristic
141	C₂×C₈	#11929613	1	1	yes	characteristic
142	C₂×C₈	#11929613	1	1	yes	characteristic
143	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
144	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
145	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
146	C₄²	C₂×C₇₉⋊C₈	1	1	yes	characteristic
147	C₄²	#11929608	1	1	yes	characteristic
148	C₁₆	—	316	2	no	—
149	C₁₆	—	316	2	no	—
150	C₂³	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
154	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
155	C₂×C₄	#14579072	1	1	yes	characteristic
156	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
157	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
158	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
159	C₂×C₄	#14579093	1	1	yes	characteristic
160	C₂×C₄	#14579095	1	1	yes	characteristic
161	C₂×C₄	#14579097	1	1	yes	characteristic
162	C₂×C₄	#14579097	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	4	2	no	—
166	C₈	—	4	2	no	—
167	C₈	#14579083	1	1	yes	characteristic
168	C₈	#14579083	1	1	yes	characteristic
169	C₈	#14579086	1	1	yes	characteristic
170	C₈	#14579086	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#19098314	1	1	yes	characteristic
174	C₂²	#19098316	1	1	yes	characteristic
175	C₂²	#19098318	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#19098314	1	1	yes	characteristic
180	C₄	#19098316	1	1	yes	characteristic
181	C₄	#19098326	1	1	yes	characteristic
182	C₄	#19098328	1	1	yes	characteristic
183	C₄	#19098347	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#23761235	1	1	yes	characteristic
187	C₂	#23761486	1	1	yes	characteristic
188	C₂	#23761489	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	C₇₉×#1450	C₂
socle	C₂²×C₇₉	#336
prime radical	C₂³×C₇₉	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	#1450	D₇₉
hypocenter	C₇₉	#7153
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
98. C₇₉	#7153	semidirect

Maximal subgroups

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

subgroup	index
#23765156	2
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₇₉×#1450	2
#7153	79