Subgroups — #34175157

Order 20224.

200 conjugacy classes of subgroups.

The subgroup lattice has 176 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×#19098384	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579101	—	2	1	no	—
5	C₂×#14579101	—	2	1	no	—
6	#19098384	—	2	1	no	—
7	#19098384	—	2	1	no	—
8	#19098384	C₄	1	2	yes	—
9	#19098384	C₄	1	2	yes	—
10	C₂×M₃₂×C₇₉	C₄	1	1	yes	characteristic
11	C₂×M₃₂×C₇₉	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
13	#346×C₇₉	—	2	1	no	—
14	C₄×C₁₆×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	—	4	1	no	—
20	C₂×C₇₉⋊C₁₆	—	4	1	no	—
21	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
24	M₃₂×C₇₉	—	2	1	no	—
25	M₃₂×C₇₉	—	2	1	no	—
26	M₃₂×C₇₉	C₈	1	2	yes	—
27	M₃₂×C₇₉	C₈	1	2	yes	—
28	C₂×C₁₆×C₇₉	—	2	1	no	—
29	C₂×C₁₆×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	—	2	1	no	—
37	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
50	C₄²×C₇₉	—	2	1	no	—
51	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
53	C₁₆×C₇₉	—	2	2	no	—
54	C₁₆×C₇₉	—	2	2	no	—
55	C₂³×C₇₉	#108	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	4	2	no	—
59	C₂×C₄×C₇₉	#95	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#108	1	1	yes	characteristic
65	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
66	C₈×C₇₉	#97	1	1	yes	characteristic
67	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
68	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₈×C₇₉	#102	1	1	yes	characteristic
70	C₈×C₇₉	#102	1	1	yes	characteristic
71	C₈×C₇₉	#103	1	1	yes	characteristic
72	C₈×C₇₉	#103	1	1	yes	characteristic
73	C₂²×C₇₉	—	4	1	no	—
74	C₂²×C₇₉	#334	1	1	yes	characteristic
75	C₂²×C₇₉	#335	1	1	yes	characteristic
76	C₂²×C₇₉	#341	1	1	yes	characteristic
77	C₄×C₇₉	—	2	2	no	—
78	C₄×C₇₉	—	4	1	no	—
79	C₄×C₇₉	#334	1	1	yes	characteristic
80	C₄×C₇₉	#335	1	1	yes	characteristic
81	C₄×C₇₉	#336	1	1	yes	characteristic
82	C₄×C₇₉	#340	1	1	yes	characteristic
83	#7162	—	79	1	no	maximal, Sylow
84	C₂×C₇₉	—	2	1	no	—
85	C₂×C₇₉	—	2	1	no	—
86	C₂×C₇₉	—	2	1	no	—
87	C₂×C₇₉	#1277	1	1	yes	characteristic
88	C₂×#364	—	79	1	no	—
89	C₂×#364	—	79	1	no	—
90	C₄×M₃₂	D₇₉	1	1	yes	characteristic
91	C₇₉	#7162	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	158	1	no	—
93	C₂×M₃₂	—	158	1	no	—
94	C₂×M₃₂	Dic₇₉	1	1	yes	characteristic
95	C₂×M₃₂	Dic₇₉	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
97	#364	—	79	2	no	—
98	#364	—	79	2	no	—
99	#364	—	158	1	no	—
100	#364	—	158	1	no	—
101	#346	—	2	1	no	—
102	C₄×C₁₆	—	2	1	no	—
103	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
105	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
106	M₃₂	—	2	1	no	—
107	M₃₂	—	2	1	no	—
108	M₃₂	—	316	1	no	—
109	M₃₂	—	316	1	no	—
110	M₃₂	—	158	2	no	—
111	M₃₂	—	158	2	no	—
112	M₃₂	C₇₉⋊C₈	1	2	yes	—
113	M₃₂	C₇₉⋊C₈	1	2	yes	—
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	—	316	1	no	—
117	C₂×C₁₆	—	316	1	no	—
118	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
124	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
125	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
126	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
128	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
129	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
130	C₂×C₈	#11929608	1	1	yes	characteristic
131	C₂×C₈	#11929608	1	1	yes	characteristic
132	C₂×C₈	D₆₃₂	1	1	yes	characteristic
133	C₂×C₈	#11929613	1	1	yes	characteristic
134	C₂×C₈	#11929613	1	1	yes	characteristic
135	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
138	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	316	2	no	—
142	C₁₆	—	316	2	no	—
143	C₂³	#14579097	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	C₇₉⋊#95	1	1	yes	characteristic
148	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
149	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
150	C₂×C₄	#14579093	1	1	yes	characteristic
151	C₂×C₄	#14579095	1	1	yes	characteristic
152	C₂×C₄	#14579097	1	1	yes	characteristic
153	C₈	C₄×C₇₉⋊C₈	1	1	yes	characteristic
154	C₈	#14579072	1	1	yes	characteristic
155	C₈	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
156	C₈	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
157	C₈	#14579083	1	1	yes	characteristic
158	C₈	#14579083	1	1	yes	characteristic
159	C₈	#14579086	1	1	yes	characteristic
160	C₈	#14579086	1	1	yes	characteristic
161	C₂²	—	4	1	no	—
162	C₂²	#19098314	1	1	yes	characteristic
163	C₂²	#19098316	1	1	yes	characteristic
164	C₂²	#19098328	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	4	1	no	—
167	C₄	#19098314	1	1	yes	characteristic
168	C₄	#19098316	1	1	yes	characteristic
169	C₄	#19098318	1	1	yes	characteristic
170	C₄	#19098326	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#23761235	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579083
commutator	C₈×C₇₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₉
Fitting	C₄×M₃₂×C₇₉	C₂
socle	C₂×C₇₉	#1277
prime radical	C₂³×C₇₉	#108
cosocle	C₂×C₄×C₈×C₇₉	C₂²
hypercenter	C₄×M₃₂	D₇₉
hypocenter	C₇₉	#7162
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
91. C₇₉	#7162	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
C₄×M₃₂×C₇₉	2
#7162	79