Subgroups — #34199816

Order 20224.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 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	#23761683	C₂	1	1	yes	maximal, characteristic
3	#23761463	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	—	2	1	no	—
6	#19098616	C₄	1	1	yes	characteristic
7	#19098616	C₄	1	1	yes	characteristic
8	C₄⋊(C₇₉⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
10	#19098284	C₄	1	2	yes	—
11	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₇₉	—	2	1	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₇₉	—	2	1	no	—
15	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
16	#14579101	—	4	2	no	—
17	C₂×C₇₉⋊C₁₆	—	2	2	no	—
18	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
19	C₄⋊(C₇₉⋊C₈)	C₈	1	2	yes	—
20	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₇₉⋊C₈)	C₈	1	2	yes	—
22	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₉	—	2	2	no	—
28	C₂×C₁₆×C₇₉	—	4	2	no	—
29	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
33	C₂×C₇₉⋊C₈	—	2	2	no	—
34	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
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	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
43	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
44	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₉	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
49	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₇₉	—	4	2	no	—
51	C₂³×C₇₉	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	2	2	no	—
53	C₂×C₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#106	1	2	yes	—
58	C₂×C₄×C₇₉	#108	1	2	yes	—
59	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₇₉	—	2	1	no	—
63	C₈×C₇₉	—	2	1	no	—
64	C₈×C₇₉	#106	1	2	yes	—
65	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₈×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₇₉⋊C₈	—	2	4	no	—
68	C₇₉⋊C₈	—	4	2	no	—
69	C₂²×C₇₉	—	2	1	no	—
70	C₂²×C₇₉	—	2	1	no	—
71	C₂²×C₇₉	#334	1	2	yes	—
72	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₇₉	—	2	1	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	—	2	1	no	—
76	C₄×C₇₉	#334	1	2	yes	—
77	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
78	C₄×C₇₉	C₂×#108	1	1	yes	characteristic
79	C₄×C₇₉	#504	1	2	yes	—
80	#11997	—	79	1	no	maximal, Sylow
81	C₂×C₇₉	—	2	2	no	—
82	C₂×C₇₉	C₂×#334	1	1	yes	characteristic
83	C₂×C₇₉	#2151	1	1	yes	characteristic
84	C₂×C₇₉	#2153	1	1	yes	characteristic
85	#2150	—	79	1	no	—
86	C₂×C₄⋊C₁₆	D₇₉	1	1	yes	characteristic
87	#1450	—	79	1	no	—
88	#1367	—	79	4	no	—
89	C₇₉	#11997	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₃₂	—	158	1	no	—
91	C₂²×C₁₆	—	2	1	no	—
92	#431	—	79	1	no	—
93	#431	—	79	1	no	—
94	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
95	C₄⋊C₁₆	—	2	1	no	—
96	C₄⋊C₁₆	—	79	2	no	—
97	C₄⋊C₁₆	—	158	1	no	—
98	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
99	#322	—	79	2	no	—
100	C₈²	—	79	2	no	—
101	C₂²×C₈	Dic₁₅₈	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
104	M₃₂	—	316	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	4	2	no	—
107	C₂×C₁₆	—	158	2	no	—
108	C₄⋊C₈	—	79	2	no	—
109	C₄⋊C₈	—	158	1	no	—
110	C₂²⋊C₈	—	79	2	no	—
111	C₂²⋊C₈	—	158	1	no	—
112	C₄×C₈	—	79	2	no	—
113	C₄×C₈	—	79	2	no	—
114	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
115	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
118	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
119	C₂²×C₄	C₈×D₇₉	1	1	yes	characteristic
120	C₂²×C₄	#11929607	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	79	4	no	—
123	C₂×C₈	—	158	2	no	—
124	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
125	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
126	C₂×C₈	#11929611	1	2	yes	—
127	C₂×C₈	#11929612	1	2	yes	—
128	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
130	C₄²	Dic₇₉⋊C₄	1	1	yes	characteristic
131	C₄²	C₈×D₇₉	1	1	yes	characteristic
132	C₄²	#11929607	1	1	yes	characteristic
133	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
134	C₁₆	—	4	2	no	—
135	C₁₆	—	316	2	no	—
136	C₂³	Dic₇₉⋊C₈	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
140	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
141	C₂×C₄	Dic₇₉⋊C₈	1	1	yes	characteristic
142	C₂×C₄	#14579092	1	2	yes	—
143	C₂×C₄	#14579096	1	2	yes	—
144	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
146	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	316	2	no	—
150	C₈	—	158	4	no	—
151	C₈	#14579092	1	2	yes	—
152	C₈	C₂×#11929611	1	1	yes	characteristic
153	C₈	C₂×#11929612	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#19098313	1	2	yes	—
157	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#19098313	1	2	yes	—
162	C₄	C₂×#14579092	1	1	yes	characteristic
163	C₄	C₂×#14579096	1	1	yes	characteristic
164	C₄	#19098979	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#19098313	1	1	yes	characteristic
167	C₂	#23765158	1	1	yes	characteristic
168	C₂	#23765165	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄⋊C₈
prime radical	C₂³×C₇₉	C₄⋊C₈
cosocle	C₄×C₈×C₇₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#11997
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #23761463	C₂	semidirect
89. C₇₉	#11997	semidirect

Maximal subgroups

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

subgroup	index
#23765156	2
#23761683	2
#23761463	2
C₂×C₄⋊C₁₆×C₇₉	2
#11997	79