Subgroups — #34203648

Order 20224.

308 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	#23765173	C₂	1	2	yes	maximal
2	#23763279	C₂	1	2	yes	maximal
3	#23761504	C₂	1	1	yes	maximal, characteristic
4	#23761502	C₂	1	1	yes	maximal, characteristic
5	C₄×M₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	C₂²	1	2	yes	—
7	#19098601	C₂²	1	2	yes	—
8	#19098518	—	2	2	no	—
9	#19098386	C₄	1	8	yes	complemented
10	#19098323	—	2	2	no	—
11	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#346×C₇₉	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₇₉	C₄	1	2	yes	complemented
16	C₂×#11929608	—	2	2	no	—
17	#14579120	—	2	2	no	—
18	#14579097	—	2	2	no	—
19	#14579097	C₂×C₄	1	4	yes	—
20	#14579095	—	2	2	no	—
21	#14579093	C₂×C₄	1	4	yes	—
22	#14579071	—	4	2	no	—
23	C₈×Dic₇₉	—	4	2	no	—
24	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
27	M₃₂×C₇₉	—	2	2	no	—
28	M₃₂×C₇₉	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
35	#11929608	—	2	4	no	—
36	#11929608	—	4	2	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₄⋊Dic₇₉	—	2	4	no	—
39	C₂²⋊Dic₇₉	—	4	2	no	—
40	C₄×Dic₇₉	—	4	2	no	—
41	C₂²×C₄×C₇₉	—	2	1	no	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
47	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₇₉	#41	1	2	yes	—
51	C₄²×C₇₉	—	2	1	no	—
52	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₇₉	—	2	2	no	—
55	C₁₆×C₇₉	C₄²	1	4	yes	—
56	C₂³×C₇₉	#135	1	1	yes	characteristic
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	2	1	no	—
59	C₂×C₄×C₇₉	—	4	2	no	—
60	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
63	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
64	C₂×Dic₇₉	—	4	4	no	—
65	C₈×C₇₉	#106	1	2	yes	—
66	C₈×C₇₉	#106	1	1	yes	characteristic
67	C₈×C₇₉	#106	1	1	yes	characteristic
68	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₇₉	#117	1	1	yes	characteristic
70	C₈×C₇₉	C₄×Q₈	1	2	yes	—
71	C₇₉⋊C₈	—	4	4	no	—
72	C₂²×C₇₉	—	4	1	no	—
73	C₂²×C₇₉	#427	1	1	yes	characteristic
74	C₂²×C₇₉	C₄×SD₁₆	1	2	yes	—
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	—	4	1	no	—
77	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
79	C₄×C₇₉	C₄×SD₁₆	1	2	yes	—
80	Dic₇₉	—	8	4	no	—
81	#12668	—	79	1	no	maximal, Sylow
82	C₂×C₇₉	—	2	2	no	—
83	C₂×C₇₉	—	2	1	no	—
84	C₂×C₇₉	C₄×#106	1	1	yes	characteristic
85	#2155	—	79	2	no	—
86	C₄×M₃₂	D₇₉	1	1	yes	characteristic
87	#1772	—	79	2	no	—
88	#1385	—	79	1	no	—
89	#1384	—	79	1	no	—
90	C₇₉	#12668	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
92	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
93	C₂×#108	—	79	2	no	—
94	#427	—	79	2	no	—
95	#405	—	158	2	no	—
96	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
97	#365	—	79	8	no	—
98	#346	Dic₇₉	1	2	yes	complemented
99	C₄×C₁₆	Dic₇₉	1	2	yes	complemented
100	#339	—	158	2	no	—
101	C₂×M₁₆	—	158	2	no	—
102	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
104	#117	—	158	2	no	—
105	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	C₄×D₇₉	1	4	yes	—
108	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
109	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
110	#108	—	79	4	no	—
111	#108	—	158	2	no	—
112	#107	—	158	2	no	—
113	#106	—	79	4	no	—
114	#97	—	316	2	no	—
115	C₄×C₈	—	316	2	no	—
116	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
117	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
122	M₁₆	—	316	2	no	—
123	M₁₆	—	158	4	no	—
124	C₂×C₈	—	316	2	no	—
125	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
127	C₂×C₈	#11929613	1	2	yes	—
128	C₂×C₈	#11929613	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
130	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
131	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
132	C₂×C₈	#11929624	1	2	yes	—
133	C₄⋊C₄	—	158	4	no	—
134	C₂²⋊C₄	—	316	2	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	316	2	no	—
137	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	C₄×Dic₇₉	1	4	yes	—
141	C₂³	#14579183	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	316	4	no	—
146	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
147	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
148	C₂×C₄	C₂×#11929613	1	2	yes	—
149	C₂×C₄	#14579183	1	1	yes	characteristic
150	C₈	—	316	4	no	—
151	C₈	#14579093	1	2	yes	—
152	C₈	#14579093	1	1	yes	characteristic
153	C₈	#14579093	1	1	yes	characteristic
154	C₈	C₂×C₄×Dic₇₉	1	1	yes	characteristic
155	C₈	#14579120	1	1	yes	characteristic
156	C₈	C₄×Dic₁₅₈	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#19098601	1	1	yes	characteristic
159	C₂²	C₄×#11929613	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	632	4	no	—
163	C₄	C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
164	C₄	C₂×#14579093	1	1	yes	characteristic
165	C₄	C₄×#11929613	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×#14579093	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579093
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₄×M₃₂×C₇₉	C₂
socle	C₂×C₇₉	C₄×#106
prime radical	C₂³×C₇₉	#135
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₄×M₃₂	D₇₉
hypocenter	C₇₉	#12668
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
9. #19098386	C₄	semidirect
14. #346×C₇₉	C₄	semidirect
15. C₄×C₁₆×C₇₉	C₄	semidirect
90. C₇₉	#12668	semidirect
98. #346	Dic₇₉	semidirect
99. C₄×C₁₆	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23765173	2
#23763279	2
#23761504	2
#23761502	2
C₄×M₃₂×C₇₉	2
#12668	79