Subgroups — #34203658

Order 20224.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098393	C₂	1	1	yes	maximal, characteristic
2	#23765184	C₂	1	1	yes	maximal, characteristic
3	#23763279	C₂	1	2	yes	maximal
4	#23761496	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×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	#19098393	C₄	1	4	yes	complemented
10	#19098391	C₄	1	2	yes	complemented
11	#19098389	C₄	1	2	yes	complemented
12	#19098323	—	2	2	no	—
13	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₄×C₁₆×C₇₉	C₄	1	4	yes	complemented
17	C₂×#11929608	—	2	2	no	—
18	#14579120	—	2	2	no	—
19	#14579097	—	2	2	no	—
20	#14579097	C₂×C₄	1	4	yes	—
21	#14579095	C₂×C₄	1	4	yes	—
22	#14579093	—	2	2	no	—
23	#14579071	—	4	2	no	—
24	C₈×Dic₇₉	—	4	2	no	—
25	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	—	2	1	no	—
29	C₂×C₁₆×C₇₉	—	2	1	no	—
30	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₇₉	Q₈	1	2	yes	—
37	#11929608	—	2	4	no	—
38	#11929608	—	4	2	no	—
39	C₂×C₇₉⋊C₈	—	4	2	no	—
40	C₄⋊Dic₇₉	—	2	4	no	—
41	C₂²⋊Dic₇₉	—	4	2	no	—
42	C₄×Dic₇₉	—	4	2	no	—
43	C₂²×C₄×C₇₉	—	2	1	no	—
44	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
49	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
55	C₄²×C₇₉	—	2	1	no	—
56	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
57	C₁₆×C₇₉	—	2	2	no	—
58	C₁₆×C₇₉	C₄²	1	2	yes	—
59	C₁₆×C₇₉	C₄²	1	2	yes	—
60	C₂³×C₇₉	#135	1	1	yes	characteristic
61	C₂×C₄×C₇₉	—	2	4	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
69	C₂×Dic₇₉	—	4	4	no	—
70	C₈×C₇₉	#107	1	2	yes	—
71	C₈×C₇₉	#107	1	2	yes	—
72	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
73	C₈×C₇₉	#117	1	1	yes	characteristic
74	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
76	C₇₉⋊C₈	—	4	4	no	—
77	C₂²×C₇₉	—	2	2	no	—
78	C₂²×C₇₉	#427	1	1	yes	characteristic
79	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
80	C₂²×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
81	C₄×C₇₉	—	2	4	no	—
82	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
84	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
85	C₄×C₇₉	C₄×Q₁₆	1	1	yes	characteristic
86	Dic₇₉	—	8	4	no	—
87	#12670	—	79	1	no	maximal, Sylow
88	C₂×C₇₉	—	2	1	no	—
89	C₂×C₇₉	—	2	1	no	—
90	C₂×C₇₉	C₄×#107	1	1	yes	characteristic
91	C₂×C₇₉	#2178	1	2	yes	—
92	C₂×#368	—	79	1	no	—
93	#2158	—	79	1	no	—
94	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
95	#1772	—	79	2	no	—
96	#1382	—	79	2	no	—
97	C₇₉	#12670	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
99	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
100	C₂×#108	—	79	2	no	—
101	#427	—	79	2	no	—
102	#405	—	158	2	no	—
103	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
104	#368	—	79	4	no	—
105	#367	—	79	2	no	—
106	#366	—	79	2	no	—
107	C₄×C₁₆	Dic₇₉	1	4	yes	complemented
108	#339	—	158	2	no	—
109	C₂×M₁₆	—	158	2	no	—
110	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
112	#117	—	158	2	no	—
113	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
117	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
118	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
120	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
121	#108	—	79	4	no	—
122	#108	—	158	2	no	—
123	#107	—	79	4	no	—
124	#106	—	158	2	no	—
125	#97	—	316	2	no	—
126	C₄×C₈	—	316	2	no	—
127	C₄×C₈	Dic₁₅₈	1	2	yes	—
128	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
131	M₁₆	—	316	2	no	—
132	M₁₆	—	158	4	no	—
133	C₂×C₈	—	316	2	no	—
134	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
135	C₂×C₈	D₆₃₂	1	1	yes	characteristic
136	C₂×C₈	D₆₃₂	1	1	yes	characteristic
137	C₂×C₈	Dic₃₁₆	1	2	yes	—
138	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
142	C₂×C₈	#11929624	1	1	yes	characteristic
143	C₂×C₈	#11929624	1	1	yes	characteristic
144	C₄⋊C₄	—	158	4	no	—
145	C₂²⋊C₄	—	316	2	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	316	2	no	—
148	C₄²	C₄⋊Dic₇₉	1	2	yes	—
149	C₁₆	—	2	2	no	—
150	C₁₆	C₄×Dic₇₉	1	2	yes	—
151	C₁₆	C₄×Dic₇₉	1	2	yes	—
152	C₂³	#14579183	1	1	yes	characteristic
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	316	4	no	—
156	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
157	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
158	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
159	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
161	C₂×C₄	#14579183	1	1	yes	characteristic
162	C₈	—	316	4	no	—
163	C₈	#14579095	1	2	yes	—
164	C₈	#14579095	1	2	yes	—
165	C₈	C₂×C₄×Dic₇₉	1	1	yes	characteristic
166	C₈	#14579120	1	1	yes	characteristic
167	C₈	C₄×Dic₁₅₈	1	1	yes	characteristic
168	C₈	C₄×Dic₁₅₈	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	#19098601	1	1	yes	characteristic
171	C₂²	C₄×D₆₃₂	1	1	yes	characteristic
172	C₂²	C₄×Dic₃₁₆	1	1	yes	characteristic
173	C₄	—	2	4	no	—
174	C₄	—	632	4	no	—
175	C₄	C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
176	C₄	C₂×#14579095	1	1	yes	characteristic
177	C₄	C₄×D₆₃₂	1	1	yes	characteristic
178	C₄	C₄×Dic₃₁₆	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×#14579095	1	1	yes	characteristic
182	C₂	#23765282	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₆₃₂
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₄×D₈
prime radical	C₂³×C₇₉	#135
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#12670
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. #19098393	C₄	semidirect
10. #19098391	C₄	semidirect
11. #19098389	C₄	semidirect
16. C₄×C₁₆×C₇₉	C₄	semidirect
97. C₇₉	#12670	semidirect
107. 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
C₂×#19098393	2
#23765184	2
#23763279	2
#23761496	2
C₂×C₄×C₁₆×C₇₉	2
#12670	79