Subgroups — #34203633

Order 20224.

326 conjugacy classes of subgroups.

The subgroup lattice has 188 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	#23764996	C₂	1	2	yes	maximal
3	#23763279	C₂	1	1	yes	maximal, characteristic
4	#23761488	C₂	1	2	yes	maximal
5	C₂×#346×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	C₂×#14579097	C₄	1	1	yes	characteristic
8	#19098601	C₄	1	1	yes	characteristic
9	#19098518	C₂²	1	2	yes	—
10	#19098384	C₄	1	4	yes	complemented
11	#19098338	—	2	2	no	—
12	C₁₆×Dic₇₉	—	2	2	no	—
13	#19098323	C₄	1	2	yes	—
14	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	#346×C₇₉	C₄	1	4	yes	complemented
18	C₂×#11929608	C₂×C₄	1	2	yes	—
19	#14579120	C₂×C₄	1	2	yes	—
20	#14579101	—	2	2	no	—
21	#14579101	C₂×C₄	1	4	yes	—
22	C₂×C₇₉⋊C₁₆	—	2	2	no	—
23	#14579097	C₈	1	4	yes	—
24	#14579095	C₈	1	2	yes	—
25	#14579093	C₈	1	2	yes	—
26	#14579071	—	2	2	no	—
27	C₈×Dic₇₉	—	2	2	no	—
28	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₉	—	2	1	no	—
32	C₂×C₁₆×C₇₉	—	2	1	no	—
33	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
35	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
36	C₂×C₁₆×C₇₉	Q₈	1	2	yes	—
37	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
39	#11929608	—	2	2	no	—
40	#11929608	C₂×C₈	1	4	yes	—
41	C₂×C₇₉⋊C₈	—	2	2	no	—
42	C₄⋊Dic₇₉	C₂×C₈	1	4	yes	—
43	C₂²⋊Dic₇₉	—	2	2	no	—
44	C₄×Dic₇₉	—	2	2	no	—
45	C₇₉⋊C₁₆	—	2	4	no	—
46	C₂²×C₄×C₇₉	—	2	1	no	—
47	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₄²	1	2	yes	—
49	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
57	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
58	C₄²×C₇₉	—	2	1	no	—
59	C₄²×C₇₉	C₄²	1	2	yes	—
60	C₁₆×C₇₉	—	2	2	no	—
61	C₁₆×C₇₉	C₄⋊C₄	1	4	yes	—
62	C₂³×C₇₉	#131	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	4	2	no	—
65	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
71	C₂×Dic₇₉	—	2	4	no	—
72	C₈×C₇₉	C₄×C₈	1	2	yes	—
73	C₈×C₇₉	C₄×C₈	1	2	yes	—
74	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₇₉	#117	1	1	yes	characteristic
76	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
77	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
78	C₇₉⋊C₈	—	2	4	no	—
79	C₂²×C₇₉	—	2	2	no	—
80	C₂²×C₇₉	#405	1	1	yes	characteristic
81	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
82	C₂²×C₇₉	C₈×Q₈	1	1	yes	characteristic
83	C₄×C₇₉	—	4	2	no	—
84	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
87	C₄×C₇₉	C₈×Q₈	1	1	yes	characteristic
88	Dic₇₉	—	4	4	no	—
89	#12665	—	79	1	no	maximal, Sylow
90	C₂×C₇₉	—	2	1	no	—
91	C₂×C₇₉	—	2	1	no	—
92	C₂×C₇₉	C₈×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₇₉	#2171	1	2	yes	—
94	C₂×#364	—	79	1	no	—
95	#2108	—	79	2	no	—
96	C₂×#346	D₇₉	1	1	yes	characteristic
97	#1772	—	79	1	no	—
98	#1379	—	79	2	no	—
99	C₇₉	#12665	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₃₂	—	79	2	no	—
101	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
103	C₂×#108	—	79	1	no	—
104	#427	—	79	1	no	—
105	#405	—	79	2	no	—
106	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
107	#364	—	79	4	no	—
108	#346	—	158	2	no	—
109	#346	Dic₇₉	1	4	yes	complemented
110	C₄×C₁₆	—	158	2	no	—
111	#339	—	79	2	no	—
112	C₂×M₁₆	—	79	2	no	—
113	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
115	#117	—	79	2	no	—
116	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
117	M₃₂	—	79	4	no	—
118	M₃₂	—	158	2	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	158	2	no	—
122	C₂×C₁₆	Dic₁₅₈	1	2	yes	—
123	C₂×C₁₆	D₃₁₆	1	2	yes	—
124	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
125	C₂×C₁₆	C₂×Dic₇₉	1	2	yes	—
126	#108	—	79	4	no	—
127	#107	—	79	2	no	—
128	#106	—	79	2	no	—
129	#97	—	158	2	no	—
130	C₄×C₈	—	158	2	no	—
131	C₄×C₈	C₄×D₇₉	1	2	yes	—
132	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
135	M₁₆	—	79	4	no	—
136	M₁₆	—	158	2	no	—
137	C₂×C₈	—	158	2	no	—
138	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
139	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
140	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
141	C₂×C₈	C₈×D₇₉	1	2	yes	—
142	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
143	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
144	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
146	C₂×C₈	#11929624	1	1	yes	characteristic
147	C₂×C₈	#11929624	1	1	yes	characteristic
148	C₄⋊C₄	—	79	4	no	—
149	C₂²⋊C₄	—	158	2	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	158	2	no	—
152	C₄²	C₄×Dic₇₉	1	2	yes	—
153	C₁₆	—	2	2	no	—
154	C₁₆	—	158	4	no	—
155	C₁₆	C₄⋊Dic₇₉	1	4	yes	—
156	C₂³	#14579167	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	158	4	no	—
160	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
161	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
162	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
163	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
164	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
165	C₂×C₄	#14579167	1	1	yes	characteristic
166	C₈	—	158	4	no	—
167	C₈	C₈×Dic₇₉	1	2	yes	—
168	C₈	C₈×Dic₇₉	1	2	yes	—
169	C₈	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
170	C₈	#14579120	1	1	yes	characteristic
171	C₈	C₄×D₃₁₆	1	1	yes	characteristic
172	C₈	C₄×Dic₁₅₈	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	#19098518	1	1	yes	characteristic
175	C₂²	C₈×D₃₁₆	1	1	yes	characteristic
176	C₂²	C₈×Dic₁₅₈	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	316	4	no	—
179	C₄	C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
180	C₄	C₂×C₈×Dic₇₉	1	1	yes	characteristic
181	C₄	C₈×D₃₁₆	1	1	yes	characteristic
182	C₄	C₈×Dic₁₅₈	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	C₈×C₄⋊Dic₇₉	1	1	yes	characteristic
186	C₂	#23765243	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₁₆
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#346×C₇₉	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂³×C₇₉	#131
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#346	D₇₉
hypocenter	C₇₉	#12665
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
10. #19098384	C₄	semidirect
17. #346×C₇₉	C₄	semidirect
99. C₇₉	#12665	semidirect
109. #346	Dic₇₉	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
#23764996	2
#23763279	2
#23761488	2
C₂×#346×C₇₉	2
#12665	79