Subgroups — #34182016

Order 20224.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763517	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579072	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098315	C₂	1	2	yes	maximal
4	C₂×#19098313	C₂	1	2	yes	maximal
5	C₇₉×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#14579072	C₄	1	2	yes	—
9	C₂×#14579072	C₄	1	1	yes	characteristic
10	C₂×#14579072	C₄	1	1	yes	characteristic
11	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
12	#19098318	—	2	2	no	—
13	#19098315	—	2	4	no	—
14	#19098313	—	2	4	no	—
15	C₂×C₄⋊C₈×C₇₉	—	2	2	no	—
16	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
17	C₄³×C₇₉	C₂²	1	1	yes	characteristic
18	C₂²×C₇₉⋊C₈	—	2	1	no	—
19	C₂²×C₇₉⋊C₈	—	2	1	no	—
20	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	#14579072	—	2	2	no	—
25	#14579072	—	2	4	no	—
26	#14579072	C₈	1	4	yes	complemented
27	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
28	C₄×C₇₉⋊C₈	Q₈	1	4	yes	—
29	C₂²×C₈×C₇₉	—	2	2	no	—
30	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	D₄	1	2	yes	—
33	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
34	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₉	—	2	2	no	—
36	C₄⋊C₈×C₇₉	—	2	2	no	—
37	C₄⋊C₈×C₇₉	—	4	4	no	—
38	C₂×C₇₉⋊C₈	—	2	2	no	—
39	C₂×C₇₉⋊C₈	—	2	2	no	—
40	C₂×C₇₉⋊C₈	—	4	1	no	—
41	C₂×C₇₉⋊C₈	—	4	1	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₂×C₇₉⋊C₈	C₄⋊C₄	1	4	yes	—
45	C₂×C₇₉⋊C₈	C₄⋊C₄	1	4	yes	—
46	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
47	C₂×C₇₉⋊C₈	M₁₆	1	4	yes	—
48	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₉	—	4	2	no	—
54	C₂×C₈×C₇₉	—	4	4	no	—
55	C₄²×C₇₉	—	2	2	no	—
56	C₄²×C₇₉	—	2	2	no	—
57	C₄²×C₇₉	—	2	2	no	—
58	C₄²×C₇₉	—	4	2	no	—
59	C₄²×C₇₉	C₂×C₈	1	2	yes	—
60	C₄²×C₇₉	M₁₆	1	2	yes	—
61	C₄²×C₇₉	C₂×D₄	1	2	yes	—
62	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
63	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	2	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₇₉	#137	1	2	yes	—
80	C₈×C₇₉	—	8	2	no	—
81	C₇₉⋊C₈	—	2	4	no	—
82	C₇₉⋊C₈	—	8	1	no	—
83	C₇₉⋊C₈	—	8	1	no	—
84	C₇₉⋊C₈	C₄⋊C₈	1	8	yes	complemented
85	C₂²×C₇₉	#384	1	1	yes	characteristic
86	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₇₉	#423	1	1	yes	characteristic
88	C₂²×C₇₉	#428	1	2	yes	—
89	C₂²×C₇₉	#430	1	2	yes	—
90	C₄×C₇₉	—	2	4	no	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	C₂×C₄⋊C₈	1	2	yes	—
94	C₄×C₇₉	#423	1	2	yes	—
95	C₄×C₇₉	#496	1	1	yes	characteristic
96	C₄×C₇₉	#497	1	1	yes	characteristic
97	C₄×C₇₉	#501	1	2	yes	—
98	#8808	—	79	1	no	maximal, Sylow
99	C₂×C₇₉	#1565	1	2	yes	—
100	C₂×C₇₉	#1569	1	2	yes	—
101	C₂×C₇₉	#1842	1	1	yes	characteristic
102	C₂×C₇₉	#1847	1	1	yes	characteristic
103	C₂×C₇₉	#1849	1	1	yes	characteristic
104	#1842	D₇₉	1	1	yes	characteristic
105	#1831	—	79	1	no	—
106	C₄×#97	—	79	1	no	—
107	C₂×#335	—	79	2	no	—
108	C₂×#334	—	79	2	no	—
109	C₇₉	#8808	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊C₈	—	2	2	no	—
111	C₂×C₄⋊C₈	—	79	1	no	—
112	C₂×C₄⋊C₈	—	79	1	no	—
113	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
114	C₂×#97	—	79	1	no	—
115	C₂×#97	—	79	2	no	—
116	C₂×#97	—	79	1	no	—
117	C₂×C₄×C₈	—	79	2	no	—
118	C₄³	C₂×D₇₉	1	1	yes	characteristic
119	#336	—	158	2	no	—
120	#335	—	158	4	no	—
121	#334	—	158	4	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	79	2	no	—
124	C₂²×C₈	—	79	2	no	—
125	C₂²×C₈	—	158	1	no	—
126	C₂²×C₈	—	158	1	no	—
127	C₂×C₄²	Dic₁₅₈	1	2	yes	—
128	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
129	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
131	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	4	4	no	—
135	C₄⋊C₈	—	158	2	no	—
136	C₄⋊C₈	—	158	2	no	—
137	#97	—	79	4	no	—
138	#97	—	158	4	no	—
139	#97	—	158	2	no	—
140	C₄×C₈	—	79	4	no	—
141	C₄×C₈	—	79	4	no	—
142	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
143	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	4	no	—
149	C₂×C₈	—	316	1	no	—
150	C₂×C₈	—	316	1	no	—
151	C₂×C₈	—	316	2	no	—
152	C₂×C₈	—	316	2	no	—
153	C₂×C₈	—	79	4	no	—
154	C₂×C₈	—	79	4	no	—
155	C₂×C₈	—	79	4	no	—
156	C₂×C₈	—	79	4	no	—
157	C₂×C₈	—	158	2	no	—
158	C₂×C₈	—	158	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₈×D₇₉	1	2	yes	—
164	C₄²	#11929607	1	2	yes	—
165	C₄²	D₄×D₇₉	1	2	yes	—
166	C₄²	Q₈×D₇₉	1	2	yes	—
167	C₂³	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	Dic₇₉⋊C₈	1	4	yes	—
175	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
176	C₂×C₄	#14579157	1	1	yes	characteristic
177	C₂×C₄	#14579159	1	1	yes	characteristic
178	C₂×C₄	#14579160	1	2	yes	—
179	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
180	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
181	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
182	C₂×C₄	#14579187	1	2	yes	—
183	C₂×C₄	#14579192	1	2	yes	—
184	C₈	—	8	2	no	—
185	C₈	—	79	8	no	—
186	C₈	—	632	1	no	—
187	C₈	—	632	1	no	—
188	C₈	—	158	4	no	—
189	C₂²	#19098432	1	1	yes	characteristic
190	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
191	C₂²	#19098586	1	1	yes	characteristic
192	C₂²	#19098603	1	2	yes	—
193	C₂²	#19098611	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	C₄⋊C₈×D₇₉	1	2	yes	—
198	C₄	#19098585	1	2	yes	—
199	C₄	#19098950	1	1	yes	characteristic
200	C₄	#19098955	1	1	yes	characteristic
201	C₄	#19098969	1	2	yes	—
202	C₂	#23762150	1	2	yes	—
203	C₂	#23762165	1	2	yes	—
204	C₂	#23763551	1	1	yes	characteristic
205	C₂	#23763566	1	1	yes	characteristic
206	C₂	#23763574	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1842	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1842	D₇₉
hypocenter	C₇₉	#8808
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
26. #14579072	C₈	semidirect
84. C₇₉⋊C₈	C₄⋊C₈	semidirect
109. C₇₉	#8808	semidirect

Maximal subgroups

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

subgroup	index
#23763517	2
C₄×#14579072	2
C₂×#19098315	2
C₂×#19098313	2
C₇₉×#1842	2
#8808	79