Subgroups — #34203593

Order 20224.

358 conjugacy classes of subgroups.

The subgroup lattice has 210 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	M₃₂×Dic₇₉	C₂	1	2	yes	maximal
3	#23763086	C₂	1	1	yes	maximal, characteristic
4	#23761481	C₂	1	2	yes	maximal
5	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	C₂×#14579072	C₄	1	1	yes	characteristic
8	C₂×#14579071	C₄	1	1	yes	characteristic
9	C₂×#14579071	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
12	#19098409	C₄	1	1	yes	characteristic
13	#19098384	C₄	1	4	yes	complemented
14	#19098338	—	2	2	no	—
15	C₁₆×Dic₇₉	—	2	2	no	—
16	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	#364×C₇₉	C₄	1	4	yes	complemented
19	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
23	#14579101	—	2	2	no	—
24	#14579101	C₂×C₄	1	4	yes	—
25	C₂×C₇₉⋊C₁₆	—	2	2	no	—
26	#14579072	—	2	2	no	—
27	#14579071	C₈	1	4	yes	—
28	#14579071	C₈	1	4	yes	—
29	C₈×Dic₇₉	—	2	1	no	—
30	C₈×Dic₇₉	—	2	1	no	—
31	C₈×Dic₇₉	D₄	1	2	yes	—
32	C₈×Dic₇₉	Q₈	1	2	yes	—
33	C₇₉⋊#95	—	2	2	no	—
34	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₇₉	—	2	2	no	—
38	M₃₂×C₇₉	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₇₉	—	2	2	no	—
40	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
42	C₂²×Dic₇₉	—	2	1	no	—
43	C₂²×Dic₇₉	M₁₆	1	2	yes	—
44	C₂×C₇₉⋊C₈	—	2	1	no	—
45	C₂×C₇₉⋊C₈	—	2	1	no	—
46	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
48	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
49	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
50	C₄×Dic₇₉	—	2	1	no	—
51	C₄×Dic₇₉	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
53	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
54	C₇₉⋊C₁₆	—	2	4	no	—
55	C₂²×C₄×C₇₉	—	2	1	no	—
56	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₄²	1	2	yes	—
58	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
65	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
66	C₄²×C₇₉	—	2	1	no	—
67	C₄²×C₇₉	C₄²	1	2	yes	—
68	C₁₆×C₇₉	—	2	4	no	—
69	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	4	2	no	—
72	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₇₉	—	2	2	no	—
79	C₂×Dic₇₉	—	4	2	no	—
80	C₂×Dic₇₉	—	4	2	no	—
81	C₂×Dic₇₉	C₄⋊C₈	1	4	yes	—
82	C₈×C₇₉	C₄×C₈	1	4	yes	—
83	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
84	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
85	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
87	C₇₉⋊C₈	—	2	2	no	—
88	C₇₉⋊C₈	C₄⋊C₈	1	4	yes	—
89	C₂²×C₇₉	—	2	2	no	—
90	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
91	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₇₉	#431	1	1	yes	characteristic
93	C₄×C₇₉	—	4	2	no	—
94	C₄×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₇₉	#431	1	1	yes	characteristic
98	Dic₇₉	—	2	4	no	—
99	Dic₇₉	—	4	2	no	—
100	#12656	—	79	1	no	maximal, Sylow
101	C₂×C₇₉	—	2	1	no	—
102	C₂×C₇₉	—	2	1	no	—
103	C₂×C₇₉	C₄×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₇₉	#2153	1	2	yes	—
105	C₂×#364	—	79	1	no	—
106	C₂×#364	D₇₉	1	1	yes	characteristic
107	C₄×M₃₂	—	79	2	no	—
108	#1726	—	79	1	no	—
109	#1376	—	79	2	no	—
110	C₇₉	#12656	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₃₂	—	79	2	no	—
112	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
113	C₂×#97	—	79	1	no	—
114	C₂×#97	—	79	1	no	—
115	C₂×#97	—	79	1	no	—
116	C₂×C₄×C₈	—	79	1	no	—
117	C₂×C₄×C₈	—	79	1	no	—
118	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
119	#376	—	79	1	no	—
120	#364	—	79	4	no	—
121	#364	Dic₇₉	1	4	yes	complemented
122	#346	—	158	2	no	—
123	C₄×C₁₆	—	158	2	no	—
124	C₂²×C₈	—	79	1	no	—
125	C₂²×C₈	—	79	1	no	—
126	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
128	C₂×C₄²	—	79	1	no	—
129	C₂×C₄²	—	79	1	no	—
130	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	79	4	no	—
133	M₃₂	—	158	2	no	—
134	M₃₂	C₂×Dic₇₉	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	158	2	no	—
137	#97	—	79	4	no	—
138	#97	—	79	4	no	—
139	#97	—	158	2	no	—
140	C₄×C₈	—	79	2	no	—
141	C₄×C₈	—	79	2	no	—
142	C₄×C₈	—	158	1	no	—
143	C₄×C₈	—	158	1	no	—
144	C₄×C₈	C₄×D₇₉	1	2	yes	—
145	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
146	#95	—	158	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	79	2	no	—
149	C₂²×C₄	—	158	1	no	—
150	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
151	C₂×C₈	—	79	2	no	—
152	C₂×C₈	—	79	2	no	—
153	C₂×C₈	—	79	2	no	—
154	C₂×C₈	—	79	2	no	—
155	C₂×C₈	—	158	1	no	—
156	C₂×C₈	—	158	1	no	—
157	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
158	C₂×C₈	C₈×D₇₉	1	2	yes	—
159	C₂×C₈	C₈×D₇₉	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
161	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
162	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
164	C₂×C₈	#11929623	1	1	yes	characteristic
165	C₂×C₈	#11929625	1	1	yes	characteristic
166	C₄²	—	2	1	no	—
167	C₄²	—	79	2	no	—
168	C₄²	—	79	2	no	—
169	C₄²	—	79	2	no	—
170	C₄²	—	158	1	no	—
171	C₄²	C₄×Dic₇₉	1	2	yes	—
172	C₁₆	—	2	4	no	—
173	C₁₆	—	158	4	no	—
174	C₂³	M₁₆×D₇₉	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	316	2	no	—
178	C₂×C₄	—	316	2	no	—
179	C₂×C₄	—	79	4	no	—
180	C₂×C₄	—	158	2	no	—
181	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
183	C₂×C₄	#14579119	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
187	C₈	—	79	4	no	—
188	C₈	—	158	2	no	—
189	C₈	C₈×Dic₇₉	1	4	yes	—
190	C₈	#14579123	1	1	yes	characteristic
191	C₈	D₄×Dic₇₉	1	1	yes	characteristic
192	C₈	#14579126	1	1	yes	characteristic
193	C₈	Q₈×Dic₇₉	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	M₁₆×Dic₇₉	1	1	yes	characteristic
196	C₂²	C₄⋊C₈×D₇₉	1	1	yes	characteristic
197	C₂²	#19098615	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	316	2	no	—
200	C₄	—	158	4	no	—
201	C₄	C₄⋊C₄×Dic₇₉	1	1	yes	characteristic
202	C₄	C₂×C₈×Dic₇₉	1	1	yes	characteristic
203	C₄	C₄⋊C₈×D₇₉	1	1	yes	characteristic
204	C₄	#19098615	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₄⋊C₈×Dic₇₉	1	1	yes	characteristic
208	C₂	#23765164	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄⋊C₈
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	#12656
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
13. #19098384	C₄	semidirect
18. #364×C₇₉	C₄	semidirect
110. C₇₉	#12656	semidirect
121. #364	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
M₃₂×Dic₇₉	2
#23763086	2
#23761481	2
C₂×#364×C₇₉	2
#12656	79