Subgroups — #34206821

Order 20224.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773914	C₂	1	1	yes	maximal, characteristic
2	#23763279	C₂	1	1	yes	maximal, characteristic
3	#23761515	C₂	1	2	yes	maximal
4	#23761502	C₂	1	2	yes	maximal
5	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19099120	—	2	1	no	—
7	#19098968	—	2	1	no	—
8	#19098965	—	2	1	no	—
9	#19098965	C₄	1	2	yes	complemented
10	#19098961	C₄	1	2	yes	complemented
11	C₂×#14579097	C₂²	1	1	yes	characteristic
12	#19098601	—	2	1	no	—
13	#19098601	C₂²	1	2	yes	—
14	#19098601	C₂²	1	1	yes	characteristic
15	#19098518	—	2	1	no	—
16	#19098323	—	2	1	no	—
17	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#364×C₇₉	—	2	1	no	—
20	#364×C₇₉	—	2	1	no	—
21	C₂×#11929608	—	2	1	no	—
22	#14579161	—	2	2	no	—
23	#14579153	—	2	2	no	—
24	#14579140	—	4	2	no	—
25	C₄×Dic₁₅₈	—	4	2	no	—
26	#14579120	—	2	1	no	—
27	#14579120	—	2	2	no	—
28	#14579097	—	2	1	no	—
29	#14579097	Q₈	1	2	yes	complemented
30	#14579095	—	2	2	no	—
31	#14579095	—	2	2	no	—
32	#14579095	—	2	1	no	—
33	#14579093	—	2	2	no	—
34	#14579093	C₂×C₄	1	4	yes	—
35	#14579093	Q₈	1	2	yes	complemented
36	#14579071	—	4	1	no	—
37	C₈×Dic₇₉	—	4	1	no	—
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	M₃₂×C₇₉	—	2	2	no	—
42	M₃₂×C₇₉	—	2	2	no	—
43	C₂×C₁₆×C₇₉	—	2	2	no	—
44	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂×Dic₁₅₈	—	4	2	no	—
49	#11929608	—	2	2	no	—
50	#11929608	—	4	1	no	—
51	C₂×C₇₉⋊C₈	—	4	1	no	—
52	C₄⋊Dic₇₉	—	2	4	no	—
53	C₄⋊Dic₇₉	—	2	2	no	—
54	C₄⋊Dic₇₉	—	4	2	no	—
55	Dic₇₉⋊C₄	—	4	4	no	—
56	C₂²⋊Dic₇₉	—	4	1	no	—
57	C₂²⋊Dic₇₉	—	4	2	no	—
58	C₄×Dic₇₉	—	4	1	no	—
59	C₄×Dic₇₉	—	4	2	no	—
60	C₂²×C₄×C₇₉	—	2	1	no	—
61	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
65	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
67	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
68	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
74	C₄²×C₇₉	—	2	1	no	—
75	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₁₆×C₇₉	—	4	2	no	—
78	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₇₉	—	2	1	no	—
80	C₂×C₄×C₇₉	—	2	1	no	—
81	C₂×C₄×C₇₉	—	4	2	no	—
82	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×Dic₇₉	—	4	4	no	—
89	C₂×Dic₇₉	—	4	2	no	—
90	Dic₁₅₈	—	8	4	no	—
91	C₈×C₇₉	#103	1	1	yes	characteristic
92	C₈×C₇₉	#103	1	1	yes	characteristic
93	C₈×C₇₉	#103	1	1	yes	characteristic
94	C₈×C₇₉	#103	1	1	yes	characteristic
95	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
96	C₈×C₇₉	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₇₉	#124	1	1	yes	characteristic
98	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
99	C₇₉⋊C₈	—	4	2	no	—
100	C₂²×C₇₉	—	4	1	no	—
101	C₂²×C₇₉	#418	1	1	yes	characteristic
102	C₂²×C₇₉	#476	1	1	yes	characteristic
103	C₂²×C₇₉	#477	1	1	yes	characteristic
104	C₄×C₇₉	—	2	2	no	—
105	C₄×C₇₉	—	4	1	no	—
106	C₄×C₇₉	#391	1	1	yes	characteristic
107	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
108	C₄×C₇₉	#476	1	1	yes	characteristic
109	C₄×C₇₉	#477	1	1	yes	characteristic
110	Dic₇₉	—	8	4	no	—
111	Dic₇₉	—	8	2	no	—
112	#13271	—	79	1	no	maximal, Sylow
113	C₂×C₇₉	—	2	1	no	—
114	C₂×C₇₉	—	2	1	no	—
115	C₂×C₇₉	—	2	1	no	—
116	C₂×C₇₉	#1987	1	1	yes	characteristic
117	#3160	—	79	1	no	—
118	C₂×#364	D₇₉	1	1	yes	characteristic
119	#1772	—	79	1	no	—
120	#1389	—	79	2	no	—
121	#1384	—	79	2	no	—
122	C₇₉	#13271	1	1	yes	characteristic, complemented, Sylow
123	#533	—	158	1	no	—
124	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
125	#500	—	158	1	no	—
126	#499	—	79	2	no	—
127	#499	—	158	1	no	—
128	#498	—	79	2	no	—
129	C₂×#108	—	79	1	no	—
130	#427	—	79	2	no	—
131	#427	—	79	1	no	—
132	#427	—	158	1	no	—
133	#405	—	158	1	no	—
134	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
135	#364	—	2	1	no	—
136	#364	—	2	1	no	—
137	#339	—	158	1	no	—
138	C₂×M₁₆	—	158	1	no	—
139	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
141	C₄⋊Q₈	—	158	2	no	—
142	#125	—	158	2	no	—
143	C₂²⋊Q₈	—	316	2	no	—
144	C₄×Q₈	—	316	2	no	—
145	#117	—	158	2	no	—
146	#117	—	158	1	no	—
147	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	79	2	no	—
152	#108	—	158	1	no	—
153	#107	—	158	2	no	—
154	#107	—	158	2	no	—
155	#107	—	158	1	no	—
156	#106	—	79	4	no	—
157	#106	—	79	2	no	—
158	#106	—	158	2	no	—
159	#97	—	316	1	no	—
160	C₄×C₈	—	316	1	no	—
161	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
162	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
165	C₂×Q₈	—	316	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
168	M₁₆	—	316	1	no	—
169	M₁₆	—	158	2	no	—
170	C₂×C₈	—	316	1	no	—
171	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
172	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	#11929613	1	1	yes	characteristic
174	C₂×C₈	#11929613	1	1	yes	characteristic
175	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
176	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
179	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
180	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
181	C₂×C₈	#11929623	1	1	yes	characteristic
182	C₂×C₈	#11929623	1	1	yes	characteristic
183	C₄⋊C₄	—	316	4	no	—
184	C₄⋊C₄	—	316	2	no	—
185	C₄⋊C₄	—	158	4	no	—
186	C₄⋊C₄	—	158	2	no	—
187	C₂²⋊C₄	—	316	1	no	—
188	C₂²⋊C₄	—	316	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	316	1	no	—
191	C₄²	—	316	2	no	—
192	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
193	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₂³	#14579189	1	1	yes	characteristic
196	Q₈	—	632	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	316	4	no	—
201	C₂×C₄	—	316	2	no	—
202	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#14579141	1	1	yes	characteristic
204	C₂×C₄	#14579141	1	1	yes	characteristic
205	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
206	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
207	C₂×C₄	#14579189	1	1	yes	characteristic
208	C₈	—	316	2	no	—
209	C₈	#14579086	1	1	yes	characteristic
210	C₈	#14579086	1	1	yes	characteristic
211	C₈	#14579086	1	1	yes	characteristic
212	C₈	#14579086	1	1	yes	characteristic
213	C₈	#14579126	1	1	yes	characteristic
214	C₈	#14579126	1	1	yes	characteristic
215	C₈	#14579148	1	1	yes	characteristic
216	C₈	#14579159	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#19098569	1	1	yes	characteristic
219	C₂²	#19098854	1	1	yes	characteristic
220	C₂²	#19098859	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	632	4	no	—
224	C₄	—	632	2	no	—
225	C₄	#19098468	1	1	yes	characteristic
226	C₄	C₂×#14579086	1	1	yes	characteristic
227	C₄	#19098854	1	1	yes	characteristic
228	C₄	#19098859	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#23764347	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19098854
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂×C₇₉	#1987
prime radical	C₂³×C₇₉	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	#13271
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
9. #19098965	C₄	semidirect
10. #19098961	C₄	semidirect
29. #14579097	Q₈	semidirect
35. #14579093	Q₈	semidirect
122. C₇₉	#13271	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
#23763279	2
#23761515	2
#23761502	2
C₂×#364×C₇₉	2
#13271	79