Subgroups — #34186988

Order 20224.

322 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763572	C₂	1	1	yes	maximal, characteristic
2	#23762174	C₂	1	2	yes	maximal
3	#23762148	C₂	1	1	yes	maximal, characteristic
4	#23761303	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19098616	C₂²	1	2	yes	—
7	C₂×#14579097	—	2	1	no	—
8	C₂×#14579097	—	2	1	no	—
9	#19098587	C₂²	1	2	yes	—
10	#19098316	C₄	1	2	yes	complemented
11	#19098315	—	2	4	no	—
12	#19098314	C₄	1	2	yes	complemented
13	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₉	—	2	1	no	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
17	#336×C₇₉	—	2	1	no	—
18	C₂×#11929608	—	2	2	no	—
19	#14579097	—	4	2	no	—
20	#14579097	—	4	2	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
24	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₇₉	—	2	1	no	—
27	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
30	C₂×C₄²×C₇₉	—	2	1	no	—
31	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₇₉	—	2	1	no	—
33	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
34	C₄×C₈×C₇₉	—	4	2	no	—
35	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
39	#95×C₇₉	Q₈	1	2	yes	—
40	#11929608	—	4	2	no	—
41	#11929608	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	2	4	no	—
43	C₂×C₇₉⋊C₈	—	4	2	no	—
44	C₂²×C₄×C₇₉	—	2	2	no	—
45	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₉	—	2	2	no	—
49	C₂×C₈×C₇₉	—	2	1	no	—
50	C₂×C₈×C₇₉	—	2	1	no	—
51	C₂×C₈×C₇₉	—	4	1	no	—
52	C₂×C₈×C₇₉	—	4	1	no	—
53	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	D₈	1	2	yes	—
58	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
59	C₄⋊C₄×C₇₉	—	2	4	no	—
60	C₄²×C₇₉	—	4	1	no	—
61	C₄²×C₇₉	—	4	1	no	—
62	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₇₉	#41	1	1	yes	characteristic
66	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	4	no	—
69	C₂×C₄×C₇₉	—	4	1	no	—
70	C₂×C₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	—	4	2	no	—
72	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
82	C₈×C₇₉	—	2	1	no	—
83	C₈×C₇₉	—	2	1	no	—
84	C₈×C₇₉	—	4	2	no	—
85	C₈×C₇₉	#107	1	2	yes	—
86	C₈×C₇₉	#107	1	2	yes	—
87	C₇₉⋊C₈	—	4	4	no	—
88	C₇₉⋊C₈	—	8	2	no	—
89	C₂²×C₇₉	—	2	1	no	—
90	C₂²×C₇₉	—	2	1	no	—
91	C₂²×C₇₉	#389	1	1	yes	characteristic
92	C₂²×C₇₉	#474	1	1	yes	characteristic
93	C₂²×C₇₉	#477	1	1	yes	characteristic
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	—	2	1	no	—
97	C₄×C₇₉	—	4	4	no	—
98	C₄×C₇₉	C₂×#107	1	1	yes	characteristic
99	C₄×C₇₉	#428	1	1	yes	characteristic
100	C₄×C₇₉	#443	1	2	yes	—
101	C₄×C₇₉	#459	1	1	yes	characteristic
102	C₄×C₇₉	#462	1	1	yes	characteristic
103	#9627	—	79	1	no	maximal, Sylow
104	C₂×C₇₉	—	2	1	no	—
105	C₂×C₇₉	—	2	1	no	—
106	C₂×C₇₉	#1945	1	1	yes	characteristic
107	C₂×C₇₉	#1947	1	1	yes	characteristic
108	C₂×C₇₉	#1953	1	1	yes	characteristic
109	#1848	—	79	1	no	—
110	C₈×C₄⋊C₄	D₇₉	1	1	yes	characteristic
111	#1571	—	79	2	no	—
112	#1564	—	79	1	no	—
113	#1303	—	79	2	no	—
114	C₇₉	#9627	1	1	yes	characteristic, complemented, Sylow
115	#431	—	79	2	no	—
116	C₂×#108	—	158	1	no	—
117	C₂×#108	—	158	1	no	—
118	#423	—	79	2	no	—
119	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
120	C₂×C₄×C₈	—	2	1	no	—
121	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
122	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
123	#336	—	2	1	no	—
124	#335	—	79	2	no	—
125	#335	—	158	4	no	—
126	#334	—	79	2	no	—
127	C₂×M₁₆	—	158	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
134	#108	—	316	2	no	—
135	#108	—	316	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	79	4	no	—
138	C₄⋊C₈	—	158	2	no	—
139	C₄⋊C₈	—	158	2	no	—
140	C₄⋊C₈	D₃₁₆	1	2	yes	—
141	C₂²⋊C₈	—	158	2	no	—
142	C₄×C₈	—	4	2	no	—
143	C₄×C₈	—	158	2	no	—
144	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
145	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
146	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
147	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
148	#95	Dic₁₅₈	1	2	yes	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
152	C₂²×C₄	#11929624	1	1	yes	characteristic
153	M₁₆	—	316	2	no	—
154	M₁₆	—	316	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	316	2	no	—
161	C₂×C₈	—	158	4	no	—
162	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
163	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
164	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
165	C₂×C₈	Dic₇₉⋊C₄	1	1	yes	characteristic
166	C₂×C₈	#11929609	1	2	yes	—
167	C₂×C₈	#11929614	1	2	yes	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
172	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
173	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
174	C₄²	#11929624	1	1	yes	characteristic
175	C₂³	#14579140	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
182	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
183	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
184	C₂×C₄	#14579135	1	1	yes	characteristic
185	C₂×C₄	#14579153	1	1	yes	characteristic
186	C₂×C₄	#14579161	1	1	yes	characteristic
187	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
188	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
189	C₂×C₄	#14579186	1	1	yes	characteristic
190	C₂×C₄	#14579191	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	2	no	—
194	C₈	—	316	4	no	—
195	C₈	—	632	2	no	—
196	C₈	#14579094	1	2	yes	—
197	C₈	#14579094	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#19098461	1	1	yes	characteristic
201	C₂²	#19098843	1	1	yes	characteristic
202	C₂²	#19098858	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	4	4	no	—
207	C₄	C₂×#14579094	1	1	yes	characteristic
208	C₄	#19098606	1	1	yes	characteristic
209	C₄	#19098681	1	2	yes	—
210	C₄	#19098758	1	1	yes	characteristic
211	C₄	#19098773	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#23764148	1	1	yes	characteristic
215	C₂	#23764158	1	1	yes	characteristic
216	C₂	#23764190	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579135
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₈×C₄⋊C₄×C₇₉	C₂
socle	C₂²×C₇₉	#389
prime radical	C₂³×C₇₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₇₉
hypocenter	C₇₉	#9627
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #19098316	C₄	semidirect
12. #19098314	C₄	semidirect
114. C₇₉	#9627	semidirect

Maximal subgroups

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

subgroup	index
#23763572	2
#23762174	2
#23762148	2
#23761303	2
C₈×C₄⋊C₄×C₇₉	2
#9627	79