Subgroups — #34174802

Order 20224.

294 conjugacy classes of subgroups.

The subgroup lattice has 224 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	#23763146	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
5	C₂×C₄×C₇₉⋊C₈	C₄	1	2	yes	—
6	#19098318	—	2	1	no	—
7	#19098318	—	2	1	no	—
8	#19098318	—	2	1	no	—
9	#19098318	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
11	C₄³×C₇₉	C₂²	1	1	yes	characteristic
12	#336×C₇₉	—	2	1	no	—
13	#336×C₇₉	C₄	1	2	yes	—
14	C₂²×C₇₉⋊C₈	—	2	2	no	—
15	C₂²×C₇₉⋊C₈	—	2	2	no	—
16	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
17	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
18	C₄×C₇₉⋊C₈	—	2	2	no	—
19	C₄×C₇₉⋊C₈	C₈	1	4	yes	complemented
20	C₂²×C₈×C₇₉	—	2	2	no	—
21	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₇₉	—	4	1	no	—
29	C₄⋊C₈×C₇₉	—	4	1	no	—
30	C₂×C₇₉⋊C₈	—	2	4	no	—
31	C₂×C₇₉⋊C₈	—	4	2	no	—
32	C₂×C₇₉⋊C₈	—	4	2	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	—	4	2	no	—
44	C₂×C₈×C₇₉	—	4	2	no	—
45	C₂×C₈×C₇₉	—	4	2	no	—
46	C₄²×C₇₉	—	2	2	no	—
47	C₄²×C₇₉	—	4	1	no	—
48	C₄²×C₇₉	—	4	1	no	—
49	C₄²×C₇₉	—	4	1	no	—
50	C₄²×C₇₉	—	4	1	no	—
51	C₄²×C₇₉	C₂×C₈	1	2	yes	—
52	C₄²×C₇₉	M₁₆	1	2	yes	—
53	C₄²×C₇₉	D₈	1	1	yes	characteristic
54	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
55	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
56	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
57	C₂³×C₇₉	#95	1	1	yes	characteristic
58	C₂×C₄×C₇₉	—	2	1	no	—
59	C₂×C₄×C₇₉	—	2	1	no	—
60	C₂×C₄×C₇₉	—	2	1	no	—
61	C₂×C₄×C₇₉	—	2	1	no	—
62	C₂×C₄×C₇₉	—	2	1	no	—
63	C₂×C₄×C₇₉	—	2	1	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	2	1	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	2	1	no	—
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	1	no	—
71	C₂×C₄×C₇₉	—	2	1	no	—
72	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
77	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
78	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#108	1	2	yes	—
83	C₈×C₇₉	—	8	2	no	—
84	C₇₉⋊C₈	—	4	4	no	—
85	C₇₉⋊C₈	—	8	2	no	—
86	C₂²×C₇₉	#336	1	1	yes	characteristic
87	C₂²×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
88	C₂²×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
89	C₂²×C₇₉	#340	1	1	yes	characteristic
90	C₂²×C₇₉	#341	1	1	yes	characteristic
91	C₂²×C₇₉	#344	1	1	yes	characteristic
92	C₂²×C₇₉	#344	1	1	yes	characteristic
93	C₄×C₇₉	—	2	1	no	—
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	—	2	1	no	—
97	C₄×C₇₉	—	2	1	no	—
98	C₄×C₇₉	—	2	1	no	—
99	C₄×C₇₉	—	2	2	no	—
100	C₄×C₇₉	—	4	1	no	—
101	C₄×C₇₉	—	4	1	no	—
102	C₄×C₇₉	#334	1	2	yes	—
103	C₄×C₇₉	#335	1	2	yes	—
104	#7018	—	79	1	no	maximal, Sylow
105	C₂×C₇₉	#1274	1	1	yes	characteristic
106	C₂×C₇₉	#1277	1	1	yes	characteristic
107	C₂×C₇₉	#1281	1	1	yes	characteristic
108	C₂×C₇₉	#1286	1	1	yes	characteristic
109	C₂×C₇₉	#1286	1	1	yes	characteristic
110	C₂×C₇₉	#1287	1	1	yes	characteristic
111	C₂×C₇₉	#1287	1	1	yes	characteristic
112	#1831	—	79	1	no	—
113	#1831	D₇₉	1	1	yes	characteristic
114	#1744	—	79	1	no	—
115	C₇₉	#7018	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₄⋊C₈	—	2	1	no	—
117	C₂×C₄⋊C₈	—	158	1	no	—
118	C₂×C₄×C₈	—	79	2	no	—
119	C₄³	C₂×D₇₉	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#336	—	79	2	no	—
122	#336	—	158	1	no	—
123	#336	—	158	1	no	—
124	#336	—	158	1	no	—
125	#336	Dic₇₉	1	2	yes	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	158	2	no	—
128	C₂²×C₈	—	158	2	no	—
129	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
130	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
132	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
133	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
136	C₄⋊C₈	—	4	1	no	—
137	C₄⋊C₈	—	4	1	no	—
138	C₄⋊C₈	—	316	1	no	—
139	C₄⋊C₈	—	316	1	no	—
140	C₄×C₈	—	79	4	no	—
141	C₄×C₈	—	158	2	no	—
142	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
143	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
146	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	316	2	no	—
153	C₂×C₈	—	316	2	no	—
154	C₂×C₈	—	316	2	no	—
155	C₂×C₈	—	316	2	no	—
156	C₂×C₈	—	316	2	no	—
157	C₂×C₈	—	158	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	4	1	no	—
163	C₄²	C₈×D₇₉	1	2	yes	—
164	C₄²	#11929607	1	2	yes	—
165	C₄²	#11929609	1	1	yes	characteristic
166	C₄²	#11929611	1	1	yes	characteristic
167	C₄²	#11929612	1	1	yes	characteristic
168	C₄²	#11929614	1	1	yes	characteristic
169	C₂³	C₇₉⋊#95	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	C₈×Dic₇₉	1	1	yes	characteristic
185	C₂×C₄	#14579071	1	1	yes	characteristic
186	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
187	C₂×C₄	#14579081	1	1	yes	characteristic
188	C₂×C₄	#14579082	1	1	yes	characteristic
189	C₂×C₄	#14579084	1	1	yes	characteristic
190	C₂×C₄	#14579085	1	1	yes	characteristic
191	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
192	C₂×C₄	#14579092	1	1	yes	characteristic
193	C₂×C₄	#14579094	1	1	yes	characteristic
194	C₂×C₄	#14579096	1	2	yes	—
195	C₈	—	8	2	no	—
196	C₈	—	316	4	no	—
197	C₈	—	632	2	no	—
198	C₂²	#19098317	1	1	yes	characteristic
199	C₂²	#19098319	1	1	yes	characteristic
200	C₂²	#19098320	1	1	yes	characteristic
201	C₂²	#19098325	1	1	yes	characteristic
202	C₂²	#19098327	1	1	yes	characteristic
203	C₂²	#19098333	1	1	yes	characteristic
204	C₂²	#19098335	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	#19098313	1	2	yes	—
215	C₄	#19098315	1	2	yes	—
216	C₂	#23761228	1	1	yes	characteristic
217	C₂	#23761234	1	1	yes	characteristic
218	C₂	#23761244	1	1	yes	characteristic
219	C₂	#23761258	1	1	yes	characteristic
220	C₂	#23761260	1	1	yes	characteristic
221	C₂	#23761261	1	1	yes	characteristic
222	C₂	#23761263	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₄³	C₂×D₇₉
Fitting	C₇₉×#1831	C₂
socle, prime radical	C₂³×C₇₉	#95
cosocle	C₄³×C₇₉	C₂²
hypercenter	#1831	D₇₉
hypocenter	C₇₉	#7018
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
19. C₄×C₇₉⋊C₈	C₈	semidirect
115. C₇₉	#7018	semidirect

Maximal subgroups

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

subgroup	index
#23763517	2
#23763146	2
C₇₉×#1831	2
#7018	79