Subgroups — #34174834

Order 20224.

344 conjugacy classes of subgroups.

The subgroup lattice has 212 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	C₄×#97×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
5	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
6	#19098318	—	2	1	no	—
7	#19098318	—	2	1	no	—
8	#19098318	—	2	1	no	—
9	#19098318	C₄	1	2	yes	—
10	C₂×#97×C₇₉	—	2	1	no	—
11	C₂×#97×C₇₉	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₇₉	—	2	1	no	—
13	C₄³×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×C₇₉⋊C₈	—	2	2	no	—
15	C₂²×C₇₉⋊C₈	—	2	2	no	—
16	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
17	C₄×C₇₉⋊C₈	—	2	4	no	—
18	C₂²×C₈×C₇₉	—	2	2	no	—
19	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
26	#97×C₇₉	—	2	4	no	—
27	#97×C₇₉	—	2	2	no	—
28	#97×C₇₉	C₈	1	4	yes	complemented
29	C₄×C₈×C₇₉	—	2	4	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	—	2	4	no	—
44	C₂×C₈×C₇₉	—	2	4	no	—
45	C₂×C₈×C₇₉	—	2	4	no	—
46	C₄²×C₇₉	—	2	1	no	—
47	C₄²×C₇₉	—	2	1	no	—
48	C₄²×C₇₉	—	2	2	no	—
49	C₄²×C₇₉	—	2	2	no	—
50	C₄²×C₇₉	—	2	2	no	—
51	C₄²×C₇₉	—	2	2	no	—
52	C₄²×C₇₉	—	4	1	no	—
53	C₄²×C₇₉	C₂×C₈	1	2	yes	—
54	C₄²×C₇₉	M₁₆	1	2	yes	—
55	C₂³×C₇₉	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	2	1	no	—
57	C₂×C₄×C₇₉	—	2	1	no	—
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	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#104	1	2	yes	—
75	C₂×C₄×C₇₉	#104	1	2	yes	—
76	C₂×C₄×C₇₉	#104	1	2	yes	—
77	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
78	C₂×C₄×C₇₉	#108	1	2	yes	—
79	C₈×C₇₉	—	2	8	no	—
80	C₇₉⋊C₈	—	4	4	no	—
81	C₇₉⋊C₈	—	8	2	no	—
82	C₂²×C₇₉	#336	1	1	yes	characteristic
83	C₂²×C₇₉	#339	1	1	yes	characteristic
84	C₂²×C₇₉	#339	1	1	yes	characteristic
85	C₂²×C₇₉	#339	1	1	yes	characteristic
86	C₂²×C₇₉	#341	1	1	yes	characteristic
87	C₂²×C₇₉	#343	1	1	yes	characteristic
88	C₂²×C₇₉	#344	1	1	yes	characteristic
89	C₄×C₇₉	—	2	1	no	—
90	C₄×C₇₉	—	2	1	no	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	2	2	no	—
93	C₄×C₇₉	—	2	2	no	—
94	C₄×C₇₉	—	2	2	no	—
95	C₄×C₇₉	—	4	1	no	—
96	C₄×C₇₉	#329	1	2	yes	—
97	C₄×C₇₉	#330	1	2	yes	—
98	#7030	—	79	1	no	maximal, Sylow
99	C₂×C₇₉	#1276	1	1	yes	characteristic
100	C₂×C₇₉	#1278	1	1	yes	characteristic
101	C₂×C₇₉	#1283	1	1	yes	characteristic
102	C₂×C₇₉	#1295	1	1	yes	characteristic
103	C₂×C₇₉	#1299	1	1	yes	characteristic
104	C₂×C₇₉	#1300	1	1	yes	characteristic
105	C₂×C₇₉	#1302	1	1	yes	characteristic
106	#1831	—	79	1	no	—
107	#1744	—	79	1	no	—
108	C₄×#97	D₇₉	1	1	yes	characteristic
109	C₇₉	#7030	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄⋊C₈	—	79	2	no	—
111	C₂×#97	—	2	1	no	—
112	C₂×#97	Dic₇₉	1	2	yes	—
113	C₂×C₄×C₈	—	2	1	no	—
114	C₂×C₄×C₈	—	158	1	no	—
115	C₄³	C₂×D₇₉	1	1	yes	characteristic
116	#336	—	79	2	no	—
117	#336	—	158	1	no	—
118	#336	—	158	1	no	—
119	#336	—	158	1	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	158	2	no	—
122	C₂²×C₈	—	158	2	no	—
123	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
124	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
126	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
127	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
128	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
130	C₄⋊C₈	—	316	2	no	—
131	#97	—	2	4	no	—
132	#97	—	2	2	no	—
133	#97	C₇₉⋊C₈	1	4	yes	complemented
134	C₄×C₈	—	2	4	no	—
135	C₄×C₈	—	158	4	no	—
136	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
137	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
138	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
140	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
141	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	316	2	no	—
147	C₂×C₈	—	316	2	no	—
148	C₂×C₈	—	316	2	no	—
149	C₂×C₈	—	316	2	no	—
150	C₂×C₈	—	316	2	no	—
151	C₂×C₈	—	158	4	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	4	1	no	—
159	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
160	C₄²	#11929608	1	2	yes	—
161	C₂³	C₇₉⋊#95	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
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	C₄×C₇₉⋊C₈	1	1	yes	characteristic
176	C₂×C₄	#14579072	1	1	yes	characteristic
177	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
178	C₂×C₄	#14579077	1	1	yes	characteristic
179	C₂×C₄	#14579079	1	1	yes	characteristic
180	C₂×C₄	#14579087	1	2	yes	—
181	C₂×C₄	#14579089	1	2	yes	—
182	C₂×C₄	#14579089	1	2	yes	—
183	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
184	C₂×C₄	#14579096	1	2	yes	—
185	C₈	—	2	8	no	—
186	C₈	—	316	4	no	—
187	C₈	—	632	2	no	—
188	C₂²	#19098318	1	1	yes	characteristic
189	C₂²	#19098323	1	1	yes	characteristic
190	C₂²	#19098323	1	1	yes	characteristic
191	C₂²	#19098324	1	1	yes	characteristic
192	C₂²	#19098327	1	1	yes	characteristic
193	C₂²	#19098331	1	1	yes	characteristic
194	C₂²	#19098335	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	#19098302	1	2	yes	—
203	C₄	#19098304	1	2	yes	—
204	C₂	#23761233	1	1	yes	characteristic
205	C₂	#23761238	1	1	yes	characteristic
206	C₂	#23761251	1	1	yes	characteristic
207	C₂	#23761282	1	1	yes	characteristic
208	C₂	#23761290	1	1	yes	characteristic
209	C₂	#23761294	1	1	yes	characteristic
210	C₂	#23761299	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₄³	C₂×D₇₉
Fitting	C₄×#97×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#95
cosocle	C₄³×C₇₉	C₂²
hypercenter	C₄×#97	D₇₉
hypocenter	C₇₉	#7030
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
28. #97×C₇₉	C₈	semidirect
109. C₇₉	#7030	semidirect
133. #97	C₇₉⋊C₈	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
C₄×#97×C₇₉	2
#7030	79