Subgroups — #34174813

Order 20224.

320 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	#23763517	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
6	C₂×C₄×C₇₉⋊C₈	C₄	1	2	yes	—
7	#19098318	—	2	1	no	—
8	#19098318	—	2	1	no	—
9	#19098318	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₇₉	C₄	1	2	yes	—
11	C₂×#97×C₇₉	—	2	1	no	—
12	C₄³×C₇₉	C₂²	1	1	yes	characteristic
13	#336×C₇₉	—	2	1	no	—
14	C₂²×C₇₉⋊C₈	—	2	2	no	—
15	C₂²×C₇₉⋊C₈	—	2	2	no	—
16	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
17	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
18	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
19	C₄×C₇₉⋊C₈	—	4	2	no	—
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	—	2	2	no	—
29	C₄⋊C₈×C₇₉	C₈	1	4	yes	complemented
30	#97×C₇₉	—	4	1	no	—
31	#97×C₇₉	—	4	1	no	—
32	C₂×C₇₉⋊C₈	—	2	4	no	—
33	C₂×C₇₉⋊C₈	—	4	2	no	—
34	C₂×C₇₉⋊C₈	—	4	2	no	—
35	C₂×C₇₉⋊C₈	—	4	2	no	—
36	C₂×C₇₉⋊C₈	—	4	2	no	—
37	C₂×C₇₉⋊C₈	—	4	2	no	—
38	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₉	—	2	4	no	—
46	C₂×C₈×C₇₉	—	4	2	no	—
47	C₂×C₈×C₇₉	—	4	2	no	—
48	C₄²×C₇₉	—	2	1	no	—
49	C₄²×C₇₉	—	2	1	no	—
50	C₄²×C₇₉	—	2	2	no	—
51	C₄²×C₇₉	—	4	1	no	—
52	C₄²×C₇₉	—	4	1	no	—
53	C₄²×C₇₉	—	4	1	no	—
54	C₄²×C₇₉	C₂×C₈	1	2	yes	—
55	C₄²×C₇₉	M₁₆	1	2	yes	—
56	C₄²×C₇₉	D₈	1	1	yes	characteristic
57	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
58	C₄²×C₇₉	SD₁₆	1	1	yes	characteristic
59	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
60	C₂³×C₇₉	#95	1	1	yes	characteristic
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	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
78	C₂×C₄×C₇₉	#101	1	1	yes	characteristic
79	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#103	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#104	1	2	yes	—
84	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
87	C₈×C₇₉	—	4	4	no	—
88	C₇₉⋊C₈	—	4	4	no	—
89	C₇₉⋊C₈	—	8	2	no	—
90	C₂²×C₇₉	#336	1	1	yes	characteristic
91	C₂²×C₇₉	H(3,ℤ/4)	1	1	yes	characteristic
92	C₂²×C₇₉	#338	1	1	yes	characteristic
93	C₂²×C₇₉	#339	1	1	yes	characteristic
94	C₂²×C₇₉	#340	1	1	yes	characteristic
95	C₂²×C₇₉	#343	1	1	yes	characteristic
96	C₂²×C₇₉	#344	1	1	yes	characteristic
97	C₄×C₇₉	—	2	1	no	—
98	C₄×C₇₉	—	2	1	no	—
99	C₄×C₇₉	—	2	1	no	—
100	C₄×C₇₉	—	2	1	no	—
101	C₄×C₇₉	—	2	1	no	—
102	C₄×C₇₉	—	2	1	no	—
103	C₄×C₇₉	—	2	1	no	—
104	C₄×C₇₉	—	2	1	no	—
105	C₄×C₇₉	—	2	2	no	—
106	C₄×C₇₉	—	4	1	no	—
107	C₄×C₇₉	D₄⋊C₈	1	2	yes	—
108	C₄×C₇₉	Q₈⋊C₈	1	2	yes	—
109	#7022	—	79	1	no	maximal, Sylow
110	C₂×C₇₉	#1275	1	1	yes	characteristic
111	C₂×C₇₉	#1279	1	1	yes	characteristic
112	C₂×C₇₉	#1283	1	1	yes	characteristic
113	C₂×C₇₉	#1285	1	1	yes	characteristic
114	C₂×C₇₉	#1286	1	1	yes	characteristic
115	C₂×C₇₉	#1291	1	1	yes	characteristic
116	C₂×C₇₉	#1292	1	1	yes	characteristic
117	#1831	—	79	1	no	—
118	#1767	D₇₉	1	1	yes	characteristic
119	C₄×C₄⋊C₈	—	79	1	no	—
120	C₇₉	#7022	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄⋊C₈	—	158	1	no	—
122	C₂×C₄⋊C₈	—	158	1	no	—
123	C₂×C₄⋊C₈	Dic₇₉	1	2	yes	—
124	C₂×#97	—	2	1	no	—
125	C₂×C₄×C₈	—	79	2	no	—
126	C₄³	C₂×D₇₉	1	1	yes	characteristic
127	#336	—	2	1	no	—
128	#336	—	79	2	no	—
129	#336	—	158	1	no	—
130	#336	—	158	1	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	158	2	no	—
133	C₂²×C₈	—	158	2	no	—
134	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
135	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
136	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
137	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
138	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	316	1	no	—
143	C₄⋊C₈	—	316	1	no	—
144	C₄⋊C₈	—	158	4	no	—
145	C₄⋊C₈	C₇₉⋊C₈	1	4	yes	complemented
146	#97	—	4	1	no	—
147	#97	—	4	1	no	—
148	C₄×C₈	—	316	2	no	—
149	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
150	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
151	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
153	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
154	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	316	2	no	—
160	C₂×C₈	—	316	2	no	—
161	C₂×C₈	—	316	2	no	—
162	C₂×C₈	—	316	2	no	—
163	C₂×C₈	—	316	2	no	—
164	C₂×C₈	—	158	4	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	4	1	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
172	C₄²	#11929608	1	2	yes	—
173	C₄²	#11929609	1	1	yes	characteristic
174	C₄²	#11929611	1	1	yes	characteristic
175	C₄²	#11929612	1	1	yes	characteristic
176	C₄²	#11929614	1	1	yes	characteristic
177	C₂³	C₇₉⋊#95	1	1	yes	characteristic
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	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
192	C₂×C₄	#14579072	1	1	yes	characteristic
193	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
194	C₂×C₄	#14579077	1	1	yes	characteristic
195	C₂×C₄	#14579079	1	1	yes	characteristic
196	C₂×C₄	#14579081	1	1	yes	characteristic
197	C₂×C₄	#14579082	1	1	yes	characteristic
198	C₂×C₄	#14579084	1	1	yes	characteristic
199	C₂×C₄	#14579085	1	1	yes	characteristic
200	C₂×C₄	#14579088	1	2	yes	—
201	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
202	C₂×C₄	#14579092	1	1	yes	characteristic
203	C₂×C₄	#14579094	1	1	yes	characteristic
204	C₈	—	4	4	no	—
205	C₈	—	316	4	no	—
206	C₈	—	632	2	no	—
207	C₂²	#19098318	1	1	yes	characteristic
208	C₂²	#19098320	1	1	yes	characteristic
209	C₂²	C₇₉⋊#338	1	1	yes	characteristic
210	C₂²	#19098322	1	1	yes	characteristic
211	C₂²	#19098325	1	1	yes	characteristic
212	C₂²	#19098331	1	1	yes	characteristic
213	C₂²	#19098335	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	1	no	—
224	C₄	D₄⋊(C₇₉⋊C₈)	1	2	yes	—
225	C₄	Q₈⋊(C₇₉⋊C₈)	1	2	yes	—
226	C₂	#23761231	1	1	yes	characteristic
227	C₂	#23761241	1	1	yes	characteristic
228	C₂	#23761251	1	1	yes	characteristic
229	C₂	#23761257	1	1	yes	characteristic
230	C₂	#23761260	1	1	yes	characteristic
231	C₂	#23761271	1	1	yes	characteristic
232	C₂	#23761274	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₇₉⋊#95
commutator	C₂×C₄×C₇₉	C₄×C₈
Frattini	C₄³	C₂×D₇₉
Fitting	C₇₉×#1767	C₂
socle, prime radical	C₂³×C₇₉	#95
cosocle	C₄³×C₇₉	C₂²
hypercenter	#1767	D₇₉
hypocenter	C₇₉	#7022
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
29. C₄⋊C₈×C₇₉	C₈	semidirect
120. C₇₉	#7022	semidirect
145. C₄⋊C₈	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
C₄×C₄⋊(C₇₉⋊C₈)	2
C₇₉×#1767	2
#7022	79