Subgroups — #34188406

Order 20224.

430 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771763	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#23761284	C₂	1	2	yes	maximal
4	#23761284	C₂	1	2	yes	maximal
5	C₇₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579153	C₄	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
9	#19098461	—	2	1	no	—
10	#19098461	—	2	1	no	—
11	#19098461	—	2	1	no	—
12	#19098461	—	2	1	no	—
13	#19098433	C₂²	1	1	yes	characteristic
14	#19098433	C₂²	1	1	yes	characteristic
15	#19098429	C₄	1	2	yes	—
16	#19098318	—	2	1	no	—
17	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
18	C₄³×C₇₉	C₂²	1	1	yes	characteristic
19	#336×C₇₉	—	2	1	no	—
20	#336×C₇₉	—	2	1	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	#14579153	—	4	4	no	—
23	C₂×C₄⋊Dic₇₉	—	2	1	no	—
24	C₂×C₄⋊Dic₇₉	—	2	1	no	—
25	C₂×C₄⋊Dic₇₉	—	2	1	no	—
26	C₂×C₄⋊Dic₇₉	—	2	1	no	—
27	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
29	C₂×Dic₇₉⋊C₄	—	2	2	no	—
30	C₂×Dic₇₉⋊C₄	—	2	2	no	—
31	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	Q₈	1	4	yes	—
33	C₄×C₇₉⋊C₈	—	4	2	no	—
34	C₇₉⋊#95	—	2	2	no	—
35	C₇₉⋊#95	—	2	2	no	—
36	C₂²×C₈×C₇₉	—	2	2	no	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₇₉	—	2	4	no	—
45	C₂²×Dic₇₉	—	2	2	no	—
46	C₂²×Dic₇₉	—	2	2	no	—
47	C₂×C₇₉⋊C₈	—	2	4	no	—
48	C₂×C₇₉⋊C₈	—	4	2	no	—
49	C₂×C₇₉⋊C₈	—	4	2	no	—
50	C₄⋊Dic₇₉	—	4	2	no	—
51	C₄⋊Dic₇₉	—	4	2	no	—
52	C₄⋊Dic₇₉	—	4	2	no	—
53	C₄⋊Dic₇₉	—	4	2	no	—
54	C₄⋊Dic₇₉	—	4	2	no	—
55	C₄⋊Dic₇₉	—	4	1	no	—
56	C₄⋊Dic₇₉	—	4	1	no	—
57	Dic₇₉⋊C₄	—	4	4	no	—
58	Dic₇₉⋊C₄	—	4	4	no	—
59	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	—	4	2	no	—
67	C₂×C₈×C₇₉	—	4	2	no	—
68	C₂×C₈×C₇₉	—	4	2	no	—
69	C₄²×C₇₉	—	2	2	no	—
70	C₄²×C₇₉	—	2	2	no	—
71	C₄²×C₇₉	—	2	2	no	—
72	C₄²×C₇₉	—	2	2	no	—
73	C₄²×C₇₉	—	2	2	no	—
74	C₄²×C₇₉	—	2	1	no	—
75	C₄²×C₇₉	—	2	1	no	—
76	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
77	C₄²×C₇₉	#41	1	2	yes	—
78	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₉	—	2	1	no	—
80	C₂×C₄×C₇₉	—	2	1	no	—
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	#100	1	2	yes	—
92	C₂×C₄×C₇₉	#101	1	2	yes	—
93	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₇₉	#135	1	2	yes	—
98	C₂×C₄×C₇₉	#135	1	2	yes	—
99	C₂×C₄×C₇₉	#135	1	2	yes	—
100	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
102	C₂×Dic₇₉	—	4	2	no	—
103	C₂×Dic₇₉	—	4	2	no	—
104	C₂×Dic₇₉	—	4	2	no	—
105	C₂×Dic₇₉	—	4	2	no	—
106	C₂×Dic₇₉	—	4	2	no	—
107	C₂×Dic₇₉	—	4	2	no	—
108	C₈×C₇₉	—	8	2	no	—
109	C₇₉⋊C₈	—	4	4	no	—
110	C₂²×C₇₉	#391	1	1	yes	characteristic
111	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
112	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
113	C₂²×C₇₉	#416	1	1	yes	characteristic
114	C₂²×C₇₉	#416	1	1	yes	characteristic
115	C₂²×C₇₉	#416	1	1	yes	characteristic
116	C₂²×C₇₉	#417	1	1	yes	characteristic
117	C₄×C₇₉	—	2	2	no	—
118	C₄×C₇₉	—	2	2	no	—
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	—	4	1	no	—
121	C₄×C₇₉	—	4	1	no	—
122	C₄×C₇₉	#478	1	2	yes	—
123	C₄×C₇₉	#479	1	2	yes	—
124	C₄×C₇₉	#488	1	2	yes	—
125	C₄×C₇₉	#499	1	2	yes	—
126	Dic₇₉	—	8	2	no	—
127	Dic₇₉	—	8	2	no	—
128	#9901	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	#1543	1	1	yes	characteristic
130	C₂×C₇₉	#1544	1	1	yes	characteristic
131	C₂×C₇₉	#1551	1	1	yes	characteristic
132	C₂×C₇₉	#1552	1	1	yes	characteristic
133	C₂×C₇₉	#1989	1	1	yes	characteristic
134	C₂×C₇₉	#1991	1	1	yes	characteristic
135	C₂×C₇₉	#1997	1	1	yes	characteristic
136	#2859	—	79	1	no	—
137	#1831	D₇₉	1	1	yes	characteristic
138	C₄×C₄⋊C₈	—	79	1	no	—
139	#1296	—	79	2	no	—
140	#1296	—	79	2	no	—
141	C₇₉	#9901	1	1	yes	characteristic, complemented, Sylow
142	C₂×#125	—	79	2	no	—
143	C₂×C₄⋊C₈	—	79	2	no	—
144	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
145	C₂×C₄×C₈	—	158	1	no	—
146	#389	—	158	1	no	—
147	#389	—	158	1	no	—
148	#389	—	158	1	no	—
149	#389	—	158	1	no	—
150	#384	—	79	1	no	—
151	#384	—	79	1	no	—
152	#382	—	79	2	no	—
153	C₄³	C₂×D₇₉	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#336	—	2	1	no	—
156	#336	—	158	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	158	2	no	—
159	#125	—	316	4	no	—
160	C₂×C₄⋊C₄	—	79	2	no	—
161	C₂×C₄⋊C₄	—	79	2	no	—
162	C₂×C₄⋊C₄	—	158	2	no	—
163	C₂×C₄⋊C₄	—	158	2	no	—
164	C₂×C₄⋊C₄	—	158	1	no	—
165	C₂×C₄⋊C₄	—	158	1	no	—
166	C₂×C₄⋊C₄	—	158	1	no	—
167	C₂×C₄⋊C₄	—	158	1	no	—
168	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
170	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
171	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	79	4	no	—
177	C₄⋊C₈	—	158	2	no	—
178	C₄×C₈	—	316	2	no	—
179	#95	—	158	2	no	—
180	#95	—	158	2	no	—
181	C₂²×C₄	—	158	2	no	—
182	C₂²×C₄	—	158	2	no	—
183	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	316	2	no	—
194	C₂×C₈	—	316	2	no	—
195	C₂×C₈	—	158	4	no	—
196	C₄⋊C₄	—	316	2	no	—
197	C₄⋊C₄	—	316	2	no	—
198	C₄⋊C₄	—	316	4	no	—
199	C₄⋊C₄	—	316	4	no	—
200	C₄⋊C₄	—	316	2	no	—
201	C₄⋊C₄	—	316	2	no	—
202	C₄⋊C₄	—	316	2	no	—
203	C₄⋊C₄	—	316	1	no	—
204	C₄⋊C₄	—	316	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	Q₈×D₇₉	1	2	yes	—
213	C₄²	#11929623	1	2	yes	—
214	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	316	2	no	—
228	C₂×C₄	—	316	2	no	—
229	C₂×C₄	—	316	2	no	—
230	C₂×C₄	—	316	2	no	—
231	C₂×C₄	—	316	2	no	—
232	C₂×C₄	—	316	2	no	—
233	C₂×C₄	#14579077	1	2	yes	—
234	C₂×C₄	#14579079	1	2	yes	—
235	C₂×C₄	#14579126	1	2	yes	—
236	C₂×C₄	#14579141	1	2	yes	—
237	C₂×C₄	#14579148	1	1	yes	characteristic
238	C₂×C₄	#14579159	1	1	yes	characteristic
239	C₂×C₄	#14579181	1	2	yes	—
240	C₂×C₄	#14579181	1	2	yes	—
241	C₂×C₄	#14579183	1	2	yes	—
242	C₂×C₄	#14579189	1	1	yes	characteristic
243	C₂×C₄	#14579194	1	1	yes	characteristic
244	C₈	—	8	2	no	—
245	C₈	—	316	4	no	—
246	C₂²	#19098468	1	1	yes	characteristic
247	C₂²	C₂×#14579077	1	1	yes	characteristic
248	C₂²	C₂×#14579079	1	1	yes	characteristic
249	C₂²	#19098557	1	1	yes	characteristic
250	C₂²	#19098557	1	1	yes	characteristic
251	C₂²	#19098560	1	1	yes	characteristic
252	C₂²	#19098565	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	632	2	no	—
259	C₄	—	632	2	no	—
260	C₄	#19098864	1	2	yes	—
261	C₄	#19098871	1	2	yes	—
262	C₄	#19098917	1	2	yes	—
263	C₄	#19098962	1	2	yes	—
264	C₂	#23762060	1	1	yes	characteristic
265	C₂	#23762064	1	1	yes	characteristic
266	C₂	#23762096	1	1	yes	characteristic
267	C₂	#23762101	1	1	yes	characteristic
268	C₂	#23764355	1	1	yes	characteristic
269	C₂	#23764364	1	1	yes	characteristic
270	C₂	#23764388	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1831	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1831	D₇₉
hypocenter	C₇₉	#9901
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
141. C₇₉	#9901	semidirect

Maximal subgroups

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

subgroup	index
#23771763	2
C₄×C₄⋊(C₇₉⋊C₈)	2
#23761284	2
C₇₉×#1831	2
#9901	79