Subgroups — #34177753

Order 20224.

630 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098620	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#23761235	C₂	1	4	yes	maximal, complemented
4	C₂×#431×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇₉⋊C₈	—	2	1	no	—
6	#19098620	—	2	2	no	—
7	#19098620	C₄	1	4	yes	—
8	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	4	yes	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
14	C₂×#14579072	—	2	1	no	—
15	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	#431×C₇₉	—	2	2	no	—
17	#431×C₇₉	—	2	2	no	—
18	C₂×C₄⋊C₈×C₇₉	C₄	1	2	yes	—
19	C₂×C₂²⋊C₈×C₇₉	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
21	C₂²×C₇₉⋊C₈	—	2	4	no	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	C₂²×C₇₉⋊C₈	—	4	4	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
26	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
28	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
29	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
30	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
31	#14579072	—	2	4	no	—
32	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₇₉	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	D₄	1	2	yes	—
40	C₂×C₄²×C₇₉	D₄	1	2	yes	—
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₇₉	—	2	4	no	—
45	C₂²⋊C₈×C₇₉	—	2	4	no	—
46	C₄×C₈×C₇₉	—	2	2	no	—
47	C₄×C₈×C₇₉	—	2	2	no	—
48	C₂×C₇₉⋊C₈	—	2	4	no	—
49	C₂×C₇₉⋊C₈	—	2	4	no	—
50	C₂×C₇₉⋊C₈	—	4	8	no	—
51	C₂×C₇₉⋊C₈	—	4	2	no	—
52	C₂×C₇₉⋊C₈	—	4	4	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₂⁴×C₇₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	—	2	4	no	—
56	C₂²×C₄×C₇₉	—	2	4	no	—
57	C₂²×C₄×C₇₉	—	2	2	no	—
58	C₂²×C₄×C₇₉	—	2	2	no	—
59	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₇₉	—	2	4	no	—
72	C₂×C₈×C₇₉	—	2	8	no	—
73	C₄²×C₇₉	C₄²	1	2	yes	—
74	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
79	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₇₉	—	2	2	no	—
84	C₂³×C₇₉	—	2	2	no	—
85	C₂³×C₇₉	#95	1	2	yes	—
86	C₂³×C₇₉	#108	1	4	yes	—
87	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₇₉	—	2	4	no	—
89	C₂×C₄×C₇₉	—	2	4	no	—
90	C₂×C₄×C₇₉	—	2	2	no	—
91	C₂×C₄×C₇₉	—	2	2	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	—	2	4	no	—
95	C₂×C₄×C₇₉	—	2	4	no	—
96	C₂×C₄×C₇₉	—	2	4	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	—	2	2	no	—
99	C₂×C₄×C₇₉	#95	1	2	yes	—
100	C₂×C₄×C₇₉	#95	1	2	yes	—
101	C₂×C₄×C₇₉	#95	1	2	yes	—
102	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	#131	1	4	yes	—
109	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₇₉	#137	1	2	yes	—
111	C₈×C₇₉	—	4	4	no	—
112	C₇₉⋊C₈	—	4	4	no	—
113	C₇₉⋊C₈	—	4	4	no	—
114	C₂²×C₇₉	—	2	8	no	—
115	C₂²×C₇₉	—	2	2	no	—
116	C₂²×C₇₉	—	2	2	no	—
117	C₂²×C₇₉	#341	1	4	yes	—
118	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
119	C₂²×C₇₉	#424	1	2	yes	—
120	C₂²×C₇₉	#428	1	2	yes	—
121	C₂²×C₇₉	C₂×#108	1	2	yes	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	—	2	2	no	—
124	C₄×C₇₉	—	2	2	no	—
125	C₄×C₇₉	—	2	1	no	—
126	C₄×C₇₉	—	2	1	no	—
127	C₄×C₇₉	#405	1	2	yes	—
128	C₄×C₇₉	#408	1	2	yes	—
129	C₄×C₇₉	#417	1	2	yes	—
130	C₄×C₇₉	#418	1	1	yes	characteristic
131	C₄×C₇₉	#419	1	1	yes	characteristic
132	#7879	—	79	1	no	maximal, Sylow
133	C₂×C₇₉	—	2	4	no	—
134	C₂×C₇₉	#1573	1	4	yes	—
135	C₂×C₇₉	#1729	1	1	yes	characteristic
136	C₂×C₇₉	#1736	1	1	yes	characteristic
137	C₂×C₇₉	C₂×#341	1	1	yes	characteristic
138	C₂×#432	—	79	1	no	—
139	C₂×#431	D₇₉	1	1	yes	characteristic
140	C₂²×C₄⋊C₈	—	79	1	no	—
141	#1277	—	79	4	no	—
142	C₇₉	#7879	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₈	—	158	1	no	—
144	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
145	#432	—	79	4	no	—
146	#432	—	158	2	no	—
147	#431	—	2	2	no	—
148	#431	—	2	2	no	—
149	C₂×C₄⋊C₈	—	79	2	no	—
150	C₂×C₄⋊C₈	—	79	2	no	—
151	C₂×C₄⋊C₈	—	79	4	no	—
152	C₂×C₄⋊C₈	—	158	2	no	—
153	C₂×C₄⋊C₈	—	158	1	no	—
154	C₂×C₄⋊C₈	Dic₇₉	1	2	yes	—
155	C₂×C₂²⋊C₈	—	158	1	no	—
156	C₂×C₂²⋊C₈	Dic₇₉	1	2	yes	—
157	C₂×#97	—	158	1	no	—
158	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
159	C₂³×C₄	Dic₁₅₈	1	1	yes	characteristic
160	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
162	C₂²×C₈	—	316	4	no	—
163	C₂²×C₈	—	158	4	no	—
164	C₂²×C₈	—	158	2	no	—
165	C₂²×C₈	—	158	2	no	—
166	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
167	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₇₉	1	2	yes	—
169	C₂×C₄²	C₄×D₇₉	1	2	yes	—
170	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
173	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
174	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	79	4	no	—
177	C₄⋊C₈	—	79	4	no	—
178	C₄⋊C₈	—	158	4	no	—
179	C₄⋊C₈	—	158	2	no	—
180	C₄⋊C₈	—	158	2	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	158	4	no	—
183	#97	—	158	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₂⁴	C₄⋊Dic₇₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
192	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
193	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
194	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
195	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
196	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
197	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	—	316	8	no	—
206	C₂×C₈	—	316	4	no	—
207	C₂×C₈	—	316	2	no	—
208	C₂×C₈	—	316	2	no	—
209	C₂×C₈	—	158	4	no	—
210	C₂×C₈	—	158	4	no	—
211	C₄²	C₄×Dic₇₉	1	2	yes	—
212	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
213	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
214	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
215	C₄²	Dic₇₉⋊C₄	1	2	yes	—
216	C₄²	Dic₇₉⋊C₄	1	2	yes	—
217	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
218	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
219	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₇₉⋊#95	1	2	yes	—
224	C₂³	#14579097	1	4	yes	—
225	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	C₇₉⋊#95	1	2	yes	—
238	C₂×C₄	C₇₉⋊#95	1	2	yes	—
239	C₂×C₄	C₇₉⋊#95	1	2	yes	—
240	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
241	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
242	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
244	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
245	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
246	C₂×C₄	#14579168	1	4	yes	—
247	C₂×C₄	#14579189	1	2	yes	—
248	C₂×C₄	#14579194	1	2	yes	—
249	C₈	—	4	4	no	—
250	C₈	—	316	4	no	—
251	C₈	—	316	4	no	—
252	C₂²	—	2	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#19098328	1	4	yes	—
256	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
257	C₂²	#19098590	1	2	yes	—
258	C₂²	#19098605	1	2	yes	—
259	C₂²	C₂×#14579097	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	#19098519	1	2	yes	—
266	C₄	#19098530	1	2	yes	—
267	C₄	#19098565	1	2	yes	—
268	C₄	#19098569	1	1	yes	characteristic
269	C₄	#19098573	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	#23762185	1	4	yes	—
272	C₂	#23763097	1	1	yes	characteristic
273	C₂	#23763120	1	1	yes	characteristic
274	C₂	C₂×#19098328	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#431×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#431	D₇₉
hypocenter	C₇₉	#7879
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #23761235	C₂	semidirect
142. C₇₉	#7879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098620	2
C₂²×C₄⋊(C₇₉⋊C₈)	2
#23761235	2
C₂×#431×C₇₉	2
#7879	79