Subgroups — #34178350

Order 20224.

662 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579072	C₂	1	1	yes	maximal, characteristic
2	#23765798	C₂	1	1	yes	maximal, characteristic
3	#23761282	C₂	1	4	yes	maximal, complemented
4	C₂×#432×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇₉⋊C₈	—	2	1	no	—
6	C₂×#14579072	—	2	2	no	—
7	C₂×#14579072	—	2	1	no	—
8	C₂×#14579072	C₄	1	4	yes	complemented
9	C₂×#14579072	C₂²	1	2	yes	—
10	#19098431	—	2	1	no	—
11	#19098429	C₂²	1	2	yes	—
12	#19098426	—	2	1	no	—
13	#19098423	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₇₉	C₄	1	2	yes	—
15	C₄×C₂²⋊Dic₇₉	—	2	1	no	—
16	#19098409	—	2	1	no	—
17	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
18	#432×C₇₉	—	2	2	no	—
19	#432×C₇₉	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₇₉	—	2	1	no	—
22	C₂×#97×C₇₉	C₂²	1	2	yes	—
23	C₂²×C₇₉⋊C₈	—	2	8	no	—
24	C₂²×C₇₉⋊C₈	—	2	4	no	—
25	C₂²×C₇₉⋊C₈	—	2	2	no	—
26	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
28	C₂×C₄×Dic₇₉	—	2	2	no	—
29	#14579072	—	2	4	no	—
30	#14579072	—	2	2	no	—
31	#14579072	C₂×C₄	1	4	yes	—
32	C₇₉⋊#95	—	2	2	no	—
33	C₇₉⋊#95	—	2	2	no	—
34	C₇₉⋊#95	—	2	2	no	—
35	C₇₉⋊#95	—	2	2	no	—
36	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	—	2	2	no	—
40	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	D₄	1	2	yes	—
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
47	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₇₉	—	2	4	no	—
49	C₂²⋊C₈×C₇₉	—	2	4	no	—
50	#97×C₇₉	—	2	2	no	—
51	#97×C₇₉	C₂×C₄	1	4	yes	—
52	C₂²×Dic₇₉	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	2	16	no	—
54	C₂×C₇₉⋊C₈	—	2	8	no	—
55	C₂×C₇₉⋊C₈	—	2	4	no	—
56	C₄⋊Dic₇₉	—	4	4	no	—
57	C₂²⋊Dic₇₉	—	4	4	no	—
58	C₄×Dic₇₉	—	4	4	no	—
59	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	—	2	4	no	—
61	C₂²×C₄×C₇₉	—	4	2	no	—
62	C₂²×C₄×C₇₉	—	4	2	no	—
63	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₉	—	2	4	no	—
76	C₂×C₈×C₇₉	—	4	4	no	—
77	C₄²×C₇₉	C₄²	1	2	yes	—
78	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
81	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
82	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
83	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
87	C₂³×C₇₉	—	2	2	no	—
88	C₂³×C₇₉	—	2	2	no	—
89	C₂³×C₇₉	#95	1	2	yes	—
90	C₂³×C₇₉	#104	1	4	yes	—
91	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₉	—	2	4	no	—
93	C₂×C₄×C₇₉	—	2	4	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	—	2	2	no	—
96	C₂×C₄×C₇₉	—	2	2	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	—	2	2	no	—
99	C₂×C₄×C₇₉	—	2	2	no	—
100	C₂×C₄×C₇₉	—	4	2	no	—
101	C₂×C₄×C₇₉	—	4	2	no	—
102	C₂×C₄×C₇₉	—	4	2	no	—
103	C₂×C₄×C₇₉	—	4	2	no	—
104	C₂×C₄×C₇₉	#95	1	2	yes	—
105	C₂×C₄×C₇₉	#95	1	2	yes	—
106	C₂×C₄×C₇₉	#95	1	2	yes	—
107	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×Dic₇₉	—	4	4	no	—
114	C₂×Dic₇₉	—	4	4	no	—
115	C₂×Dic₇₉	—	4	4	no	—
116	C₈×C₇₉	—	4	4	no	—
117	C₇₉⋊C₈	—	2	8	no	—
118	C₂²×C₇₉	—	2	2	no	—
119	C₂²×C₇₉	—	2	2	no	—
120	C₂²×C₇₉	—	2	8	no	—
121	C₂²×C₇₉	#339	1	4	yes	—
122	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
123	C₂²×C₇₉	#413	1	2	yes	—
124	C₂²×C₇₉	C₂×#104	1	2	yes	—
125	C₂²×C₇₉	#421	1	2	yes	—
126	C₄×C₇₉	—	2	2	no	—
127	C₄×C₇₉	—	2	1	no	—
128	C₄×C₇₉	—	2	1	no	—
129	C₄×C₇₉	—	2	2	no	—
130	C₄×C₇₉	—	2	1	no	—
131	C₄×C₇₉	—	2	1	no	—
132	C₄×C₇₉	—	2	2	no	—
133	C₄×C₇₉	—	2	1	no	—
134	C₄×C₇₉	—	2	1	no	—
135	Dic₇₉	—	8	4	no	—
136	#8036	—	79	1	no	maximal, Sylow
137	C₂×C₇₉	—	2	4	no	—
138	C₂×C₇₉	#1532	1	4	yes	—
139	C₂×C₇₉	C₂×#339	1	1	yes	characteristic
140	C₂×C₇₉	#1733	1	1	yes	characteristic
141	C₂×C₇₉	#1742	1	1	yes	characteristic
142	C₂×#432	D₇₉	1	1	yes	characteristic
143	C₂²×#97	—	79	1	no	—
144	#2283	—	79	1	no	—
145	#1295	—	79	4	no	—
146	C₇₉	#8036	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₈	—	158	1	no	—
148	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
149	#432	—	2	2	no	—
150	#432	Dic₇₉	1	4	yes	complemented
151	C₂×C₄⋊C₈	—	2	1	no	—
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×#97	—	79	2	no	—
154	C₂×#97	—	79	4	no	—
155	C₂×#97	—	158	2	no	—
156	C₂×#97	—	158	1	no	—
157	C₂×#97	C₂×D₇₉	1	2	yes	—
158	#383	—	158	1	no	—
159	#382	—	79	2	no	—
160	#381	—	158	1	no	—
161	#380	—	79	2	no	—
162	C₄×C₄⋊C₄	—	79	2	no	—
163	C₄×C₂²⋊C₄	—	158	1	no	—
164	#376	—	158	1	no	—
165	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
166	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
167	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	158	8	no	—
170	C₂²×C₈	—	158	4	no	—
171	C₂²×C₈	—	158	2	no	—
172	C₂×C₄⋊C₄	—	79	4	no	—
173	C₂×C₂²⋊C₄	—	158	2	no	—
174	C₂×C₄²	—	158	2	no	—
175	C₂×C₄²	Dic₁₅₈	1	2	yes	—
176	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₇₉	1	2	yes	—
178	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
181	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
183	C₄⋊C₈	—	2	4	no	—
184	C₂²⋊C₈	—	2	4	no	—
185	#97	—	2	2	no	—
186	#97	—	79	4	no	—
187	#97	—	158	4	no	—
188	#97	—	158	2	no	—
189	#97	C₂×Dic₇₉	1	4	yes	—
190	#95	—	158	2	no	—
191	#95	—	158	2	no	—
192	#95	—	158	2	no	—
193	#95	—	158	2	no	—
194	C₂⁴	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	158	4	no	—
199	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
200	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
204	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
205	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
206	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
207	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	—	158	8	no	—
214	C₂×C₈	—	158	16	no	—
215	C₂×C₈	—	158	4	no	—
216	C₄⋊C₄	—	316	4	no	—
217	C₂²⋊C₄	—	316	4	no	—
218	C₄²	—	316	4	no	—
219	C₄²	C₄×Dic₇₉	1	2	yes	—
220	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
221	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
222	C₄²	Dic₇₉⋊C₄	1	2	yes	—
223	C₄²	Dic₇₉⋊C₄	1	2	yes	—
224	C₄²	C₄⋊Dic₇₉	1	2	yes	—
225	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
226	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
228	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₇₉⋊#95	1	2	yes	—
232	C₂³	#14579087	1	4	yes	—
233	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	316	4	no	—
247	C₂×C₄	—	316	4	no	—
248	C₂×C₄	—	316	4	no	—
249	C₂×C₄	C₇₉⋊#95	1	2	yes	—
250	C₂×C₄	C₇₉⋊#95	1	2	yes	—
251	C₂×C₄	C₇₉⋊#95	1	2	yes	—
252	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
253	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
255	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
258	C₈	—	4	4	no	—
259	C₈	—	158	8	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	#19098324	1	4	yes	—
264	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
265	C₂²	#19098545	1	2	yes	—
266	C₂²	C₂×#14579087	1	2	yes	—
267	C₂²	#19098581	1	2	yes	—
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	632	4	no	—
278	C₂	—	2	4	no	—
279	C₂	#23762021	1	4	yes	—
280	C₂	C₂×#19098324	1	1	yes	characteristic
281	C₂	#23763110	1	1	yes	characteristic
282	C₂	#23763138	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#432×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#432	D₇₉
hypocenter	C₇₉	#8036
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #23761282	C₂	semidirect
8. C₂×#14579072	C₄	semidirect
19. #432×C₇₉	C₄	semidirect
146. C₇₉	#8036	semidirect
150. #432	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579072	2
#23765798	2
#23761282	2
C₂×#432×C₇₉	2
#8036	79