Subgroups — #34260309

Order 20224.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773979	C₂	1	1	yes	maximal, characteristic
2	#23772466	C₂	1	2	yes	maximal, complemented
3	#23762315	C₂	1	2	yes	maximal, complemented
4	#23762231	C₂	1	4	yes	maximal, complemented
5	#23762225	C₂	1	2	yes	maximal, complemented
6	#23762148	C₂	1	1	yes	maximal, characteristic, complemented
7	#23761818	C₂	1	2	yes	maximal, complemented
8	C₇₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	#19099113	C₂²	1	2	yes	—
10	#19098968	C₄	1	1	yes	characteristic
11	#19098940	C₄	1	1	yes	characteristic
12	#19098910	C₄	1	2	yes	—
13	#19098799	C₄	1	2	yes	—
14	#19098799	C₄	1	2	yes	—
15	C₄×D₆₃₂	C₂²	1	2	yes	—
16	C₄×D₆₃₂	C₂²	1	2	yes	—
17	#19098642	C₂²	1	4	yes	—
18	#19098640	—	2	2	no	—
19	C₈×D₃₁₆	C₂²	1	4	yes	—
20	#19098628	—	2	2	no	—
21	#19098616	C₂²	1	2	yes	—
22	#19098528	—	2	4	no	—
23	#19098316	C₂²	1	2	yes	—
24	#19098314	C₂²	1	1	yes	characteristic
25	#19098314	C₂²	1	1	yes	characteristic
26	#19098292	C₂²	1	4	yes	—
27	#19098292	C₂²	1	4	yes	—
28	#431×C₇₉	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
31	#322×C₇₉	C₂²	1	2	yes	—
32	#322×C₇₉	C₂²	1	2	yes	—
33	C₂×#11929624	—	2	2	no	—
34	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
35	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
36	C₂×#11929607	—	2	4	no	—
37	C₂×C₈×D₇₉	—	2	4	no	—
38	#14579161	C₂×C₄	1	2	yes	—
39	#14579158	C₂×C₄	1	2	yes	—
40	#14579149	—	2	2	no	—
41	#14579135	—	2	2	no	—
42	#14579135	C₂×C₄	1	4	yes	—
43	C₄×D₃₁₆	—	2	2	no	—
44	C₄×D₃₁₆	C₂³	1	4	yes	—
45	#14579095	C₂×C₄	1	2	yes	—
46	#14579095	C₂×C₄	1	2	yes	—
47	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
49	#14579083	C₂×C₄	1	4	yes	—
50	#14579083	C₂×C₄	1	4	yes	—
51	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
52	(C₂×D₇₉)⋊C₈	—	2	4	no	—
53	(C₂×D₇₉)⋊C₈	—	2	4	no	—
54	C₄×C₇₉⋊C₈	—	2	2	no	—
55	C₂×M₁₆×C₇₉	—	2	1	no	—
56	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
60	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
61	#97×C₇₉	—	2	1	no	—
62	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₇₉	C₂³	1	2	yes	—
65	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₉	C₂³	1	2	yes	—
68	#11929624	—	8	2	no	—
69	#11929624	—	8	2	no	—
70	C₂×Dic₁₅₈	—	2	2	no	—
71	C₂×D₃₁₆	—	2	2	no	—
72	C₂×D₃₁₆	—	2	2	no	—
73	C₂×D₃₁₆	C₂²×C₄	1	4	yes	—
74	C₂×C₂²⋊D₇₉	—	2	4	no	—
75	C₂×C₄×D₇₉	—	2	4	no	—
76	D₆₃₂	—	2	4	no	—
77	D₆₃₂	—	4	2	no	—
78	#11929607	—	8	4	no	—
79	C₂×C₇₉⋊C₈	—	2	2	no	—
80	C₂×C₇₉⋊C₈	—	2	2	no	—
81	C₈×D₇₉	—	8	4	no	—
82	C₄⋊Dic₇₉	—	2	2	no	—
83	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
84	(C₂×D₇₉)⋊C₄	—	2	4	no	—
85	(C₂×D₇₉)⋊C₄	—	2	4	no	—
86	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
89	M₁₆×C₇₉	—	4	2	no	—
90	C₂×C₈×C₇₉	—	2	2	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	4	1	no	—
93	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
98	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₉	#41	1	1	yes	characteristic
101	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
102	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
103	C₂²×D₇₉	—	2	4	no	—
104	C₂×C₄×C₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	2	1	no	—
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	—	2	2	no	—
108	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
114	C₂²⋊D₇₉	—	8	4	no	—
115	C₂×Dic₇₉	—	2	4	no	—
116	D₃₁₆	—	2	4	no	—
117	D₃₁₆	—	2	4	no	—
118	D₃₁₆	—	4	4	no	—
119	D₃₁₆	—	8	2	no	—
120	C₄×D₇₉	—	4	4	no	—
121	C₄×D₇₉	—	4	4	no	—
122	Dic₁₅₈	—	8	2	no	—
123	C₈×C₇₉	—	2	1	no	—
124	C₈×C₇₉	—	2	1	no	—
125	C₈×C₇₉	—	2	1	no	—
126	C₈×C₇₉	—	2	4	no	—
127	C₈×C₇₉	—	4	2	no	—
128	C₈×C₇₉	#131	1	2	yes	—
129	C₇₉⋊C₈	—	4	2	no	—
130	C₇₉⋊C₈	—	4	2	no	—
131	C₂²×C₇₉	—	2	1	no	—
132	C₂²×C₇₉	—	2	1	no	—
133	C₂²×C₇₉	—	2	1	no	—
134	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×D₇₉	—	4	4	no	—
136	C₂×D₇₉	—	4	4	no	—
137	C₂×D₇₉	—	4	4	no	—
138	C₄×C₇₉	—	2	1	no	—
139	C₄×C₇₉	—	2	1	no	—
140	C₄×C₇₉	—	2	1	no	—
141	C₄×C₇₉	#444	1	2	yes	—
142	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
143	C₄×C₇₉	#568	1	1	yes	characteristic
144	C₄×C₇₉	#576	1	2	yes	—
145	Dic₇₉	—	4	4	no	—
146	#18091	—	79	1	no	maximal, Sylow
147	C₂×C₇₉	—	4	1	no	—
148	C₂×C₇₉	#2935	1	1	yes	characteristic
149	C₂×C₇₉	#2948	1	1	yes	characteristic
150	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
151	D₇₉	—	8	4	no	—
152	#3169	—	79	1	no	—
153	#2979	—	79	2	no	—
154	#1594	—	79	2	no	—
155	#1581	—	79	4	no	—
156	#1580	—	79	2	no	—
157	#1564	—	79	1	no	—
158	#1484	—	79	2	no	—
159	#1451	D₇₉	1	1	yes	characteristic, complemented
160	C₇₉	#18091	1	1	yes	characteristic, complemented, Sylow
161	#532	—	79	2	no	—
162	#500	—	79	1	no	—
163	#493	—	79	1	no	—
164	#486	—	79	2	no	—
165	#466	—	79	2	no	—
166	#466	—	79	2	no	—
167	C₄×D₈	—	79	2	no	—
168	C₄×D₈	—	79	2	no	—
169	#436	—	79	4	no	—
170	#436	—	158	2	no	—
171	C₈×D₄	—	79	4	no	—
172	C₈×D₄	—	158	2	no	—
173	#431	—	79	2	no	—
174	#431	C₂×D₇₉	1	1	yes	characteristic
175	#408	—	158	4	no	—
176	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
178	#335	—	79	2	no	—
179	#334	—	79	1	no	—
180	#334	—	79	1	no	—
181	D₄⋊C₈	—	79	4	no	—
182	D₄⋊C₈	—	79	4	no	—
183	#322	C₂×D₇₉	1	2	yes	—
184	#322	C₂×D₇₉	1	2	yes	—
185	C₂×#41	—	158	2	no	—
186	C₂×D₈	—	79	2	no	—
187	C₂×D₈	—	79	2	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	158	4	no	—
190	C₂²×C₈	—	158	4	no	—
191	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
192	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
193	C₄⋊Q₈	—	79	2	no	—
194	#127	—	79	2	no	—
195	#124	—	158	2	no	—
196	#121	—	79	4	no	—
197	#121	—	158	2	no	—
198	C₄×D₄	—	79	4	no	—
199	C₄×D₄	—	158	2	no	—
200	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
201	#107	—	79	2	no	—
202	#107	—	79	2	no	—
203	C₄⋊C₈	—	79	2	no	—
204	C₄⋊C₈	—	79	2	no	—
205	C₄⋊C₈	D₃₁₆	1	2	yes	—
206	#102	—	79	4	no	—
207	#102	—	79	4	no	—
208	C₂²⋊C₈	—	158	2	no	—
209	C₂²⋊C₈	—	158	4	no	—
210	C₂²⋊C₈	—	158	4	no	—
211	C₂²⋊C₈	D₃₁₆	1	2	yes	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	158	2	no	—
214	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
215	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₇₉	1	2	yes	—
217	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₇₉	1	2	yes	—
220	#41	—	632	2	no	—
221	#41	—	632	2	no	—
222	C₂×Q₈	—	158	2	no	—
223	C₂×D₄	—	79	4	no	—
224	C₂×D₄	—	158	4	no	—
225	C₂×D₄	—	158	2	no	—
226	C₂×D₄	—	158	2	no	—
227	C₂²×C₄	—	158	4	no	—
228	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
230	C₂²×C₄	#11929624	1	1	yes	characteristic
231	D₈	—	316	2	no	—
232	D₈	—	158	4	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	632	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	632	4	no	—
239	C₂×C₈	—	158	2	no	—
240	C₂×C₈	—	158	2	no	—
241	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
245	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
246	C₄⋊C₄	—	79	4	no	—
247	C₄⋊C₄	—	158	2	no	—
248	C₂²⋊C₄	—	158	4	no	—
249	C₂²⋊C₄	—	158	4	no	—
250	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
251	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
252	C₄²	#11929624	1	1	yes	characteristic
253	C₄²	C₂³×D₇₉	1	1	yes	characteristic
254	C₂³	—	158	4	no	—
255	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
256	Q₈	—	632	2	no	—
257	D₄	—	316	4	no	—
258	D₄	—	632	2	no	—
259	D₄	—	632	4	no	—
260	D₄	—	158	4	no	—
261	D₄	—	158	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	316	4	no	—
267	C₂×C₄	—	316	4	no	—
268	C₂×C₄	—	158	4	no	—
269	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
270	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
271	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
274	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₈	—	316	2	no	—
281	C₈	—	316	2	no	—
282	C₈	#14579169	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	316	4	no	—
287	C₂²	—	316	4	no	—
288	C₂²	—	316	4	no	—
289	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	316	4	no	—
294	C₄	#19098686	1	2	yes	—
295	C₄	C₂×#14579169	1	1	yes	characteristic
296	C₄	#19099329	1	1	yes	characteristic
297	C₄	#19099368	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	—	632	4	no	—
300	C₂	#23772146	1	1	yes	characteristic
301	C₂	#23772253	1	1	yes	characteristic
302	C₂	C₂×#19098686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1451	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#23773979	C₂
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1451	D₇₉
hypocenter	C₇₉	#18091
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #23772466	C₂	semidirect
3. #23762315	C₂	semidirect
4. #23762231	C₂	semidirect
5. #23762225	C₂	semidirect
6. #23762148	C₂	semidirect
7. #23761818	C₂	semidirect
8. C₇₉×#1451	C₂	semidirect
159. #1451	D₇₉	semidirect
160. C₇₉	#18091	semidirect

Maximal subgroups

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

subgroup	index
#23773979	2
#23772466	2
#23762315	2
#23762231	2
#23762225	2
#23762148	2
#23761818	2
C₇₉×#1451	2
#18091	79