Subgroups — #34182915

Order 20224.

586 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772424	C₂	1	1	yes	maximal, characteristic
2	#23761795	C₂	1	2	yes	maximal, complemented
3	#23761721	C₂	1	1	yes	maximal, characteristic
4	#23761233	C₂	1	2	yes	maximal, complemented
5	C₄×#124×C₇₉	C₂	1	1	yes	maximal, characteristic
6	D₄×C₇₉⋊C₈	—	2	4	no	—
7	#19098616	C₂²	1	2	yes	—
8	#19098616	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
10	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
11	#19098287	—	2	2	no	—
12	C₂×#124×C₇₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
15	#391×C₇₉	C₄	1	1	yes	characteristic
16	#390×C₇₉	C₄	1	1	yes	characteristic
17	#388×C₇₉	—	2	2	no	—
18	#382×C₇₉	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊C₄×C₇₉	—	2	2	no	—
20	C₄³×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	—	2	4	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
24	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
27	C₄×C₇₉⋊C₈	—	2	2	no	—
28	C₄×C₇₉⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₇₉	D₄	1	2	yes	—
32	#124×C₇₉	—	2	2	no	—
33	#124×C₇₉	C₈	1	4	yes	complemented
34	C₄×Q₈×C₇₉	—	2	2	no	—
35	C₄×Q₈×C₇₉	—	2	2	no	—
36	C₄×D₄×C₇₉	—	2	2	no	—
37	C₄×D₄×C₇₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
44	C₂×C₄²×C₇₉	—	2	1	no	—
45	C₂×C₄²×C₇₉	—	2	1	no	—
46	C₂×C₄²×C₇₉	—	2	1	no	—
47	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	#95×C₇₉	—	2	2	no	—
51	#95×C₇₉	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₂×C₇₉⋊C₈	—	8	4	no	—
55	C₂⁴×C₇₉	C₂²⋊C₄	1	2	yes	—
56	C₂×Q₈×C₇₉	—	2	1	no	—
57	C₂×Q₈×C₇₉	—	2	2	no	—
58	C₂×Q₈×C₇₉	—	2	2	no	—
59	C₂×Q₈×C₇₉	C₂×C₈	1	2	yes	complemented
60	C₂×D₄×C₇₉	—	2	2	no	—
61	C₂×D₄×C₇₉	—	2	2	no	—
62	C₂×D₄×C₇₉	C₂×C₈	1	2	yes	—
63	C₂×D₄×C₇₉	M₁₆	1	2	yes	—
64	C₂²×C₄×C₇₉	—	2	1	no	—
65	C₂²×C₄×C₇₉	—	2	1	no	—
66	C₂²×C₄×C₇₉	—	2	1	no	—
67	C₂²×C₄×C₇₉	—	2	2	no	—
68	C₂²×C₄×C₇₉	—	2	2	no	—
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	—	4	4	no	—
71	C₂²×C₄×C₇₉	C₂²⋊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	C₄⋊C₄×C₇₉	—	2	2	no	—
76	C₄⋊C₄×C₇₉	—	2	2	no	—
77	C₄⋊C₄×C₇₉	—	4	2	no	—
78	C₂²⋊C₄×C₇₉	—	2	8	no	—
79	C₂²⋊C₄×C₇₉	—	2	4	no	—
80	C₂²⋊C₄×C₇₉	—	4	4	no	—
81	C₄²×C₇₉	—	2	1	no	—
82	C₄²×C₇₉	—	2	2	no	—
83	C₄²×C₇₉	—	4	2	no	—
84	C₄²×C₇₉	—	4	2	no	—
85	C₄²×C₇₉	—	4	1	no	—
86	C₄²×C₇₉	—	4	1	no	—
87	C₄²×C₇₉	C₂×C₈	1	2	yes	complemented
88	C₄²×C₇₉	C₂×D₄	1	2	yes	—
89	C₄²×C₇₉	#41	1	2	yes	—
90	C₂³×C₇₉	—	2	2	no	—
91	C₂³×C₇₉	—	2	2	no	—
92	C₂³×C₇₉	—	4	2	no	—
93	C₂³×C₇₉	—	4	2	no	—
94	C₂³×C₇₉	C₂²⋊C₈	1	4	yes	—
95	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	Q₈×C₇₉	—	2	4	no	—
97	Q₈×C₇₉	—	4	2	no	—
98	D₄×C₇₉	—	2	4	no	—
99	D₄×C₇₉	—	2	4	no	—
100	C₂×C₄×C₇₉	—	2	2	no	—
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	—	4	1	no	—
106	C₂×C₄×C₇₉	—	4	1	no	—
107	C₂×C₄×C₇₉	—	4	1	no	—
108	C₂×C₄×C₇₉	—	4	1	no	—
109	C₂×C₄×C₇₉	—	4	1	no	—
110	C₂×C₄×C₇₉	—	4	1	no	—
111	C₂×C₄×C₇₉	—	4	1	no	—
112	C₂×C₄×C₇₉	—	4	4	no	—
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	—	4	1	no	—
116	C₂×C₄×C₇₉	—	8	2	no	—
117	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
119	C₂×C₄×C₇₉	#121	1	2	yes	—
120	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#131	1	2	yes	—
124	C₇₉⋊C₈	—	4	4	no	—
125	C₇₉⋊C₈	—	8	2	no	—
126	C₂²×C₇₉	—	2	1	no	—
127	C₂²×C₇₉	—	2	1	no	—
128	C₂²×C₇₉	—	2	4	no	—
129	C₂²×C₇₉	—	4	4	no	—
130	C₂²×C₇₉	—	4	2	no	—
131	C₂²×C₇₉	—	4	2	no	—
132	C₂²×C₇₉	—	8	2	no	—
133	C₂²×C₇₉	#386	1	1	yes	characteristic
134	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₂²×C₇₉	#408	1	1	yes	characteristic
136	C₄×C₇₉	—	2	2	no	—
137	C₄×C₇₉	—	2	1	no	—
138	C₄×C₇₉	—	4	2	no	—
139	C₄×C₇₉	—	4	2	no	—
140	C₄×C₇₉	—	4	2	no	—
141	C₄×C₇₉	—	4	2	no	—
142	C₄×C₇₉	—	8	1	no	—
143	C₄×C₇₉	C₈×D₄	1	2	yes	—
144	C₄×C₇₉	#435	1	2	yes	—
145	C₄×C₇₉	#455	1	2	yes	—
146	#8957	—	79	1	no	maximal, Sylow
147	C₂×C₇₉	—	2	1	no	—
148	C₂×C₇₉	—	2	1	no	—
149	C₂×C₇₉	—	4	4	no	—
150	C₂×C₇₉	#1869	1	1	yes	characteristic
151	C₂×C₇₉	#1884	1	1	yes	characteristic
152	C₂×C₇₉	#1885	1	1	yes	characteristic
153	#2972	—	79	1	no	—
154	C₄×#124	D₇₉	1	1	yes	characteristic
155	#1479	—	79	2	no	—
156	#1460	—	79	1	no	—
157	#1276	—	79	2	no	—
158	C₇₉	#8957	1	1	yes	characteristic, complemented, Sylow
159	C₂×#124	Dic₇₉	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
161	C₂×C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
162	C₈×D₄	—	158	4	no	—
163	#431	—	79	2	no	—
164	#431	—	79	2	no	—
165	C₂×C₂²⋊C₈	—	158	2	no	—
166	#391	Dic₇₉	1	1	yes	characteristic
167	#390	Dic₇₉	1	1	yes	characteristic
168	#388	—	2	2	no	—
169	#382	Dic₇₉	1	1	yes	characteristic
170	C₄×C₂²⋊C₄	—	2	2	no	—
171	C₄³	C₂×D₇₉	1	1	yes	characteristic
172	Q₈⋊C₈	—	158	4	no	—
173	#323	—	158	2	no	—
174	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
175	C₂²×D₄	C₂×Dic₇₉	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₇₉	1	2	yes	—
177	C₂²×C₈	—	158	4	no	—
178	#124	—	2	2	no	—
179	#124	C₇₉⋊C₈	1	4	yes	complemented
180	C₄×Q₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	C₂²⋊D₇₉	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
194	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
196	C₄⋊C₈	—	79	4	no	—
197	C₄⋊C₈	—	158	2	no	—
198	C₂²⋊C₈	—	158	2	no	—
199	C₂²⋊C₈	—	158	4	no	—
200	C₂²⋊C₈	—	158	2	no	—
201	C₄×C₈	—	158	2	no	—
202	C₄×C₈	—	158	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂²⋊Dic₇₉	1	2	yes	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂×C₇₉⋊C₈	1	2	yes	complemented
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂×C₇₉⋊C₈	1	2	yes	—
213	C₂×D₄	#11929608	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
225	C₂×C₈	—	316	2	no	—
226	C₂×C₈	—	632	4	no	—
227	C₂×C₈	—	158	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	2	8	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	C₂×C₇₉⋊C₈	1	2	yes	complemented
241	C₄²	D₄×D₇₉	1	2	yes	—
242	C₄²	#11929623	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	C₂²⋊(C₇₉⋊C₈)	1	4	yes	—
248	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	4	2	no	—
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
271	C₂×C₄	#14579129	1	1	yes	characteristic
272	C₂×C₄	#14579133	1	2	yes	—
273	C₂×C₄	#14579142	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
275	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
276	C₂×C₄	#14579168	1	2	yes	—
277	C₈	—	316	4	no	—
278	C₈	—	632	2	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	#19098443	1	1	yes	characteristic
287	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
288	C₂²	#19098530	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	1	no	—
296	C₄	D₄×C₇₉⋊C₈	1	2	yes	—
297	C₄	#19098635	1	2	yes	—
298	C₄	#19098740	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	4	no	—
302	C₂	#23763693	1	1	yes	characteristic
303	C₂	#23763784	1	1	yes	characteristic
304	C₂	#23763789	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#124×C₇₉	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂²×D₄×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#124	D₇₉
hypocenter	C₇₉	#8957
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
2. #23761795	C₂	semidirect
4. #23761233	C₂	semidirect
33. #124×C₇₉	C₈	semidirect
59. C₂×Q₈×C₇₉	C₂×C₈	semidirect
87. C₄²×C₇₉	C₂×C₈	semidirect
158. C₇₉	#8957	semidirect
179. #124	C₇₉⋊C₈	semidirect
209. C₂×Q₈	C₂×C₇₉⋊C₈	semidirect
240. C₄²	C₂×C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#23772424	2
#23761795	2
#23761721	2
#23761233	2
C₄×#124×C₇₉	2
#8957	79