Subgroups — #34181128

Order 20224.

462 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098519	C₂	1	1	yes	maximal, characteristic
2	#23762146	C₂	1	2	yes	maximal, complemented
3	#23761714	C₂	1	1	yes	maximal, characteristic
4	#23761230	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₄	1	1	yes	characteristic
7	C₂³×C₇₉⋊C₈	C₄	1	1	yes	characteristic
8	#19098616	C₂²	1	2	yes	—
9	#19098519	—	2	2	no	—
10	#19098519	—	2	2	no	—
11	C₂×#14579072	C₄	1	1	yes	characteristic
12	C₂×#14579072	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	#19098315	—	2	2	no	—
16	#19098313	—	2	2	no	—
17	#19098289	—	2	1	no	—
18	#19098287	—	2	1	no	—
19	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
20	#431×C₇₉	C₂²	1	2	yes	—
21	#324×C₇₉	—	2	1	no	—
22	#323×C₇₉	—	2	1	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	—	2	2	no	—
25	C₂×#11929608	—	2	2	no	—
26	C₂²×C₇₉⋊C₈	—	2	1	no	—
27	C₂²×C₇₉⋊C₈	—	2	1	no	—
28	C₂²×C₇₉⋊C₈	—	2	2	no	—
29	C₂²×C₇₉⋊C₈	C₈	1	4	yes	complemented
30	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
35	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
36	#14579072	—	2	2	no	—
37	#14579072	—	2	2	no	—
38	C₄×C₇₉⋊C₈	—	2	1	no	—
39	C₄×C₇₉⋊C₈	—	2	1	no	—
40	C₄×C₇₉⋊C₈	—	2	2	no	—
41	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
42	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
43	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
44	#117×C₇₉	—	2	2	no	—
45	#117×C₇₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₉	Q₈	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₇₉	D₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₇₉	Q₈	1	1	yes	characteristic
50	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₇₉	—	2	2	no	—
53	C₂²⋊C₈×C₇₉	—	2	2	no	—
54	C₄×C₈×C₇₉	—	2	2	no	—
55	#11929608	—	2	8	no	—
56	C₂×C₇₉⋊C₈	—	2	4	no	—
57	C₂×C₇₉⋊C₈	—	2	4	no	—
58	C₂×C₇₉⋊C₈	—	2	1	no	—
59	C₂×C₇₉⋊C₈	—	2	1	no	—
60	C₂×C₇₉⋊C₈	—	2	1	no	—
61	C₂×C₇₉⋊C₈	—	2	1	no	—
62	C₂×C₇₉⋊C₈	—	2	1	no	—
63	C₂×C₇₉⋊C₈	—	2	1	no	—
64	C₂×C₇₉⋊C₈	—	4	2	no	—
65	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
69	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
70	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
71	C₂×C₇₉⋊C₈	M₁₆	1	2	yes	—
72	C₂⁴×C₇₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	—	2	1	no	—
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	—	2	1	no	—
77	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
82	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₇₉	—	4	2	no	—
86	C₄⋊C₄×C₇₉	—	2	2	no	—
87	C₄⋊C₄×C₇₉	—	2	2	no	—
88	C₂²⋊C₄×C₇₉	—	2	2	no	—
89	C₂²⋊C₄×C₇₉	—	2	2	no	—
90	C₄²×C₇₉	—	2	1	no	—
91	C₄²×C₇₉	—	2	1	no	—
92	C₄²×C₇₉	C₂×D₄	1	2	yes	—
93	C₄²×C₇₉	#41	1	2	yes	—
94	C₂³×C₇₉	—	2	1	no	—
95	C₂³×C₇₉	—	2	1	no	—
96	C₂³×C₇₉	—	4	1	no	—
97	C₂³×C₇₉	—	4	1	no	—
98	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
99	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	—	2	1	no	—
101	C₂×C₄×C₇₉	—	2	1	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	—	2	2	no	—
106	C₂×C₄×C₇₉	—	2	1	no	—
107	C₂×C₄×C₇₉	—	2	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	C₄⋊C₈	1	2	yes	—
113	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₇₉	#121	1	2	yes	—
116	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
121	C₈×C₇₉	—	8	2	no	—
122	C₇₉⋊C₈	—	2	2	no	—
123	C₇₉⋊C₈	—	2	2	no	—
124	C₇₉⋊C₈	—	2	2	no	—
125	C₇₉⋊C₈	—	8	2	no	—
126	C₇₉⋊C₈	C₂²⋊C₈	1	2	yes	—
127	C₇₉⋊C₈	C₂²⋊C₈	1	2	yes	—
128	C₂²×C₇₉	—	2	1	no	—
129	C₂²×C₇₉	—	2	1	no	—
130	C₂²×C₇₉	—	2	2	no	—
131	C₂²×C₇₉	—	4	2	no	—
132	C₂²×C₇₉	—	4	2	no	—
133	C₂²×C₇₉	—	4	1	no	—
134	C₂²×C₇₉	—	4	1	no	—
135	C₂²×C₇₉	#380	1	1	yes	characteristic
136	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₂²×C₇₉	#423	1	1	yes	characteristic
138	C₄×C₇₉	—	2	2	no	—
139	C₄×C₇₉	—	2	2	no	—
140	C₄×C₇₉	—	2	2	no	—
141	C₄×C₇₉	—	4	1	no	—
142	C₄×C₇₉	—	4	1	no	—
143	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₄×C₇₉	#407	1	1	yes	characteristic
145	C₄×C₇₉	#431	1	1	yes	characteristic
146	C₄×C₇₉	#432	1	1	yes	characteristic
147	#8601	—	79	1	no	maximal, Sylow
148	C₂×C₇₉	—	2	1	no	—
149	C₂×C₇₉	—	2	1	no	—
150	C₂×C₇₉	—	4	2	no	—
151	C₂×C₇₉	#1797	1	1	yes	characteristic
152	C₂×C₇₉	#1812	1	1	yes	characteristic
153	C₂×C₇₉	#1813	1	1	yes	characteristic
154	C₂×#405	—	79	1	no	—
155	#1564	—	79	2	no	—
156	#1458	—	79	1	no	—
157	#1458	D₇₉	1	1	yes	characteristic
158	#1275	—	79	2	no	—
159	C₇₉	#8601	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	79	1	no	—
161	C₂³×C₈	—	79	1	no	—
162	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
163	#431	—	79	2	no	—
164	#431	C₂×D₇₉	1	2	yes	—
165	#405	—	158	2	no	—
166	#405	—	158	2	no	—
167	C₂×#97	—	79	1	no	—
168	C₂×#97	—	79	1	no	—
169	C₂×C₄×C₈	—	79	1	no	—
170	C₂×C₄×C₈	—	79	1	no	—
171	#335	—	158	2	no	—
172	#334	—	158	2	no	—
173	#324	—	2	1	no	—
174	#324	—	158	1	no	—
175	#323	—	2	1	no	—
176	#323	—	158	1	no	—
177	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
178	C₂×M₁₆	—	158	2	no	—
179	C₂×M₁₆	—	158	2	no	—
180	C₂×M₁₆	—	158	2	no	—
181	C₂²×C₈	—	79	1	no	—
182	C₂²×C₈	—	79	4	no	—
183	C₂²×C₈	—	79	1	no	—
184	C₂²×C₈	—	79	1	no	—
185	C₂²×C₈	—	79	1	no	—
186	C₂²×C₈	—	158	1	no	—
187	C₂²×C₈	—	158	1	no	—
188	C₂²×C₈	—	158	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	Dic₁₅₈	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	Dic₁₅₈	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	158	2	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	158	2	no	—
201	#97	—	158	2	no	—
202	#97	—	158	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	—	79	2	no	—
205	C₄×C₈	—	79	2	no	—
206	C₄×C₈	—	158	1	no	—
207	C₄×C₈	—	158	1	no	—
208	C₄×C₈	—	158	2	no	—
209	C₂⁴	Dic₇₉⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	C₈×D₇₉	1	2	yes	—
218	C₂²×C₄	#11929607	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
222	M₁₆	—	158	8	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	316	2	no	—
225	C₂×C₈	—	79	1	no	—
226	C₂×C₈	—	79	1	no	—
227	C₂×C₈	—	79	2	no	—
228	C₂×C₈	—	79	2	no	—
229	C₂×C₈	—	79	2	no	—
230	C₂×C₈	—	79	2	no	—
231	C₂×C₈	—	79	2	no	—
232	C₂×C₈	—	158	4	no	—
233	C₂×C₈	—	158	4	no	—
234	C₂×C₈	—	158	1	no	—
235	C₂×C₈	—	158	1	no	—
236	C₂×C₈	—	158	1	no	—
237	C₂×C₈	—	158	1	no	—
238	C₂×C₈	—	158	1	no	—
239	C₂×C₈	—	158	1	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	D₄×D₇₉	1	2	yes	—
247	C₄²	#11929623	1	2	yes	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	Dic₇₉⋊C₈	1	2	yes	—
253	C₂³	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	Dic₇₉⋊C₈	1	2	yes	—
267	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
268	C₂×C₄	#14579117	1	1	yes	characteristic
269	C₂×C₄	#14579133	1	2	yes	—
270	C₂×C₄	#14579138	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
273	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
274	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
275	C₈	—	8	2	no	—
276	C₈	—	79	2	no	—
277	C₈	—	79	2	no	—
278	C₈	—	632	2	no	—
279	C₈	—	158	2	no	—
280	C₈	—	158	2	no	—
281	C₈	—	158	2	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	#19098421	1	1	yes	characteristic
290	C₂²	C₂×Dic₇₉⋊C₈	1	1	yes	characteristic
291	C₂²	#19098586	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
298	C₄	#19098525	1	1	yes	characteristic
299	C₄	#19098613	1	1	yes	characteristic
300	C₄	#19098617	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	2	no	—
304	C₂	#23763378	1	1	yes	characteristic
305	C₂	#23763444	1	1	yes	characteristic
306	C₂	#23763448	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579117
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1458	C₂
socle	C₂²×C₇₉	#380
prime radical	C₂⁴×C₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1458	D₇₉
hypocenter	C₇₉	#8601
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
2. #23762146	C₂	semidirect
4. #23761230	C₂	semidirect
29. C₂²×C₇₉⋊C₈	C₈	semidirect
159. C₇₉	#8601	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098519	2
#23762146	2
#23761714	2
#23761230	2
C₇₉×#1458	2
#8601	79