Subgroups — #34181472

Order 20224.

434 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772111	C₂	1	1	yes	maximal, characteristic
2	#23763562	C₂	1	1	yes	maximal, characteristic, complemented
3	#23762146	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761961	C₂	1	1	yes	maximal, characteristic
5	#23761959	C₂	1	1	yes	maximal, characteristic
6	#23761260	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1286	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579169	—	2	1	no	—
9	#19099113	C₂²	1	1	yes	characteristic
10	#19098640	—	2	1	no	—
11	#19098640	C₄	1	2	yes	complemented
12	#19098628	—	2	1	no	—
13	#19098628	C₄	1	2	yes	complemented
14	#19098616	C₂²	1	1	yes	characteristic
15	C₂×#14579094	—	2	1	no	—
16	C₂×#14579092	—	2	1	no	—
17	#19098531	—	2	1	no	—
18	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
19	#19098429	C₂²	1	1	yes	characteristic
20	#19098315	—	2	1	no	—
21	#19098313	—	2	1	no	—
22	#19098298	—	2	2	no	—
23	#19098296	—	2	2	no	—
24	#431×C₇₉	C₂²	1	1	yes	characteristic
25	#423×C₇₉	C₂²	1	1	yes	characteristic
26	#382×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#11929624	—	2	1	no	—
28	#14579169	—	4	2	no	—
29	#14579169	—	4	2	no	—
30	C₂×#11929607	—	2	2	no	—
31	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₈×D₇₉	—	2	2	no	—
34	#14579161	Q₈	1	1	yes	characteristic
35	#14579158	Q₈	1	1	yes	characteristic
36	#14579149	—	2	1	no	—
37	#14579135	—	2	1	no	—
38	#14579135	D₄	1	2	yes	—
39	C₄×D₃₁₆	—	2	1	no	—
40	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊Dic₇₉	—	2	1	no	—
42	#14579094	—	4	2	no	—
43	#14579092	—	4	2	no	—
44	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
45	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
46	(C₂×D₇₉)⋊C₈	—	2	2	no	—
47	(C₂×D₇₉)⋊C₈	—	2	2	no	—
48	C₄×C₇₉⋊C₈	—	2	1	no	—
49	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
52	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
53	C₇₉⋊#95	—	2	1	no	—
54	C₇₉⋊#95	—	2	1	no	—
55	C₂×M₁₆×C₇₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
57	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₇₉	—	2	1	no	—
59	C₄⋊C₈×C₇₉	—	2	1	no	—
60	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₇₉	—	2	1	no	—
62	C₄×C₈×C₇₉	—	2	1	no	—
63	#95×C₇₉	—	2	1	no	—
64	#95×C₇₉	—	2	1	no	—
65	#11929624	—	4	1	no	—
66	#11929624	—	4	1	no	—
67	#11929624	—	4	1	no	—
68	#11929624	—	4	1	no	—
69	C₂×Dic₁₅₈	—	2	1	no	—
70	C₂×D₃₁₆	—	2	1	no	—
71	C₂×D₃₁₆	—	2	1	no	—
72	C₂×D₃₁₆	C₄⋊C₄	1	2	yes	—
73	C₂×C₂²⋊D₇₉	—	2	2	no	—
74	C₂²×Dic₇₉	—	4	1	no	—
75	C₂×C₄×D₇₉	—	2	2	no	—
76	#11929607	—	4	4	no	—
77	C₂×C₇₉⋊C₈	—	2	1	no	—
78	C₂×C₇₉⋊C₈	—	2	1	no	—
79	C₂×C₇₉⋊C₈	—	4	1	no	—
80	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
85	C₂×C₇₉⋊C₈	SD₁₆	1	2	yes	—
86	C₈×D₇₉	—	4	4	no	—
87	C₄⋊Dic₇₉	—	2	1	no	—
88	C₄⋊Dic₇₉	—	4	1	no	—
89	C₄⋊Dic₇₉	—	4	1	no	—
90	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
91	(C₂×D₇₉)⋊C₄	—	2	2	no	—
92	(C₂×D₇₉)⋊C₄	—	2	2	no	—
93	C₂²×C₄×C₇₉	—	4	1	no	—
94	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
97	M₁₆×C₇₉	—	4	1	no	—
98	M₁₆×C₇₉	—	4	1	no	—
99	C₂×C₈×C₇₉	—	2	2	no	—
100	C₂×C₈×C₇₉	—	4	1	no	—
101	C₄⋊C₄×C₇₉	—	4	1	no	—
102	C₄⋊C₄×C₇₉	—	4	1	no	—
103	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₇₉	#41	1	1	yes	characteristic
107	C₂³×C₇₉	#121	1	1	yes	characteristic
108	C₂²×D₇₉	—	2	2	no	—
109	C₂×C₄×C₇₉	—	4	1	no	—
110	C₂×C₄×C₇₉	—	4	1	no	—
111	C₂×C₄×C₇₉	—	8	1	no	—
112	C₂×C₄×C₇₉	—	8	1	no	—
113	C₂×C₄×C₇₉	—	8	1	no	—
114	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
124	C₂²⋊D₇₉	—	4	2	no	—
125	C₂²⋊D₇₉	—	4	2	no	—
126	C₂×Dic₇₉	—	2	2	no	—
127	C₂×Dic₇₉	—	8	1	no	—
128	C₂×Dic₇₉	—	8	1	no	—
129	C₂×Dic₇₉	—	8	1	no	—
130	D₃₁₆	—	4	2	no	—
131	D₃₁₆	—	4	2	no	—
132	D₃₁₆	—	4	1	no	—
133	D₃₁₆	—	4	1	no	—
134	C₄×D₇₉	—	4	2	no	—
135	C₄×D₇₉	—	4	2	no	—
136	Dic₁₅₈	—	4	1	no	—
137	Dic₁₅₈	—	4	1	no	—
138	C₈×C₇₉	—	4	2	no	—
139	C₈×C₇₉	—	8	1	no	—
140	C₇₉⋊C₈	—	2	1	no	—
141	C₇₉⋊C₈	—	2	1	no	—
142	C₇₉⋊C₈	—	8	1	no	—
143	C₇₉⋊C₈	#102	1	2	yes	—
144	C₇₉⋊C₈	#102	1	2	yes	—
145	C₂²×C₇₉	—	2	1	no	—
146	C₂²×C₇₉	—	2	1	no	—
147	C₂²×C₇₉	#380	1	1	yes	characteristic
148	C₂²×C₇₉	#459	1	1	yes	characteristic
149	C₂²×C₇₉	#461	1	1	yes	characteristic
150	C₂×D₇₉	—	4	2	no	—
151	C₂×D₇₉	—	4	2	no	—
152	C₂×D₇₉	—	4	2	no	—
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	—	2	1	no	—
156	C₄×C₇₉	—	2	1	no	—
157	C₄×C₇₉	—	16	1	no	—
158	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
159	C₄×C₇₉	#417	1	1	yes	characteristic
160	C₄×C₇₉	#474	1	1	yes	characteristic
161	C₄×C₇₉	#476	1	1	yes	characteristic
162	Dic₇₉	—	4	2	no	—
163	Dic₇₉	—	16	1	no	—
164	#8674	—	79	1	no	maximal, Sylow
165	C₂×C₇₉	—	2	1	no	—
166	C₂×C₇₉	—	2	1	no	—
167	C₂×C₇₉	#1802	1	1	yes	characteristic
168	C₂×C₇₉	#1812	1	1	yes	characteristic
169	C₂×C₇₉	#1814	1	1	yes	characteristic
170	D₇₉	—	8	2	no	—
171	#2931	—	79	1	no	—
172	#1846	—	79	1	no	—
173	#1564	—	79	1	no	—
174	#1517	—	79	1	no	—
175	#1517	—	79	1	no	—
176	#1286	—	79	1	no	—
177	#1286	D₇₉	1	1	yes	characteristic, complemented
178	C₇₉	#8674	1	1	yes	characteristic, complemented, Sylow
179	C₂×#131	—	158	1	no	—
180	#532	—	79	1	no	—
181	#436	—	79	2	no	—
182	#436	—	158	1	no	—
183	C₈×D₄	—	79	2	no	—
184	C₈×D₄	—	158	1	no	—
185	#431	—	79	1	no	—
186	#431	C₂×D₇₉	1	1	yes	characteristic
187	C₂×#107	—	158	1	no	—
188	C₂×#106	—	158	1	no	—
189	#423	C₂×D₇₉	1	1	yes	characteristic
190	#408	—	158	1	no	—
191	C₂×C₄×C₈	—	79	1	no	—
192	#382	—	79	1	no	—
193	#382	C₂×D₇₉	1	1	yes	characteristic
194	#335	—	158	1	no	—
195	#334	—	158	1	no	—
196	#327	—	158	2	no	—
197	#327	—	158	2	no	—
198	C₂×#41	—	158	1	no	—
199	#131	—	316	2	no	—
200	#131	—	316	2	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	158	2	no	—
203	C₂²×C₈	—	79	1	no	—
204	C₂²×C₈	—	79	1	no	—
205	C₂²×C₈	—	158	2	no	—
206	C₄⋊Q₈	—	79	1	no	—
207	#127	—	79	1	no	—
208	#124	—	158	1	no	—
209	#121	—	79	2	no	—
210	#121	—	158	1	no	—
211	C₄×D₄	—	79	2	no	—
212	C₄×D₄	—	158	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	158	1	no	—
215	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
216	#107	—	316	2	no	—
217	#106	—	316	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	158	1	no	—
221	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	158	2	no	—
224	C₂²⋊C₈	—	158	2	no	—
225	C₂²⋊C₈	—	158	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	79	1	no	—
228	C₄×C₈	—	79	1	no	—
229	C₄×C₈	—	79	1	no	—
230	C₄×C₈	—	79	1	no	—
231	C₄×C₈	—	158	1	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	#95	—	158	1	no	—
235	#95	—	158	1	no	—
236	#41	—	316	1	no	—
237	#41	—	316	1	no	—
238	#41	—	316	1	no	—
239	#41	—	316	1	no	—
240	C₂×Q₈	—	158	1	no	—
241	C₂×D₄	—	79	2	no	—
242	C₂×D₄	—	158	2	no	—
243	C₂×D₄	—	158	1	no	—
244	C₂×D₄	—	158	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	316	1	no	—
247	C₂²×C₄	—	158	2	no	—
248	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
250	C₂²×C₄	#11929624	1	1	yes	characteristic
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	316	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	316	4	no	—
257	C₂×C₈	—	316	1	no	—
258	C₂×C₈	—	79	1	no	—
259	C₂×C₈	—	79	1	no	—
260	C₂×C₈	—	79	2	no	—
261	C₂×C₈	—	79	2	no	—
262	C₂×C₈	—	79	1	no	—
263	C₂×C₈	—	79	1	no	—
264	C₂×C₈	—	158	1	no	—
265	C₂×C₈	—	158	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	316	1	no	—
269	C₄⋊C₄	—	316	1	no	—
270	C₄⋊C₄	—	79	2	no	—
271	C₄⋊C₄	—	158	1	no	—
272	C₂²⋊C₄	—	158	2	no	—
273	C₂²⋊C₄	—	158	2	no	—
274	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
275	C₄²	D₄×D₇₉	1	1	yes	characteristic
276	C₄²	Q₈×D₇₉	1	1	yes	characteristic
277	C₄²	#11929624	1	1	yes	characteristic
278	C₂³	—	158	2	no	—
279	C₂³	#14579134	1	1	yes	characteristic
280	Q₈	—	316	1	no	—
281	Q₈	—	316	1	no	—
282	D₄	—	316	2	no	—
283	D₄	—	316	2	no	—
284	D₄	—	316	2	no	—
285	D₄	—	316	2	no	—
286	D₄	—	316	1	no	—
287	D₄	—	316	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	316	2	no	—
294	C₂×C₄	—	316	2	no	—
295	C₂×C₄	—	632	1	no	—
296	C₂×C₄	—	632	1	no	—
297	C₂×C₄	—	632	1	no	—
298	C₂×C₄	—	158	2	no	—
299	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
300	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
301	C₂×C₄	#14579118	1	1	yes	characteristic
302	C₂×C₄	#14579134	1	1	yes	characteristic
303	C₂×C₄	#14579139	1	1	yes	characteristic
304	C₂×C₄	#14579139	1	1	yes	characteristic
305	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
307	C₂×C₄	#14579188	1	1	yes	characteristic
308	C₂×C₄	#14579193	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	8	1	no	—
311	C₈	—	79	2	no	—
312	C₈	—	79	2	no	—
313	C₈	—	632	1	no	—
314	C₈	—	158	1	no	—
315	C₈	—	158	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	316	2	no	—
319	C₂²	—	316	2	no	—
320	C₂²	—	316	2	no	—
321	C₂²	#19098422	1	1	yes	characteristic
322	C₂²	#19098757	1	1	yes	characteristic
323	C₂²	#19098767	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	16	1	no	—
329	C₄	—	316	2	no	—
330	C₄	—	1264	1	no	—
331	C₄	#102×D₇₉	1	1	yes	characteristic
332	C₄	#19098563	1	1	yes	characteristic
333	C₄	#19098842	1	1	yes	characteristic
334	C₄	#19098852	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	632	2	no	—
338	C₂	#23763399	1	1	yes	characteristic
339	C₂	#23763445	1	1	yes	characteristic
340	C₂	#23763453	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098422
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1286	C₂
socle	C₂²×C₇₉	#380
prime radical	#19099113	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1286	D₇₉
hypocenter	C₇₉	#8674
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1286

Complements

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

N	H ≅ G/N	type
2. #23763562	C₂	semidirect
3. #23762146	C₂	semidirect
6. #23761260	C₂	semidirect
7. C₇₉×#1286	C₂	semidirect
11. #19098640	C₄	semidirect
13. #19098628	C₄	semidirect
177. #1286	D₇₉	semidirect
178. C₇₉	#8674	semidirect

Maximal subgroups

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

subgroup	index
#23772111	2
#23763562	2
#23762146	2
#23761961	2
#23761959	2
#23761260	2
C₇₉×#1286	2
#8674	79