Subgroups — #34206491

Order 20224.

514 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098948	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098365	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098363	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763563	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#358×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11929611	—	2	1	no	—
9	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
10	C₂×#14579150	—	2	1	no	—
11	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#19098948	—	2	2	no	—
13	#19098948	C₄	1	4	yes	complemented
14	C₂×#14579095	C₂²	1	1	yes	characteristic
15	C₂×#14579092	—	2	1	no	—
16	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
17	#19098428	—	2	1	no	—
18	#19098367	—	2	2	no	—
19	#19098365	—	2	2	no	—
20	#19098363	—	2	2	no	—
21	C₁₆×Dic₇₉	—	2	2	no	—
22	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
25	#358×C₇₉	—	2	2	no	—
26	C₂²×D₃₁₆	—	2	1	no	—
27	C₂×#11929611	—	2	4	no	—
28	C₂×#11929611	—	4	2	no	—
29	C₂×#11929611	—	4	2	no	—
30	C₂×D₆₃₂	—	2	1	no	—
31	C₂×D₆₃₂	—	2	1	no	—
32	C₂×D₆₃₂	—	2	1	no	—
33	C₂×D₆₃₂	—	2	1	no	—
34	C₂×D₆₃₂	—	2	1	no	—
35	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
36	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
37	#14579150	—	2	4	no	—
38	#14579150	—	4	2	no	—
39	C₂×C₄⋊Dic₇₉	—	2	1	no	—
40	C₂×(C₂×D₇₉)⋊C₄	—	4	2	no	—
41	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
42	C₂×C₇₉⋊C₁₆	—	2	1	no	—
43	C₂×C₇₉⋊C₁₆	—	2	1	no	—
44	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
45	#14579095	—	2	1	no	—
46	#14579095	—	2	1	no	—
47	#14579092	—	4	2	no	—
48	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
49	C₈×Dic₇₉	D₄	1	2	yes	—
50	C₇₉⋊#95	—	4	2	no	—
51	C₂²×Q₈×C₇₉	—	2	1	no	—
52	C₂×Q₁₆×C₇₉	—	2	1	no	—
53	C₂×Q₁₆×C₇₉	—	2	1	no	—
54	C₂×Q₁₆×C₇₉	—	2	1	no	—
55	C₂×Q₁₆×C₇₉	—	2	1	no	—
56	C₂×Q₁₆×C₇₉	—	2	1	no	—
57	C₂×Q₁₆×C₇₉	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
60	C₂×C₁₆×C₇₉	—	2	1	no	—
61	C₂×C₁₆×C₇₉	—	2	1	no	—
62	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
63	#107×C₇₉	—	2	1	no	—
64	#107×C₇₉	—	2	1	no	—
65	C₂³×D₇₉	—	4	1	no	—
66	C₂×D₃₁₆	—	2	2	no	—
67	C₂×D₃₁₆	—	4	1	no	—
68	C₂×D₃₁₆	—	4	1	no	—
69	C₂×D₃₁₆	—	4	1	no	—
70	C₂×D₃₁₆	—	4	1	no	—
71	C₂×D₃₁₆	—	4	1	no	—
72	C₂²×Dic₇₉	—	4	1	no	—
73	C₂²×Dic₇₉	D₈	1	2	yes	—
74	#11929611	—	4	8	no	—
75	D₆₃₂	—	2	2	no	—
76	D₆₃₂	—	2	2	no	—
77	D₆₃₂	—	4	1	no	—
78	D₆₃₂	—	4	1	no	—
79	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
81	C₂×C₇₉⋊C₈	SD₁₆	1	2	yes	—
82	C₄⋊Dic₇₉	—	4	1	no	—
83	C₄⋊Dic₇₉	—	4	1	no	—
84	(C₂×D₇₉)⋊C₄	—	4	8	no	—
85	C₄×Dic₇₉	C₂²⋊C₄	1	2	yes	—
86	C₄×Dic₇₉	SD₁₆	1	2	yes	—
87	C₇₉⋊C₁₆	—	2	2	no	—
88	C₂×Q₈×C₇₉	—	2	2	no	—
89	C₂×Q₈×C₇₉	—	4	1	no	—
90	C₂×Q₈×C₇₉	—	4	1	no	—
91	C₂×Q₈×C₇₉	—	4	1	no	—
92	C₂×Q₈×C₇₉	—	4	1	no	—
93	C₂×Q₈×C₇₉	—	4	1	no	—
94	C₂²×C₄×C₇₉	—	4	1	no	—
95	C₂²×C₄×C₇₉	—	4	1	no	—
96	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
97	Q₁₆×C₇₉	—	2	2	no	—
98	Q₁₆×C₇₉	—	2	2	no	—
99	Q₁₆×C₇₉	—	4	1	no	—
100	Q₁₆×C₇₉	—	4	1	no	—
101	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
106	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
107	C₄⋊C₄×C₇₉	—	4	1	no	—
108	C₄⋊C₄×C₇₉	—	4	1	no	—
109	C₁₆×C₇₉	—	2	2	no	—
110	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
111	C₂²×D₇₉	—	4	2	no	—
112	C₂²×D₇₉	—	8	1	no	—
113	C₂²×D₇₉	—	8	1	no	—
114	C₂²×D₇₉	—	8	2	no	—
115	C₂²×D₇₉	—	8	2	no	—
116	Q₈×C₇₉	—	4	2	no	—
117	Q₈×C₇₉	—	4	2	no	—
118	Q₈×C₇₉	—	8	1	no	—
119	Q₈×C₇₉	—	8	1	no	—
120	C₂×C₄×C₇₉	—	4	2	no	—
121	C₂×C₄×C₇₉	—	8	1	no	—
122	C₂×C₄×C₇₉	—	8	1	no	—
123	C₂×C₄×C₇₉	—	8	1	no	—
124	C₂×C₄×C₇₉	—	8	2	no	—
125	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
127	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×Dic₇₉	—	2	2	no	—
132	C₂×Dic₇₉	—	2	2	no	—
133	C₂×Dic₇₉	—	8	1	no	—
134	C₂×Dic₇₉	—	8	2	no	—
135	C₂×Dic₇₉	#102	1	4	yes	—
136	D₃₁₆	—	4	2	no	—
137	D₃₁₆	—	4	2	no	—
138	D₃₁₆	—	8	1	no	—
139	D₃₁₆	—	8	1	no	—
140	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
141	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
142	C₈×C₇₉	#121	1	1	yes	characteristic
143	C₈×C₇₉	#121	1	1	yes	characteristic
144	C₇₉⋊C₈	#102	1	4	yes	—
145	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
146	C₂²×C₇₉	#486	1	1	yes	characteristic
147	C₂²×C₇₉	#493	1	1	yes	characteristic
148	C₂²×C₇₉	#508	1	2	yes	—
149	C₂²×C₇₉	#508	1	2	yes	—
150	C₂×D₇₉	—	8	4	no	—
151	C₂×D₇₉	—	8	2	no	—
152	C₂×D₇₉	—	16	2	no	—
153	C₂×D₇₉	—	16	1	no	—
154	C₂×D₇₉	—	16	1	no	—
155	C₄×C₇₉	—	8	2	no	—
156	C₄×C₇₉	—	16	1	no	—
157	C₄×C₇₉	#390	1	1	yes	characteristic
158	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
159	C₄×C₇₉	#486	1	1	yes	characteristic
160	C₄×C₇₉	#492	1	1	yes	characteristic
161	Dic₇₉	—	2	4	no	—
162	Dic₇₉	—	16	1	no	—
163	#13204	—	79	1	no	maximal, Sylow
164	C₂×C₇₉	#1957	1	1	yes	characteristic
165	C₂×C₇₉	#2138	1	2	yes	—
166	C₂×C₇₉	#2242	1	2	yes	—
167	C₂×C₇₉	#2249	1	1	yes	characteristic
168	C₂×C₇₉	#2249	1	1	yes	characteristic
169	D₇₉	—	16	2	no	—
170	C₂×#495	—	79	1	no	—
171	C₂×#358	—	79	1	no	—
172	C₂×#358	D₇₉	1	1	yes	characteristic, complemented
173	C₂×#357	—	79	1	no	—
174	C₂×#357	—	79	1	no	—
175	C₂×C₄×C₁₆	—	79	1	no	—
176	#1846	—	79	1	no	—
177	C₇₉	#13204	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
179	C₂²×SD₁₆	—	158	1	no	—
180	C₂²×D₈	—	79	1	no	—
181	C₂×#124	—	158	1	no	—
182	C₂²×C₁₆	—	79	1	no	—
183	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
184	#495	—	79	4	no	—
185	#495	—	158	2	no	—
186	C₂×#107	—	79	1	no	—
187	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
188	C₂×#106	—	158	1	no	—
189	C₂×C₄×C₈	—	79	1	no	—
190	#382	—	158	1	no	—
191	#358	—	2	2	no	—
192	#358	—	158	2	no	—
193	#357	—	158	2	no	—
194	#357	—	158	2	no	—
195	C₄×C₁₆	—	158	2	no	—
196	C₂²×Q₈	—	2	1	no	—
197	C₂²×D₄	—	158	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₄×D₇₉	1	2	yes	—
204	C₂×SD₁₆	—	316	2	no	—
205	C₂×SD₁₆	—	316	2	no	—
206	C₂×SD₁₆	—	158	4	no	—
207	C₂×D₈	—	79	2	no	—
208	C₂×D₈	—	158	1	no	—
209	C₂×D₈	—	158	1	no	—
210	C₂×D₈	—	158	1	no	—
211	C₂×D₈	—	158	1	no	—
212	C₂×D₈	—	158	1	no	—
213	C₂²×C₈	—	79	1	no	—
214	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
215	#124	—	316	2	no	—
216	#124	—	158	4	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	158	1	no	—
219	C₂×C₂²⋊C₄	—	316	2	no	—
220	C₂×C₄²	—	79	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	79	2	no	—
224	C₂×C₁₆	—	158	1	no	—
225	C₂×C₁₆	—	158	1	no	—
226	C₂×C₁₆	C₂²⋊D₇₉	1	2	yes	—
227	#107	—	2	1	no	—
228	#107	—	2	1	no	—
229	#107	—	158	1	no	—
230	#107	—	158	1	no	—
231	#106	—	316	2	no	—
232	C₄×C₈	—	79	2	no	—
233	C₄×C₈	—	79	2	no	—
234	#95	—	316	2	no	—
235	C₂⁴	—	316	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×D₄	—	316	1	no	—
243	C₂×D₄	—	316	1	no	—
244	C₂×D₄	—	316	1	no	—
245	C₂×D₄	—	316	1	no	—
246	C₂×D₄	—	316	1	no	—
247	C₂×D₄	—	158	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	316	1	no	—
251	C₂²×C₄	—	79	2	no	—
252	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	4	1	no	—
256	Q₁₆	—	4	1	no	—
257	SD₁₆	—	316	8	no	—
258	D₈	—	316	1	no	—
259	D₈	—	316	1	no	—
260	D₈	—	158	2	no	—
261	D₈	—	158	2	no	—
262	C₂×C₈	—	79	2	no	—
263	C₂×C₈	—	79	2	no	—
264	C₂×C₈	—	79	2	no	—
265	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
266	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
267	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
269	C₂×C₈	#11929623	1	1	yes	characteristic
270	C₂×C₈	#11929624	1	1	yes	characteristic
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	316	1	no	—
274	C₄⋊C₄	—	316	1	no	—
275	C₂²⋊C₄	—	316	8	no	—
276	C₄²	—	79	2	no	—
277	C₄²	—	79	2	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	158	2	no	—
280	C₂³	—	316	2	no	—
281	C₂³	—	632	1	no	—
282	C₂³	—	632	1	no	—
283	C₂³	—	632	2	no	—
284	C₂³	—	632	2	no	—
285	C₂³	D₈×D₇₉	1	1	yes	characteristic
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	1	no	—
289	Q₈	—	8	1	no	—
290	D₄	—	316	2	no	—
291	D₄	—	316	2	no	—
292	D₄	—	632	1	no	—
293	D₄	—	632	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	79	4	no	—
300	C₂×C₄	—	632	1	no	—
301	C₂×C₄	—	632	2	no	—
302	C₂×C₄	—	158	2	no	—
303	C₂×C₄	—	158	2	no	—
304	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
305	C₂×C₄	#14579147	1	1	yes	characteristic
306	C₂×C₄	#14579157	1	1	yes	characteristic
307	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
308	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
309	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
310	C₈	—	79	4	no	—
311	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
312	C₈	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
313	C₈	#14579133	1	1	yes	characteristic
314	C₈	#14579134	1	1	yes	characteristic
315	C₂²	—	632	4	no	—
316	C₂²	—	632	2	no	—
317	C₂²	—	1264	2	no	—
318	C₂²	—	1264	1	no	—
319	C₂²	—	1264	1	no	—
320	C₂²	#102×D₇₉	1	1	yes	characteristic
321	C₂²	#19098908	1	1	yes	characteristic
322	C₂²	#19098938	1	1	yes	characteristic
323	C₂²	#19098994	1	2	yes	—
324	C₂²	#19098996	1	2	yes	—
325	C₄	—	8	2	no	—
326	C₄	—	16	1	no	—
327	C₄	—	158	4	no	—
328	C₄	—	1264	1	no	—
329	C₄	#19098463	1	1	yes	characteristic
330	C₄	#102×D₇₉	1	1	yes	characteristic
331	C₄	#19098908	1	1	yes	characteristic
332	C₄	#19098934	1	1	yes	characteristic
333	C₂	—	1264	2	no	—
334	C₂	#23764204	1	1	yes	characteristic
335	C₂	#23765106	1	2	yes	—
336	C₂	#23765633	1	2	yes	—
337	C₂	#23765660	1	1	yes	characteristic
338	C₂	#23765662	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#358×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₆₃₂	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#358	D₇₉
hypocenter	C₇₉	#13204
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
2. C₂×#19098367	C₂	semidirect
5. C₂×C₁₆×Dic₇₉	C₂	semidirect
6. #23763563	C₂	semidirect
7. C₂×#358×C₇₉	C₂	semidirect
13. #19098948	C₄	semidirect
172. C₂×#358	D₇₉	semidirect
177. C₇₉	#13204	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098948	2
C₂×#19098367	2
C₂×#19098365	2
C₂×#19098363	2
C₂×C₁₆×Dic₇₉	2
#23763563	2
C₂×#358×C₇₉	2
#13204	79