Subgroups — #34204309

Order 20224.

510 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098645	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₉⋊C₁₆	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	#23763558	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
9	C₂×#14579123	—	2	1	no	—
10	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#19098645	—	2	2	no	—
12	#19098645	C₄	1	4	yes	complemented
13	C₂×#14579095	C₂²	1	1	yes	characteristic
14	C₂×#14579094	—	2	1	no	—
15	C₂×#14579081	—	2	1	no	—
16	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
17	#19098432	—	2	1	no	—
18	Dic₇₉⋊C₁₆	—	2	2	no	—
19	#19098365	—	2	2	no	—
20	#19098363	—	2	2	no	—
21	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
24	C₂²×D₃₁₆	—	2	1	no	—
25	C₂²×C₄×D₇₉	—	4	1	no	—
26	C₂×D₆₃₂	—	2	1	no	—
27	C₂×D₆₃₂	—	2	1	no	—
28	C₂×D₆₃₂	—	2	1	no	—
29	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
30	C₂×D₆₃₂	D₄	1	2	yes	—
31	C₂×D₆₃₂	Q₈	1	2	yes	—
32	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
33	#14579123	—	2	4	no	—
34	#14579123	—	4	2	no	—
35	C₂×C₄⋊Dic₇₉	—	2	1	no	—
36	C₂×Dic₇₉⋊C₄	—	4	1	no	—
37	C₂×Dic₇₉⋊C₄	—	4	1	no	—
38	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
39	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
40	C₂×C₇₉⋊C₁₆	—	2	1	no	—
41	C₂×C₇₉⋊C₁₆	—	2	1	no	—
42	C₂×C₇₉⋊C₁₆	—	2	1	no	—
43	#14579095	—	2	1	no	—
44	#14579095	—	2	1	no	—
45	#14579094	—	4	2	no	—
46	#14579081	—	2	4	no	—
47	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
48	C₈×Dic₇₉	D₄	1	2	yes	—
49	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
52	C₂×C₁₆×C₇₉	—	2	1	no	—
53	C₂×C₁₆×C₇₉	—	2	1	no	—
54	C₂×C₁₆×C₇₉	—	2	1	no	—
55	#107×C₇₉	—	2	1	no	—
56	#107×C₇₉	—	2	1	no	—
57	#107×C₇₉	—	2	1	no	—
58	#107×C₇₉	C₂×C₄	1	2	yes	—
59	C₂³×D₇₉	—	4	1	no	—
60	C₂×D₃₁₆	—	2	2	no	—
61	C₂×D₃₁₆	—	2	2	no	—
62	C₂×D₃₁₆	—	2	2	no	—
63	C₂×D₃₁₆	—	4	1	no	—
64	C₂×D₃₁₆	—	4	1	no	—
65	C₂×D₃₁₆	—	4	1	no	—
66	C₂²×Dic₇₉	—	4	1	no	—
67	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
68	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
69	C₂×C₄×D₇₉	—	4	4	no	—
70	C₂×C₄×D₇₉	—	8	2	no	—
71	C₂×C₄×D₇₉	—	8	2	no	—
72	D₆₃₂	—	2	2	no	—
73	D₆₃₂	—	2	2	no	—
74	D₆₃₂	—	2	2	no	—
75	D₆₃₂	C₄⋊C₄	1	4	yes	complemented
76	C₂×C₇₉⋊C₈	—	2	1	no	—
77	C₂×C₇₉⋊C₈	—	2	1	no	—
78	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
79	C₄⋊Dic₇₉	—	4	1	no	—
80	C₄⋊Dic₇₉	—	4	1	no	—
81	Dic₇₉⋊C₄	—	8	2	no	—
82	Dic₇₉⋊C₄	—	8	2	no	—
83	(C₂×D₇₉)⋊C₄	—	4	4	no	—
84	C₄×Dic₇₉	C₂²⋊C₄	1	2	yes	—
85	C₄×Dic₇₉	SD₁₆	1	2	yes	—
86	C₇₉⋊C₁₆	—	4	1	no	—
87	C₂²×C₄×C₇₉	—	4	1	no	—
88	C₂²×C₄×C₇₉	—	4	1	no	—
89	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₇₉	—	2	2	no	—
97	C₄⋊C₄×C₇₉	—	4	1	no	—
98	C₄⋊C₄×C₇₉	—	4	1	no	—
99	C₄⋊C₄×C₇₉	—	4	1	no	—
100	C₁₆×C₇₉	—	4	1	no	—
101	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₇₉	—	4	2	no	—
103	C₂²×D₇₉	—	4	2	no	—
104	C₂²×D₇₉	—	4	2	no	—
105	C₂²×D₇₉	—	8	1	no	—
106	C₂²×D₇₉	—	8	1	no	—
107	C₂²×D₇₉	—	8	1	no	—
108	C₂²×D₇₉	—	8	1	no	—
109	C₂×C₄×C₇₉	—	4	2	no	—
110	C₂×C₄×C₇₉	—	8	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	—	8	1	no	—
115	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
121	C₂×Dic₇₉	—	2	1	no	—
122	C₂×Dic₇₉	—	2	1	no	—
123	C₂×Dic₇₉	—	8	1	no	—
124	C₂×Dic₇₉	—	8	1	no	—
125	C₂×Dic₇₉	—	8	1	no	—
126	C₂×Dic₇₉	#102	1	2	yes	—
127	C₂×Dic₇₉	#102	1	2	yes	—
128	C₂×Dic₇₉	D₁₆	1	2	yes	—
129	C₂×Dic₇₉	SD₃₂	1	2	yes	—
130	D₃₁₆	—	2	4	no	—
131	D₃₁₆	—	4	2	no	—
132	D₃₁₆	—	4	2	no	—
133	D₃₁₆	—	4	2	no	—
134	C₄×D₇₉	—	8	8	no	—
135	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₇₉	#117	1	1	yes	characteristic
137	C₈×C₇₉	#121	1	1	yes	characteristic
138	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
139	C₇₉⋊C₈	—	2	2	no	—
140	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
141	C₂²×C₇₉	#459	1	1	yes	characteristic
142	C₂²×C₇₉	#474	1	1	yes	characteristic
143	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
144	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
145	C₂²×C₇₉	#509	1	1	yes	characteristic
146	C₂²×C₇₉	#510	1	1	yes	characteristic
147	C₂×D₇₉	—	4	4	no	—
148	C₂×D₇₉	—	8	2	no	—
149	C₂×D₇₉	—	8	2	no	—
150	C₂×D₇₉	—	8	2	no	—
151	C₂×D₇₉	—	8	2	no	—
152	C₂×D₇₉	—	8	2	no	—
153	C₂×D₇₉	—	8	2	no	—
154	C₄×C₇₉	—	8	2	no	—
155	C₄×C₇₉	—	16	1	no	—
156	C₄×C₇₉	#380	1	1	yes	characteristic
157	C₄×C₇₉	#417	1	1	yes	characteristic
158	C₄×C₇₉	#461	1	1	yes	characteristic
159	C₄×C₇₉	#476	1	1	yes	characteristic
160	Dic₇₉	—	2	2	no	—
161	Dic₇₉	—	16	1	no	—
162	Dic₇₉	#357	1	4	yes	complemented
163	#12811	—	79	1	no	maximal, Sylow
164	C₂×C₇₉	#1802	1	1	yes	characteristic
165	C₂×C₇₉	C₂×#357	1	1	yes	characteristic
166	C₂×C₇₉	#2139	1	1	yes	characteristic
167	C₂×C₇₉	#2206	1	1	yes	characteristic
168	C₂×C₇₉	#2208	1	1	yes	characteristic
169	C₂×C₇₉	#2224	1	1	yes	characteristic
170	C₂×C₇₉	#2226	1	1	yes	characteristic
171	D₇₉	—	8	4	no	—
172	C₂×C₄×D₈	—	79	1	no	—
173	C₂×C₄⋊C₁₆	—	79	1	no	—
174	C₂×#357	—	79	1	no	—
175	C₂×#357	—	79	1	no	—
176	#1845	—	79	1	no	—
177	#1380	—	79	1	no	—
178	#1380	D₇₉	1	1	yes	characteristic, complemented
179	C₇₉	#12811	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₈	—	79	1	no	—
181	C₂×C₄×D₄	—	158	1	no	—
182	C₂²×C₁₆	—	79	1	no	—
183	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
184	C₄×D₈	—	79	4	no	—
185	C₄×D₈	—	158	2	no	—
186	C₂×#107	—	79	1	no	—
187	C₂×#107	—	158	1	no	—
188	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
189	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
190	C₂×#102	—	158	1	no	—
191	C₂×C₄×C₈	—	79	1	no	—
192	#384	—	158	1	no	—
193	C₄⋊C₁₆	—	158	2	no	—
194	#357	—	158	2	no	—
195	#357	—	158	2	no	—
196	C₂²×D₄	—	158	1	no	—
197	C₂³×C₄	—	316	1	no	—
198	C₂×D₈	—	79	2	no	—
199	C₂×D₈	—	79	2	no	—
200	C₂×D₈	—	79	2	no	—
201	C₂×D₈	—	158	1	no	—
202	C₂×D₈	—	158	1	no	—
203	C₂×D₈	—	158	1	no	—
204	C₂²×C₈	—	79	1	no	—
205	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
206	C₄×D₄	—	316	2	no	—
207	C₄×D₄	—	158	4	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	316	1	no	—
211	C₂×C₄⋊C₄	—	316	1	no	—
212	C₂×C₄⋊C₄	—	158	1	no	—
213	C₂×C₂²⋊C₄	—	316	1	no	—
214	C₂×C₄²	—	79	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	158	1	no	—
219	C₂×C₁₆	—	158	1	no	—
220	C₂×C₁₆	—	158	1	no	—
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	—	316	2	no	—
225	#107	—	158	1	no	—
226	#107	—	158	1	no	—
227	#107	C₄×D₇₉	1	2	yes	—
228	#102	—	158	4	no	—
229	C₄×C₈	—	79	2	no	—
230	C₄×C₈	—	79	2	no	—
231	C₂⁴	—	316	1	no	—
232	C₂×D₄	—	316	1	no	—
233	C₂×D₄	—	316	1	no	—
234	C₂×D₄	—	316	1	no	—
235	C₂×D₄	—	158	2	no	—
236	C₂×D₄	—	158	2	no	—
237	C₂×D₄	—	158	2	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	316	4	no	—
241	C₂²×C₄	—	316	1	no	—
242	C₂²×C₄	—	79	1	no	—
243	C₂²×C₄	—	79	1	no	—
244	C₂²×C₄	—	632	2	no	—
245	C₂²×C₄	—	632	2	no	—
246	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
247	D₈	—	79	4	no	—
248	D₈	—	158	2	no	—
249	D₈	—	158	2	no	—
250	D₈	—	158	2	no	—
251	C₂×C₈	—	79	2	no	—
252	C₂×C₈	—	158	1	no	—
253	C₂×C₈	—	158	1	no	—
254	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
255	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
256	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
257	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
258	C₂×C₈	#11929624	1	1	yes	characteristic
259	C₂×C₈	#11929624	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	316	1	no	—
265	C₄⋊C₄	—	316	1	no	—
266	C₄⋊C₄	—	632	2	no	—
267	C₄⋊C₄	—	632	2	no	—
268	C₂²⋊C₄	—	316	4	no	—
269	C₄²	—	79	2	no	—
270	C₄²	—	79	2	no	—
271	C₁₆	—	4	1	no	—
272	C₁₆	—	316	1	no	—
273	C₂³	—	316	2	no	—
274	C₂³	—	316	2	no	—
275	C₂³	—	316	2	no	—
276	C₂³	—	632	1	no	—
277	C₂³	—	632	1	no	—
278	C₂³	—	632	1	no	—
279	C₂³	—	632	1	no	—
280	C₂³	D₈×D₇₉	1	1	yes	characteristic
281	D₄	—	316	2	no	—
282	D₄	—	316	2	no	—
283	D₄	—	316	2	no	—
284	D₄	—	158	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	79	2	no	—
292	C₂×C₄	—	79	2	no	—
293	C₂×C₄	—	79	2	no	—
294	C₂×C₄	—	79	2	no	—
295	C₂×C₄	—	632	8	no	—
296	C₂×C₄	—	632	1	no	—
297	C₂×C₄	—	632	1	no	—
298	C₂×C₄	—	632	1	no	—
299	C₂×C₄	—	158	1	no	—
300	C₂×C₄	—	158	1	no	—
301	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
302	C₂×C₄	#14579134	1	1	yes	characteristic
303	C₂×C₄	#14579139	1	1	yes	characteristic
304	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
305	C₂×C₄	#14579188	1	1	yes	characteristic
306	C₂×C₄	#14579193	1	1	yes	characteristic
307	C₈	—	158	2	no	—
308	C₈	C₄⋊C₄×D₇₉	1	1	yes	characteristic
309	C₈	#14579118	1	1	yes	characteristic
310	C₈	#14579134	1	1	yes	characteristic
311	C₈	#14579139	1	1	yes	characteristic
312	C₂²	—	316	4	no	—
313	C₂²	—	632	2	no	—
314	C₂²	—	632	2	no	—
315	C₂²	—	632	2	no	—
316	C₂²	—	632	2	no	—
317	C₂²	—	632	2	no	—
318	C₂²	—	632	2	no	—
319	C₂²	#102×D₇₉	1	1	yes	characteristic
320	C₂²	#19098757	1	1	yes	characteristic
321	C₂²	#19098842	1	1	yes	characteristic
322	C₂²	D₁₆×D₇₉	1	1	yes	characteristic
323	C₂²	SD₃₂×D₇₉	1	1	yes	characteristic
324	C₂²	#19099000	1	1	yes	characteristic
325	C₂²	#19099005	1	1	yes	characteristic
326	C₄	—	8	2	no	—
327	C₄	—	16	1	no	—
328	C₄	—	79	4	no	—
329	C₄	—	158	2	no	—
330	C₄	—	1264	1	no	—
331	C₄	#19098422	1	1	yes	characteristic
332	C₄	#19098563	1	1	yes	characteristic
333	C₄	#19098767	1	1	yes	characteristic
334	C₄	#19098852	1	1	yes	characteristic
335	C₂	—	632	4	no	—
336	C₂	#23763399	1	1	yes	characteristic
337	C₂	#357×D₇₉	1	1	yes	characteristic
338	C₂	#23765111	1	1	yes	characteristic
339	C₂	#23765434	1	1	yes	characteristic
340	C₂	#23765444	1	1	yes	characteristic
341	C₂	#23765536	1	1	yes	characteristic
342	C₂	#23765546	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1380	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₆₃₂	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1380	D₇₉
hypocenter	C₇₉	#12811
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₉⋊C₁₆	C₂	semidirect
5. #23763558	C₂	semidirect
6. #23761491	C₂	semidirect
7. C₇₉×#1380	C₂	semidirect
12. #19098645	C₄	semidirect
75. D₆₃₂	C₄⋊C₄	semidirect
162. Dic₇₉	#357	semidirect
178. #1380	D₇₉	semidirect
179. C₇₉	#12811	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098645	2
C₂×Dic₇₉⋊C₁₆	2
C₂×#19098365	2
C₂×#19098363	2
#23763558	2
#23761491	2
C₇₉×#1380	2
#12811	79