Subgroups — #34279797

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776268	C₂	1	1	yes	maximal, characteristic, complemented
2	#23775748	C₂	1	1	yes	maximal, characteristic
3	#23775718	C₂	1	2	yes	maximal, complemented
4	#23763010	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762976	C₂	1	2	yes	maximal, complemented
6	#23762799	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762794	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762616	C₂	1	2	yes	maximal, complemented
9	#23762553	C₂	1	2	yes	maximal, complemented
10	#23762286	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099273	C₂²	1	2	yes	—
13	#19099217	C₂²	1	2	yes	—
14	#19098968	C₂²	1	1	yes	characteristic
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098965	—	2	1	no	—
17	#19098914	C₂²	1	2	yes	—
18	#19098914	C₂²	1	2	yes	—
19	#19098895	—	2	2	no	—
20	#19098877	—	2	2	no	—
21	#19098864	—	2	2	no	—
22	#19098854	C₂²	1	2	yes	—
23	#19098853	C₂²	1	2	yes	—
24	#19098844	C₂²	1	2	yes	—
25	#19098820	—	2	2	no	—
26	#19098799	—	2	1	no	—
27	#19098773	C₂²	1	2	yes	—
28	#19098722	—	2	2	no	—
29	C₄×D₆₃₂	—	2	1	no	—
30	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
31	#19098642	C₂²	1	2	yes	—
32	C₂×#14579095	—	2	1	no	—
33	#19098561	—	2	2	no	—
34	#19098316	C₂²	1	1	yes	characteristic
35	#19098311	C₂²	1	2	yes	—
36	#19098308	C₂²	1	2	yes	—
37	#19098292	C₂²	1	2	yes	—
38	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
41	#334×C₇₉	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
44	C₈²×C₇₉	C₂²	1	1	yes	characteristic
45	C₂×#11929623	—	2	2	no	—
46	C₂×#11929614	—	2	2	no	—
47	C₂×D₆₃₂	D₄	1	1	yes	characteristic
48	C₂×#11929607	—	2	2	no	—
49	#14579161	C₂³	1	2	yes	—
50	#14579161	C₂³	1	2	yes	—
51	#14579153	—	2	2	no	—
52	#14579145	—	4	2	no	—
53	#14579141	—	2	2	no	—
54	#14579139	—	4	2	no	—
55	#14579137	—	2	2	no	—
56	#14579135	—	2	2	no	—
57	C₄×D₃₁₆	—	2	2	no	—
58	C₄×D₃₁₆	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₇₉	—	2	2	no	—
60	C₄⋊C₄×D₇₉	—	2	2	no	—
61	#14579095	—	2	1	no	—
62	#14579095	—	2	1	no	—
63	#14579095	—	2	1	no	—
64	#14579095	—	2	1	no	—
65	#14579095	—	2	2	no	—
66	#14579095	D₄	1	1	yes	characteristic
67	#14579093	—	2	2	no	—
68	#14579092	—	2	2	no	—
69	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
70	#14579086	—	2	2	no	—
71	#14579086	—	2	2	no	—
72	#14579086	—	4	2	no	—
73	#14579084	—	2	2	no	—
74	#14579083	—	2	2	no	—
75	#14579083	—	2	2	no	—
76	#14579083	D₄	1	2	yes	—
77	#14579082	—	2	2	no	—
78	(C₂×D₇₉)⋊C₈	—	2	2	no	—
79	C₂×SD₁₆×C₇₉	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₇₉	—	2	1	no	—
81	C₄×Q₈×C₇₉	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₇₉	C₂³	1	1	yes	characteristic
83	#106×C₇₉	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₇₉	—	2	1	no	—
85	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
87	#103×C₇₉	D₄	1	1	yes	characteristic
88	#102×C₇₉	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₇₉	—	2	1	no	—
90	C₄×C₈×C₇₉	—	2	1	no	—
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
94	#11929623	—	4	4	no	—
95	#11929623	—	8	2	no	—
96	C₂×Dic₁₅₈	—	2	2	no	—
97	C₂×Dic₁₅₈	—	2	2	no	—
98	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
99	C₂×C₂²⋊D₇₉	—	4	2	no	—
100	C₂²×Dic₇₉	—	4	2	no	—
101	C₂×C₄×D₇₉	—	2	2	no	—
102	C₂×C₄×D₇₉	—	4	2	no	—
103	#11929614	—	8	2	no	—
104	D₆₃₂	—	2	2	no	—
105	#11929607	—	8	2	no	—
106	C₂×C₇₉⋊C₈	—	2	2	no	—
107	C₄⋊Dic₇₉	—	2	2	no	—
108	C₄⋊Dic₇₉	—	2	2	no	—
109	C₄⋊Dic₇₉	—	2	2	no	—
110	C₄⋊Dic₇₉	—	2	2	no	—
111	C₄⋊Dic₇₉	—	4	2	no	—
112	C₄⋊Dic₇₉	C₂×D₄	1	2	yes	—
113	Dic₇₉⋊C₄	—	4	2	no	—
114	Dic₇₉⋊C₄	—	4	2	no	—
115	(C₂×D₇₉)⋊C₄	—	2	2	no	—
116	(C₂×D₇₉)⋊C₄	—	4	2	no	—
117	(C₂×D₇₉)⋊C₄	—	4	2	no	—
118	C₂×Q₈×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×D₄×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₇₉	—	2	1	no	—
121	SD₁₆×C₇₉	—	2	2	no	—
122	C₂×C₈×C₇₉	—	2	1	no	—
123	C₂×C₈×C₇₉	—	2	1	no	—
124	C₂×C₈×C₇₉	—	2	1	no	—
125	C₂×C₈×C₇₉	—	4	2	no	—
126	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₇₉	—	2	1	no	—
131	C₄⋊C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
133	C₂²⋊C₄×C₇₉	—	2	1	no	—
134	C₄²×C₇₉	—	2	1	no	—
135	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₇₉	—	2	1	no	—
137	C₂²×D₇₉	—	2	2	no	—
138	Q₈×C₇₉	—	2	1	no	—
139	Q₈×C₇₉	—	2	1	no	—
140	D₄×C₇₉	—	2	1	no	—
141	D₄×C₇₉	#135	1	2	yes	—
142	C₂×C₄×C₇₉	—	2	1	no	—
143	C₂×C₄×C₇₉	—	2	1	no	—
144	C₂×C₄×C₇₉	—	2	1	no	—
145	C₂×C₄×C₇₉	—	4	1	no	—
146	C₂×C₄×C₇₉	—	4	1	no	—
147	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²⋊D₇₉	—	8	4	no	—
151	C₂×Dic₇₉	—	2	2	no	—
152	C₂×Dic₇₉	—	4	2	no	—
153	C₂×Dic₇₉	—	4	2	no	—
154	C₂×Dic₇₉	—	8	4	no	—
155	D₃₁₆	—	2	2	no	—
156	D₃₁₆	—	2	2	no	—
157	C₄×D₇₉	—	4	2	no	—
158	C₄×D₇₉	—	4	2	no	—
159	C₄×D₇₉	—	16	2	no	—
160	Dic₁₅₈	—	4	2	no	—
161	Dic₁₅₈	—	4	2	no	—
162	Dic₁₅₈	—	8	2	no	—
163	C₈×C₇₉	—	2	1	no	—
164	C₈×C₇₉	—	2	1	no	—
165	C₈×C₇₉	—	2	1	no	—
166	C₈×C₇₉	—	4	1	no	—
167	C₈×C₇₉	—	4	1	no	—
168	C₈×C₇₉	—	4	1	no	—
169	C₈×C₇₉	#137	1	2	yes	—
170	C₇₉⋊C₈	—	4	2	no	—
171	C₂²×C₇₉	—	2	2	no	—
172	C₂²×C₇₉	—	4	1	no	—
173	C₂²×C₇₉	—	4	1	no	—
174	C₂²×C₇₉	D₄²	1	1	yes	characteristic
175	C₂×D₇₉	—	4	2	no	—
176	C₂×D₇₉	—	4	2	no	—
177	C₂×D₇₉	—	4	2	no	—
178	C₄×C₇₉	—	2	1	no	—
179	C₄×C₇₉	—	4	1	no	—
180	C₄×C₇₉	—	4	1	no	—
181	C₄×C₇₉	—	4	1	no	—
182	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
183	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
184	C₄×C₇₉	#576	1	1	yes	characteristic
185	C₄×C₇₉	#578	1	1	yes	characteristic
186	Dic₇₉	—	4	2	no	—
187	Dic₇₉	—	8	2	no	—
188	Dic₇₉	—	8	2	no	—
189	#19903	—	79	1	no	maximal, Sylow
190	C₂×C₇₉	—	4	2	no	—
191	C₂×C₇₉	#3276	1	1	yes	characteristic
192	C₂×C₇₉	#3280	1	1	yes	characteristic
193	C₂×C₇₉	#3290	1	1	yes	characteristic
194	D₇₉	—	8	2	no	—
195	#3380	—	79	1	no	—
196	#3334	—	79	1	no	—
197	#3331	—	79	2	no	—
198	#1710	—	79	1	no	—
199	#1703	—	79	2	no	—
200	#1675	—	79	1	no	—
201	#1674	—	79	1	no	—
202	#1644	—	79	2	no	—
203	#1635	—	79	2	no	—
204	#1589	—	79	1	no	—
205	C₈×SD₁₆	D₇₉	1	1	yes	characteristic, complemented
206	C₇₉	#19903	1	1	yes	characteristic, complemented, Sylow
207	#554	—	79	2	no	—
208	#547	—	79	2	no	—
209	#500	—	79	1	no	—
210	#500	—	79	1	no	—
211	#499	—	158	1	no	—
212	#487	—	79	2	no	—
213	#487	—	79	2	no	—
214	#483	—	158	2	no	—
215	#480	—	158	2	no	—
216	#478	—	158	2	no	—
217	#476	—	79	2	no	—
218	#476	—	79	2	no	—
219	#474	—	79	2	no	—
220	#470	—	158	2	no	—
221	#466	—	158	1	no	—
222	#462	—	79	2	no	—
223	#452	—	158	2	no	—
224	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
225	C₄×SD₁₆	C₂×D₇₉	1	1	yes	characteristic
226	C₄×D₈	—	79	1	no	—
227	C₄×D₈	—	158	1	no	—
228	#436	—	79	2	no	—
229	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
230	C₂×#107	—	158	1	no	—
231	#417	—	158	2	no	—
232	#335	—	79	1	no	—
233	#334	C₂×D₇₉	1	1	yes	characteristic
234	#333	—	79	2	no	—
235	#332	—	79	2	no	—
236	Q₈⋊C₈	C₂×D₇₉	1	1	yes	characteristic
237	D₄⋊C₈	—	79	2	no	—
238	D₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
239	C₈²	C₂×D₇₉	1	1	yes	characteristic
240	C₂×#41	—	158	2	no	—
241	C₂×Q₁₆	—	158	2	no	—
242	C₂×SD₁₆	D₃₁₆	1	1	yes	characteristic
243	C₂×D₈	—	79	1	no	—
244	C₂×M₁₆	—	158	2	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	79	2	no	—
247	C₄⋊Q₈	—	79	2	no	—
248	#125	—	158	2	no	—
249	#123	—	316	2	no	—
250	C₂²⋊Q₈	—	316	2	no	—
251	C₂²⋊Q₈	—	158	2	no	—
252	C₂²⋊Q₈	—	158	2	no	—
253	#121	—	158	2	no	—
254	C₄×Q₈	C₂²×D₇₉	1	1	yes	characteristic
255	C₄×D₄	—	79	2	no	—
256	C₄×D₄	—	158	2	no	—
257	C₄×D₄	C₂²×D₇₉	1	1	yes	characteristic
258	C₂×C₄⋊C₄	—	158	2	no	—
259	C₂×C₄⋊C₄	—	158	2	no	—
260	#107	—	79	1	no	—
261	#107	—	158	2	no	—
262	#107	—	158	1	no	—
263	#107	—	158	1	no	—
264	#107	—	158	1	no	—
265	#107	—	158	1	no	—
266	#106	—	158	2	no	—
267	#106	—	158	2	no	—
268	#106	D₃₁₆	1	1	yes	characteristic
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	79	2	no	—
271	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
272	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
273	#103	—	316	2	no	—
274	#103	—	158	2	no	—
275	#103	—	158	2	no	—
276	#103	—	158	2	no	—
277	#103	D₃₁₆	1	1	yes	characteristic
278	#102	—	79	2	no	—
279	#102	—	158	2	no	—
280	#102	—	158	2	no	—
281	#102	—	158	2	no	—
282	#102	D₃₁₆	1	1	yes	characteristic
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	158	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
289	#41	—	316	4	no	—
290	#41	—	632	2	no	—
291	C₂×Q₈	—	158	2	no	—
292	C₂×Q₈	—	158	2	no	—
293	C₂×Q₈	C₂×D₃₁₆	1	1	yes	characteristic
294	C₂×D₄	—	316	2	no	—
295	C₂×D₄	—	79	2	no	—
296	C₂×D₄	C₂×D₃₁₆	1	1	yes	characteristic
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	316	2	no	—
299	C₂²×C₄	—	316	2	no	—
300	C₂²×C₄	—	158	2	no	—
301	Q₁₆	—	632	2	no	—
302	SD₁₆	—	2	2	no	—
303	D₈	—	158	2	no	—
304	M₁₆	—	632	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	158	2	no	—
310	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
311	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
312	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
313	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	316	2	no	—
316	C₄⋊C₄	—	316	2	no	—
317	C₄⋊C₄	—	316	2	no	—
318	C₄⋊C₄	—	79	2	no	—
319	C₄⋊C₄	—	158	2	no	—
320	C₄⋊C₄	—	158	2	no	—
321	C₄⋊C₄	—	158	2	no	—
322	C₄⋊C₄	—	158	2	no	—
323	C₄⋊C₄	C₂×D₃₁₆	1	1	yes	characteristic
324	C₄⋊C₄	C₂×D₃₁₆	1	1	yes	characteristic
325	C₂²⋊C₄	—	2	1	no	—
326	C₂²⋊C₄	—	316	2	no	—
327	C₂²⋊C₄	—	316	2	no	—
328	C₂²⋊C₄	—	158	2	no	—
329	C₄²	—	2	1	no	—
330	C₄²	C₂³×D₇₉	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	158	2	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	2	1	no	—
335	Q₈	—	316	2	no	—
336	Q₈	—	316	2	no	—
337	Q₈	—	632	2	no	—
338	D₄	—	2	1	no	—
339	D₄	—	632	4	no	—
340	D₄	—	158	2	no	—
341	D₄	—	158	2	no	—
342	D₄	#14579183	1	2	yes	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	316	2	no	—
349	C₂×C₄	—	316	2	no	—
350	C₂×C₄	—	316	2	no	—
351	C₂×C₄	—	316	2	no	—
352	C₂×C₄	—	632	4	no	—
353	C₂×C₄	—	158	2	no	—
354	C₂×C₄	—	1264	2	no	—
355	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
357	C₂×C₄	#14579207	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	316	2	no	—
365	C₈	#14579190	1	2	yes	—
366	C₂²	—	2	2	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	316	2	no	—
370	C₂²	—	316	2	no	—
371	C₂²	—	316	2	no	—
372	C₂²	D₄×D₃₁₆	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	4	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	316	2	no	—
378	C₄	—	632	2	no	—
379	C₄	—	632	2	no	—
380	C₄	C₂×#14579183	1	1	yes	characteristic
381	C₄	C₂×#14579190	1	1	yes	characteristic
382	C₄	#19099368	1	1	yes	characteristic
383	C₄	#19099377	1	1	yes	characteristic
384	C₂	—	4	2	no	—
385	C₂	—	632	2	no	—
386	C₂	#23775047	1	1	yes	characteristic
387	C₂	#23775087	1	1	yes	characteristic
388	C₂	#23775187	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₁₆
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₈×SD₁₆×C₇₉	C₂
socle	C₂²×C₇₉	D₄²
prime radical	#23775748	C₂
hypercenter	C₈×SD₁₆	D₇₉
hypocenter	C₇₉	#19903
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #23776268	C₂	semidirect
3. #23775718	C₂	semidirect
4. #23763010	C₂	semidirect
5. #23762976	C₂	semidirect
6. #23762799	C₂	semidirect
7. #23762794	C₂	semidirect
8. #23762616	C₂	semidirect
9. #23762553	C₂	semidirect
10. #23762286	C₂	semidirect
11. C₈×SD₁₆×C₇₉	C₂	semidirect
205. C₈×SD₁₆	D₇₉	semidirect
206. C₇₉	#19903	semidirect

Maximal subgroups

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

subgroup	index
#23776268	2
#23775748	2
#23775718	2
#23763010	2
#23762976	2
#23762799	2
#23762794	2
#23762616	2
#23762553	2
#23762286	2
C₈×SD₁₆×C₇₉	2
#19903	79