Subgroups — #34442909

Order 20224.

546 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	C₄×#14579183	C₂	1	1	yes	maximal, characteristic
2	#23765561	C₂	1	2	yes	maximal, complemented
3	#23765554	C₂	1	2	yes	maximal, complemented
4	#23765549	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765544	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765539	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765534	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765182	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765105	C₂	1	2	yes	maximal
11	C₇₉×#3160	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579183	C₂²	1	1	yes	characteristic
13	C₄×#11929624	—	2	1	no	—
14	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
15	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
16	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
17	C₄×#11929613	—	2	1	no	—
18	C₄×#11929613	—	2	1	no	—
19	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
20	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
21	#19098601	C₂²	1	1	yes	characteristic
22	#19098560	—	2	1	no	—
23	#19098390	C₂²	1	2	yes	—
24	#19098390	C₂²	1	2	yes	—
25	#19098388	C₂²	1	2	yes	—
26	#19098388	C₂²	1	2	yes	—
27	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
28	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
30	#19098366	C₂²	1	2	yes	—
31	#19098366	C₂²	1	2	yes	—
32	#19098363	C₂²	1	2	yes	—
33	#19098363	C₂²	1	2	yes	—
34	#533×C₇₉	—	2	1	no	—
35	#500×C₇₉	C₂²	1	1	yes	characteristic
36	#500×C₇₉	C₂²	1	1	yes	characteristic
37	#499×C₇₉	—	2	1	no	—
38	#499×C₇₉	C₂²	1	2	yes	—
39	#498×C₇₉	—	2	1	no	—
40	#427×C₇₉	—	2	1	no	—
41	#427×C₇₉	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×#11929624	—	2	1	no	—
44	#14579183	—	2	2	no	—
45	#14579183	Q₈	1	4	yes	complemented
46	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
47	C₂×#11929613	—	2	1	no	—
48	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
49	C₄×Dic₁₅₈	—	2	1	no	—
50	C₄×Dic₁₅₈	—	2	1	no	—
51	C₄×D₃₁₆	—	2	1	no	—
52	C₄×D₃₁₆	—	2	1	no	—
53	C₄×C₂²⋊D₇₉	—	4	1	no	—
54	C₄×C₂²⋊D₇₉	—	4	1	no	—
55	#14579120	—	2	1	no	—
56	#14579118	—	4	1	no	—
57	C₄²×D₇₉	—	4	1	no	—
58	C₂×C₇₉⋊C₁₆	—	2	4	no	—
59	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
60	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
61	#14579095	C₂³	1	1	yes	characteristic
62	#14579095	C₂³	1	1	yes	characteristic
63	#14579093	—	2	1	no	—
64	#14579086	—	2	1	no	—
65	#14579086	—	2	1	no	—
66	#14579083	—	2	1	no	—
67	#14579083	—	2	1	no	—
68	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₇₉	—	2	1	no	—
71	C₄⋊Q₈×C₇₉	—	2	1	no	—
72	#125×C₇₉	—	2	2	no	—
73	C₂²⋊Q₈×C₇₉	—	4	1	no	—
74	C₂²⋊Q₈×C₇₉	—	4	1	no	—
75	C₄×Q₈×C₇₉	—	4	1	no	—
76	C₄×Q₈×C₇₉	—	4	1	no	—
77	#117×C₇₉	—	2	2	no	—
78	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
79	#107×C₇₉	—	2	1	no	—
80	#107×C₇₉	—	2	1	no	—
81	#107×C₇₉	C₂³	1	2	yes	—
82	#107×C₇₉	C₂³	1	2	yes	—
83	#106×C₇₉	—	2	2	no	—
84	#106×C₇₉	—	2	2	no	—
85	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
87	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
89	#11929624	—	2	4	no	—
90	#11929624	—	4	2	no	—
91	C₂×Dic₁₅₈	—	2	1	no	—
92	C₂×D₃₁₆	—	2	1	no	—
93	C₂×C₂²⋊D₇₉	—	4	1	no	—
94	C₂×C₄×D₇₉	—	4	1	no	—
95	C₂×C₄×D₇₉	—	4	1	no	—
96	C₂×C₄×D₇₉	—	4	1	no	—
97	Dic₃₁₆	—	2	1	no	—
98	Dic₃₁₆	C₂×Q₈	1	2	yes	—
99	#11929613	—	2	4	no	—
100	D₆₃₂	—	2	1	no	—
101	D₆₃₂	C₂×Q₈	1	2	yes	—
102	C₄⋊Dic₇₉	—	2	1	no	—
103	C₄⋊Dic₇₉	—	2	1	no	—
104	Dic₇₉⋊C₄	—	4	1	no	—
105	Dic₇₉⋊C₄	—	4	1	no	—
106	C₂²⋊Dic₇₉	—	4	1	no	—
107	(C₂×D₇₉)⋊C₄	—	4	1	no	—
108	(C₂×D₇₉)⋊C₄	—	4	1	no	—
109	C₄×Dic₇₉	—	4	1	no	—
110	C₄×Dic₇₉	—	4	1	no	—
111	C₄×Dic₇₉	—	4	1	no	—
112	C₇₉⋊C₁₆	—	2	2	no	—
113	C₇₉⋊C₁₆	—	2	2	no	—
114	C₂×Q₈×C₇₉	—	4	1	no	—
115	C₂×Q₈×C₇₉	—	4	1	no	—
116	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
117	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
118	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	—	2	2	no	—
120	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
124	C₂×C₈×C₇₉	#41	1	2	yes	—
125	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
126	C₄⋊C₄×C₇₉	—	2	2	no	—
127	C₄⋊C₄×C₇₉	—	2	2	no	—
128	C₄⋊C₄×C₇₉	—	4	2	no	—
129	C₄⋊C₄×C₇₉	—	4	2	no	—
130	C₄⋊C₄×C₇₉	—	4	1	no	—
131	C₄⋊C₄×C₇₉	—	4	1	no	—
132	C₂²⋊C₄×C₇₉	—	4	2	no	—
133	C₄²×C₇₉	—	4	2	no	—
134	C₄²×C₇₉	D₈	1	1	yes	characteristic
135	C₄²×C₇₉	D₈	1	1	yes	characteristic
136	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
138	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₇₉	—	4	1	no	—
140	Q₈×C₇₉	—	8	2	no	—
141	Q₈×C₇₉	—	8	2	no	—
142	C₂×C₄×C₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	—	4	1	no	—
144	C₂×C₄×C₇₉	—	4	2	no	—
145	C₂×C₄×C₇₉	—	4	2	no	—
146	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₇₉	#137	1	2	yes	—
153	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
154	C₂²⋊D₇₉	—	4	4	no	—
155	C₂×Dic₇₉	—	4	1	no	—
156	C₂×Dic₇₉	—	4	1	no	—
157	C₂×Dic₇₉	—	4	1	no	—
158	D₃₁₆	—	2	2	no	—
159	D₃₁₆	—	4	1	no	—
160	C₄×D₇₉	—	4	4	no	—
161	C₄×D₇₉	—	8	2	no	—
162	C₄×D₇₉	—	8	2	no	—
163	Dic₁₅₈	—	2	2	no	—
164	Dic₁₅₈	—	4	1	no	—
165	C₈×C₇₉	—	2	1	no	—
166	C₈×C₇₉	—	2	1	no	—
167	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
168	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
169	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
170	C₂²×C₇₉	—	2	1	no	—
171	C₂²×C₇₉	—	2	1	no	—
172	C₂²×C₇₉	#474	1	2	yes	—
173	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₇₉	—	4	2	no	—
175	C₂×D₇₉	—	8	1	no	—
176	C₂×D₇₉	—	8	1	no	—
177	C₄×C₇₉	—	2	1	no	—
178	C₄×C₇₉	—	2	1	no	—
179	C₄×C₇₉	—	2	1	no	—
180	C₄×C₇₉	—	8	2	no	—
181	C₄×C₇₉	—	8	2	no	—
182	C₄×C₇₉	#474	1	2	yes	—
183	C₄×C₇₉	#508	1	2	yes	—
184	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
185	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
186	Dic₇₉	—	4	2	no	—
187	Dic₇₉	—	8	1	no	—
188	Dic₇₉	—	8	1	no	—
189	#33626	—	79	1	no	maximal, Sylow
190	C₂×C₇₉	—	2	2	no	—
191	C₂×C₇₉	C₂×#474	1	1	yes	characteristic
192	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
193	C₂×C₇₉	#3414	1	1	yes	characteristic
194	D₇₉	—	8	2	no	—
195	#3160	D₇₉	1	1	yes	characteristic, complemented
196	C₄×#135	—	79	1	no	—
197	#2229	—	79	2	no	—
198	#2228	—	79	2	no	—
199	#2227	—	79	1	no	—
200	#2226	—	79	1	no	—
201	#2225	—	79	1	no	—
202	#2224	—	79	1	no	—
203	#2158	—	79	2	no	—
204	C₂×C₄⋊C₁₆	—	79	1	no	—
205	#2138	—	79	2	no	—
206	C₇₉	#33626	1	1	yes	characteristic, complemented, Sylow
207	C₂×#135	—	79	1	no	—
208	#533	—	2	1	no	—
209	C₄×#41	—	158	1	no	—
210	C₂²×C₁₆	—	79	2	no	—
211	#500	C₂×D₇₉	1	1	yes	characteristic
212	#500	C₂×D₇₉	1	1	yes	characteristic
213	#499	—	2	1	no	—
214	#499	C₂×D₇₉	1	2	yes	—
215	#498	—	2	1	no	—
216	C₄×Q₁₆	—	79	1	no	—
217	C₄×Q₁₆	—	79	1	no	—
218	C₄×SD₁₆	—	158	1	no	—
219	C₄×SD₁₆	—	158	1	no	—
220	C₄×D₈	—	79	1	no	—
221	C₄×D₈	—	79	1	no	—
222	#427	—	2	1	no	—
223	#427	—	79	1	no	—
224	#427	C₂×D₇₉	1	2	yes	—
225	#416	—	158	1	no	—
226	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
227	#367	—	79	2	no	—
228	#367	—	79	2	no	—
229	#366	—	79	2	no	—
230	#366	—	79	2	no	—
231	C₄⋊C₁₆	—	79	1	no	—
232	C₄⋊C₁₆	—	79	1	no	—
233	C₄⋊C₁₆	—	79	2	no	—
234	#358	—	79	2	no	—
235	#358	—	79	2	no	—
236	#357	—	79	2	no	—
237	#357	—	79	2	no	—
238	C₂×#41	—	158	1	no	—
239	#135	—	79	4	no	—
240	#135	—	158	2	no	—
241	C₂×Q₁₆	—	79	1	no	—
242	C₂×SD₁₆	—	158	1	no	—
243	C₂×D₈	—	79	1	no	—
244	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	#125	—	2	2	no	—
249	C₂²⋊Q₈	—	4	1	no	—
250	C₂²⋊Q₈	—	4	1	no	—
251	C₄×Q₈	—	4	1	no	—
252	C₄×Q₈	—	4	1	no	—
253	C₄×Q₈	—	158	1	no	—
254	C₄×Q₈	—	158	1	no	—
255	C₄×D₄	—	316	1	no	—
256	C₄×D₄	—	316	1	no	—
257	C₄×D₄	—	158	1	no	—
258	C₄×D₄	—	158	1	no	—
259	#117	—	2	2	no	—
260	#117	—	316	1	no	—
261	#117	—	158	1	no	—
262	C₂×C₄²	—	316	1	no	—
263	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
264	C₂×C₁₆	—	79	2	no	—
265	C₂×C₁₆	—	79	2	no	—
266	C₂×C₁₆	—	158	4	no	—
267	#107	—	2	1	no	—
268	#107	—	2	1	no	—
269	#107	—	79	1	no	—
270	#107	—	79	1	no	—
271	#107	C₂²×D₇₉	1	2	yes	—
272	#107	C₂²×D₇₉	1	2	yes	—
273	#106	—	2	2	no	—
274	#106	—	2	2	no	—
275	#106	—	158	1	no	—
276	#103	—	158	1	no	—
277	#103	—	158	1	no	—
278	#102	—	158	1	no	—
279	#102	—	158	1	no	—
280	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
281	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
284	#41	—	316	2	no	—
285	#41	—	158	4	no	—
286	C₂×Q₈	—	4	1	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	—	158	1	no	—
289	C₂×D₄	—	316	1	no	—
290	C₂×D₄	—	158	1	no	—
291	C₂²×C₄	—	316	1	no	—
292	C₂²×C₄	—	316	1	no	—
293	C₂²×C₄	—	316	1	no	—
294	C₂²×C₄	#11929609	1	1	yes	characteristic
295	C₂²×C₄	#11929609	1	1	yes	characteristic
296	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
297	Q₁₆	—	79	2	no	—
298	Q₁₆	—	158	1	no	—
299	SD₁₆	—	158	4	no	—
300	D₈	—	79	2	no	—
301	D₈	—	158	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
304	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₂×C₈	Q₈×D₇₉	1	2	yes	—
307	C₂×C₈	#11929625	1	2	yes	—
308	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	316	1	no	—
316	C₄⋊C₄	—	316	1	no	—
317	C₄⋊C₄	—	158	1	no	—
318	C₄⋊C₄	—	158	1	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	316	1	no	—
321	C₂²⋊C₄	—	316	1	no	—
322	C₂²⋊C₄	—	316	1	no	—
323	C₄²	—	4	2	no	—
324	C₄²	—	316	1	no	—
325	C₄²	—	316	1	no	—
326	C₄²	—	316	1	no	—
327	C₄²	#11929609	1	1	yes	characteristic
328	C₄²	#11929609	1	1	yes	characteristic
329	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
330	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
331	C₁₆	—	158	2	no	—
332	C₁₆	—	158	2	no	—
333	C₂³	—	316	1	no	—
334	C₂³	C₂×#11929609	1	1	yes	characteristic
335	Q₈	—	8	2	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	316	1	no	—
338	Q₈	—	158	2	no	—
339	D₄	—	316	4	no	—
340	D₄	—	316	1	no	—
341	D₄	—	158	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	316	1	no	—
347	C₂×C₄	—	316	4	no	—
348	C₂×C₄	—	316	1	no	—
349	C₂×C₄	—	316	1	no	—
350	C₂×C₄	—	632	2	no	—
351	C₂×C₄	—	632	2	no	—
352	C₂×C₄	#14579137	1	2	yes	—
353	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
354	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
355	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
356	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
357	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
358	C₂×C₄	#14579192	1	2	yes	—
359	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	#14579137	1	2	yes	—
363	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
364	C₈	C₂×#11929625	1	1	yes	characteristic
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	316	2	no	—
368	C₂²	—	632	1	no	—
369	C₂²	—	632	1	no	—
370	C₂²	#19098840	1	2	yes	—
371	C₂²	C₂²×#11929609	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	8	2	no	—
377	C₄	—	316	2	no	—
378	C₄	—	632	1	no	—
379	C₄	—	632	1	no	—
380	C₄	#19098840	1	2	yes	—
381	C₄	#19098993	1	2	yes	—
382	C₄	C₂×#14579137	1	1	yes	characteristic
383	C₄	C₂×#14579192	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	632	2	no	—
386	C₂	C₂×#19098840	1	1	yes	characteristic
387	C₂	C₂×#19098993	1	1	yes	characteristic
388	C₂	#23776534	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3160	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#14579183	C₂²
hypercenter	#3160	D₇₉
hypocenter	C₇₉	#33626
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #23765561	C₂	semidirect
3. #23765554	C₂	semidirect
4. #23765549	C₂	semidirect
5. #23765544	C₂	semidirect
6. #23765539	C₂	semidirect
7. #23765534	C₂	semidirect
8. #23765182	C₂	semidirect
9. C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	semidirect
11. C₇₉×#3160	C₂	semidirect
45. #14579183	Q₈	semidirect
195. #3160	D₇₉	semidirect
206. C₇₉	#33626	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579183	2
#23765561	2
#23765554	2
#23765549	2
#23765544	2
#23765539	2
#23765534	2
#23765182	2
C₂×C₄⋊(C₇₉⋊C₁₆)	2
#23765105	2
C₇₉×#3160	2
#33626	79