Subgroups — #34442917

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	#23773914	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14579183	C₂	1	1	yes	maximal, characteristic
3	#23765565	C₂	1	2	yes	maximal, complemented
4	#23765558	C₂	1	2	yes	maximal, complemented
5	#23765553	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765548	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765543	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765538	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765184	C₂	1	2	yes	maximal, complemented
10	#23765108	C₂	1	2	yes	maximal
11	C₂×C₄⋊C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579183	C₂²	1	1	yes	characteristic
13	#19099120	—	2	1	no	—
14	C₄×#11929624	—	2	1	no	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098965	—	2	1	no	—
18	#19098965	C₂²	1	2	yes	—
19	#19098961	—	2	1	no	—
20	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
22	C₄×#11929613	—	2	1	no	—
23	C₄×#11929613	—	2	1	no	—
24	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
25	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
26	#19098601	—	2	1	no	—
27	#19098601	C₂²	1	1	yes	characteristic
28	#19098601	C₂²	1	2	yes	—
29	#19098560	—	2	1	no	—
30	#19098391	C₂²	1	2	yes	—
31	#19098391	C₂²	1	2	yes	—
32	#19098389	C₂²	1	2	yes	—
33	#19098389	C₂²	1	2	yes	—
34	#19098368	C₂²	1	2	yes	—
35	#19098368	C₂²	1	2	yes	—
36	#19098365	C₂²	1	2	yes	—
37	#19098365	C₂²	1	2	yes	—
38	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
40	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
42	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
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	#14579161	—	2	1	no	—
50	#14579161	—	2	1	no	—
51	#14579153	—	2	2	no	—
52	#14579140	—	4	1	no	—
53	#14579140	—	4	1	no	—
54	C₄×Dic₁₅₈	—	2	1	no	—
55	C₄×Dic₁₅₈	—	2	1	no	—
56	C₄×Dic₁₅₈	—	4	1	no	—
57	C₄×Dic₁₅₈	—	4	1	no	—
58	C₄×D₃₁₆	—	2	1	no	—
59	C₄×D₃₁₆	—	2	1	no	—
60	C₄×C₂²⋊D₇₉	—	4	1	no	—
61	C₄×C₂²⋊D₇₉	—	4	1	no	—
62	#14579120	—	2	1	no	—
63	#14579120	—	2	2	no	—
64	#14579118	—	4	1	no	—
65	C₄²×D₇₉	—	4	1	no	—
66	#14579095	—	2	1	no	—
67	#14579095	—	2	1	no	—
68	#14579095	C₂³	1	2	yes	—
69	#14579095	C₂³	1	2	yes	—
70	#14579095	C₂³	1	1	yes	characteristic
71	#14579095	C₂³	1	1	yes	characteristic
72	#14579093	—	2	1	no	—
73	#14579093	—	2	2	no	—
74	#14579093	—	2	2	no	—
75	#14579086	—	2	1	no	—
76	#14579086	—	2	1	no	—
77	#14579083	—	2	1	no	—
78	#14579083	—	2	1	no	—
79	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
82	C₂×C₁₆×C₇₉	—	2	4	no	—
83	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
84	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
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×Dic₁₅₈	—	4	1	no	—
93	C₂×Dic₁₅₈	—	4	1	no	—
94	C₂×D₃₁₆	—	2	1	no	—
95	C₂×C₂²⋊D₇₉	—	4	1	no	—
96	C₂×C₄×D₇₉	—	4	1	no	—
97	C₂×C₄×D₇₉	—	4	1	no	—
98	C₂×C₄×D₇₉	—	4	1	no	—
99	Dic₃₁₆	—	2	1	no	—
100	Dic₃₁₆	C₂×Q₈	1	2	yes	—
101	#11929613	—	2	4	no	—
102	D₆₃₂	—	2	1	no	—
103	D₆₃₂	C₂×Q₈	1	2	yes	—
104	C₄⋊Dic₇₉	—	2	1	no	—
105	C₄⋊Dic₇₉	—	2	1	no	—
106	C₄⋊Dic₇₉	—	2	2	no	—
107	C₄⋊Dic₇₉	—	2	2	no	—
108	C₄⋊Dic₇₉	—	4	1	no	—
109	C₄⋊Dic₇₉	—	4	1	no	—
110	Dic₇₉⋊C₄	—	4	1	no	—
111	Dic₇₉⋊C₄	—	4	1	no	—
112	Dic₇₉⋊C₄	—	4	2	no	—
113	Dic₇₉⋊C₄	—	4	2	no	—
114	C₂²⋊Dic₇₉	—	4	1	no	—
115	C₂²⋊Dic₇₉	—	4	2	no	—
116	(C₂×D₇₉)⋊C₄	—	4	1	no	—
117	(C₂×D₇₉)⋊C₄	—	4	1	no	—
118	C₄×Dic₇₉	—	4	1	no	—
119	C₄×Dic₇₉	—	4	1	no	—
120	C₄×Dic₇₉	—	4	1	no	—
121	C₄×Dic₇₉	—	4	2	no	—
122	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
123	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
124	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₇₉	—	2	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×Q₈	1	2	yes	—
130	C₂×C₈×C₇₉	#41	1	2	yes	—
131	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
132	C₄²×C₇₉	D₈	1	1	yes	characteristic
133	C₄²×C₇₉	D₈	1	1	yes	characteristic
134	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
136	C₁₆×C₇₉	—	2	2	no	—
137	C₁₆×C₇₉	—	2	2	no	—
138	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₇₉	—	4	1	no	—
140	C₂×C₄×C₇₉	—	2	2	no	—
141	C₂×C₄×C₇₉	—	4	1	no	—
142	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₇₉	#137	1	2	yes	—
149	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₇₉	—	4	4	no	—
151	C₂×Dic₇₉	—	4	1	no	—
152	C₂×Dic₇₉	—	4	1	no	—
153	C₂×Dic₇₉	—	4	1	no	—
154	C₂×Dic₇₉	—	4	2	no	—
155	C₂×Dic₇₉	—	4	2	no	—
156	D₃₁₆	—	2	2	no	—
157	D₃₁₆	—	4	1	no	—
158	C₄×D₇₉	—	4	4	no	—
159	C₄×D₇₉	—	8	2	no	—
160	C₄×D₇₉	—	8	2	no	—
161	Dic₁₅₈	—	2	2	no	—
162	Dic₁₅₈	—	4	1	no	—
163	Dic₁₅₈	—	8	2	no	—
164	Dic₁₅₈	—	8	2	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	#474	1	2	yes	—
181	C₄×C₇₉	#508	1	2	yes	—
182	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
183	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
184	Dic₇₉	—	4	2	no	—
185	Dic₇₉	—	8	2	no	—
186	Dic₇₉	—	8	1	no	—
187	Dic₇₉	—	8	2	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	—	79	1	no	—
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	D₇₉	1	1	yes	characteristic, complemented
205	#2138	—	79	2	no	—
206	C₇₉	#33626	1	1	yes	characteristic, complemented, Sylow
207	C₂×#135	—	79	1	no	—
208	#533	—	158	1	no	—
209	C₄×#41	—	158	1	no	—
210	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
211	#500	—	79	1	no	—
212	#500	—	79	1	no	—
213	#499	—	79	2	no	—
214	#499	—	158	1	no	—
215	#498	—	158	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	—	79	2	no	—
223	#427	—	79	1	no	—
224	#427	—	158	1	no	—
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	C₂×D₇₉	1	1	yes	characteristic
232	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
233	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
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	D₃₁₆	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
246	C₄⋊Q₈	—	158	1	no	—
247	C₄⋊Q₈	—	158	1	no	—
248	#125	—	158	2	no	—
249	C₂²⋊Q₈	—	316	1	no	—
250	C₂²⋊Q₈	—	316	1	no	—
251	C₄×Q₈	—	316	1	no	—
252	C₄×Q₈	—	316	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	—	316	1	no	—
260	#117	—	158	1	no	—
261	#117	—	158	2	no	—
262	C₂×C₄²	—	316	1	no	—
263	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
264	C₂×C₁₆	—	2	4	no	—
265	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
266	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
267	#107	—	79	2	no	—
268	#107	—	79	2	no	—
269	#107	—	79	1	no	—
270	#107	—	79	1	no	—
271	#107	—	158	1	no	—
272	#107	—	158	1	no	—
273	#106	—	158	1	no	—
274	#106	—	158	2	no	—
275	#106	—	158	2	no	—
276	#103	—	158	1	no	—
277	#103	—	158	1	no	—
278	#102	—	158	1	no	—
279	#102	—	158	1	no	—
280	C₄×C₈	D₃₁₆	1	1	yes	characteristic
281	C₄×C₈	D₃₁₆	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	—	316	1	no	—
287	C₂×Q₈	—	316	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	D₆₃₂	1	1	yes	characteristic
295	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₃₁₆	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×D₃₁₆	1	1	yes	characteristic
304	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
305	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
306	C₂×C₈	Q₈×D₇₉	1	2	yes	—
307	C₂×C₈	#11929623	1	2	yes	—
308	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
309	C₄⋊C₄	—	316	1	no	—
310	C₄⋊C₄	—	316	1	no	—
311	C₄⋊C₄	—	316	2	no	—
312	C₄⋊C₄	—	316	2	no	—
313	C₄⋊C₄	—	316	1	no	—
314	C₄⋊C₄	—	316	1	no	—
315	C₄⋊C₄	—	158	1	no	—
316	C₄⋊C₄	—	158	1	no	—
317	C₄⋊C₄	—	158	2	no	—
318	C₄⋊C₄	—	158	2	no	—
319	C₂²⋊C₄	—	316	1	no	—
320	C₂²⋊C₄	—	316	2	no	—
321	C₂²⋊C₄	—	316	1	no	—
322	C₂²⋊C₄	—	316	1	no	—
323	C₄²	—	316	1	no	—
324	C₄²	—	316	1	no	—
325	C₄²	—	316	1	no	—
326	C₄²	—	316	2	no	—
327	C₄²	D₆₃₂	1	1	yes	characteristic
328	C₄²	D₆₃₂	1	1	yes	characteristic
329	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
330	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	2	2	no	—
333	C₂³	—	316	1	no	—
334	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
335	Q₈	—	316	1	no	—
336	Q₈	—	632	2	no	—
337	Q₈	—	632	2	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	—	316	1	no	—
345	C₂×C₄	—	316	4	no	—
346	C₂×C₄	—	316	1	no	—
347	C₂×C₄	—	316	1	no	—
348	C₂×C₄	—	316	2	no	—
349	C₂×C₄	—	316	2	no	—
350	C₂×C₄	—	632	2	no	—
351	C₂×C₄	—	632	2	no	—
352	C₂×C₄	#14579141	1	2	yes	—
353	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
354	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
355	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
356	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
357	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
358	C₂×C₄	#14579194	1	2	yes	—
359	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	#14579141	1	2	yes	—
363	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
364	C₈	C₂×#11929623	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	#19098844	1	2	yes	—
371	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	316	2	no	—
376	C₄	—	632	1	no	—
377	C₄	—	632	2	no	—
378	C₄	—	632	1	no	—
379	C₄	—	632	2	no	—
380	C₄	#19098844	1	2	yes	—
381	C₄	#19098995	1	2	yes	—
382	C₄	C₂×#14579141	1	1	yes	characteristic
383	C₄	C₂×#14579194	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	632	2	no	—
386	C₂	C₂×#19098844	1	1	yes	characteristic
387	C₂	C₂×#19098995	1	1	yes	characteristic
388	C₂	#23776536	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×C₄⋊C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#14579183	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₇₉
hypocenter	C₇₉	#33626
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. #23773914	C₂	semidirect
3. #23765565	C₂	semidirect
4. #23765558	C₂	semidirect
5. #23765553	C₂	semidirect
6. #23765548	C₂	semidirect
7. #23765543	C₂	semidirect
8. #23765538	C₂	semidirect
9. #23765184	C₂	semidirect
11. C₂×C₄⋊C₁₆×C₇₉	C₂	semidirect
45. #14579183	Q₈	semidirect
204. C₂×C₄⋊C₁₆	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
#23773914	2
C₄×#14579183	2
#23765565	2
#23765558	2
#23765553	2
#23765548	2
#23765543	2
#23765538	2
#23765184	2
#23765108	2
C₂×C₄⋊C₁₆×C₇₉	2
#33626	79