Subgroups — #34441505

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×D₇₉	C₂	1	2	yes	maximal, complemented
2	#23772298	C₂	1	1	yes	maximal, characteristic
3	#23765242	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765241	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765239	C₂	1	2	yes	maximal, complemented
6	#23765238	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765235	C₂	1	2	yes	maximal, complemented
8	#23765164	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765026	C₂	1	2	yes	maximal, complemented
10	#23764995	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099360	C₄	1	1	yes	characteristic
13	#131×D₇₉	C₂²	1	2	yes	—
14	#19098981	—	2	2	no	—
15	#19098981	C₂²	1	2	yes	complemented
16	#19098980	C₂²	1	2	yes	complemented
17	#19098979	—	2	2	no	—
18	#19098979	C₂²	1	2	yes	complemented
19	C₂×#14579100	C₂²	1	1	yes	characteristic
20	M₃₂×D₇₉	—	2	2	no	—
21	M₃₂×D₇₉	C₂²	1	2	yes	—
22	C₂×C₁₆×D₇₉	—	2	2	no	—
23	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
24	#19098685	C₄	1	1	yes	characteristic
25	#19098684	C₄	1	1	yes	characteristic
26	#19098682	C₄	1	2	yes	—
27	#19098681	C₂²	1	1	yes	characteristic
28	#19098678	C₂²	1	2	yes	—
29	#19098610	C₄	1	1	yes	characteristic
30	#19098578	C₄	1	2	yes	—
31	#19098517	C₂²	1	1	yes	characteristic
32	#19098384	C₂²	1	1	yes	characteristic, complemented
33	#19098383	C₂²	1	2	yes	complemented
34	#19098348	C₂²	1	2	yes	complemented
35	#19098347	C₂²	1	2	yes	complemented
36	#19098346	C₂²	1	2	yes	complemented
37	#19098339	C₂²	1	1	yes	characteristic, complemented
38	#19098338	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic, complemented
40	#504×C₇₉	C₂²	1	2	yes	complemented
41	#443×C₇₉	C₂²	1	1	yes	characteristic
42	#364×C₇₉	C₂²	1	1	yes	characteristic, complemented
43	#350×C₇₉	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
45	#41×D₇₉	C₂×C₄	1	2	yes	—
46	#14579193	C₈	1	2	yes	—
47	#14579190	C₈	1	2	yes	—
48	#14579188	C₈	1	2	yes	—
49	#14579185	C₈	1	2	yes	—
50	#14579183	C₂×C₄	1	1	yes	characteristic
51	#14579181	C₂×C₄	1	2	yes	—
52	#14579169	—	2	2	no	—
53	#14579169	C₂×C₄	1	2	yes	—
54	#14579168	C₂×C₄	1	2	yes	—
55	#14579167	—	2	2	no	—
56	#14579167	C₂³	1	2	yes	—
57	C₂×#11929607	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₇₉	—	2	2	no	—
59	M₁₆×D₇₉	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₇₉	—	2	2	no	—
61	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
62	#14579118	C₂×C₄	1	1	yes	characteristic
63	#14579101	—	2	2	no	—
64	#14579101	C₂³	1	2	yes	—
65	#14579100	—	2	2	no	—
66	#14579100	—	2	2	no	—
67	#14579100	—	2	2	no	—
68	#14579100	—	2	2	no	—
69	C₂×C₇₉⋊C₁₆	—	2	2	no	—
70	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₁₆×D₇₉	—	2	2	no	—
73	C₁₆×D₇₉	—	2	2	no	—
74	C₁₆×D₇₉	—	2	2	no	—
75	C₁₆×D₇₉	D₄	1	2	yes	—
76	C₁₆×D₇₉	D₄	1	2	yes	—
77	#14579097	C₂×C₄	1	1	yes	characteristic
78	#14579096	C₂×C₄	1	2	yes	—
79	#14579089	C₂×C₄	1	2	yes	—
80	#14579088	C₂×C₄	1	2	yes	—
81	#14579087	C₂×C₄	1	2	yes	—
82	#14579072	C₂×C₄	1	1	yes	characteristic
83	#14579071	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
85	#135×C₇₉	C₂×C₄	1	1	yes	characteristic
86	#131×C₇₉	C₂³	1	2	yes	—
87	M₃₂×C₇₉	—	2	2	no	—
88	M₃₂×C₇₉	C₂³	1	2	yes	—
89	C₂×C₁₆×C₇₉	—	2	2	no	—
90	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
92	#108×C₇₉	C₂×C₄	1	1	yes	characteristic
93	#104×C₇₉	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
95	#11929625	C₂×C₈	1	2	yes	—
96	#11929624	—	2	2	no	—
97	#11929624	C₂²×C₄	1	2	yes	—
98	#11929623	—	2	2	no	—
99	#11929623	C₂×C₈	1	2	yes	—
100	Q₈×D₇₉	C₂×C₈	1	2	yes	—
101	D₄×D₇₉	—	2	2	no	—
102	D₄×D₇₉	C₂×C₈	1	2	yes	—
103	C₂×C₄×D₇₉	—	2	2	no	—
104	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
105	Dic₃₁₆	C₂×C₈	1	1	yes	characteristic
106	#11929614	C₂×C₈	1	2	yes	—
107	#11929613	C₂×C₈	1	2	yes	—
108	#11929612	C₂×C₈	1	2	yes	—
109	#11929611	C₂×C₈	1	2	yes	—
110	D₆₃₂	C₂×C₈	1	1	yes	characteristic
111	#11929609	C₂×C₈	1	2	yes	—
112	#11929608	—	2	2	no	—
113	#11929608	C₂²×C₄	1	2	yes	—
114	#11929607	—	2	2	no	—
115	#11929607	—	2	2	no	—
116	#11929607	—	2	2	no	—
117	#11929607	C₂×C₈	1	2	yes	—
118	#11929607	C₂×C₈	1	2	yes	—
119	C₂×C₇₉⋊C₈	—	2	2	no	—
120	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₈×D₇₉	—	2	2	no	—
123	C₈×D₇₉	—	2	2	no	—
124	C₈×D₇₉	—	2	2	no	—
125	C₈×D₇₉	C₂×D₄	1	2	yes	—
126	C₈×D₇₉	#41	1	2	yes	—
127	Dic₇₉⋊C₄	—	2	1	no	—
128	(C₂×D₇₉)⋊C₄	—	2	1	no	—
129	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
130	C₇₉⋊C₁₆	—	2	2	no	—
131	C₇₉⋊C₁₆	—	2	1	no	—
132	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
133	#41×C₇₉	C₂²×C₄	1	2	yes	—
134	Q₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
135	SD₁₆×C₇₉	C₂×C₈	1	2	yes	—
136	D₈×C₇₉	C₂×C₈	1	1	yes	characteristic
137	M₁₆×C₇₉	—	2	2	no	—
138	M₁₆×C₇₉	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₇₉	—	2	2	no	—
140	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
142	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
143	C₁₆×C₇₉	—	2	2	no	—
144	C₁₆×C₇₉	—	2	1	no	—
145	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
146	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
147	C₂²×D₇₉	—	2	2	no	—
148	C₂²×D₇₉	#131	1	1	yes	characteristic
149	Q₈×C₇₉	C₂²×C₈	1	2	yes	—
150	D₄×C₇₉	—	2	2	no	—
151	D₄×C₇₉	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₇₉	—	2	2	no	—
153	C₂×C₄×C₇₉	—	2	1	no	—
154	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₇₉	—	2	2	no	—
156	C₂²⋊D₇₉	—	2	2	no	—
157	C₂²⋊D₇₉	—	2	2	no	—
158	C₂×Dic₇₉	—	2	2	no	—
159	C₂×Dic₇₉	—	2	1	no	—
160	C₂×Dic₇₉	#131	1	1	yes	characteristic
161	D₃₁₆	—	2	2	no	—
162	D₃₁₆	C₂²×C₈	1	2	yes	—
163	C₄×D₇₉	—	2	2	no	—
164	C₄×D₇₉	—	2	2	no	—
165	C₄×D₇₉	—	2	2	no	—
166	C₄×D₇₉	C₄×D₄	1	2	yes	—
167	C₄×D₇₉	C₂²×C₈	1	2	yes	—
168	Dic₁₅₈	—	2	2	no	—
169	Dic₁₅₈	C₂²×C₈	1	2	yes	—
170	C₈×C₇₉	—	2	2	no	—
171	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₇₉	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
175	C₇₉⋊C₈	—	2	2	no	—
176	C₇₉⋊C₈	C₄×D₄	1	2	yes	—
177	C₇₉⋊C₈	C₂²×C₈	1	2	yes	—
178	C₂²×C₇₉	—	2	2	no	—
179	C₂²×C₇₉	C₂×#131	1	1	yes	characteristic
180	C₂×D₇₉	—	2	2	no	—
181	C₂×D₇₉	—	4	1	no	—
182	C₂×D₇₉	—	4	4	no	—
183	C₂×D₇₉	C₈×D₄	1	2	yes	—
184	C₄×C₇₉	—	2	2	no	—
185	C₄×C₇₉	—	4	1	no	—
186	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₇₉	C₂³×C₈	1	1	yes	characteristic
188	Dic₇₉	—	2	2	no	—
189	Dic₇₉	—	4	1	no	—
190	Dic₇₉	C₈×D₄	1	2	yes	—
191	#33441	—	79	1	no	maximal, Sylow
192	C₂×C₇₉	—	2	1	no	—
193	C₂×C₇₉	—	4	2	no	—
194	C₂×C₇₉	C₂×C₈×D₄	1	1	yes	characteristic
195	D₇₉	—	2	2	no	—
196	D₇₉	—	4	2	no	—
197	C₂×#504	—	79	2	no	—
198	#2955	—	79	1	no	—
199	#2171	—	79	1	no	—
200	#2171	—	79	1	no	—
201	#2171	—	79	2	no	—
202	#2170	—	79	2	no	—
203	#2170	—	79	1	no	—
204	#2170	D₇₉	1	1	yes	characteristic, complemented
205	#2153	—	79	1	no	—
206	#2117	—	79	2	no	—
207	#2108	—	79	1	no	—
208	C₇₉	#33441	1	1	yes	characteristic, complemented, Sylow
209	#575	—	79	1	no	—
210	C₂×#131	—	79	2	no	—
211	#504	—	79	2	no	—
212	#504	—	79	2	no	—
213	#504	—	79	2	no	—
214	#504	—	158	2	no	—
215	#504	—	158	2	no	—
216	#504	C₂×D₇₉	1	2	yes	complemented
217	C₂×M₃₂	—	79	1	no	—
218	C₂×M₃₂	—	79	2	no	—
219	C₂×M₃₂	—	158	2	no	—
220	C₂²×C₁₆	—	79	1	no	—
221	C₂²×C₁₆	—	158	2	no	—
222	#444	—	79	1	no	—
223	#444	—	79	1	no	—
224	#444	—	79	2	no	—
225	#443	—	79	2	no	—
226	#443	—	79	1	no	—
227	#443	C₂×D₇₉	1	1	yes	characteristic
228	#430	—	79	1	no	—
229	#421	—	79	2	no	—
230	#405	—	79	1	no	—
231	#364	—	79	2	no	—
232	#364	—	79	1	no	—
233	#364	C₂×D₇₉	1	1	yes	characteristic, complemented
234	#350	—	79	2	no	—
235	#350	—	79	2	no	—
236	#350	—	79	2	no	—
237	#350	C₂×D₇₉	1	2	yes	complemented
238	#346	—	79	1	no	—
239	#346	—	79	1	no	—
240	C₄×C₁₆	—	79	1	no	—
241	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
242	C₂×#41	—	79	2	no	—
243	#137	—	79	2	no	—
244	#137	—	79	2	no	—
245	AGL(1,ℤ/8)	—	79	2	no	—
246	AGL(1,ℤ/8)	—	79	2	no	—
247	#135	—	79	2	no	—
248	#135	—	79	1	no	—
249	#135	C₄×D₇₉	1	1	yes	characteristic
250	#131	—	79	2	no	—
251	#131	—	79	2	no	—
252	#131	—	79	2	no	—
253	#131	—	158	2	no	—
254	#131	—	158	2	no	—
255	#131	C₂²×D₇₉	1	2	yes	—
256	C₂×M₁₆	—	79	1	no	—
257	C₂×M₁₆	—	79	2	no	—
258	C₂×M₁₆	—	158	2	no	—
259	C₂²×C₈	—	79	1	no	—
260	C₂²×C₈	—	158	2	no	—
261	#117	—	79	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	79	2	no	—
264	M₃₂	—	158	2	no	—
265	M₃₂	—	158	2	no	—
266	M₃₂	—	158	2	no	—
267	M₃₂	—	158	2	no	—
268	M₃₂	—	158	2	no	—
269	M₃₂	C₂²×D₇₉	1	2	yes	—
270	C₂×C₁₆	—	2	2	no	—
271	C₂×C₁₆	—	79	1	no	—
272	C₂×C₁₆	—	79	2	no	—
273	C₂×C₁₆	—	79	2	no	—
274	C₂×C₁₆	—	79	1	no	—
275	C₂×C₁₆	—	158	2	no	—
276	C₂×C₁₆	—	158	2	no	—
277	C₂×C₁₆	—	158	2	no	—
278	C₂×C₁₆	—	158	2	no	—
279	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
281	#108	—	79	2	no	—
282	#108	—	79	1	no	—
283	#108	C₄×D₇₉	1	1	yes	characteristic
284	#104	—	79	2	no	—
285	#104	—	79	2	no	—
286	#104	—	79	2	no	—
287	#104	C₄×D₇₉	1	2	yes	—
288	#97	—	79	1	no	—
289	#97	—	79	1	no	—
290	C₄×C₈	—	79	1	no	—
291	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
292	#41	—	79	2	no	—
293	#41	—	79	2	no	—
294	#41	—	79	2	no	—
295	#41	—	158	2	no	—
296	#41	—	158	2	no	—
297	#41	C₂×C₄×D₇₉	1	2	yes	—
298	C₂×Q₈	—	79	2	no	—
299	C₂×D₄	—	79	2	no	—
300	C₂×D₄	—	158	2	no	—
301	C₂²×C₄	—	79	1	no	—
302	C₂²×C₄	—	158	2	no	—
303	Q₁₆	—	79	2	no	—
304	Q₁₆	—	79	1	no	—
305	Q₁₆	C₈×D₇₉	1	1	yes	characteristic
306	SD₁₆	—	79	2	no	—
307	SD₁₆	—	79	2	no	—
308	SD₁₆	—	79	2	no	—
309	SD₁₆	C₈×D₇₉	1	2	yes	—
310	D₈	—	79	2	no	—
311	D₈	—	79	1	no	—
312	D₈	C₈×D₇₉	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	79	2	no	—
315	M₁₆	—	79	2	no	—
316	M₁₆	—	79	2	no	—
317	M₁₆	—	158	2	no	—
318	M₁₆	—	158	2	no	—
319	M₁₆	—	158	2	no	—
320	M₁₆	—	158	2	no	—
321	M₁₆	C₂×C₄×D₇₉	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	79	1	no	—
324	C₂×C₈	—	79	1	no	—
325	C₂×C₈	—	79	2	no	—
326	C₂×C₈	—	79	2	no	—
327	C₂×C₈	—	158	2	no	—
328	C₂×C₈	—	158	2	no	—
329	C₂×C₈	—	158	2	no	—
330	C₂×C₈	—	158	2	no	—
331	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
333	C₄⋊C₄	—	158	1	no	—
334	C₂²⋊C₄	—	158	1	no	—
335	C₄²	—	79	1	no	—
336	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
337	C₁₆	—	2	1	no	—
338	C₁₆	—	2	2	no	—
339	C₁₆	—	79	2	no	—
340	C₁₆	—	158	1	no	—
341	C₁₆	—	158	2	no	—
342	C₁₆	D₄×D₇₉	1	1	yes	characteristic
343	C₁₆	D₄×D₇₉	1	1	yes	characteristic
344	C₂³	—	79	1	no	—
345	C₂³	—	158	2	no	—
346	Q₈	—	79	2	no	—
347	Q₈	—	158	2	no	—
348	Q₈	C₂×C₈×D₇₉	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	79	2	no	—
351	D₄	—	158	2	no	—
352	D₄	—	158	2	no	—
353	D₄	—	158	2	no	—
354	D₄	—	158	2	no	—
355	D₄	C₂×C₈×D₇₉	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	79	1	no	—
359	C₂×C₄	—	79	2	no	—
360	C₂×C₄	—	79	2	no	—
361	C₂×C₄	—	158	2	no	—
362	C₂×C₄	—	158	2	no	—
363	C₂×C₄	—	158	1	no	—
364	C₂×C₄	—	158	2	no	—
365	C₂×C₄	—	158	2	no	—
366	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	79	2	no	—
369	C₈	—	79	2	no	—
370	C₈	—	158	2	no	—
371	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
372	C₈	C₂×C₈×D₇₉	1	1	yes	characteristic
373	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
374	C₈	#41×D₇₉	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	316	1	no	—
377	C₂²	—	316	4	no	—
378	C₂²	—	79	2	no	—
379	C₂²	—	158	2	no	—
380	C₂²	#131×D₇₉	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	316	1	no	—
384	C₄	—	79	2	no	—
385	C₄	—	158	2	no	—
386	C₄	C₄×D₄×D₇₉	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₇₉	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	316	2	no	—
391	C₂	—	158	2	no	—
392	C₂	C₈×D₄×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₄×D₇₉
commutator	C₄×C₇₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2170	C₂
socle	C₂×C₇₉	C₂×C₈×D₄
prime radical	#19099360	C₄
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	#2170	D₇₉
hypocenter	C₇₉	#33441
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #504×D₇₉	C₂	semidirect
3. #23765242	C₂	semidirect
4. #23765241	C₂	semidirect
5. #23765239	C₂	semidirect
6. #23765238	C₂	semidirect
7. #23765235	C₂	semidirect
8. #23765164	C₂	semidirect
9. #23765026	C₂	semidirect
10. #23764995	C₂	semidirect
11. C₇₉×#2170	C₂	semidirect
15. #19098981	C₂²	semidirect
16. #19098980	C₂²	semidirect
18. #19098979	C₂²	semidirect
32. #19098384	C₂²	semidirect
33. #19098383	C₂²	semidirect
34. #19098348	C₂²	semidirect
35. #19098347	C₂²	semidirect
36. #19098346	C₂²	semidirect
37. #19098339	C₂²	semidirect
38. #19098338	C₂²	semidirect
39. C₁₆×Dic₇₉	C₂²	semidirect
40. #504×C₇₉	C₂²	semidirect
42. #364×C₇₉	C₂²	semidirect
43. #350×C₇₉	C₂²	semidirect
44. C₄×C₁₆×C₇₉	C₂²	semidirect
204. #2170	D₇₉	semidirect
208. C₇₉	#33441	semidirect
216. #504	C₂×D₇₉	semidirect
233. #364	C₂×D₇₉	semidirect
237. #350	C₂×D₇₉	semidirect
241. C₄×C₁₆	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#504×D₇₉	2
#23772298	2
#23765242	2
#23765241	2
#23765239	2
#23765238	2
#23765235	2
#23765164	2
#23765026	2
#23764995	2
C₇₉×#2170	2
#33441	79