Subgroups — #34443605

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773940	C₂	1	1	yes	maximal, characteristic, complemented
2	#23773887	C₂	1	1	yes	maximal, characteristic
3	#23765640	C₂	1	2	yes	maximal, complemented
4	#23765638	C₂	1	2	yes	maximal, complemented
5	#23765633	C₂	1	2	yes	maximal, complemented
6	#23765622	C₂	1	2	yes	maximal, complemented
7	#23765115	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765112	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19098365	C₂	1	1	yes	maximal, characteristic, complemented
10	#23764996	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2138	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579186	—	2	1	no	—
13	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
14	#19099191	—	2	1	no	—
15	#19099127	—	2	1	no	—
16	C₂×#14579101	C₂²	1	1	yes	characteristic
17	#19098970	—	2	2	no	—
18	#19098967	C₂²	1	2	yes	complemented
19	#19098964	C₂²	1	2	yes	complemented
20	#19098949	—	2	2	no	—
21	#19098948	C₂²	1	2	yes	—
22	#19098938	C₂²	1	2	yes	—
23	#19098606	—	2	1	no	—
24	C₂×#14579095	C₂²	1	1	yes	characteristic
25	#19098518	C₂²	1	1	yes	characteristic
26	#19098367	C₂²	1	2	yes	complemented
27	#19098365	C₂²	1	2	yes	—
28	#19098365	C₂²	1	2	yes	—
29	#19098364	C₂²	1	2	yes	complemented
30	#19098363	C₂²	1	2	yes	—
31	#19098363	C₂²	1	2	yes	—
32	#19098338	C₂²	1	2	yes	complemented
33	C₁₆×Dic₇₉	C₂²	1	2	yes	complemented
34	C₂×#135×C₇₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
36	#427×C₇₉	C₂²	1	1	yes	characteristic
37	#358×C₇₉	C₂²	1	1	yes	characteristic, complemented
38	#358×C₇₉	C₂²	1	1	yes	characteristic, complemented
39	#357×C₇₉	C₂²	1	1	yes	characteristic, complemented
40	#357×C₇₉	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₃₁₆	—	2	1	no	—
42	#14579186	—	4	4	no	—
43	C₂×#11929611	—	2	2	no	—
44	C₂×D₆₃₂	—	2	2	no	—
45	C₂×D₆₃₂	—	2	2	no	—
46	C₂×D₆₃₂	—	2	1	no	—
47	C₂×D₆₃₂	C₂³	1	2	yes	—
48	C₂×#11929609	—	2	2	no	—
49	C₂×#11929608	D₄	1	1	yes	characteristic
50	#14579161	—	4	1	no	—
51	#14579160	—	2	2	no	—
52	#14579157	—	2	2	no	—
53	#14579153	—	4	1	no	—
54	#14579152	—	2	2	no	—
55	#14579150	—	2	2	no	—
56	#14579138	—	4	2	no	—
57	#14579135	—	4	1	no	—
58	#14579135	—	4	1	no	—
59	#14579129	—	4	2	no	—
60	#14579120	D₄	1	1	yes	characteristic
61	C₂×C₄⋊Dic₇₉	—	2	1	no	—
62	#14579101	—	2	2	no	—
63	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
64	#14579095	—	2	1	no	—
65	#14579095	C₂³	1	2	yes	—
66	#14579094	—	2	2	no	—
67	#14579092	—	2	2	no	—
68	#14579071	D₄	1	2	yes	—
69	C₈×Dic₇₉	C₂³	1	2	yes	—
70	C₂×#41×C₇₉	—	2	1	no	—
71	#135×C₇₉	—	2	2	no	—
72	#135×C₇₉	—	2	2	no	—
73	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×SD₁₆×C₇₉	—	2	1	no	—
75	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
77	#117×C₇₉	—	2	1	no	—
78	C₂×C₁₆×C₇₉	—	2	2	no	—
79	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
81	#107×C₇₉	C₂³	1	1	yes	characteristic
82	#107×C₇₉	C₂³	1	1	yes	characteristic
83	#106×C₇₉	—	2	1	no	—
84	C₂³×D₇₉	—	4	1	no	—
85	C₂×Dic₁₅₈	—	4	2	no	—
86	C₂×D₃₁₆	—	2	2	no	—
87	C₂×D₃₁₆	—	4	2	no	—
88	C₂×D₃₁₆	—	4	1	no	—
89	C₂×D₃₁₆	—	4	2	no	—
90	C₂×C₂²⋊D₇₉	—	4	2	no	—
91	C₂×C₂²⋊D₇₉	—	4	2	no	—
92	C₂²×Dic₇₉	—	4	1	no	—
93	#11929611	—	4	4	no	—
94	D₆₃₂	—	2	2	no	—
95	D₆₃₂	—	2	2	no	—
96	D₆₃₂	—	4	2	no	—
97	#11929609	—	4	4	no	—
98	#11929608	SD₁₆	1	4	yes	—
99	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
100	C₄⋊Dic₇₉	—	2	2	no	—
101	C₄⋊Dic₇₉	—	4	2	no	—
102	C₄⋊Dic₇₉	—	4	1	no	—
103	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
104	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
105	Dic₇₉⋊C₄	—	4	2	no	—
106	Dic₇₉⋊C₄	—	4	2	no	—
107	C₂²⋊Dic₇₉	D₈	1	2	yes	—
108	(C₂×D₇₉)⋊C₄	—	4	2	no	—
109	(C₂×D₇₉)⋊C₄	—	4	2	no	—
110	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
111	C₇₉⋊C₁₆	—	2	2	no	—
112	#41×C₇₉	—	4	2	no	—
113	#41×C₇₉	—	4	2	no	—
114	C₂×Q₈×C₇₉	—	2	1	no	—
115	C₂×D₄×C₇₉	—	2	1	no	—
116	C₂×D₄×C₇₉	—	4	1	no	—
117	C₂²×C₄×C₇₉	—	4	1	no	—
118	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
119	Q₁₆×C₇₉	—	2	1	no	—
120	Q₁₆×C₇₉	—	2	1	no	—
121	SD₁₆×C₇₉	—	4	2	no	—
122	D₈×C₇₉	—	2	1	no	—
123	D₈×C₇₉	—	2	1	no	—
124	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₇₉	#41	1	2	yes	—
126	C₂×C₈×C₇₉	#41	1	2	yes	—
127	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₇₉	—	2	1	no	—
129	C₄⋊C₄×C₇₉	—	2	1	no	—
130	C₂²⋊C₄×C₇₉	—	4	1	no	—
131	C₄²×C₇₉	—	4	1	no	—
132	C₁₆×C₇₉	—	2	1	no	—
133	C₁₆×C₇₉	—	2	1	no	—
134	C₂³×C₇₉	—	4	1	no	—
135	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₇₉	—	4	2	no	—
137	C₂²×D₇₉	—	8	2	no	—
138	C₂²×D₇₉	—	16	1	no	—
139	C₂²×D₇₉	—	16	1	no	—
140	Q₈×C₇₉	—	4	1	no	—
141	Q₈×C₇₉	—	4	1	no	—
142	D₄×C₇₉	—	4	1	no	—
143	D₄×C₇₉	—	4	1	no	—
144	D₄×C₇₉	—	8	2	no	—
145	C₂×C₄×C₇₉	—	4	1	no	—
146	C₂×C₄×C₇₉	—	4	1	no	—
147	C₂×C₄×C₇₉	—	4	1	no	—
148	C₂×C₄×C₇₉	—	8	2	no	—
149	C₂×C₄×C₇₉	#124	1	2	yes	—
150	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
152	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
153	C₂²⋊D₇₉	—	16	2	no	—
154	C₂²⋊D₇₉	—	16	2	no	—
155	C₂×Dic₇₉	—	4	2	no	—
156	C₂×Dic₇₉	—	16	1	no	—
157	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
158	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
159	D₃₁₆	—	4	2	no	—
160	D₃₁₆	—	4	2	no	—
161	D₃₁₆	—	8	2	no	—
162	Dic₁₅₈	—	16	2	no	—
163	C₈×C₇₉	#124	1	2	yes	—
164	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
165	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
166	C₇₉⋊C₈	C₂×SD₁₆	1	4	yes	—
167	C₂²×C₇₉	—	2	1	no	—
168	C₂²×C₇₉	—	2	1	no	—
169	C₂²×C₇₉	—	8	1	no	—
170	C₂²×C₇₉	—	8	1	no	—
171	C₂²×C₇₉	—	8	1	no	—
172	C₂²×C₇₉	#486	1	2	yes	—
173	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₇₉	—	8	2	no	—
175	C₂×D₇₉	—	8	2	no	—
176	C₂×D₇₉	—	8	2	no	—
177	C₂×D₇₉	—	16	4	no	—
178	C₄×C₇₉	—	8	1	no	—
179	C₄×C₇₉	—	8	1	no	—
180	C₄×C₇₉	—	8	1	no	—
181	C₄×C₇₉	#486	1	2	yes	—
182	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
183	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
184	Dic₇₉	—	2	2	no	—
185	Dic₇₉	—	2	2	no	—
186	Dic₇₉	—	8	2	no	—
187	#33702	—	79	1	no	maximal, Sylow
188	C₂×C₇₉	—	2	2	no	—
189	C₂×C₇₉	—	16	1	no	—
190	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
191	C₂×C₇₉	#3415	1	1	yes	characteristic
192	C₂×C₇₉	#3416	1	1	yes	characteristic
193	D₇₉	—	16	2	no	—
194	#3164	—	79	1	no	—
195	#3155	—	79	1	no	—
196	#2244	—	79	2	no	—
197	#2243	—	79	2	no	—
198	#2242	—	79	2	no	—
199	#2240	—	79	2	no	—
200	#2140	—	79	1	no	—
201	#2139	—	79	1	no	—
202	#2138	D₇₉	1	1	yes	characteristic, complemented
203	C₂×#357	—	79	1	no	—
204	#2108	—	79	1	no	—
205	C₇₉	#33702	1	1	yes	characteristic, complemented, Sylow
206	C₂×AGL(1,ℤ/8)	—	158	1	no	—
207	C₂×#135	C₂×D₇₉	1	1	yes	characteristic
208	C₂²×D₈	—	79	1	no	—
209	#544	—	158	1	no	—
210	#535	—	158	1	no	—
211	C₂×M₃₂	—	79	1	no	—
212	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
213	#501	—	158	2	no	—
214	#500	—	79	2	no	—
215	#499	—	79	2	no	—
216	#496	—	158	2	no	—
217	#495	—	79	2	no	—
218	#493	—	79	2	no	—
219	#428	—	158	1	no	—
220	#427	C₂×D₇₉	1	1	yes	characteristic
221	C₂×#107	—	79	1	no	—
222	#405	—	79	1	no	—
223	#358	—	79	2	no	—
224	#358	C₂×D₇₉	1	1	yes	characteristic, complemented
225	#358	C₂×D₇₉	1	1	yes	characteristic, complemented
226	#357	—	79	2	no	—
227	#357	—	79	2	no	—
228	#357	—	79	2	no	—
229	#357	—	79	2	no	—
230	#357	—	79	2	no	—
231	#357	C₂×D₇₉	1	1	yes	characteristic, complemented
232	#357	C₂×D₇₉	1	1	yes	characteristic, complemented
233	#346	—	79	2	no	—
234	C₄×C₁₆	—	79	2	no	—
235	C₂×#41	—	2	1	no	—
236	C₂²×D₄	—	158	1	no	—
237	AGL(1,ℤ/8)	—	316	4	no	—
238	#135	—	2	2	no	—
239	#135	—	2	2	no	—
240	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
241	C₂×SD₁₆	—	2	1	no	—
242	C₂×SD₁₆	—	158	2	no	—
243	C₂×D₈	—	79	2	no	—
244	C₂×D₈	—	158	2	no	—
245	C₂×D₈	—	158	1	no	—
246	C₂×D₈	—	158	2	no	—
247	C₂×D₈	—	158	2	no	—
248	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
249	C₂×M₁₆	—	79	1	no	—
250	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
251	C₄⋊Q₈	—	316	1	no	—
252	C₄⋊Q₈	—	158	2	no	—
253	#127	—	158	2	no	—
254	#125	—	316	1	no	—
255	#125	—	158	2	no	—
256	#124	—	158	2	no	—
257	C₂²⋊Q₈	—	316	2	no	—
258	#121	—	316	1	no	—
259	#121	—	316	1	no	—
260	#120	—	316	2	no	—
261	#117	—	2	1	no	—
262	#117	—	79	1	no	—
263	C₂×C₄⋊C₄	—	158	1	no	—
264	M₃₂	—	158	2	no	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	—	79	2	no	—
267	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
269	#107	—	79	2	no	—
270	#107	—	158	2	no	—
271	#107	—	158	1	no	—
272	#107	C₂²×D₇₉	1	1	yes	characteristic
273	#107	C₂²×D₇₉	1	1	yes	characteristic
274	#106	—	2	1	no	—
275	#106	—	158	2	no	—
276	#97	—	79	2	no	—
277	C₄×C₈	—	79	2	no	—
278	C₂⁴	—	316	1	no	—
279	#41	—	4	2	no	—
280	#41	—	4	2	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	316	2	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	316	2	no	—
286	C₂×D₄	—	316	2	no	—
287	C₂×D₄	—	316	2	no	—
288	C₂×D₄	—	316	1	no	—
289	C₂×D₄	—	316	2	no	—
290	C₂×D₄	—	158	2	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	316	1	no	—
293	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
294	Q₁₆	—	2	1	no	—
295	Q₁₆	—	2	1	no	—
296	SD₁₆	—	4	2	no	—
297	SD₁₆	—	316	4	no	—
298	D₈	—	2	1	no	—
299	D₈	—	2	1	no	—
300	D₈	—	316	4	no	—
301	D₈	—	316	2	no	—
302	D₈	—	158	2	no	—
303	D₈	—	158	2	no	—
304	M₁₆	—	79	4	no	—
305	C₂×C₈	—	79	2	no	—
306	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
307	C₂×C₈	#11929623	1	2	yes	—
308	C₂×C₈	#11929624	1	2	yes	—
309	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	316	2	no	—
313	C₄⋊C₄	—	316	2	no	—
314	C₄⋊C₄	—	316	2	no	—
315	C₄⋊C₄	—	316	1	no	—
316	C₄⋊C₄	—	79	1	no	—
317	C₄⋊C₄	—	79	1	no	—
318	C₄⋊C₄	—	158	2	no	—
319	C₂²⋊C₄	—	4	1	no	—
320	C₂²⋊C₄	—	316	2	no	—
321	C₂²⋊C₄	—	316	2	no	—
322	C₂²⋊C₄	—	79	2	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	79	2	no	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	—	158	2	no	—
328	C₂³	—	4	1	no	—
329	C₂³	—	316	2	no	—
330	C₂³	—	632	2	no	—
331	C₂³	—	1264	1	no	—
332	C₂³	—	1264	1	no	—
333	C₂³	D₈×D₇₉	1	1	yes	characteristic
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	1264	2	no	—
337	D₄	—	4	1	no	—
338	D₄	—	4	1	no	—
339	D₄	—	8	2	no	—
340	D₄	—	316	2	no	—
341	D₄	—	316	2	no	—
342	D₄	—	632	2	no	—
343	D₄	—	1264	2	no	—
344	D₄	—	1264	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	316	2	no	—
350	C₂×C₄	—	79	2	no	—
351	C₂×C₄	—	79	2	no	—
352	C₂×C₄	—	1264	1	no	—
353	C₂×C₄	#14579147	1	2	yes	—
354	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
355	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
357	C₈	—	79	4	no	—
358	C₈	#14579147	1	2	yes	—
359	C₈	C₂×#11929623	1	1	yes	characteristic
360	C₈	C₂×#11929624	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	632	2	no	—
367	C₂²	—	632	2	no	—
368	C₂²	—	632	2	no	—
369	C₂²	—	1264	4	no	—
370	C₂²	#19098908	1	2	yes	—
371	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
372	C₄	—	8	1	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	1	no	—
375	C₄	—	632	2	no	—
376	C₄	—	158	2	no	—
377	C₄	—	158	2	no	—
378	C₄	#19098908	1	2	yes	—
379	C₄	C₂×#14579147	1	1	yes	characteristic
380	C₄	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	16	1	no	—
383	C₂	—	1264	2	no	—
384	C₂	C₂×#19098908	1	1	yes	characteristic
385	C₂	#23776538	1	1	yes	characteristic
386	C₂	#23776543	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2138	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23773887	C₂
hypercenter	#2138	D₇₉
hypocenter	C₇₉	#33702
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2138

Complements

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

N	H ≅ G/N	type
1. #23773940	C₂	semidirect
3. #23765640	C₂	semidirect
4. #23765638	C₂	semidirect
5. #23765633	C₂	semidirect
6. #23765622	C₂	semidirect
7. #23765115	C₂	semidirect
8. #23765112	C₂	semidirect
9. C₂×#19098365	C₂	semidirect
10. #23764996	C₂	semidirect
11. C₇₉×#2138	C₂	semidirect
18. #19098967	C₂²	semidirect
19. #19098964	C₂²	semidirect
26. #19098367	C₂²	semidirect
29. #19098364	C₂²	semidirect
32. #19098338	C₂²	semidirect
33. C₁₆×Dic₇₉	C₂²	semidirect
37. #358×C₇₉	C₂²	semidirect
38. #358×C₇₉	C₂²	semidirect
39. #357×C₇₉	C₂²	semidirect
40. #357×C₇₉	C₂²	semidirect
202. #2138	D₇₉	semidirect
205. C₇₉	#33702	semidirect
224. #358	C₂×D₇₉	semidirect
225. #358	C₂×D₇₉	semidirect
231. #357	C₂×D₇₉	semidirect
232. #357	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23773940	2
#23773887	2
#23765640	2
#23765638	2
#23765633	2
#23765622	2
#23765115	2
#23765112	2
C₂×#19098365	2
#23764996	2
C₇₉×#2138	2
#33702	79