Subgroups — #34203186

Order 20224.

586 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098598	C₂	1	1	yes	maximal, characteristic
3	#427×D₇₉	C₂	1	1	yes	maximal, characteristic
4	#23761507	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761506	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761505	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1386	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
9	C₂×#14579119	—	2	1	no	—
10	#117×D₇₉	—	2	1	no	—
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	C₂×#14579100	—	2	1	no	—
13	M₃₂×D₇₉	—	2	2	no	—
14	M₃₂×D₇₉	C₄	1	4	yes	complemented
15	C₂×C₁₆×D₇₉	—	2	1	no	—
16	#19098601	C₂²	1	1	yes	characteristic
17	#19098600	C₄	1	2	yes	complemented
18	#19098599	—	2	1	no	—
19	#19098598	—	2	1	no	—
20	#19098598	—	2	2	no	—
21	#19098598	C₄	1	4	yes	complemented
22	C₂×#14579094	—	2	1	no	—
23	#107×D₇₉	C₄	1	2	yes	complemented
24	C₂×#14579093	C₂²	1	1	yes	characteristic
25	#106×D₇₉	—	2	1	no	—
26	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
27	#427×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₇₉	D₄	1	1	yes	characteristic
30	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
36	C₂×C₈×D₇₉	D₄	1	2	yes	—
37	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
38	C₂×C₈×D₇₉	Q₈	1	2	yes	—
39	#14579120	—	2	1	no	—
40	#14579119	—	2	2	no	—
41	#14579119	—	2	4	no	—
42	#14579119	—	4	2	no	—
43	#14579118	—	4	1	no	—
44	#14579117	—	4	1	no	—
45	C₂×C₄⋊Dic₇₉	—	2	1	no	—
46	C₄⋊C₄×D₇₉	—	2	2	no	—
47	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
48	C₂²⋊C₄×D₇₉	—	4	1	no	—
49	C₂×C₄×Dic₇₉	—	4	1	no	—
50	C₄²×D₇₉	—	4	1	no	—
51	#14579101	—	2	1	no	—
52	#14579101	C₂×C₄	1	2	yes	complemented
53	#14579100	—	2	4	no	—
54	C₂×C₇₉⋊C₁₆	—	2	1	no	—
55	C₁₆×D₇₉	—	2	4	no	—
56	#14579095	C₂×C₄	1	2	yes	—
57	#14579094	—	2	2	no	—
58	#14579094	—	2	4	no	—
59	#14579093	—	2	1	no	—
60	#14579093	—	2	1	no	—
61	#14579093	C₂×C₄	1	2	yes	complemented
62	#14579092	—	2	2	no	—
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	#117×C₇₉	—	2	1	no	—
65	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
66	M₃₂×C₇₉	—	2	1	no	—
67	M₃₂×C₇₉	C₂×C₄	1	2	yes	complemented
68	C₂×C₁₆×C₇₉	—	2	1	no	—
69	#107×C₇₉	C₂×C₄	1	2	yes	—
70	#106×C₇₉	—	2	1	no	—
71	#106×C₇₉	—	2	1	no	—
72	#106×C₇₉	C₂×C₄	1	2	yes	complemented
73	C₂³×D₇₉	D₈	1	1	yes	characteristic
74	C₂²×Dic₇₉	—	4	1	no	—
75	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
76	C₂×C₄×D₇₉	—	4	2	no	—
77	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×D₇₉	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₇₉	SD₁₆	1	1	yes	characteristic
82	C₂×C₄×D₇₉	SD₁₆	1	1	yes	characteristic
83	C₂×C₄×D₇₉	SD₁₆	1	2	yes	—
84	C₂×C₄×D₇₉	Q₁₆	1	2	yes	—
85	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
89	C₈×D₇₉	C₄²	1	2	yes	—
90	C₈×D₇₉	C₄²	1	2	yes	—
91	C₈×D₇₉	C₂²⋊C₄	1	2	yes	—
92	C₈×D₇₉	C₂²⋊C₄	1	2	yes	—
93	C₈×D₇₉	C₄⋊C₄	1	2	yes	—
94	C₈×D₇₉	C₄⋊C₄	1	2	yes	—
95	C₈×D₇₉	C₄⋊C₄	1	2	yes	—
96	C₈×D₇₉	C₄⋊C₄	1	2	yes	—
97	C₄⋊Dic₇₉	—	2	2	no	—
98	C₄⋊Dic₇₉	—	2	2	no	—
99	C₄⋊Dic₇₉	—	4	1	no	—
100	Dic₇₉⋊C₄	—	4	2	no	—
101	C₂²⋊Dic₇₉	—	4	1	no	—
102	(C₂×D₇₉)⋊C₄	—	4	2	no	—
103	(C₂×D₇₉)⋊C₄	—	4	4	no	—
104	C₄×Dic₇₉	—	4	2	no	—
105	C₄×Dic₇₉	—	4	4	no	—
106	C₄×Dic₇₉	—	4	1	no	—
107	C₇₉⋊C₁₆	—	2	2	no	—
108	C₂²×C₄×C₇₉	—	4	1	no	—
109	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
116	C₄⋊C₄×C₇₉	—	2	2	no	—
117	C₄⋊C₄×C₇₉	—	2	2	no	—
118	C₄⋊C₄×C₇₉	—	4	1	no	—
119	C₂²⋊C₄×C₇₉	—	4	1	no	—
120	C₄²×C₇₉	—	4	1	no	—
121	C₁₆×C₇₉	—	2	2	no	—
122	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₇₉	—	4	2	no	—
124	C₂²×D₇₉	#102	1	1	yes	characteristic
125	C₂²×D₇₉	#102	1	1	yes	characteristic
126	C₂²×D₇₉	#102	1	2	yes	—
127	C₂²×D₇₉	#107	1	2	yes	—
128	C₂×C₄×C₇₉	—	4	2	no	—
129	C₂×C₄×C₇₉	—	4	2	no	—
130	C₂×C₄×C₇₉	—	8	2	no	—
131	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×Dic₇₉	—	4	2	no	—
138	C₂×Dic₇₉	—	4	2	no	—
139	C₂×Dic₇₉	—	8	2	no	—
140	C₂×Dic₇₉	#102	1	2	yes	—
141	C₂×Dic₇₉	#102	1	1	yes	characteristic
142	C₂×Dic₇₉	#102	1	1	yes	characteristic
143	C₂×Dic₇₉	#107	1	2	yes	—
144	C₄×D₇₉	—	8	4	no	—
145	C₄×D₇₉	#95	1	2	yes	—
146	C₄×D₇₉	#95	1	2	yes	—
147	C₄×D₇₉	#103	1	2	yes	—
148	C₄×D₇₉	#103	1	2	yes	—
149	C₄×D₇₉	#103	1	2	yes	—
150	C₄×D₇₉	#103	1	2	yes	—
151	C₄×D₇₉	#106	1	2	yes	—
152	C₄×D₇₉	#106	1	2	yes	—
153	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
154	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₇₉⋊C₈	#95	1	2	yes	—
158	C₇₉⋊C₈	#95	1	2	yes	—
159	C₂²×C₇₉	—	2	2	no	—
160	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
161	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
162	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
163	C₂×D₇₉	—	2	2	no	—
164	C₂×D₇₉	—	2	2	no	—
165	C₂×D₇₉	—	4	2	no	—
166	C₂×D₇₉	—	4	2	no	—
167	C₂×D₇₉	#340	1	2	yes	—
168	C₂×D₇₉	#340	1	2	yes	—
169	C₄×C₇₉	—	8	2	no	—
170	C₄×C₇₉	—	8	2	no	—
171	C₄×C₇₉	C₂×#95	1	1	yes	characteristic
172	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
173	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
174	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
175	Dic₇₉	—	8	2	no	—
176	Dic₇₉	—	8	2	no	—
177	Dic₇₉	#340	1	2	yes	—
178	Dic₇₉	#340	1	2	yes	—
179	#12567	—	79	1	no	maximal, Sylow
180	C₂×C₇₉	—	2	1	no	—
181	C₂×C₇₉	—	2	1	no	—
182	C₂×C₇₉	C₂×#340	1	1	yes	characteristic
183	C₂×C₇₉	#2145	1	2	yes	—
184	D₇₉	—	2	2	no	—
185	D₇₉	—	2	2	no	—
186	C₂²×M₃₂	—	79	1	no	—
187	C₂×#427	—	79	1	no	—
188	C₂×#427	—	79	1	no	—
189	#1386	—	79	1	no	—
190	#1386	—	79	1	no	—
191	#1386	—	79	1	no	—
192	#1386	D₇₉	1	1	yes	characteristic, complemented
193	C₇₉	#12567	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₈	—	79	1	no	—
195	C₂×#117	—	158	1	no	—
196	C₂×#117	—	158	1	no	—
197	C₂×M₃₂	—	79	1	no	—
198	C₂×M₃₂	—	79	4	no	—
199	C₂×M₃₂	—	158	1	no	—
200	C₂×M₃₂	—	158	2	no	—
201	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
202	C₂²×C₁₆	—	158	1	no	—
203	#427	—	79	4	no	—
204	#427	—	79	1	no	—
205	#427	—	79	2	no	—
206	#427	—	158	1	no	—
207	#427	—	158	2	no	—
208	#427	—	158	1	no	—
209	#427	C₂×D₇₉	1	1	yes	characteristic
210	C₂×#107	—	79	2	no	—
211	C₂×#107	—	158	1	no	—
212	C₂×#106	—	79	1	no	—
213	C₂×#106	—	158	1	no	—
214	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
215	C₂³×C₄	—	79	1	no	—
216	C₂²×C₈	—	79	1	no	—
217	C₂²×C₈	—	79	1	no	—
218	C₂²×C₈	—	79	1	no	—
219	C₂²×C₈	—	79	2	no	—
220	C₂²×C₈	—	79	2	no	—
221	C₂²×C₈	—	79	1	no	—
222	C₂²×C₈	—	79	1	no	—
223	C₂²×C₈	—	79	2	no	—
224	C₂²×C₈	—	79	2	no	—
225	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
226	#117	—	2	1	no	—
227	#117	—	316	1	no	—
228	#117	—	316	1	no	—
229	#117	—	316	2	no	—
230	#117	—	158	2	no	—
231	#117	—	158	4	no	—
232	#117	—	158	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	158	2	no	—
235	C₂×C₄⋊C₄	—	158	1	no	—
236	C₂×C₂²⋊C₄	—	316	1	no	—
237	C₂×C₂²⋊C₄	—	316	1	no	—
238	C₂×C₄²	—	316	1	no	—
239	C₂×C₄²	—	316	1	no	—
240	M₃₂	—	2	1	no	—
241	M₃₂	—	79	2	no	—
242	M₃₂	—	158	4	no	—
243	M₃₂	—	158	1	no	—
244	M₃₂	C₄×D₇₉	1	2	yes	complemented
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	158	4	no	—
247	C₂×C₁₆	—	158	1	no	—
248	#107	—	79	2	no	—
249	#107	—	158	2	no	—
250	#107	—	158	4	no	—
251	#107	C₄×D₇₉	1	2	yes	—
252	#106	—	2	1	no	—
253	#106	—	2	1	no	—
254	#106	—	79	2	no	—
255	#106	—	158	2	no	—
256	#106	—	158	1	no	—
257	#106	—	158	1	no	—
258	#106	C₄×D₇₉	1	2	yes	complemented
259	C₂⁴	—	79	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	316	2	no	—
262	C₂²×C₄	—	316	1	no	—
263	C₂²×C₄	—	79	1	no	—
264	C₂²×C₄	—	79	1	no	—
265	C₂²×C₄	—	79	2	no	—
266	C₂²×C₄	—	79	1	no	—
267	C₂²×C₄	—	79	2	no	—
268	C₂²×C₄	—	79	1	no	—
269	C₂²×C₄	—	79	2	no	—
270	C₂²×C₄	—	79	1	no	—
271	C₂²×C₄	—	79	2	no	—
272	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
273	C₂×C₈	—	79	1	no	—
274	C₂×C₈	—	79	2	no	—
275	C₂×C₈	—	79	1	no	—
276	C₂×C₈	—	79	2	no	—
277	C₂×C₈	—	79	2	no	—
278	C₂×C₈	—	79	2	no	—
279	C₂×C₈	—	79	2	no	—
280	C₂×C₈	—	79	2	no	—
281	C₂×C₈	—	79	2	no	—
282	C₂×C₈	—	79	2	no	—
283	C₂×C₈	—	79	2	no	—
284	C₂×C₈	—	79	2	no	—
285	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
286	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
288	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
289	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
290	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	316	2	no	—
295	C₄⋊C₄	—	316	1	no	—
296	C₄⋊C₄	—	158	2	no	—
297	C₄⋊C₄	—	158	2	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	316	4	no	—
300	C₂²⋊C₄	—	316	2	no	—
301	C₂²⋊C₄	—	316	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	316	4	no	—
304	C₄²	—	316	2	no	—
305	C₄²	—	316	1	no	—
306	C₁₆	—	2	2	no	—
307	C₁₆	—	158	2	no	—
308	C₂³	—	316	2	no	—
309	C₂³	—	79	2	no	—
310	C₂³	—	79	2	no	—
311	C₂³	—	79	1	no	—
312	C₂³	—	79	1	no	—
313	C₂³	D₈×D₇₉	1	1	yes	characteristic
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	316	2	no	—
318	C₂×C₄	—	316	2	no	—
319	C₂×C₄	—	79	2	no	—
320	C₂×C₄	—	79	1	no	—
321	C₂×C₄	—	79	1	no	—
322	C₂×C₄	—	79	2	no	—
323	C₂×C₄	—	79	2	no	—
324	C₂×C₄	—	79	2	no	—
325	C₂×C₄	—	79	2	no	—
326	C₂×C₄	—	79	2	no	—
327	C₂×C₄	—	79	2	no	—
328	C₂×C₄	—	79	2	no	—
329	C₂×C₄	—	79	2	no	—
330	C₂×C₄	—	79	2	no	—
331	C₂×C₄	—	632	4	no	—
332	C₂×C₄	—	632	2	no	—
333	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
334	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
335	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
336	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
337	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
338	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
339	C₈	—	79	2	no	—
340	C₈	—	79	2	no	—
341	C₈	C₄²×D₇₉	1	1	yes	characteristic
342	C₈	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
343	C₈	C₄⋊C₄×D₇₉	1	1	yes	characteristic
344	C₈	C₄⋊C₄×D₇₉	1	1	yes	characteristic
345	C₂²	—	2	2	no	—
346	C₂²	—	316	2	no	—
347	C₂²	—	316	2	no	—
348	C₂²	—	79	2	no	—
349	C₂²	—	79	2	no	—
350	C₂²	—	158	2	no	—
351	C₂²	—	158	2	no	—
352	C₂²	#102×D₇₉	1	1	yes	characteristic
353	C₂²	#102×D₇₉	1	1	yes	characteristic
354	C₂²	#107×D₇₉	1	1	yes	characteristic
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	79	2	no	—
358	C₄	—	79	2	no	—
359	C₄	—	632	2	no	—
360	C₄	—	632	2	no	—
361	C₄	#95×D₇₉	1	1	yes	characteristic
362	C₄	#103×D₇₉	1	1	yes	characteristic
363	C₄	#103×D₇₉	1	1	yes	characteristic
364	C₄	#106×D₇₉	1	1	yes	characteristic
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	158	2	no	—
368	C₂	—	158	2	no	—
369	C₂	#340×D₇₉	1	1	yes	characteristic
370	C₂	#23765134	1	2	yes	—

	subgroup	G/H
center	C₂²	#102×D₇₉
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1386	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×D₇₉	D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1386	D₇₉
hypocenter	C₇₉	#12567
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1386

Complements

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

N	H ≅ G/N	type
4. #23761507	C₂	semidirect
5. #23761506	C₂	semidirect
6. #23761505	C₂	semidirect
7. C₇₉×#1386	C₂	semidirect
14. M₃₂×D₇₉	C₄	semidirect
17. #19098600	C₄	semidirect
21. #19098598	C₄	semidirect
23. #107×D₇₉	C₄	semidirect
52. #14579101	C₂×C₄	semidirect
61. #14579093	C₂×C₄	semidirect
67. M₃₂×C₇₉	C₂×C₄	semidirect
72. #106×C₇₉	C₂×C₄	semidirect
192. #1386	D₇₉	semidirect
193. C₇₉	#12567	semidirect
244. M₃₂	C₄×D₇₉	semidirect
258. #106	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₇₉	2
C₂×#19098598	2
#427×D₇₉	2
#23761507	2
#23761506	2
#23761505	2
C₇₉×#1386	2
#12567	79