Subgroups — #34258798

Order 20224.

614 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773983	C₂	1	1	yes	maximal, characteristic
2	#23772331	C₂	1	2	yes	maximal
3	#23762298	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762295	C₂	1	2	yes	maximal, complemented
5	Q₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762206	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762204	C₂	1	2	yes	maximal, complemented
8	#23762157	C₂	1	1	yes	maximal, characteristic
9	#23761853	C₂	1	2	yes	maximal
10	C₈²×D₇₉	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099117	C₂²	1	2	yes	—
13	#19098967	C₄	1	1	yes	characteristic
14	#19098941	C₄	1	1	yes	characteristic
15	#19098912	C₄	1	2	yes	—
16	#19098805	C₄	1	1	yes	characteristic
17	#19098803	C₄	1	2	yes	—
18	#19098801	C₄	1	1	yes	characteristic
19	#19098693	C₂²	1	2	yes	—
20	#19098691	—	2	2	no	—
21	Q₈×C₇₉⋊C₈	C₂²	1	2	yes	—
22	#19098687	—	2	2	no	—
23	#19098687	C₂²	1	2	yes	—
24	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
25	C₄×#11929614	C₂²	1	2	yes	—
26	#19098588	—	2	2	no	—
27	#19098585	C₂²	1	2	yes	—
28	#19098520	—	2	2	no	—
29	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
32	#19098316	C₂²	1	1	yes	characteristic
33	#19098315	C₂²	1	2	yes	—
34	#19098295	C₂²	1	2	yes	—
35	Q₈⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
36	#19098293	C₂²	1	2	yes	—
37	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
38	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
40	#446×C₇₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
42	#335×C₇₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₉	C₂²	1	2	yes	—
44	C₈²×C₇₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11929614	C₂×C₄	1	2	yes	—
47	C₂×#11929607	—	2	2	no	—
48	C₂×C₈×D₇₉	—	2	1	no	—
49	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
52	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
53	#14579160	C₂×C₄	1	2	yes	—
54	#14579159	C₂×C₄	1	2	yes	—
55	#14579151	—	2	2	no	—
56	#14579141	C₂×C₄	1	2	yes	—
57	#14579139	—	2	2	no	—
58	#14579137	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₅₈	—	2	2	no	—
60	C₄×Dic₁₅₈	C₂³	1	2	yes	—
61	#14579118	—	2	2	no	—
62	C₄²×D₇₉	C₂³	1	1	yes	characteristic
63	#14579095	C₂×C₄	1	1	yes	characteristic
64	#14579094	C₂×C₄	1	2	yes	—
65	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
66	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
67	Dic₇₉⋊C₈	—	2	2	no	—
68	Dic₇₉⋊C₈	—	2	2	no	—
69	Dic₇₉⋊C₈	—	2	2	no	—
70	#14579086	C₂×C₄	1	2	yes	—
71	#14579085	C₂×C₄	1	2	yes	—
72	#14579084	C₂×C₄	1	2	yes	—
73	#14579071	—	2	2	no	—
74	C₄×C₇₉⋊C₈	—	2	2	no	—
75	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×Dic₇₉	—	2	2	no	—
79	C₈×Dic₇₉	—	2	1	no	—
80	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₇₉	D₄	1	1	yes	characteristic
83	C₈×Dic₇₉	D₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₇₉	C₂³	1	2	yes	—
86	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₇₉	—	2	2	no	—
88	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
89	#103×C₇₉	C₂×C₄	1	2	yes	—
90	#97×C₇₉	—	2	2	no	—
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₅₈	—	2	2	no	—
95	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₇₉	#41	1	1	yes	characteristic
99	Dic₃₁₆	—	2	2	no	—
100	#11929614	—	2	4	no	—
101	#11929607	—	4	4	no	—
102	C₂×C₇₉⋊C₈	—	2	2	no	—
103	C₂×C₇₉⋊C₈	—	2	1	no	—
104	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₈×D₇₉	—	2	4	no	—
109	C₈×D₇₉	—	2	2	no	—
110	C₈×D₇₉	—	2	2	no	—
111	C₈×D₇₉	—	2	2	no	—
112	C₈×D₇₉	Q₁₆	1	4	yes	complemented
113	C₄⋊Dic₇₉	—	2	2	no	—
114	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
115	Dic₇₉⋊C₄	—	2	2	no	—
116	Dic₇₉⋊C₄	—	2	2	no	—
117	Dic₇₉⋊C₄	—	2	2	no	—
118	Dic₇₉⋊C₄	—	2	2	no	—
119	Dic₇₉⋊C₄	—	2	2	no	—
120	(C₂×D₇₉)⋊C₄	—	2	2	no	—
121	(C₂×D₇₉)⋊C₄	—	2	2	no	—
122	C₄×Dic₇₉	—	2	2	no	—
123	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₇₉	#41	1	1	yes	characteristic
126	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
127	Q₁₆×C₇₉	—	2	2	no	—
128	C₂×C₈×C₇₉	—	2	2	no	—
129	C₂×C₈×C₇₉	—	2	1	no	—
130	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₇₉	—	2	2	no	—
135	C₄⋊C₄×C₇₉	C₂²×C₄	1	2	yes	—
136	C₄²×C₇₉	—	2	2	no	—
137	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
138	C₂²×D₇₉	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₇₉	—	2	2	no	—
140	Q₈×C₇₉	—	2	2	no	—
141	C₂×C₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₇₉	—	2	2	no	—
147	C₂×Dic₇₉	—	2	2	no	—
148	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
150	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
151	C₄×D₇₉	—	2	1	no	—
152	C₄×D₇₉	—	2	1	no	—
153	C₄×D₇₉	—	2	2	no	—
154	C₄×D₇₉	C₂×Q₁₆	1	2	yes	—
155	C₄×D₇₉	#135	1	2	yes	—
156	Dic₁₅₈	—	2	2	no	—
157	Dic₁₅₈	—	2	2	no	—
158	Dic₁₅₈	—	4	4	no	—
159	C₈×C₇₉	—	2	2	no	—
160	C₈×C₇₉	—	2	1	no	—
161	C₈×C₇₉	—	2	1	no	—
162	C₈×C₇₉	—	4	2	no	—
163	C₈×C₇₉	#131	1	2	yes	—
164	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
165	C₇₉⋊C₈	—	2	2	no	—
166	C₇₉⋊C₈	—	2	1	no	—
167	C₇₉⋊C₈	—	2	1	no	—
168	C₇₉⋊C₈	—	4	2	no	—
169	C₇₉⋊C₈	#131	1	2	yes	—
170	C₇₉⋊C₈	C₂×Q₁₆	1	2	yes	—
171	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₇₉	—	2	1	no	—
173	C₂×D₇₉	—	2	1	no	—
174	C₂×D₇₉	C₄×Q₁₆	1	2	yes	—
175	C₄×C₇₉	—	2	1	no	—
176	C₄×C₇₉	—	4	2	no	—
177	C₄×C₇₉	—	4	2	no	—
178	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
179	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
180	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
181	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
182	Dic₇₉	—	2	1	no	—
183	Dic₇₉	—	4	2	no	—
184	Dic₇₉	—	4	2	no	—
185	Dic₇₉	C₄×Q₁₆	1	2	yes	—
186	Dic₇₉	#443	1	2	yes	—
187	#17958	—	79	1	no	maximal, Sylow
188	C₂×C₇₉	#2931	1	1	yes	characteristic
189	C₂×C₇₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
190	C₂×C₇₉	C₂×#443	1	1	yes	characteristic
191	D₇₉	—	2	2	no	—
192	#3170	—	79	1	no	—
193	#2960	—	79	2	no	—
194	#1591	—	79	2	no	—
195	#1591	—	79	1	no	—
196	#1591	D₇₉	1	1	yes	characteristic, complemented
197	C₈×Q₁₆	—	79	1	no	—
198	C₈×Q₁₆	—	79	1	no	—
199	C₈×Q₁₆	—	79	2	no	—
200	#1567	—	79	1	no	—
201	#1493	—	79	2	no	—
202	C₂×C₈²	—	79	1	no	—
203	C₇₉	#17958	1	1	yes	characteristic, complemented, Sylow
204	#533	—	79	2	no	—
205	#500	—	79	1	no	—
206	#494	—	79	1	no	—
207	#487	—	79	2	no	—
208	#467	—	79	1	no	—
209	#467	—	79	1	no	—
210	#467	—	79	2	no	—
211	#446	—	79	2	no	—
212	#446	—	158	2	no	—
213	#446	C₂×D₇₉	1	2	yes	—
214	C₈×Q₈	—	79	2	no	—
215	C₈×Q₈	—	79	2	no	—
216	C₈×Q₈	—	158	2	no	—
217	C₄×Q₁₆	—	79	2	no	—
218	C₄×Q₁₆	—	79	1	no	—
219	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
220	#424	—	158	2	no	—
221	#423	—	79	2	no	—
222	#405	—	158	2	no	—
223	C₂×C₄×C₈	—	79	1	no	—
224	C₂×C₄×C₈	—	79	1	no	—
225	C₂×C₄×C₈	—	79	1	no	—
226	#335	—	79	2	no	—
227	#335	—	79	1	no	—
228	#335	C₂×D₇₉	1	1	yes	characteristic
229	Q₈⋊C₈	—	79	2	no	—
230	Q₈⋊C₈	—	79	2	no	—
231	Q₈⋊C₈	—	79	2	no	—
232	Q₈⋊C₈	C₂×D₇₉	1	2	yes	—
233	C₈²	—	79	1	no	—
234	C₈²	—	79	1	no	—
235	C₈²	—	79	1	no	—
236	C₈²	C₂×D₇₉	1	1	yes	characteristic
237	C₂×Q₁₆	—	79	2	no	—
238	C₂×Q₁₆	—	79	1	no	—
239	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
240	C₂×M₁₆	—	158	2	no	—
241	C₂²×C₈	—	79	1	no	—
242	C₂²×C₈	—	79	1	no	—
243	C₂²×C₈	—	79	1	no	—
244	C₂²×C₈	—	79	1	no	—
245	C₂²×C₈	—	158	1	no	—
246	C₄⋊Q₈	—	79	2	no	—
247	C₄⋊Q₈	—	79	2	no	—
248	#125	—	158	2	no	—
249	C₂²⋊Q₈	—	79	2	no	—
250	C₂²⋊Q₈	—	79	2	no	—
251	C₂²⋊Q₈	—	158	2	no	—
252	C₄×Q₈	—	79	2	no	—
253	C₄×Q₈	—	158	2	no	—
254	C₄×Q₈	C₂²×D₇₉	1	2	yes	—
255	#117	—	158	2	no	—
256	C₂×C₄²	—	79	1	no	—
257	#107	—	79	1	no	—
258	#107	—	79	2	no	—
259	#107	C₄×D₇₉	1	1	yes	characteristic
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	79	2	no	—
262	C₄⋊C₈	—	158	2	no	—
263	C₄⋊C₈	—	158	2	no	—
264	C₄⋊C₈	—	158	2	no	—
265	C₄⋊C₈	—	158	2	no	—
266	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
267	#103	—	79	2	no	—
268	#103	—	79	2	no	—
269	#103	—	79	2	no	—
270	#103	C₄×D₇₉	1	2	yes	—
271	#97	—	2	2	no	—
272	#97	—	158	2	no	—
273	C₄×C₈	—	79	1	no	—
274	C₄×C₈	—	79	1	no	—
275	C₄×C₈	—	79	1	no	—
276	C₄×C₈	—	79	1	no	—
277	C₄×C₈	—	79	1	no	—
278	C₄×C₈	—	79	1	no	—
279	C₄×C₈	—	79	1	no	—
280	C₄×C₈	—	158	2	no	—
281	C₄×C₈	—	158	1	no	—
282	C₄×C₈	—	158	2	no	—
283	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
286	C₂×Q₈	—	79	2	no	—
287	C₂×Q₈	—	158	2	no	—
288	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
289	C₂²×C₄	—	79	1	no	—
290	C₂²×C₄	—	79	1	no	—
291	C₂²×C₄	—	79	1	no	—
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	158	4	no	—
294	Q₁₆	—	158	2	no	—
295	M₁₆	—	316	4	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	79	1	no	—
299	C₂×C₈	—	79	1	no	—
300	C₂×C₈	—	79	4	no	—
301	C₂×C₈	—	79	1	no	—
302	C₂×C₈	—	79	1	no	—
303	C₂×C₈	—	158	2	no	—
304	C₂×C₈	—	158	2	no	—
305	C₂×C₈	—	158	4	no	—
306	C₂×C₈	—	158	2	no	—
307	C₂×C₈	—	158	1	no	—
308	C₂×C₈	—	158	2	no	—
309	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
310	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
311	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
312	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	79	2	no	—
315	C₄⋊C₄	—	158	2	no	—
316	C₄⋊C₄	—	158	2	no	—
317	C₄⋊C₄	—	158	2	no	—
318	C₄⋊C₄	—	158	2	no	—
319	C₄⋊C₄	—	158	2	no	—
320	C₄⋊C₄	—	158	2	no	—
321	C₄⋊C₄	C₂×C₄×D₇₉	1	2	yes	—
322	C₂²⋊C₄	—	158	2	no	—
323	C₂²⋊C₄	—	158	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	79	1	no	—
326	C₄²	—	79	1	no	—
327	C₄²	—	79	1	no	—
328	C₄²	—	158	2	no	—
329	C₄²	C₂³×D₇₉	1	1	yes	characteristic
330	C₂³	—	79	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	316	4	no	—
334	Q₈	—	158	2	no	—
335	Q₈	—	158	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	79	1	no	—
339	C₂×C₄	—	79	1	no	—
340	C₂×C₄	—	79	2	no	—
341	C₂×C₄	—	79	2	no	—
342	C₂×C₄	—	79	1	no	—
343	C₂×C₄	—	158	2	no	—
344	C₂×C₄	—	158	2	no	—
345	C₂×C₄	—	158	2	no	—
346	C₂×C₄	—	158	1	no	—
347	C₂×C₄	—	158	1	no	—
348	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
350	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	4	2	no	—
355	C₈	—	316	2	no	—
356	C₈	—	79	2	no	—
357	C₈	—	79	2	no	—
358	C₈	—	158	2	no	—
359	C₈	—	158	1	no	—
360	C₈	—	158	1	no	—
361	C₈	#14579169	1	2	yes	—
362	C₈	Q₁₆×D₇₉	1	2	yes	—
363	C₂²	—	79	2	no	—
364	C₂²	—	158	1	no	—
365	C₂²	—	158	1	no	—
366	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	316	2	no	—
371	C₄	—	316	2	no	—
372	C₄	—	79	2	no	—
373	C₄	—	79	2	no	—
374	C₄	—	158	1	no	—
375	C₄	#131×D₇₉	1	1	yes	characteristic
376	C₄	C₂×#14579169	1	1	yes	characteristic
377	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
378	C₄	#135×D₇₉	1	1	yes	characteristic
379	C₂	—	158	2	no	—
380	C₂	#23772109	1	1	yes	characteristic
381	C₂	C₄×Q₁₆×D₇₉	1	1	yes	characteristic
382	C₂	#443×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1591	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂²×D₇₉	C₄×D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1591	D₇₉
hypocenter	C₇₉	#17958
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #23762298	C₂	semidirect
4. #23762295	C₂	semidirect
5. Q₁₆×C₇₉⋊C₈	C₂	semidirect
6. #23762206	C₂	semidirect
7. #23762204	C₂	semidirect
11. C₇₉×#1591	C₂	semidirect
112. C₈×D₇₉	Q₁₆	semidirect
196. #1591	D₇₉	semidirect
203. C₇₉	#17958	semidirect

Maximal subgroups

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

subgroup	index
#23773983	2
#23772331	2
#23762298	2
#23762295	2
Q₁₆×C₇₉⋊C₈	2
#23762206	2
#23762204	2
#23762157	2
#23761853	2
C₈²×D₇₉	2
C₇₉×#1591	2
#17958	79