Subgroups — #34260271

Order 20224.

606 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	#23773968	C₂	1	1	yes	maximal, characteristic
2	#23772561	C₂	1	2	yes	maximal, complemented
3	#23762296	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762276	C₂	1	2	yes	maximal, complemented
5	#23762254	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#335×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#23761810	C₂	1	2	yes	maximal, complemented
10	C₈×#11929607	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1589	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099095	C₂²	1	2	yes	—
13	#19098964	C₄	1	1	yes	characteristic
14	#19098946	C₄	1	1	yes	characteristic
15	#19098916	C₄	1	2	yes	—
16	#19098805	C₄	1	1	yes	characteristic
17	#19098796	C₄	1	1	yes	characteristic
18	#19098791	C₄	1	2	yes	—
19	#19098690	C₂²	1	2	yes	—
20	C₈×Dic₁₅₈	C₂²	1	2	yes	—
21	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
22	C₄×#11929612	C₂²	1	2	yes	—
23	#19098632	—	2	2	no	—
24	D₄×C₇₉⋊C₈	C₂²	1	2	yes	—
25	(C₂²⋊D₇₉)⋊C₈	—	2	2	no	—
26	#19098621	—	2	2	no	—
27	#19098614	C₂²	1	1	yes	characteristic
28	#19098591	—	2	2	no	—
29	C₄⋊C₈×D₇₉	C₂²	1	2	yes	—
30	C₄×#11929607	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	#19098293	C₂²	1	2	yes	—
36	D₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
37	#19098285	C₂²	1	1	yes	characteristic
38	#19098284	C₂²	1	1	yes	characteristic
39	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
41	#436×C₇₉	C₂²	1	2	yes	—
42	#335×C₇₉	C₂²	1	1	yes	characteristic
43	D₄⋊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×#11929612	C₂×C₄	1	2	yes	—
47	C₂×#11929607	—	2	1	no	—
48	C₂²×C₇₉⋊C₈	—	2	2	no	—
49	C₂×C₈×D₇₉	—	2	2	no	—
50	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
52	#14579154	—	2	2	no	—
53	#14579152	C₂×C₄	1	2	yes	—
54	#14579148	C₂×C₄	1	2	yes	—
55	#14579143	—	2	2	no	—
56	#14579141	C₂×C₄	1	2	yes	—
57	#14579132	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₅₈	C₂³	1	2	yes	—
59	C₄×C₂²⋊D₇₉	—	2	2	no	—
60	#14579120	—	2	2	no	—
61	C₄²×D₇₉	C₂³	1	1	yes	characteristic
62	#14579095	C₂×C₄	1	1	yes	characteristic
63	#14579092	C₂×C₄	1	2	yes	—
64	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
65	Dic₇₉⋊C₈	—	2	2	no	—
66	Dic₇₉⋊C₈	—	2	2	no	—
67	Dic₇₉⋊C₈	—	2	2	no	—
68	Dic₇₉⋊C₈	D₄	1	1	yes	characteristic
69	Dic₇₉⋊C₈	D₄	1	1	yes	characteristic
70	#14579086	C₂×C₄	1	2	yes	—
71	#14579084	C₂×C₄	1	2	yes	—
72	#14579082	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
74	(C₂×D₇₉)⋊C₈	D₄	1	1	yes	characteristic
75	(C₂×D₇₉)⋊C₈	D₄	1	1	yes	characteristic
76	#14579071	—	2	2	no	—
77	#14579071	—	2	1	no	—
78	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₇₉	—	2	2	no	—
82	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
84	C₂×D₈×C₇₉	C₂×C₄	1	1	yes	characteristic
85	C₂×M₁₆×C₇₉	—	2	2	no	—
86	C₄×D₄×C₇₉	C₂³	1	2	yes	—
87	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
89	#102×C₇₉	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈×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	C₂²×C₄	1	2	yes	—
95	C₂×C₂²⋊D₇₉	—	2	2	no	—
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	#11929612	—	4	2	no	—
101	#11929607	—	2	4	no	—
102	C₂×C₇₉⋊C₈	—	2	2	no	—
103	C₂×C₇₉⋊C₈	—	2	1	no	—
104	C₂×C₇₉⋊C₈	—	8	2	no	—
105	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₈×D₇₉	—	2	2	no	—
110	C₈×D₇₉	—	4	1	no	—
111	C₈×D₇₉	—	8	2	no	—
112	C₄⋊Dic₇₉	—	2	2	no	—
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	C₂²⋊Dic₇₉	—	2	2	no	—
118	C₂²⋊Dic₇₉	—	2	2	no	—
119	C₂²⋊Dic₇₉	—	2	2	no	—
120	(C₂×D₇₉)⋊C₄	—	2	2	no	—
121	C₄×Dic₇₉	—	2	2	no	—
122	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₇₉	#41	1	1	yes	characteristic
125	C₂×D₄×C₇₉	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₇₉	—	2	2	no	—
127	D₈×C₇₉	—	2	2	no	—
128	M₁₆×C₇₉	—	8	2	no	—
129	C₂×C₈×C₇₉	—	2	2	no	—
130	C₂×C₈×C₇₉	—	2	1	no	—
131	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₇₉	C₂²×C₄	1	2	yes	—
136	C₂²⋊C₄×C₇₉	—	2	2	no	—
137	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
138	C₂³×C₇₉	—	2	2	no	—
139	C₂²×D₇₉	C₄×D₄	1	1	yes	characteristic
140	D₄×C₇₉	—	2	2	no	—
141	D₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	—	4	2	no	—
144	C₂×C₄×C₇₉	—	4	2	no	—
145	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₇₉	—	4	4	no	—
149	C₂×Dic₇₉	—	2	2	no	—
150	C₂×Dic₇₉	—	2	2	no	—
151	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
154	C₄×D₇₉	—	2	2	no	—
155	C₄×D₇₉	—	2	1	no	—
156	C₄×D₇₉	—	2	1	no	—
157	C₄×D₇₉	—	2	1	no	—
158	C₄×D₇₉	—	2	1	no	—
159	Dic₁₅₈	—	2	2	no	—
160	Dic₁₅₈	—	2	2	no	—
161	C₈×C₇₉	—	2	2	no	—
162	C₈×C₇₉	—	2	1	no	—
163	C₈×C₇₉	—	2	1	no	—
164	C₈×C₇₉	—	2	1	no	—
165	C₈×C₇₉	—	4	2	no	—
166	C₈×C₇₉	#131	1	2	yes	—
167	C₇₉⋊C₈	—	2	2	no	—
168	C₇₉⋊C₈	—	2	1	no	—
169	C₇₉⋊C₈	—	2	1	no	—
170	C₇₉⋊C₈	—	2	1	no	—
171	C₇₉⋊C₈	—	2	1	no	—
172	C₇₉⋊C₈	—	4	2	no	—
173	C₂²×C₇₉	—	4	2	no	—
174	C₂²×C₇₉	—	4	2	no	—
175	C₂²×C₇₉	—	4	2	no	—
176	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×D₇₉	—	2	1	no	—
178	C₂×D₇₉	—	2	1	no	—
179	C₂×D₇₉	#443	1	2	yes	—
180	C₄×C₇₉	—	2	1	no	—
181	C₄×C₇₉	—	4	2	no	—
182	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
183	C₄×C₇₉	#568	1	1	yes	characteristic
184	C₄×C₇₉	#576	1	1	yes	characteristic
185	C₄×C₇₉	#578	1	1	yes	characteristic
186	Dic₇₉	—	2	1	no	—
187	Dic₇₉	—	2	1	no	—
188	Dic₇₉	—	2	1	no	—
189	Dic₇₉	—	4	2	no	—
190	Dic₇₉	—	4	2	no	—
191	#18087	—	79	1	no	maximal, Sylow
192	C₂×C₇₉	—	8	2	no	—
193	C₂×C₇₉	#2935	1	1	yes	characteristic
194	C₂×C₇₉	#2951	1	1	yes	characteristic
195	C₂×C₇₉	C₂×#443	1	1	yes	characteristic
196	D₇₉	—	2	2	no	—
197	#3168	—	79	1	no	—
198	#2994	—	79	2	no	—
199	#1591	—	79	1	no	—
200	#1589	D₇₉	1	1	yes	characteristic, complemented
201	#1588	—	79	2	no	—
202	#1585	—	79	2	no	—
203	C₈×Q₁₆	—	79	1	no	—
204	C₈×D₈	—	79	1	no	—
205	C₂×#335	—	79	1	no	—
206	#1483	—	79	2	no	—
207	C₈×M₁₆	—	79	1	no	—
208	C₇₉	#18087	1	1	yes	characteristic, complemented, Sylow
209	#529	—	79	2	no	—
210	#499	—	79	1	no	—
211	#495	—	79	1	no	—
212	#488	—	79	2	no	—
213	#467	—	79	1	no	—
214	#466	—	79	1	no	—
215	#465	—	79	2	no	—
216	#446	—	79	2	no	—
217	C₈×Q₈	—	79	2	no	—
218	C₄×Q₁₆	—	79	1	no	—
219	C₄×SD₁₆	—	79	2	no	—
220	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
221	#436	C₂×D₇₉	1	2	yes	—
222	#435	—	158	2	no	—
223	C₈×D₄	—	79	2	no	—
224	C₈×D₄	—	158	2	no	—
225	#433	—	158	2	no	—
226	#431	—	79	1	no	—
227	#424	—	158	2	no	—
228	C₂×C₄⋊C₈	—	79	2	no	—
229	C₄×M₁₆	—	79	1	no	—
230	C₂×C₄×C₈	—	79	1	no	—
231	#335	—	79	2	no	—
232	#335	—	79	1	no	—
233	#335	C₂×D₇₉	1	1	yes	characteristic
234	Q₈⋊C₈	—	79	2	no	—
235	Q₈⋊C₈	—	79	2	no	—
236	D₄⋊C₈	—	79	2	no	—
237	D₄⋊C₈	C₂×D₇₉	1	2	yes	—
238	#322	—	79	1	no	—
239	#322	—	79	1	no	—
240	C₈²	—	79	1	no	—
241	C₈²	C₂×D₇₉	1	1	yes	characteristic
242	C₂×Q₁₆	—	79	1	no	—
243	C₂×SD₁₆	—	79	2	no	—
244	C₂×D₈	C₄×D₇₉	1	1	yes	characteristic
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	158	1	no	—
247	C₂²×C₈	—	79	1	no	—
248	C₂²×C₈	—	79	1	no	—
249	C₂²×C₈	—	158	2	no	—
250	C₂²×C₈	—	158	2	no	—
251	#126	—	158	2	no	—
252	#125	—	79	2	no	—
253	#124	—	79	2	no	—
254	#123	—	158	2	no	—
255	C₂²⋊Q₈	—	79	2	no	—
256	#121	—	79	2	no	—
257	C₄×Q₈	—	79	2	no	—
258	C₄×D₄	—	158	2	no	—
259	C₄×D₄	C₂²×D₇₉	1	2	yes	—
260	#117	—	158	2	no	—
261	C₂×C₄²	—	79	1	no	—
262	#107	—	79	1	no	—
263	#107	C₄×D₇₉	1	1	yes	characteristic
264	#106	—	79	2	no	—
265	C₄⋊C₈	—	79	1	no	—
266	C₄⋊C₈	—	79	1	no	—
267	C₄⋊C₈	—	79	2	no	—
268	C₄⋊C₈	—	158	2	no	—
269	C₄⋊C₈	—	158	2	no	—
270	C₄⋊C₈	—	158	2	no	—
271	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
272	#103	—	79	2	no	—
273	#103	—	79	2	no	—
274	#102	—	79	2	no	—
275	#102	C₄×D₇₉	1	2	yes	—
276	C₂²⋊C₈	—	2	2	no	—
277	C₂²⋊C₈	—	79	1	no	—
278	C₂²⋊C₈	—	79	1	no	—
279	C₂²⋊C₈	—	158	2	no	—
280	#97	—	158	1	no	—
281	#97	—	158	2	no	—
282	C₄×C₈	—	79	1	no	—
283	C₄×C₈	—	79	1	no	—
284	C₄×C₈	—	79	1	no	—
285	C₄×C₈	—	79	1	no	—
286	C₄×C₈	—	79	1	no	—
287	C₄×C₈	—	158	2	no	—
288	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
291	C₂×Q₈	—	79	2	no	—
292	C₂×D₄	—	158	2	no	—
293	C₂×D₄	C₂×C₄×D₇₉	1	2	yes	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	79	1	no	—
296	C₂²×C₄	—	79	1	no	—
297	C₂²×C₄	—	79	1	no	—
298	Q₁₆	—	158	2	no	—
299	SD₁₆	—	316	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	8	2	no	—
302	M₁₆	—	158	4	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	316	1	no	—
306	C₂×C₈	—	79	1	no	—
307	C₂×C₈	—	79	1	no	—
308	C₂×C₈	—	79	1	no	—
309	C₂×C₈	—	79	1	no	—
310	C₂×C₈	—	632	2	no	—
311	C₂×C₈	—	632	2	no	—
312	C₂×C₈	—	158	2	no	—
313	C₂×C₈	—	158	1	no	—
314	C₂×C₈	—	158	2	no	—
315	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
317	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
318	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
319	C₄⋊C₄	—	79	2	no	—
320	C₄⋊C₄	—	158	2	no	—
321	C₄⋊C₄	—	158	2	no	—
322	C₄⋊C₄	—	158	2	no	—
323	C₄⋊C₄	—	158	2	no	—
324	C₄⋊C₄	C₂×C₄×D₇₉	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	158	2	no	—
327	C₂²⋊C₄	—	158	2	no	—
328	C₂²⋊C₄	—	158	2	no	—
329	C₂²⋊C₄	—	158	2	no	—
330	C₄²	—	79	1	no	—
331	C₄²	—	79	1	no	—
332	C₄²	—	79	1	no	—
333	C₄²	—	158	2	no	—
334	C₄²	C₂³×D₇₉	1	1	yes	characteristic
335	C₂³	—	2	2	no	—
336	C₂³	—	79	1	no	—
337	Q₈	—	158	2	no	—
338	Q₈	—	158	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	316	4	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	79	1	no	—
346	C₂×C₄	—	79	1	no	—
347	C₂×C₄	—	79	1	no	—
348	C₂×C₄	—	158	2	no	—
349	C₂×C₄	—	158	2	no	—
350	C₂×C₄	—	158	2	no	—
351	C₂×C₄	—	158	1	no	—
352	C₂×C₄	—	158	1	no	—
353	C₂×C₄	—	158	1	no	—
354	C₂×C₄	—	158	1	no	—
355	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
357	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
358	C₈	—	2	2	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	4	2	no	—
363	C₈	—	316	2	no	—
364	C₈	—	158	2	no	—
365	C₈	—	158	1	no	—
366	C₈	—	158	1	no	—
367	C₈	—	158	1	no	—
368	C₈	—	158	1	no	—
369	C₈	#14579169	1	2	yes	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	79	2	no	—
374	C₂²	—	158	1	no	—
375	C₂²	—	158	1	no	—
376	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	316	2	no	—
380	C₄	—	316	2	no	—
381	C₄	—	158	1	no	—
382	C₄	—	158	1	no	—
383	C₄	—	158	1	no	—
384	C₄	C₂×#14579169	1	1	yes	characteristic
385	C₄	#19099328	1	1	yes	characteristic
386	C₄	#19099364	1	1	yes	characteristic
387	C₄	#19099377	1	1	yes	characteristic
388	C₂	—	8	2	no	—
389	C₂	—	158	2	no	—
390	C₂	#23772145	1	1	yes	characteristic
391	C₂	#23772270	1	1	yes	characteristic
392	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×#1589	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#19098796	C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1589	D₇₉
hypocenter	C₇₉	#18087
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
2. #23772561	C₂	semidirect
3. #23762296	C₂	semidirect
4. #23762276	C₂	semidirect
5. #23762254	C₂	semidirect
6. C₈×Dic₃₁₆	C₂	semidirect
7. D₈×C₇₉⋊C₈	C₂	semidirect
8. #335×D₇₉	C₂	semidirect
9. #23761810	C₂	semidirect
10. C₈×#11929607	C₂	semidirect
11. C₇₉×#1589	C₂	semidirect
200. #1589	D₇₉	semidirect
208. C₇₉	#18087	semidirect

Maximal subgroups

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

subgroup	index
#23773968	2
#23772561	2
#23762296	2
#23762276	2
#23762254	2
C₈×Dic₃₁₆	2
D₈×C₇₉⋊C₈	2
#335×D₇₉	2
#23761810	2
C₈×#11929607	2
C₇₉×#1589	2
#18087	79