Subgroups — #34260318

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773970	C₂	1	1	yes	maximal, characteristic
2	#23772562	C₂	1	2	yes	maximal, complemented
3	#23762304	C₂	1	2	yes	maximal, complemented
4	#23762252	C₂	1	2	yes	maximal, complemented
5	#23762231	C₂	1	2	yes	maximal, complemented
6	#23762214	C₂	1	2	yes	maximal, complemented
7	C₂×#19098316	C₂	1	1	yes	maximal, characteristic, complemented
8	#23761814	C₂	1	2	yes	maximal, complemented
9	C₇₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	#19099097	C₂²	1	2	yes	—
11	#19098965	C₄	1	1	yes	characteristic
12	#19098947	C₄	1	1	yes	characteristic
13	#19098918	C₄	1	2	yes	—
14	#19098804	C₄	1	1	yes	characteristic
15	#19098799	C₄	1	1	yes	characteristic
16	#19098792	C₄	1	2	yes	—
17	#19098693	C₂²	1	2	yes	—
18	C₈×Dic₁₅₈	C₂²	1	2	yes	—
19	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
20	C₄×#11929613	C₂²	1	2	yes	—
21	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
22	#19098642	C₂²	1	2	yes	—
23	#19098631	—	2	2	no	—
24	C₈×D₃₁₆	C₂²	1	2	yes	—
25	(C₂²⋊D₇₉)⋊C₈	—	2	2	no	—
26	#19098623	—	2	2	no	—
27	#19098588	—	2	2	no	—
28	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
29	#19098316	C₂²	1	1	yes	characteristic
30	#19098316	C₂²	1	2	yes	—
31	#19098316	C₂²	1	1	yes	characteristic
32	#19098295	C₂²	1	2	yes	—
33	#19098295	C₂²	1	2	yes	—
34	#19098292	C₂²	1	2	yes	—
35	#19098292	C₂²	1	2	yes	—
36	#431×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
39	#322×C₇₉	C₂²	1	2	yes	—
40	#322×C₇₉	C₂²	1	2	yes	—
41	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×#11929613	C₂×C₄	1	2	yes	—
43	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
44	C₂×#11929607	—	2	2	no	—
45	C₂²×C₇₉⋊C₈	—	2	2	no	—
46	C₂×C₈×D₇₉	—	2	2	no	—
47	#14579155	—	2	2	no	—
48	#14579153	C₂×C₄	1	2	yes	—
49	#14579149	C₂×C₄	1	2	yes	—
50	#14579144	—	2	2	no	—
51	#14579140	C₂×C₄	1	2	yes	—
52	#14579135	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₅₈	C₂³	1	2	yes	—
54	C₄×D₃₁₆	C₂³	1	2	yes	—
55	C₄×C₂²⋊D₇₉	—	2	2	no	—
56	#14579118	—	2	2	no	—
57	#14579095	C₂×C₄	1	1	yes	characteristic
58	#14579095	C₂×C₄	1	1	yes	characteristic
59	#14579093	C₂×C₄	1	2	yes	—
60	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
61	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
62	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
63	Dic₇₉⋊C₈	—	2	2	no	—
64	Dic₇₉⋊C₈	—	2	2	no	—
65	#14579086	C₂×C₄	1	2	yes	—
66	#14579086	C₂×C₄	1	2	yes	—
67	#14579083	C₂×C₄	1	2	yes	—
68	#14579083	C₂×C₄	1	2	yes	—
69	(C₂×D₇₉)⋊C₈	—	2	2	no	—
70	(C₂×D₇₉)⋊C₈	—	2	2	no	—
71	#14579071	—	2	2	no	—
72	C₈×Dic₇₉	—	2	2	no	—
73	C₂×M₁₆×C₇₉	—	2	1	no	—
74	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
78	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
79	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
80	#97×C₇₉	—	2	1	no	—
81	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₇₉	C₂³	1	2	yes	—
84	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₉	C₂³	1	2	yes	—
87	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
88	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
89	C₂×C₂²⋊D₇₉	—	2	2	no	—
90	C₂×C₄×D₇₉	—	2	2	no	—
91	Dic₃₁₆	—	4	1	no	—
92	#11929613	—	2	4	no	—
93	D₆₃₂	—	4	1	no	—
94	#11929607	—	8	2	no	—
95	C₂×C₇₉⋊C₈	—	2	2	no	—
96	C₂×C₇₉⋊C₈	—	2	2	no	—
97	C₂×C₇₉⋊C₈	—	8	2	no	—
98	C₈×D₇₉	—	8	2	no	—
99	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
100	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
101	Dic₇₉⋊C₄	—	2	2	no	—
102	Dic₇₉⋊C₄	—	2	2	no	—
103	Dic₇₉⋊C₄	—	2	2	no	—
104	Dic₇₉⋊C₄	—	2	2	no	—
105	C₂²⋊Dic₇₉	—	2	2	no	—
106	(C₂×D₇₉)⋊C₄	—	2	2	no	—
107	(C₂×D₇₉)⋊C₄	—	2	2	no	—
108	(C₂×D₇₉)⋊C₄	—	2	2	no	—
109	(C₂×D₇₉)⋊C₄	—	2	2	no	—
110	C₄×Dic₇₉	—	2	2	no	—
111	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
114	M₁₆×C₇₉	—	4	2	no	—
115	C₂×C₈×C₇₉	—	2	2	no	—
116	C₂×C₈×C₇₉	—	2	2	no	—
117	C₂×C₈×C₇₉	—	4	1	no	—
118	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
124	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₇₉	#41	1	1	yes	characteristic
127	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
128	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
129	C₂²×D₇₉	—	2	2	no	—
130	C₂×C₄×C₇₉	—	2	2	no	—
131	C₂×C₄×C₇₉	—	2	1	no	—
132	C₂×C₄×C₇₉	—	2	1	no	—
133	C₂×C₄×C₇₉	—	2	1	no	—
134	C₂×C₄×C₇₉	—	2	1	no	—
135	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
141	C₂²⋊D₇₉	—	4	4	no	—
142	C₂×Dic₇₉	—	2	2	no	—
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	2	2	no	—
145	D₃₁₆	—	2	2	no	—
146	D₃₁₆	—	2	2	no	—
147	C₄×D₇₉	—	4	2	no	—
148	C₄×D₇₉	—	4	2	no	—
149	Dic₁₅₈	—	2	2	no	—
150	Dic₁₅₈	—	2	2	no	—
151	C₈×C₇₉	—	2	1	no	—
152	C₈×C₇₉	—	2	1	no	—
153	C₈×C₇₉	—	2	1	no	—
154	C₈×C₇₉	—	2	2	no	—
155	C₈×C₇₉	—	2	2	no	—
156	C₈×C₇₉	—	4	2	no	—
157	C₈×C₇₉	#131	1	2	yes	—
158	C₇₉⋊C₈	—	4	2	no	—
159	C₇₉⋊C₈	—	4	2	no	—
160	C₂²×C₇₉	—	2	1	no	—
161	C₂²×C₇₉	—	2	1	no	—
162	C₂²×C₇₉	#444	1	2	yes	—
163	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₇₉	—	4	2	no	—
165	C₂×D₇₉	—	4	2	no	—
166	C₂×D₇₉	—	4	2	no	—
167	C₄×C₇₉	—	2	1	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	—	2	1	no	—
171	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
172	C₄×C₇₉	#568	1	1	yes	characteristic
173	C₄×C₇₉	#576	1	1	yes	characteristic
174	C₄×C₇₉	#578	1	1	yes	characteristic
175	Dic₇₉	—	4	2	no	—
176	Dic₇₉	—	4	2	no	—
177	Dic₇₉	—	4	2	no	—
178	#18092	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	2	2	no	—
180	C₂×C₇₉	#2935	1	1	yes	characteristic
181	C₂×C₇₉	#2951	1	1	yes	characteristic
182	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
183	D₇₉	—	8	2	no	—
184	#3168	—	79	1	no	—
185	#2994	—	79	2	no	—
186	#1592	—	79	2	no	—
187	#1584	—	79	2	no	—
188	#1581	—	79	2	no	—
189	SD₁₆⋊C₈	—	79	2	no	—
190	C₂×#335	—	79	1	no	—
191	#1483	—	79	2	no	—
192	#1451	D₇₉	1	1	yes	characteristic, complemented
193	C₇₉	#18092	1	1	yes	characteristic, complemented, Sylow
194	#529	—	79	2	no	—
195	#499	—	79	1	no	—
196	#495	—	79	1	no	—
197	#488	—	79	2	no	—
198	#467	—	79	1	no	—
199	#466	—	79	1	no	—
200	#465	—	79	2	no	—
201	#446	—	79	2	no	—
202	C₈×Q₈	—	79	2	no	—
203	C₄×Q₁₆	—	79	1	no	—
204	C₄×SD₁₆	—	79	2	no	—
205	C₄×D₈	—	79	1	no	—
206	#436	—	79	2	no	—
207	#435	—	158	2	no	—
208	C₈×D₄	—	79	2	no	—
209	C₈×D₄	—	158	2	no	—
210	#433	—	158	2	no	—
211	#431	C₂×D₇₉	1	1	yes	characteristic
212	#424	—	158	2	no	—
213	C₂×C₄⋊C₈	—	79	2	no	—
214	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
215	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
216	#335	—	79	1	no	—
217	#335	—	79	2	no	—
218	#335	—	79	1	no	—
219	Q₈⋊C₈	—	79	2	no	—
220	Q₈⋊C₈	—	79	2	no	—
221	D₄⋊C₈	—	79	2	no	—
222	D₄⋊C₈	—	79	2	no	—
223	#322	C₂×D₇₉	1	2	yes	—
224	#322	C₂×D₇₉	1	2	yes	—
225	C₂×Q₁₆	—	79	1	no	—
226	C₂×SD₁₆	—	79	2	no	—
227	C₂×D₈	—	79	1	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	158	2	no	—
230	C₂²×C₈	—	158	2	no	—
231	C₂²×C₈	—	158	2	no	—
232	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
233	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
234	#126	—	158	2	no	—
235	#125	—	79	2	no	—
236	#124	—	79	2	no	—
237	#123	—	158	2	no	—
238	C₂²⋊Q₈	—	79	2	no	—
239	#121	—	79	2	no	—
240	C₄×Q₈	—	79	2	no	—
241	C₄×D₄	—	79	2	no	—
242	C₄×D₄	—	158	2	no	—
243	#117	—	158	2	no	—
244	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
245	#107	—	79	1	no	—
246	#107	—	79	1	no	—
247	#106	—	79	2	no	—
248	C₄⋊C₈	—	79	2	no	—
249	C₄⋊C₈	—	79	2	no	—
250	C₄⋊C₈	—	158	2	no	—
251	C₄⋊C₈	—	158	2	no	—
252	C₄⋊C₈	—	158	2	no	—
253	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
254	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
255	#103	—	79	2	no	—
256	#103	—	79	2	no	—
257	#102	—	79	2	no	—
258	#102	—	79	2	no	—
259	C₂²⋊C₈	—	158	2	no	—
260	C₂²⋊C₈	—	158	2	no	—
261	C₂²⋊C₈	D₃₁₆	1	2	yes	—
262	#97	—	2	1	no	—
263	#97	—	158	2	no	—
264	C₄×C₈	—	158	2	no	—
265	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
266	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₇₉	1	2	yes	—
268	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₇₉	1	2	yes	—
271	C₂×Q₈	—	79	2	no	—
272	C₂×D₄	—	79	2	no	—
273	C₂×D₄	—	158	2	no	—
274	C₂²×C₄	—	158	2	no	—
275	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
277	C₂²×C₄	#11929624	1	1	yes	characteristic
278	Q₁₆	—	316	1	no	—
279	SD₁₆	—	158	4	no	—
280	D₈	—	316	1	no	—
281	M₁₆	—	4	2	no	—
282	M₁₆	—	632	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	632	2	no	—
287	C₂×C₈	—	632	2	no	—
288	C₂×C₈	—	158	2	no	—
289	C₂×C₈	—	158	2	no	—
290	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
294	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
295	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
296	C₄⋊C₄	—	79	2	no	—
297	C₄⋊C₄	—	79	2	no	—
298	C₄⋊C₄	—	158	2	no	—
299	C₄⋊C₄	—	158	2	no	—
300	C₄⋊C₄	—	158	2	no	—
301	C₄⋊C₄	—	158	2	no	—
302	C₂²⋊C₄	—	158	2	no	—
303	C₂²⋊C₄	—	158	2	no	—
304	C₂²⋊C₄	—	158	2	no	—
305	C₂²⋊C₄	—	158	2	no	—
306	C₂²⋊C₄	—	158	2	no	—
307	C₄²	—	158	2	no	—
308	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
309	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
310	C₄²	#11929624	1	1	yes	characteristic
311	C₄²	C₂³×D₇₉	1	1	yes	characteristic
312	C₂³	—	158	2	no	—
313	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
314	Q₈	—	158	2	no	—
315	Q₈	—	158	2	no	—
316	D₄	—	316	4	no	—
317	D₄	—	158	2	no	—
318	D₄	—	158	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	316	2	no	—
325	C₂×C₄	—	316	2	no	—
326	C₂×C₄	—	158	2	no	—
327	C₂×C₄	—	158	2	no	—
328	C₂×C₄	—	158	2	no	—
329	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
330	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
331	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
334	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	316	2	no	—
342	C₈	—	316	2	no	—
343	C₈	#14579169	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	316	2	no	—
347	C₂²	—	316	2	no	—
348	C₂²	—	316	2	no	—
349	C₂²	#19098686	1	2	yes	—
350	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	316	2	no	—
356	C₄	—	316	2	no	—
357	C₄	—	316	2	no	—
358	C₄	C₂×#14579169	1	1	yes	characteristic
359	C₄	#19099329	1	1	yes	characteristic
360	C₄	#19099368	1	1	yes	characteristic
361	C₄	#19099380	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	632	2	no	—
364	C₂	#23772146	1	1	yes	characteristic
365	C₂	#23772277	1	1	yes	characteristic
366	C₂	C₂×#19098686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1451	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#19098799	C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1451	D₇₉
hypocenter	C₇₉	#18092
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
2. #23772562	C₂	semidirect
3. #23762304	C₂	semidirect
4. #23762252	C₂	semidirect
5. #23762231	C₂	semidirect
6. #23762214	C₂	semidirect
7. C₂×#19098316	C₂	semidirect
8. #23761814	C₂	semidirect
9. C₇₉×#1451	C₂	semidirect
192. #1451	D₇₉	semidirect
193. C₇₉	#18092	semidirect

Maximal subgroups

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

subgroup	index
#23773970	2
#23772562	2
#23762304	2
#23762252	2
#23762231	2
#23762214	2
C₂×#19098316	2
#23761814	2
C₇₉×#1451	2
#18092	79