Subgroups — #34296464

Order 20224.

738 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773745	C₂	1	1	yes	maximal, characteristic
2	#23772340	C₂	1	1	yes	maximal, characteristic, complemented
3	#23764376	C₂	1	1	yes	maximal, characteristic, complemented
4	#23764229	C₂	1	1	yes	maximal, characteristic
5	#23764114	C₂	1	2	yes	maximal, complemented
6	#23764107	C₂	1	2	yes	maximal, complemented
7	#23763540	C₂	1	1	yes	maximal, characteristic, complemented
8	#23763242	C₂	1	2	yes	maximal, complemented
9	#23763235	C₂	1	2	yes	maximal, complemented
10	#97×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2932	C₂	1	1	yes	maximal, characteristic
12	#19099120	C₂²	1	1	yes	characteristic
13	#19098931	C₄	1	2	yes	complemented
14	#19098927	C₄	1	2	yes	complemented
15	#19098922	C₄	1	4	yes	complemented
16	#19098691	—	2	4	no	—
17	#19098589	C₂²	1	2	yes	—
18	#19098587	C₂²	1	1	yes	characteristic
19	C₂×#14579088	C₂²	1	2	yes	—
20	C₂×#14579088	C₂²	1	2	yes	—
21	#19098559	C₂²	1	2	yes	—
22	#19098559	C₂²	1	2	yes	—
23	#19098518	C₂²	1	2	yes	—
24	C₂×#14579072	C₂²	1	1	yes	characteristic
25	C₂×#14579071	C₂²	1	1	yes	characteristic
26	C₂×#14579071	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
29	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
30	#19098327	C₂²	1	2	yes	—
31	#19098327	C₂²	1	2	yes	—
32	#19098324	C₂²	1	2	yes	—
33	#19098324	C₂²	1	2	yes	—
34	C₂×#131×C₇₉	C₂²	1	2	yes	—
35	#532×C₇₉	C₂²	1	1	yes	characteristic
36	#435×C₇₉	—	2	4	no	—
37	#423×C₇₉	C₂²	1	1	yes	characteristic
38	#408×C₇₉	C₂²	1	2	yes	—
39	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
40	C₂×#11929608	C₂³	1	2	yes	—
41	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	#14579161	C₂×C₄	1	2	yes	—
44	#14579153	C₂×C₄	1	2	yes	—
45	#14579140	—	2	2	no	—
46	C₄×Dic₁₅₈	—	2	2	no	—
47	#14579120	C₂³	1	2	yes	—
48	C₂×C₄×Dic₇₉	—	2	2	no	—
49	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
51	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
52	Dic₇₉⋊C₈	—	2	4	no	—
53	#14579088	—	2	4	no	—
54	#14579088	—	2	4	no	—
55	#14579085	C₂×C₄	1	4	yes	—
56	#14579085	C₂×C₄	1	4	yes	—
57	#14579082	C₂×C₄	1	4	yes	—
58	#14579082	C₂×C₄	1	4	yes	—
59	#14579072	C₂×C₄	1	2	yes	—
60	#14579072	C₂×C₄	1	2	yes	—
61	#14579071	—	2	2	no	—
62	#14579071	—	2	2	no	—
63	#14579071	—	2	2	no	—
64	C₈×Dic₇₉	—	2	2	no	—
65	C₈×Dic₇₉	—	2	2	no	—
66	C₈×Dic₇₉	—	2	2	no	—
67	C₂×#41×C₇₉	C₂³	1	2	yes	—
68	#131×C₇₉	—	2	4	no	—
69	#131×C₇₉	—	2	4	no	—
70	C₂×M₁₆×C₇₉	—	2	2	no	—
71	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
72	C₂²×C₈×C₇₉	—	2	2	no	—
73	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
76	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
77	#124×C₇₉	C₂×C₄	1	2	yes	—
78	#121×C₇₉	—	2	2	no	—
79	C₄×D₄×C₇₉	—	2	2	no	—
80	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₇₉	—	2	2	no	—
82	C₂²⋊C₈×C₇₉	—	2	4	no	—
83	#97×C₇₉	D₄	1	4	yes	—
84	C₂×Dic₁₅₈	—	2	2	no	—
85	C₂²×Dic₇₉	—	2	1	no	—
86	C₂²×Dic₇₉	—	2	1	no	—
87	#11929608	—	2	2	no	—
88	#11929608	—	2	2	no	—
89	C₂×C₇₉⋊C₈	—	2	2	no	—
90	C₂×C₇₉⋊C₈	—	2	1	no	—
91	C₂×C₇₉⋊C₈	—	2	1	no	—
92	C₂×C₇₉⋊C₈	—	2	1	no	—
93	C₂×C₇₉⋊C₈	—	2	1	no	—
94	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
96	C₄⋊Dic₇₉	—	2	2	no	—
97	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
98	Dic₇₉⋊C₄	—	2	4	no	—
99	C₂²⋊Dic₇₉	—	2	2	no	—
100	C₄×Dic₇₉	—	2	2	no	—
101	C₄×Dic₇₉	—	2	1	no	—
102	C₄×Dic₇₉	—	2	1	no	—
103	C₄×Dic₇₉	—	4	4	no	—
104	C₄×Dic₇₉	#41	1	2	yes	—
105	C₄×Dic₇₉	#41	1	2	yes	—
106	#41×C₇₉	—	2	2	no	—
107	#41×C₇₉	—	2	2	no	—
108	#41×C₇₉	—	2	2	no	—
109	#41×C₇₉	—	2	2	no	—
110	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₇₉	—	2	2	no	—
112	C₂×D₄×C₇₉	—	2	2	no	—
113	C₂×D₄×C₇₉	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₇₉	—	2	2	no	—
115	C₂²×C₄×C₇₉	#41	1	2	yes	—
116	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₇₉	—	2	2	no	—
118	M₁₆×C₇₉	—	2	2	no	—
119	M₁₆×C₇₉	—	4	4	no	—
120	C₂×C₈×C₇₉	—	2	2	no	—
121	C₂×C₈×C₇₉	—	2	1	no	—
122	C₂×C₈×C₇₉	—	2	1	no	—
123	C₂×C₈×C₇₉	—	4	4	no	—
124	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₇₉	#41	1	2	yes	—
127	C₂×C₈×C₇₉	#41	1	2	yes	—
128	C₄⋊C₄×C₇₉	—	2	2	no	—
129	C₂²⋊C₄×C₇₉	—	2	4	no	—
130	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₇₉	C₂×D₄	1	2	yes	—
133	C₂³×C₇₉	—	2	2	no	—
134	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
135	Q₈×C₇₉	—	2	2	no	—
136	Q₈×C₇₉	—	2	2	no	—
137	D₄×C₇₉	—	2	2	no	—
138	D₄×C₇₉	—	2	2	no	—
139	D₄×C₇₉	—	4	2	no	—
140	D₄×C₇₉	—	4	2	no	—
141	D₄×C₇₉	—	8	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	—	4	2	no	—
146	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₇₉	—	2	4	no	—
155	C₂×Dic₇₉	—	2	2	no	—
156	C₂×Dic₇₉	—	2	4	no	—
157	C₂×Dic₇₉	—	2	2	no	—
158	C₂×Dic₇₉	—	4	2	no	—
159	Dic₁₅₈	—	8	2	no	—
160	C₈×C₇₉	—	2	2	no	—
161	C₈×C₇₉	—	4	2	no	—
162	C₈×C₇₉	#124	1	4	yes	—
163	C₇₉⋊C₈	—	2	2	no	—
164	C₇₉⋊C₈	—	2	2	no	—
165	C₇₉⋊C₈	—	4	2	no	—
166	C₂²×C₇₉	—	2	2	no	—
167	C₂²×C₇₉	—	4	4	no	—
168	C₂²×C₇₉	—	4	2	no	—
169	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₇₉	#529	1	1	yes	characteristic
171	C₂²×C₇₉	#529	1	1	yes	characteristic
172	C₄×C₇₉	—	2	2	no	—
173	C₄×C₇₉	—	2	2	no	—
174	C₄×C₇₉	—	4	2	no	—
175	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
176	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
177	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
178	C₄×C₇₉	#532	1	1	yes	characteristic
179	Dic₇₉	—	2	4	no	—
180	Dic₇₉	—	4	2	no	—
181	Dic₇₉	—	4	4	no	—
182	#21662	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	2	1	no	—
184	C₂×C₇₉	—	2	1	no	—
185	C₂×C₇₉	—	8	2	no	—
186	C₂×C₇₉	C₄×#124	1	1	yes	characteristic
187	C₂×C₇₉	#2956	1	2	yes	—
188	#3135	—	79	1	no	—
189	#2961	—	79	1	no	—
190	#2932	D₇₉	1	1	yes	characteristic
191	#1994	—	79	1	no	—
192	#1962	—	79	1	no	—
193	#1938	—	79	2	no	—
194	#1937	—	79	2	no	—
195	#1838	—	79	1	no	—
196	#1765	—	79	2	no	—
197	#1764	—	79	2	no	—
198	C₄×#97	—	79	1	no	—
199	C₇₉	#21662	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	C₂×D₇₉	1	2	yes	—
201	#533	—	79	1	no	—
202	#532	C₂×D₇₉	1	1	yes	characteristic
203	#491	—	79	2	no	—
204	#490	—	79	2	no	—
205	#489	—	79	4	no	—
206	#446	—	158	4	no	—
207	#435	—	2	4	no	—
208	#424	—	79	2	no	—
209	#423	—	79	1	no	—
210	#423	C₂×D₇₉	1	1	yes	characteristic
211	C₂×#104	—	79	2	no	—
212	C₂×#104	—	79	2	no	—
213	#416	—	79	2	no	—
214	#416	—	79	2	no	—
215	#408	C₂×D₇₉	1	2	yes	—
216	#405	—	79	2	no	—
217	C₂×#97	—	79	1	no	—
218	C₂×#97	—	79	1	no	—
219	C₂×#97	—	79	1	no	—
220	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
221	C₂×C₄×C₈	—	79	1	no	—
222	C₂×C₄×C₈	—	79	1	no	—
223	C₄³	—	79	1	no	—
224	#341	—	79	2	no	—
225	#341	—	79	2	no	—
226	#339	—	79	2	no	—
227	#339	—	79	2	no	—
228	C₂×#41	C₂²×D₇₉	1	2	yes	—
229	#131	—	2	4	no	—
230	#131	—	2	4	no	—
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	—	79	2	no	—
233	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
234	C₂²×C₈	—	2	2	no	—
235	C₂²×C₈	—	79	1	no	—
236	C₂²×C₈	—	79	1	no	—
237	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
239	C₄⋊Q₈	—	79	2	no	—
240	C₄⋊Q₈	C₄×D₇₉	1	1	yes	characteristic
241	#127	C₄×D₇₉	1	1	yes	characteristic
242	#125	—	79	2	no	—
243	#124	C₄×D₇₉	1	2	yes	—
244	C₂²⋊Q₈	—	158	2	no	—
245	#121	—	2	2	no	—
246	C₄×Q₈	—	158	2	no	—
247	C₄×D₄	—	2	2	no	—
248	#117	—	79	2	no	—
249	C₂×C₄²	—	79	1	no	—
250	C₂×C₄²	—	79	1	no	—
251	C₂×C₄²	—	158	2	no	—
252	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	158	4	no	—
255	C₄⋊C₈	—	158	2	no	—
256	#104	—	158	4	no	—
257	#104	—	158	4	no	—
258	#103	—	79	4	no	—
259	#103	—	79	4	no	—
260	#102	—	79	4	no	—
261	#102	—	79	4	no	—
262	C₂²⋊C₈	—	2	4	no	—
263	#97	—	79	2	no	—
264	#97	—	79	2	no	—
265	#97	—	158	2	no	—
266	#97	—	158	2	no	—
267	#97	—	158	2	no	—
268	#97	C₂²⋊D₇₉	1	4	yes	—
269	C₄×C₈	—	158	2	no	—
270	C₄×C₈	—	158	2	no	—
271	C₄×C₈	—	158	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	#41	—	2	2	no	—
276	C₂×Q₈	—	158	2	no	—
277	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	C₂×C₄×D₇₉	1	2	yes	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	158	1	no	—
283	C₂²×C₄	—	158	1	no	—
284	C₂²×C₄	#11929624	1	2	yes	—
285	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	4	4	no	—
289	M₁₆	—	158	2	no	—
290	M₁₆	—	158	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	4	no	—
295	C₂×C₈	—	79	2	no	—
296	C₂×C₈	—	79	2	no	—
297	C₂×C₈	—	158	2	no	—
298	C₂×C₈	—	158	1	no	—
299	C₂×C₈	—	158	1	no	—
300	C₂×C₈	—	158	1	no	—
301	C₂×C₈	—	158	1	no	—
302	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
303	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
304	C₂×C₈	#11929623	1	2	yes	—
305	C₂×C₈	#11929623	1	2	yes	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	79	4	no	—
308	C₄⋊C₄	—	158	4	no	—
309	C₄⋊C₄	—	158	2	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₂²⋊C₄	—	158	2	no	—
312	C₄²	—	316	4	no	—
313	C₄²	—	79	2	no	—
314	C₄²	—	79	2	no	—
315	C₄²	—	158	1	no	—
316	C₄²	—	158	1	no	—
317	C₄²	—	158	2	no	—
318	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
319	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
321	C₂³	—	2	2	no	—
322	C₂³	C₂×#11929624	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	632	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	8	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	316	2	no	—
336	C₂×C₄	—	158	4	no	—
337	C₂×C₄	—	158	2	no	—
338	C₂×C₄	—	158	4	no	—
339	C₂×C₄	—	158	2	no	—
340	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
341	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
342	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
345	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
346	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
347	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	316	2	no	—
351	C₈	—	158	2	no	—
352	C₈	—	158	2	no	—
353	C₈	#14579148	1	4	yes	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	C₂×C₄×C₂²⋊D₇₉	1	1	yes	characteristic
358	C₂²	#19099095	1	1	yes	characteristic
359	C₂²	#19099095	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	4	2	no	—
363	C₄	—	316	2	no	—
364	C₄	—	316	4	no	—
365	C₄	—	158	4	no	—
366	C₄	C₄×#11929623	1	1	yes	characteristic
367	C₄	C₄×#11929623	1	1	yes	characteristic
368	C₄	C₂×#14579148	1	1	yes	characteristic
369	C₄	#19099109	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	8	2	no	—
373	C₂	C₄×#14579148	1	1	yes	characteristic
374	C₂	#23772307	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#2932	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#532×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2932	D₇₉
hypocenter	C₇₉	#21662
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #2932

Complements

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

N	H ≅ G/N	type
2. #23772340	C₂	semidirect
3. #23764376	C₂	semidirect
5. #23764114	C₂	semidirect
6. #23764107	C₂	semidirect
7. #23763540	C₂	semidirect
8. #23763242	C₂	semidirect
9. #23763235	C₂	semidirect
10. #97×Dic₇₉	C₂	semidirect
13. #19098931	C₄	semidirect
14. #19098927	C₄	semidirect
15. #19098922	C₄	semidirect
199. C₇₉	#21662	semidirect

Maximal subgroups

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

subgroup	index
#23773745	2
#23772340	2
#23764376	2
#23764229	2
#23764114	2
#23764107	2
#23763540	2
#23763242	2
#23763235	2
#97×Dic₇₉	2
C₇₉×#2932	2
#21662	79