Subgroups — #34296465

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	#23773748	C₂	1	1	yes	maximal, characteristic
2	#23772341	C₂	1	1	yes	maximal, characteristic, complemented
3	#23772117	C₂	1	1	yes	maximal, characteristic
4	#23764228	C₂	1	1	yes	maximal, characteristic
5	#23764112	C₂	1	2	yes	maximal, complemented
6	#23764105	C₂	1	2	yes	maximal, complemented
7	#23763540	C₂	1	1	yes	maximal, characteristic, complemented
8	#23763241	C₂	1	2	yes	maximal, complemented
9	#23763234	C₂	1	2	yes	maximal, complemented
10	C₄×#14579072	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1994	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579169	C₂²	1	2	yes	—
13	#19099113	C₂²	1	1	yes	characteristic
14	#19098933	C₄	1	2	yes	complemented
15	#19098929	C₄	1	2	yes	complemented
16	#19098925	C₄	1	4	yes	complemented
17	#19098694	—	2	4	no	—
18	#19098633	—	2	4	no	—
19	#19098590	C₂²	1	2	yes	—
20	#19098587	C₂²	1	1	yes	characteristic
21	C₂×#14579089	C₂²	1	2	yes	—
22	C₂×#14579089	C₂²	1	2	yes	—
23	#19098557	C₂²	1	2	yes	—
24	#19098557	C₂²	1	2	yes	—
25	#19098531	C₂²	1	2	yes	—
26	#19098519	C₂²	1	2	yes	—
27	C₂×#14579072	C₂²	1	1	yes	characteristic
28	C₂×#14579072	C₂²	1	1	yes	characteristic
29	C₂×#14579072	C₂²	1	1	yes	characteristic
30	C₂×#14579072	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
33	#19098327	C₂²	1	2	yes	—
34	#19098327	C₂²	1	2	yes	—
35	#533×C₇₉	C₂²	1	1	yes	characteristic
36	#423×C₇₉	C₂²	1	1	yes	characteristic
37	C₄³×C₇₉	C₂²	1	1	yes	characteristic
38	#339×C₇₉	C₂²	1	2	yes	—
39	#339×C₇₉	C₂²	1	2	yes	—
40	C₂×#11929624	C₂³	1	2	yes	—
41	#14579169	—	2	4	no	—
42	#14579169	—	2	4	no	—
43	C₂×#11929608	C₂³	1	2	yes	—
44	C₂×#11929607	—	2	2	no	—
45	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₇₉	—	2	2	no	—
50	#14579161	C₂×C₄	1	1	yes	characteristic
51	#14579158	C₂×C₄	1	1	yes	characteristic
52	#14579149	C₂×C₄	1	2	yes	—
53	#14579135	—	2	2	no	—
54	C₄×D₃₁₆	—	2	2	no	—
55	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
56	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
57	#14579089	—	2	4	no	—
58	#14579089	—	2	4	no	—
59	#14579084	C₂×C₄	1	4	yes	—
60	#14579084	C₂×C₄	1	4	yes	—
61	#14579081	C₂×C₄	1	4	yes	—
62	#14579081	C₂×C₄	1	4	yes	—
63	(C₂×D₇₉)⋊C₈	—	2	4	no	—
64	#14579072	—	2	2	no	—
65	#14579072	—	2	2	no	—
66	#14579072	—	2	2	no	—
67	#14579072	C₂×C₄	1	2	yes	—
68	#14579072	C₂×C₄	1	2	yes	—
69	#14579072	D₄	1	4	yes	—
70	C₄×C₇₉⋊C₈	—	2	2	no	—
71	C₄×C₇₉⋊C₈	—	2	2	no	—
72	C₄×C₇₉⋊C₈	—	2	2	no	—
73	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
75	#125×C₇₉	C₂×C₄	1	2	yes	—
76	C₂²⋊Q₈×C₇₉	—	2	2	no	—
77	C₄×Q₈×C₇₉	—	2	2	no	—
78	#117×C₇₉	C₂³	1	2	yes	—
79	C₂×C₄²×C₇₉	—	2	2	no	—
80	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₇₉	—	2	2	no	—
84	#11929624	—	2	2	no	—
85	#11929624	—	2	2	no	—
86	#11929624	—	2	2	no	—
87	#11929624	—	2	2	no	—
88	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
89	C₂×D₃₁₆	—	2	2	no	—
90	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
91	C₂×C₂²⋊D₇₉	—	2	2	no	—
92	C₂×C₄×D₇₉	—	2	2	no	—
93	#11929608	—	2	2	no	—
94	#11929608	—	2	2	no	—
95	#11929607	—	4	4	no	—
96	C₂×C₇₉⋊C₈	—	2	2	no	—
97	C₂×C₇₉⋊C₈	—	2	1	no	—
98	C₂×C₇₉⋊C₈	—	2	1	no	—
99	C₂×C₇₉⋊C₈	—	2	1	no	—
100	C₂×C₇₉⋊C₈	—	2	1	no	—
101	C₂×C₇₉⋊C₈	—	2	1	no	—
102	C₂×C₇₉⋊C₈	—	2	1	no	—
103	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
106	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
107	C₂×C₇₉⋊C₈	#41	1	2	yes	—
108	C₂×C₇₉⋊C₈	#41	1	2	yes	—
109	C₈×D₇₉	—	4	4	no	—
110	C₄⋊Dic₇₉	—	2	2	no	—
111	(C₂×D₇₉)⋊C₄	—	2	4	no	—
112	C₂×Q₈×C₇₉	—	2	2	no	—
113	C₂²×C₄×C₇₉	—	2	1	no	—
114	C₂²×C₄×C₇₉	—	2	1	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	C₂×C₈×C₇₉	—	2	2	no	—
120	C₄⋊C₄×C₇₉	—	2	4	no	—
121	C₄⋊C₄×C₇₉	—	2	2	no	—
122	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
123	C₂²⋊C₄×C₇₉	—	2	2	no	—
124	C₄²×C₇₉	—	2	2	no	—
125	C₄²×C₇₉	—	2	1	no	—
126	C₄²×C₇₉	—	2	1	no	—
127	C₄²×C₇₉	—	4	4	no	—
128	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₇₉	C₂×D₄	1	2	yes	—
131	C₄²×C₇₉	#41	1	2	yes	—
132	C₄²×C₇₉	#41	1	2	yes	—
133	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
134	C₂²×D₇₉	—	2	2	no	—
135	Q₈×C₇₉	—	8	2	no	—
136	C₂×C₄×C₇₉	—	2	4	no	—
137	C₂×C₄×C₇₉	—	2	2	no	—
138	C₂×C₄×C₇₉	—	2	4	no	—
139	C₂×C₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	—	4	2	no	—
141	C₂×C₄×C₇₉	—	4	2	no	—
142	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
150	C₂²⋊D₇₉	—	4	2	no	—
151	C₂²⋊D₇₉	—	4	2	no	—
152	C₂×Dic₇₉	—	2	2	no	—
153	D₃₁₆	—	2	2	no	—
154	D₃₁₆	—	2	2	no	—
155	D₃₁₆	—	8	2	no	—
156	C₄×D₇₉	—	4	2	no	—
157	C₄×D₇₉	—	4	2	no	—
158	Dic₁₅₈	—	2	2	no	—
159	Dic₁₅₈	—	2	2	no	—
160	C₈×C₇₉	—	4	2	no	—
161	C₇₉⋊C₈	—	2	2	no	—
162	C₇₉⋊C₈	—	2	2	no	—
163	C₇₉⋊C₈	—	2	2	no	—
164	C₇₉⋊C₈	—	4	2	no	—
165	C₇₉⋊C₈	#124	1	4	yes	—
166	C₂²×C₇₉	—	2	2	no	—
167	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₇₉	#529	1	1	yes	characteristic
169	C₂²×C₇₉	#529	1	1	yes	characteristic
170	C₂×D₇₉	—	4	4	no	—
171	C₂×D₇₉	—	4	2	no	—
172	C₄×C₇₉	—	2	4	no	—
173	C₄×C₇₉	—	2	2	no	—
174	C₄×C₇₉	—	2	2	no	—
175	C₄×C₇₉	—	4	2	no	—
176	C₄×C₇₉	—	4	4	no	—
177	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
178	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
179	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
180	C₄×C₇₉	#532	1	1	yes	characteristic
181	Dic₇₉	—	4	2	no	—
182	#21662	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	2	1	no	—
184	C₂×C₇₉	—	2	1	no	—
185	C₂×C₇₉	C₄×#124	1	1	yes	characteristic
186	C₂×C₇₉	#2956	1	2	yes	—
187	D₇₉	—	8	2	no	—
188	#3135	—	79	1	no	—
189	#2961	—	79	1	no	—
190	#2932	—	79	1	no	—
191	#1994	D₇₉	1	1	yes	characteristic, complemented
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	—	79	2	no	—
201	#533	C₂×D₇₉	1	1	yes	characteristic
202	#532	—	79	1	no	—
203	#491	—	79	2	no	—
204	#490	—	79	2	no	—
205	#489	—	79	4	no	—
206	#446	—	158	4	no	—
207	#435	—	158	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	—	79	2	no	—
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	—	79	1	no	—
221	C₂×C₄×C₈	—	79	1	no	—
222	C₂×C₄×C₈	—	79	1	no	—
223	C₄³	C₂×D₇₉	1	1	yes	characteristic
224	#341	—	79	2	no	—
225	#341	—	79	2	no	—
226	#339	C₂×D₇₉	1	2	yes	—
227	#339	C₂×D₇₉	1	2	yes	—
228	C₂×#41	—	79	2	no	—
229	#131	—	158	4	no	—
230	#131	—	158	4	no	—
231	C₂×M₁₆	—	79	2	no	—
232	C₂×M₁₆	—	158	2	no	—
233	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
234	C₂²×C₈	—	79	1	no	—
235	C₂²×C₈	—	79	1	no	—
236	C₂²×C₈	—	79	1	no	—
237	C₂²×C₈	—	79	1	no	—
238	C₂²×C₈	—	158	2	no	—
239	C₄⋊Q₈	—	79	1	no	—
240	C₄⋊Q₈	C₄×D₇₉	1	2	yes	—
241	#127	—	79	1	no	—
242	#125	C₄×D₇₉	1	2	yes	—
243	#124	—	79	2	no	—
244	C₂²⋊Q₈	—	2	2	no	—
245	#121	—	158	2	no	—
246	C₄×Q₈	—	2	2	no	—
247	C₄×D₄	—	158	2	no	—
248	#117	C₂²×D₇₉	1	2	yes	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
251	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
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	—	158	4	no	—
263	#97	—	79	2	no	—
264	#97	—	79	2	no	—
265	#97	—	79	4	no	—
266	#97	—	158	2	no	—
267	#97	—	158	2	no	—
268	#97	—	158	2	no	—
269	C₄×C₈	—	158	2	no	—
270	C₄×C₈	—	158	2	no	—
271	C₄×C₈	—	158	2	no	—
272	#41	—	158	2	no	—
273	#41	—	158	2	no	—
274	#41	—	158	2	no	—
275	#41	—	158	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	79	2	no	—
278	C₂×D₄	—	79	2	no	—
279	C₂×D₄	—	158	2	no	—
280	C₂×D₄	—	158	2	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	158	2	no	—
284	C₂²×C₄	#11929623	1	2	yes	—
285	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	316	4	no	—
289	M₁₆	—	158	2	no	—
290	M₁₆	—	158	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	316	4	no	—
293	C₂×C₈	—	79	2	no	—
294	C₂×C₈	—	79	2	no	—
295	C₂×C₈	—	79	2	no	—
296	C₂×C₈	—	79	2	no	—
297	C₂×C₈	—	79	2	no	—
298	C₂×C₈	—	79	2	no	—
299	C₂×C₈	—	158	2	no	—
300	C₂×C₈	—	158	1	no	—
301	C₂×C₈	—	158	1	no	—
302	C₂×C₈	—	158	1	no	—
303	C₂×C₈	—	158	1	no	—
304	C₂×C₈	—	158	1	no	—
305	C₂×C₈	—	158	1	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	158	2	no	—
309	C₄⋊C₄	C₂×C₄×D₇₉	1	4	yes	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	—	158	4	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	4	4	no	—
316	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
318	C₄²	D₄×D₇₉	1	2	yes	—
319	C₄²	#11929625	1	2	yes	—
320	C₄²	#11929625	1	2	yes	—
321	C₂³	—	158	2	no	—
322	C₂³	C₂×#11929623	1	1	yes	characteristic
323	Q₈	—	8	2	no	—
324	Q₈	—	158	2	no	—
325	Q₈	—	158	2	no	—
326	D₄	—	316	2	no	—
327	D₄	—	316	2	no	—
328	D₄	—	632	2	no	—
329	D₄	—	158	2	no	—
330	D₄	—	158	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	316	2	no	—
338	C₂×C₄	—	316	2	no	—
339	C₂×C₄	—	158	2	no	—
340	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
341	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
342	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×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×#11929625	1	1	yes	characteristic
347	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
348	C₈	—	4	2	no	—
349	C₈	—	316	2	no	—
350	C₈	—	79	4	no	—
351	C₈	—	158	2	no	—
352	C₈	—	158	2	no	—
353	C₈	—	158	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	316	4	no	—
356	C₂²	—	316	2	no	—
357	C₂²	C₂×(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
358	C₂²	#19099099	1	1	yes	characteristic
359	C₂²	#19099099	1	1	yes	characteristic
360	C₄	—	2	4	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	4	no	—
365	C₄	—	316	2	no	—
366	C₄	C₄×#11929625	1	1	yes	characteristic
367	C₄	C₄×#11929625	1	1	yes	characteristic
368	C₄	#124×D₇₉	1	1	yes	characteristic
369	C₄	#19099114	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	632	2	no	—
373	C₂	#23765943	1	1	yes	characteristic
374	C₂	#23772306	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1994	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#19099113	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1994	D₇₉
hypocenter	C₇₉	#21662
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1994

Complements

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

N	H ≅ G/N	type
2. #23772341	C₂	semidirect
5. #23764112	C₂	semidirect
6. #23764105	C₂	semidirect
7. #23763540	C₂	semidirect
8. #23763241	C₂	semidirect
9. #23763234	C₂	semidirect
10. C₄×#14579072	C₂	semidirect
11. C₇₉×#1994	C₂	semidirect
14. #19098933	C₄	semidirect
15. #19098929	C₄	semidirect
16. #19098925	C₄	semidirect
191. #1994	D₇₉	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
#23773748	2
#23772341	2
#23772117	2
#23764228	2
#23764112	2
#23764105	2
#23763540	2
#23763241	2
#23763234	2
C₄×#14579072	2
C₇₉×#1994	2
#21662	79