Subgroups — #34205152

Order 20224.

488 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776502	C₂	1	1	yes	maximal, characteristic, complemented
2	#23772096	C₂	1	1	yes	maximal, characteristic
3	#23765020	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765010	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765010	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579167	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
10	#19099075	C₄	1	1	yes	characteristic
11	#19098981	—	2	2	no	—
12	#19098979	—	2	2	no	—
13	C₂×#14579101	C₂²	1	1	yes	characteristic
14	C₂×#14579101	C₂²	1	1	yes	characteristic
15	M₃₂×D₇₉	—	2	2	no	—
16	#19098636	—	2	2	no	—
17	C₈×D₃₁₆	—	2	2	no	—
18	#19098590	C₄	1	1	yes	characteristic
19	#19098531	C₄	1	1	yes	characteristic
20	#19098526	C₄	1	1	yes	characteristic
21	#19098344	—	2	2	no	—
22	(C₂×D₇₉)⋊C₁₆	—	2	2	no	—
23	(C₂×D₇₉)⋊C₁₆	—	2	2	no	—
24	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
26	#405×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
29	#14579167	—	2	1	no	—
30	#14579167	—	2	1	no	—
31	#14579167	—	2	1	no	—
32	#14579167	—	2	1	no	—
33	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
34	C₂×#11929607	D₄	1	2	yes	—
35	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₇₉	—	2	1	no	—
37	C₂×C₈×D₇₉	—	2	1	no	—
38	C₂×C₈×D₇₉	—	2	1	no	—
39	C₂×C₈×D₇₉	—	2	1	no	—
40	C₂×C₈×D₇₉	—	2	1	no	—
41	C₂×C₈×D₇₉	—	2	1	no	—
42	C₂×C₈×D₇₉	D₄	1	2	yes	—
43	#14579145	C₈	1	2	yes	—
44	#14579141	—	2	1	no	—
45	#14579136	—	2	1	no	—
46	#14579131	C₈	1	2	yes	—
47	C₄×D₃₁₆	—	2	2	no	—
48	#14579101	—	2	1	no	—
49	#14579101	—	2	1	no	—
50	#14579101	—	2	1	no	—
51	#14579101	—	2	1	no	—
52	#14579100	—	2	4	no	—
53	C₂×C₇₉⋊C₁₆	—	2	1	no	—
54	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	complemented
55	C₁₆×D₇₉	—	2	4	no	—
56	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
57	(C₂×D₇₉)⋊C₈	—	2	2	no	—
58	(C₂×D₇₉)⋊C₈	—	2	2	no	—
59	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
62	M₃₂×C₇₉	—	2	1	no	—
63	M₃₂×C₇₉	—	2	1	no	—
64	M₃₂×C₇₉	—	2	1	no	—
65	M₃₂×C₇₉	—	2	1	no	—
66	C₂×C₁₆×C₇₉	—	2	1	no	—
67	C₂×C₁₆×C₇₉	D₄	1	2	yes	complemented
68	#97×C₇₉	—	2	1	no	—
69	C₄×C₈×C₇₉	—	2	1	no	—
70	C₂³×D₇₉	C₂×C₈	1	1	yes	characteristic
71	#11929624	—	2	1	no	—
72	#11929624	—	2	1	no	—
73	#11929624	—	2	1	no	—
74	#11929624	—	2	1	no	—
75	C₂×Dic₁₅₈	M₁₆	1	1	yes	characteristic
76	C₂×D₃₁₆	—	2	2	no	—
77	C₂×D₃₁₆	M₁₆	1	1	yes	characteristic
78	C₂×C₂²⋊D₇₉	C₂×C₈	1	2	yes	—
79	C₂²×Dic₇₉	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₇₉	—	2	1	no	—
81	C₂×C₄×D₇₉	—	2	1	no	—
82	C₂×C₄×D₇₉	—	2	1	no	—
83	C₂×C₄×D₇₉	—	2	1	no	—
84	C₂×C₄×D₇₉	—	2	1	no	—
85	C₂×C₄×D₇₉	—	2	1	no	—
86	C₂×C₄×D₇₉	C₂²⋊C₄	1	2	yes	—
87	#11929608	—	2	1	no	—
88	#11929608	—	2	1	no	—
89	#11929607	—	2	2	no	—
90	#11929607	—	2	2	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	C₈×D₇₉	—	2	2	no	—
96	C₈×D₇₉	—	2	2	no	—
97	C₈×D₇₉	—	4	1	no	—
98	C₈×D₇₉	—	4	1	no	—
99	C₈×D₇₉	—	4	1	no	—
100	C₈×D₇₉	—	4	1	no	—
101	C₄⋊Dic₇₉	—	2	2	no	—
102	(C₂×D₇₉)⋊C₄	—	2	2	no	—
103	(C₂×D₇₉)⋊C₄	—	2	2	no	—
104	C₇₉⋊C₁₆	—	2	2	no	—
105	C₇₉⋊C₁₆	—	4	1	no	—
106	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
107	M₁₆×C₇₉	—	2	1	no	—
108	M₁₆×C₇₉	—	2	1	no	—
109	C₂×C₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
115	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
116	C₄⋊C₄×C₇₉	—	2	1	no	—
117	C₂²⋊C₄×C₇₉	C₂×C₈	1	2	yes	—
118	C₄²×C₇₉	—	2	1	no	—
119	C₁₆×C₇₉	—	2	2	no	—
120	C₁₆×C₇₉	—	4	1	no	—
121	C₂³×C₇₉	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₇₉	—	2	1	no	—
123	C₂²×D₇₉	—	2	1	no	—
124	C₂²×D₇₉	—	2	1	no	—
125	C₂²×D₇₉	—	4	2	no	—
126	C₂²×D₇₉	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₇₉	—	2	2	no	—
128	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
134	C₂²⋊D₇₉	—	2	2	no	—
135	C₂²⋊D₇₉	—	2	2	no	—
136	C₂×Dic₇₉	—	2	1	no	—
137	C₂×Dic₇₉	—	2	1	no	—
138	C₂×Dic₇₉	—	2	1	no	—
139	C₂×Dic₇₉	C₂²⋊C₈	1	2	yes	—
140	D₃₁₆	—	2	1	no	—
141	D₃₁₆	—	2	1	no	—
142	D₃₁₆	—	4	4	no	—
143	C₄×D₇₉	—	2	2	no	—
144	C₄×D₇₉	—	2	2	no	—
145	C₄×D₇₉	—	4	1	no	—
146	C₄×D₇₉	—	4	1	no	—
147	C₄×D₇₉	—	4	1	no	—
148	C₄×D₇₉	—	4	1	no	—
149	Dic₁₅₈	—	2	1	no	—
150	Dic₁₅₈	—	2	1	no	—
151	C₈×C₇₉	—	4	1	no	—
152	C₈×C₇₉	#120	1	1	yes	characteristic
153	C₈×C₇₉	#121	1	1	yes	characteristic
154	C₈×C₇₉	#121	1	1	yes	characteristic
155	C₈×C₇₉	#123	1	1	yes	characteristic
156	C₇₉⋊C₈	—	2	2	no	—
157	C₇₉⋊C₈	—	4	1	no	—
158	C₂²×C₇₉	—	4	1	no	—
159	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
161	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
162	C₂×D₇₉	—	2	2	no	—
163	C₂×D₇₉	—	2	2	no	—
164	C₂×D₇₉	—	2	2	no	—
165	C₂×D₇₉	—	2	2	no	—
166	C₂×D₇₉	—	4	1	no	—
167	C₂×D₇₉	—	4	1	no	—
168	C₂×D₇₉	—	4	1	no	—
169	C₂×D₇₉	—	4	1	no	—
170	C₂×D₇₉	—	4	2	no	—
171	C₄×C₇₉	—	4	2	no	—
172	C₄×C₇₉	#386	1	1	yes	characteristic
173	C₄×C₇₉	#408	1	1	yes	characteristic
174	C₄×C₇₉	#435	1	1	yes	characteristic
175	C₄×C₇₉	#435	1	1	yes	characteristic
176	Dic₇₉	—	2	2	no	—
177	Dic₇₉	—	4	1	no	—
178	#12981	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	2	1	no	—
180	C₂×C₇₉	—	2	1	no	—
181	C₂×C₇₉	—	2	1	no	—
182	C₂×C₇₉	#1869	1	1	yes	characteristic
183	D₇₉	—	4	2	no	—
184	D₇₉	—	4	2	no	—
185	#3407	—	79	1	no	—
186	#2929	—	79	1	no	—
187	#2115	—	79	1	no	—
188	#2112	—	79	1	no	—
189	#2112	—	79	1	no	—
190	#1378	—	79	1	no	—
191	#1378	D₇₉	1	1	yes	characteristic, complemented
192	C₇₉	#12981	1	1	yes	characteristic, complemented, Sylow
193	C₂×#131	—	79	1	no	—
194	C₂³×C₈	—	79	1	no	—
195	#525	—	79	1	no	—
196	#504	—	158	2	no	—
197	#504	—	158	2	no	—
198	C₂×M₃₂	—	79	1	no	—
199	C₂×M₃₂	—	79	1	no	—
200	C₂×M₃₂	—	158	2	no	—
201	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
202	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
203	#435	—	158	2	no	—
204	C₈×D₄	—	158	2	no	—
205	#424	—	79	1	no	—
206	#408	—	79	1	no	—
207	#407	—	79	1	no	—
208	#405	C₂×D₇₉	1	1	yes	characteristic
209	#349	—	158	2	no	—
210	C₂²⋊C₁₆	—	158	2	no	—
211	C₂²⋊C₁₆	—	158	2	no	—
212	C₂×#41	—	79	1	no	—
213	C₂³×C₄	—	79	1	no	—
214	#131	—	158	1	no	—
215	#131	—	158	1	no	—
216	#131	—	158	1	no	—
217	#131	—	158	1	no	—
218	C₂×M₁₆	—	79	1	no	—
219	C₂×M₁₆	—	79	2	no	—
220	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
221	C₂²×C₈	—	79	2	no	—
222	C₂²×C₈	—	79	1	no	—
223	C₂²×C₈	—	158	1	no	—
224	C₂²×C₈	—	158	1	no	—
225	C₂²×C₈	—	158	1	no	—
226	C₂²×C₈	—	158	1	no	—
227	C₂²×C₈	—	158	1	no	—
228	C₂²×C₈	—	158	1	no	—
229	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
230	#123	—	79	2	no	—
231	C₂²⋊Q₈	—	158	1	no	—
232	#121	—	158	1	no	—
233	#120	—	79	2	no	—
234	C₄×D₄	—	158	2	no	—
235	#117	C₄×D₇₉	1	1	yes	characteristic
236	M₃₂	—	2	1	no	—
237	M₃₂	—	2	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	2	1	no	—
240	M₃₂	—	158	4	no	—
241	M₃₂	—	158	1	no	—
242	M₃₂	—	158	1	no	—
243	M₃₂	—	158	1	no	—
244	M₃₂	—	158	1	no	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	79	2	no	—
247	C₂×C₁₆	—	158	4	no	—
248	C₂×C₁₆	—	158	1	no	—
249	C₂×C₁₆	D₃₁₆	1	2	yes	complemented
250	C₄⋊C₈	—	158	2	no	—
251	C₂²⋊C₈	—	158	2	no	—
252	C₂²⋊C₈	—	158	2	no	—
253	#97	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₂⁴	—	79	1	no	—
256	#41	—	158	1	no	—
257	#41	—	158	1	no	—
258	#41	—	158	1	no	—
259	#41	—	158	1	no	—
260	C₂×Q₈	—	79	1	no	—
261	C₂×D₄	—	79	2	no	—
262	C₂×D₄	—	79	1	no	—
263	C₂×D₄	—	158	2	no	—
264	C₂²×C₄	—	79	2	no	—
265	C₂²×C₄	—	79	1	no	—
266	C₂²×C₄	—	158	1	no	—
267	C₂²×C₄	—	158	1	no	—
268	C₂²×C₄	—	158	1	no	—
269	C₂²×C₄	—	158	1	no	—
270	C₂²×C₄	—	158	1	no	—
271	C₂²×C₄	—	158	1	no	—
272	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	158	1	no	—
276	M₁₆	—	158	1	no	—
277	M₁₆	—	158	2	no	—
278	M₁₆	—	158	2	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	316	1	no	—
281	C₂×C₈	—	316	1	no	—
282	C₂×C₈	—	316	1	no	—
283	C₂×C₈	—	316	1	no	—
284	C₂×C₈	—	79	2	no	—
285	C₂×C₈	—	158	1	no	—
286	C₂×C₈	—	158	1	no	—
287	C₂×C₈	—	158	1	no	—
288	C₂×C₈	—	158	2	no	—
289	C₂×C₈	—	158	2	no	—
290	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
291	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
292	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
293	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
294	C₂×C₈	#11929624	1	1	yes	characteristic
295	C₂×C₈	#11929625	1	1	yes	characteristic
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	158	2	no	—
298	C₂²⋊C₄	—	158	2	no	—
299	C₂²⋊C₄	—	158	2	no	—
300	C₂²⋊C₄	C₈×D₇₉	1	2	yes	—
301	C₄²	—	2	1	no	—
302	C₁₆	—	2	2	no	—
303	C₁₆	—	4	1	no	—
304	C₁₆	—	316	1	no	—
305	C₁₆	—	158	2	no	—
306	C₂³	—	316	2	no	—
307	C₂³	—	79	2	no	—
308	C₂³	—	158	1	no	—
309	C₂³	—	158	1	no	—
310	C₂³	—	158	1	no	—
311	C₂³	C₂×C₈×D₇₉	1	1	yes	characteristic
312	Q₈	—	158	1	no	—
313	Q₈	—	158	1	no	—
314	D₄	—	316	4	no	—
315	D₄	—	158	2	no	—
316	D₄	—	158	2	no	—
317	D₄	—	158	1	no	—
318	D₄	—	158	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	316	1	no	—
321	C₂×C₄	—	316	1	no	—
322	C₂×C₄	—	316	1	no	—
323	C₂×C₄	—	316	1	no	—
324	C₂×C₄	—	79	2	no	—
325	C₂×C₄	—	158	1	no	—
326	C₂×C₄	—	158	1	no	—
327	C₂×C₄	—	158	1	no	—
328	C₂×C₄	—	158	2	no	—
329	C₂×C₄	—	158	2	no	—
330	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
331	C₂×C₄	#14579123	1	1	yes	characteristic
332	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
333	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
334	C₂×C₄	#14579167	1	1	yes	characteristic
335	C₂×C₄	#14579169	1	1	yes	characteristic
336	C₈	—	4	1	no	—
337	C₈	—	316	1	no	—
338	C₈	—	158	2	no	—
339	C₈	#14579131	1	1	yes	characteristic
340	C₈	#14579134	1	1	yes	characteristic
341	C₈	#14579136	1	1	yes	characteristic
342	C₈	#14579146	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	316	2	no	—
345	C₂²	—	316	1	no	—
346	C₂²	—	316	1	no	—
347	C₂²	—	316	1	no	—
348	C₂²	—	316	1	no	—
349	C₂²	—	158	2	no	—
350	C₂²	—	158	2	no	—
351	C₂²	—	158	2	no	—
352	C₂²	—	158	2	no	—
353	C₂²	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
354	C₂²	#19098628	1	1	yes	characteristic
355	C₂²	C₈×D₃₁₆	1	1	yes	characteristic
356	C₄	—	4	2	no	—
357	C₄	—	316	1	no	—
358	C₄	—	158	2	no	—
359	C₄	#19098441	1	1	yes	characteristic
360	C₄	#19098528	1	1	yes	characteristic
361	C₄	#19098633	1	1	yes	characteristic
362	C₄	#19098636	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	316	2	no	—
367	C₂	—	316	2	no	—
368	C₂	#23763691	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579131
commutator	C₂³×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1378	C₂
socle	C₂×C₇₉	#1869
prime radical	#19099075	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1378	D₇₉
hypocenter	C₇₉	#12981
LCS	G ▷ C₂³×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
1. #23776502	C₂	semidirect
3. #23765020	C₂	semidirect
4. #23765010	C₂	semidirect
5. #23765010	C₂	semidirect
6. #23761486	C₂	semidirect
7. C₇₉×#1378	C₂	semidirect
54. C₂×C₇₉⋊C₁₆	D₄	semidirect
67. C₂×C₁₆×C₇₉	D₄	semidirect
191. #1378	D₇₉	semidirect
192. C₇₉	#12981	semidirect
249. C₂×C₁₆	D₃₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23776502	2
#23772096	2
#23765020	2
#23765010	2
#23761486	2
C₇₉×#1378	2
#12981	79