Subgroups — #34205147

Order 20224.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098981	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098681	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098348	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098348	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765015	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761478	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579183	C₄	1	1	yes	characteristic
9	C₂×#14579167	C₂²	1	1	yes	characteristic
10	C₂×#14579167	C₂²	1	1	yes	characteristic
11	#19098981	—	2	2	no	—
12	#19098981	—	2	2	no	—
13	C₂×#14579100	—	2	1	no	—
14	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₉	—	2	1	no	—
17	#19098681	—	2	2	no	—
18	#19098681	—	2	2	no	—
19	C₂×#14579097	C₄	1	1	yes	characteristic
20	C₂×#14579089	C₄	1	1	yes	characteristic
21	C₂×#14579089	C₄	1	1	yes	characteristic
22	#19098348	—	2	2	no	—
23	#19098348	—	2	2	no	—
24	(C₂×D₇₉)⋊C₁₆	—	2	1	no	—
25	(C₂×D₇₉)⋊C₁₆	—	2	1	no	—
26	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
30	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
31	#14579183	—	2	2	no	—
32	#14579183	—	2	2	no	—
33	C₂×Dic₃₁₆	C₈	1	1	yes	characteristic
34	C₂×#11929613	C₈	1	1	yes	characteristic
35	C₂×#11929613	C₈	1	1	yes	characteristic
36	C₂×D₆₃₂	C₈	1	1	yes	characteristic
37	#14579167	—	2	1	no	—
38	#14579167	—	2	1	no	—
39	#14579167	—	2	1	no	—
40	#14579167	—	2	1	no	—
41	#14579167	—	2	1	no	—
42	#14579167	—	2	1	no	—
43	#14579167	D₄	1	2	yes	—
44	#14579167	D₄	1	2	yes	—
45	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
46	C₂×#11929608	C₂×C₄	1	1	yes	characteristic
47	C₂×#11929607	—	2	1	no	—
48	C₂×#11929607	—	2	1	no	—
49	C₂×C₈×D₇₉	—	2	1	no	—
50	C₂×C₈×D₇₉	—	2	1	no	—
51	#14579100	—	2	4	no	—
52	C₂×C₇₉⋊C₁₆	—	2	1	no	—
53	C₂×C₇₉⋊C₁₆	—	2	1	no	—
54	C₂×C₇₉⋊C₁₆	—	2	1	no	—
55	C₂×C₇₉⋊C₁₆	—	2	1	no	—
56	C₂×C₇₉⋊C₁₆	—	2	1	no	—
57	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
58	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
59	C₁₆×D₇₉	—	2	4	no	—
60	#14579097	—	2	2	no	—
61	#14579089	—	2	2	no	—
62	#14579089	—	2	2	no	—
63	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
66	M₃₂×C₇₉	—	2	1	no	—
67	M₃₂×C₇₉	—	2	1	no	—
68	M₃₂×C₇₉	—	2	1	no	—
69	M₃₂×C₇₉	D₄	1	2	yes	—
70	C₂×C₁₆×C₇₉	—	2	1	no	—
71	C₂×C₁₆×C₇₉	—	2	1	no	—
72	C₄×C₈×C₇₉	—	2	1	no	—
73	C₄×C₈×C₇₉	—	2	1	no	—
74	#11929624	—	2	1	no	—
75	#11929624	—	2	1	no	—
76	#11929624	—	2	1	no	—
77	#11929624	—	2	1	no	—
78	#11929624	—	2	1	no	—
79	#11929624	—	2	1	no	—
80	#11929624	C₂²⋊C₄	1	2	yes	—
81	#11929624	M₁₆	1	2	yes	—
82	C₂×Dic₁₅₈	C₂×C₈	1	1	yes	characteristic
83	C₂×Dic₁₅₈	C₂×C₈	1	1	yes	characteristic
84	C₂×D₃₁₆	C₂×C₈	1	1	yes	characteristic
85	C₂×D₃₁₆	C₂×C₈	1	1	yes	characteristic
86	C₂×C₂²⋊D₇₉	—	2	1	no	—
87	C₂×C₂²⋊D₇₉	—	2	1	no	—
88	C₂×C₄×D₇₉	—	2	1	no	—
89	C₂×C₄×D₇₉	—	2	1	no	—
90	Dic₃₁₆	—	2	2	no	—
91	#11929613	—	2	2	no	—
92	#11929613	—	2	2	no	—
93	D₆₃₂	—	2	2	no	—
94	#11929608	—	2	1	no	—
95	#11929608	—	2	1	no	—
96	#11929608	—	2	1	no	—
97	#11929608	C₂²⋊C₄	1	2	yes	—
98	#11929607	—	2	2	no	—
99	#11929607	—	2	2	no	—
100	#11929607	—	4	1	no	—
101	#11929607	—	4	1	no	—
102	C₂×C₇₉⋊C₈	—	2	1	no	—
103	C₂×C₇₉⋊C₈	—	2	1	no	—
104	C₈×D₇₉	—	2	2	no	—
105	C₈×D₇₉	—	2	2	no	—
106	C₈×D₇₉	—	4	1	no	—
107	C₈×D₇₉	—	4	1	no	—
108	C₇₉⋊C₁₆	—	2	1	no	—
109	C₇₉⋊C₁₆	—	2	1	no	—
110	C₇₉⋊C₁₆	—	4	1	no	—
111	C₂²×C₄×C₇₉	—	2	1	no	—
112	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	—	2	1	no	—
114	C₂×C₈×C₇₉	—	2	1	no	—
115	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
117	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
122	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
123	C₄²×C₇₉	—	2	1	no	—
124	C₄²×C₇₉	—	2	1	no	—
125	C₁₆×C₇₉	—	2	2	no	—
126	C₁₆×C₇₉	—	4	1	no	—
127	C₂³×C₇₉	#131	1	1	yes	characteristic
128	C₂²×D₇₉	—	2	1	no	—
129	C₂²×D₇₉	—	2	1	no	—
130	C₂×C₄×C₇₉	—	2	2	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	1	no	—
133	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
139	C₂²⋊D₇₉	—	2	2	no	—
140	C₂²⋊D₇₉	—	2	2	no	—
141	C₂²⋊D₇₉	—	4	1	no	—
142	C₂²⋊D₇₉	—	4	1	no	—
143	C₂×Dic₇₉	—	2	1	no	—
144	C₂×Dic₇₉	—	2	1	no	—
145	D₃₁₆	—	2	1	no	—
146	D₃₁₆	—	2	1	no	—
147	D₃₁₆	—	2	1	no	—
148	D₃₁₆	C₂²⋊C₈	1	2	yes	—
149	C₄×D₇₉	—	2	2	no	—
150	C₄×D₇₉	—	2	2	no	—
151	C₄×D₇₉	—	4	1	no	—
152	C₄×D₇₉	—	4	1	no	—
153	Dic₁₅₈	—	2	1	no	—
154	Dic₁₅₈	—	2	1	no	—
155	Dic₁₅₈	—	2	1	no	—
156	Dic₁₅₈	C₂²⋊C₈	1	2	yes	—
157	C₈×C₇₉	—	2	1	no	—
158	C₈×C₇₉	—	2	1	no	—
159	C₈×C₇₉	#120	1	1	yes	characteristic
160	C₈×C₇₉	#121	1	1	yes	characteristic
161	C₈×C₇₉	#121	1	1	yes	characteristic
162	C₈×C₇₉	#123	1	1	yes	characteristic
163	C₇₉⋊C₈	—	2	2	no	—
164	C₇₉⋊C₈	—	4	1	no	—
165	C₂²×C₇₉	—	2	1	no	—
166	C₂²×C₇₉	—	2	1	no	—
167	C₂²×C₇₉	#408	1	1	yes	characteristic
168	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
169	C₂²×C₇₉	#435	1	1	yes	characteristic
170	C₂×D₇₉	—	2	2	no	—
171	C₂×D₇₉	—	4	1	no	—
172	C₂×D₇₉	—	4	1	no	—
173	C₂×D₇₉	—	4	1	no	—
174	C₂×D₇₉	—	4	1	no	—
175	C₂×D₇₉	—	4	1	no	—
176	C₄×C₇₉	—	4	2	no	—
177	C₄×C₇₉	#386	1	1	yes	characteristic
178	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
179	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
180	C₄×C₇₉	#435	1	1	yes	characteristic
181	Dic₇₉	—	2	2	no	—
182	Dic₇₉	—	4	1	no	—
183	#12980	—	79	1	no	maximal, Sylow
184	C₂×C₇₉	—	2	1	no	—
185	C₂×C₇₉	—	2	1	no	—
186	C₂×C₇₉	#1869	1	1	yes	characteristic
187	C₂×C₇₉	#2170	1	1	yes	characteristic
188	C₂×C₇₉	#2171	1	1	yes	characteristic
189	D₇₉	—	4	2	no	—
190	D₇₉	—	8	1	no	—
191	C₂×#504	—	79	1	no	—
192	C₂×#443	—	79	1	no	—
193	C₂×#350	—	79	1	no	—
194	C₂×#350	—	79	1	no	—
195	#2113	—	79	1	no	—
196	#1376	D₇₉	1	1	yes	characteristic, complemented
197	#1374	—	79	1	no	—
198	C₇₉	#12980	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	—	79	1	no	—
200	C₂×#131	—	79	1	no	—
201	C₂×#131	—	79	1	no	—
202	#504	—	158	2	no	—
203	#504	—	158	2	no	—
204	C₂×M₃₂	—	158	1	no	—
205	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
206	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
207	C₂²×C₁₆	—	79	1	no	—
208	C₂²×C₁₆	—	79	1	no	—
209	C₂²×C₁₆	—	158	1	no	—
210	#443	—	158	2	no	—
211	#443	—	158	2	no	—
212	C₂×#108	—	79	1	no	—
213	C₂×#104	—	79	1	no	—
214	C₂×#104	—	79	1	no	—
215	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
216	#350	—	158	2	no	—
217	#350	—	158	2	no	—
218	C₂²⋊C₁₆	—	158	1	no	—
219	C₂²⋊C₁₆	—	158	1	no	—
220	C₂×#41	—	79	1	no	—
221	C₂×#41	—	79	1	no	—
222	#135	—	158	2	no	—
223	#135	—	158	2	no	—
224	C₂×Q₁₆	—	79	1	no	—
225	C₂×SD₁₆	—	79	1	no	—
226	C₂×SD₁₆	—	79	1	no	—
227	C₂×D₈	—	79	1	no	—
228	#131	—	79	2	no	—
229	#131	—	79	2	no	—
230	#131	—	158	1	no	—
231	#131	—	158	1	no	—
232	#131	—	158	1	no	—
233	#131	—	158	1	no	—
234	#131	—	158	1	no	—
235	#131	—	158	1	no	—
236	C₂×M₁₆	—	79	1	no	—
237	C₂×M₁₆	—	79	1	no	—
238	C₂×M₁₆	—	158	1	no	—
239	C₂×M₁₆	—	158	1	no	—
240	C₂²×C₈	—	158	1	no	—
241	C₂²×C₈	—	158	1	no	—
242	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
243	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
244	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
245	M₃₂	—	2	1	no	—
246	M₃₂	—	2	1	no	—
247	M₃₂	—	2	1	no	—
248	M₃₂	—	158	4	no	—
249	M₃₂	D₃₁₆	1	2	yes	—
250	C₂×C₁₆	—	2	1	no	—
251	C₂×C₁₆	—	2	1	no	—
252	C₂×C₁₆	—	79	1	no	—
253	C₂×C₁₆	—	79	1	no	—
254	C₂×C₁₆	—	158	4	no	—
255	C₂×C₁₆	—	158	1	no	—
256	C₂×C₁₆	—	158	1	no	—
257	C₂×C₁₆	—	158	1	no	—
258	C₂×C₁₆	—	158	1	no	—
259	C₂×C₁₆	—	158	1	no	—
260	#108	—	158	2	no	—
261	#104	—	158	2	no	—
262	#104	—	158	2	no	—
263	C₄×C₈	—	2	1	no	—
264	C₄×C₈	—	2	1	no	—
265	#41	—	79	2	no	—
266	#41	—	79	2	no	—
267	#41	—	158	1	no	—
268	#41	—	158	1	no	—
269	#41	—	158	1	no	—
270	#41	—	158	1	no	—
271	#41	—	158	1	no	—
272	#41	—	158	1	no	—
273	C₂×Q₈	—	79	1	no	—
274	C₂×Q₈	—	79	1	no	—
275	C₂×D₄	—	79	1	no	—
276	C₂×D₄	—	79	1	no	—
277	C₂×D₄	—	158	1	no	—
278	C₂×D₄	—	158	1	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	158	1	no	—
281	C₂²×C₄	—	158	1	no	—
282	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
283	Q₁₆	—	158	2	no	—
284	SD₁₆	—	158	2	no	—
285	SD₁₆	—	158	2	no	—
286	D₈	—	158	2	no	—
287	M₁₆	—	316	1	no	—
288	M₁₆	—	316	1	no	—
289	M₁₆	—	79	2	no	—
290	M₁₆	—	158	1	no	—
291	M₁₆	—	158	1	no	—
292	M₁₆	—	158	1	no	—
293	M₁₆	—	158	2	no	—
294	M₁₆	—	158	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	316	1	no	—
298	C₂×C₈	—	316	1	no	—
299	C₂×C₈	—	158	1	no	—
300	C₂×C₈	—	158	1	no	—
301	C₂×C₈	—	158	2	no	—
302	C₂×C₈	—	158	2	no	—
303	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
304	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
305	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
306	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
307	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
308	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
309	C₂×C₈	#11929624	1	1	yes	characteristic
310	C₂×C₈	#11929625	1	1	yes	characteristic
311	C₄²	—	2	1	no	—
312	C₄²	—	2	1	no	—
313	C₁₆	—	2	2	no	—
314	C₁₆	—	4	1	no	—
315	C₁₆	—	316	1	no	—
316	C₁₆	—	158	1	no	—
317	C₁₆	—	158	1	no	—
318	C₂³	—	158	1	no	—
319	C₂³	—	158	1	no	—
320	C₂³	#14579169	1	1	yes	characteristic
321	Q₈	—	79	2	no	—
322	Q₈	—	158	1	no	—
323	Q₈	—	158	1	no	—
324	Q₈	—	158	1	no	—
325	D₄	—	316	1	no	—
326	D₄	—	316	1	no	—
327	D₄	—	79	2	no	—
328	D₄	—	158	2	no	—
329	D₄	—	158	2	no	—
330	D₄	—	158	1	no	—
331	D₄	—	158	1	no	—
332	D₄	—	158	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	316	1	no	—
337	C₂×C₄	—	316	1	no	—
338	C₂×C₄	—	158	1	no	—
339	C₂×C₄	—	158	1	no	—
340	C₂×C₄	—	158	2	no	—
341	C₂×C₄	—	158	2	no	—
342	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
343	C₂×C₄	#14579123	1	1	yes	characteristic
344	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
346	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
347	C₂×C₄	#14579167	1	1	yes	characteristic
348	C₈	—	2	1	no	—
349	C₈	—	2	1	no	—
350	C₈	—	316	1	no	—
351	C₈	—	158	2	no	—
352	C₈	#14579131	1	1	yes	characteristic
353	C₈	#14579134	1	1	yes	characteristic
354	C₈	#14579136	1	1	yes	characteristic
355	C₈	#14579146	1	1	yes	characteristic
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	316	1	no	—
359	C₂²	—	316	1	no	—
360	C₂²	—	316	1	no	—
361	C₂²	—	316	1	no	—
362	C₂²	—	316	1	no	—
363	C₂²	—	158	2	no	—
364	C₂²	#19098528	1	1	yes	characteristic
365	C₂²	#19098628	1	1	yes	characteristic
366	C₂²	#19098636	1	1	yes	characteristic
367	C₄	—	4	2	no	—
368	C₄	—	316	1	no	—
369	C₄	—	158	2	no	—
370	C₄	#19098441	1	1	yes	characteristic
371	C₄	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
372	C₄	C₈×D₃₁₆	1	1	yes	characteristic
373	C₄	#19098633	1	1	yes	characteristic
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	316	2	no	—
377	C₂	—	632	1	no	—
378	C₂	#23763691	1	1	yes	characteristic
379	C₂	#23765236	1	1	yes	characteristic
380	C₂	#23765241	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1376	C₂
socle	C₂²×C₇₉	C₈×D₄
prime radical	C₂×#14579183	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1376	D₇₉
hypocenter	C₇₉	#12980
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
1. C₂×#19098981	C₂	semidirect
3. C₂×#19098348	C₂	semidirect
4. C₂×#19098348	C₂	semidirect
5. #23765015	C₂	semidirect
6. #23761478	C₂	semidirect
7. C₇₉×#1376	C₂	semidirect
196. #1376	D₇₉	semidirect
198. C₇₉	#12980	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098981	2
C₂×#19098681	2
C₂×#19098348	2
#23765015	2
#23761478	2
C₇₉×#1376	2
#12980	79