Subgroups — #34188063

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098937	C₂	1	1	yes	maximal, characteristic
2	#23764362	C₂	1	1	yes	maximal, characteristic, complemented
3	#23764206	C₂	1	1	yes	maximal, characteristic
4	#23764205	C₂	1	1	yes	maximal, characteristic
5	#23763527	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19098284	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11929611	—	2	2	no	—
9	C₂×#14579158	C₂²	1	1	yes	characteristic
10	#19098937	—	2	2	no	—
11	#19098937	C₄	1	4	yes	complemented
12	C₂×#14579083	—	2	2	no	—
13	C₂×#14579081	—	2	2	no	—
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	#19098433	C₂²	1	1	yes	characteristic
17	#19098325	—	2	2	no	—
18	#19098325	—	2	2	no	—
19	#19098284	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	#384×C₇₉	C₂²	1	1	yes	characteristic
23	#340×C₇₉	—	2	2	no	—
24	C₂²×D₃₁₆	—	2	1	no	—
25	C₂²×D₃₁₆	—	2	1	no	—
26	C₂²×D₃₁₆	—	2	1	no	—
27	C₂×#11929611	—	2	8	no	—
28	C₂×#11929611	—	4	4	no	—
29	C₂×#11929611	—	4	4	no	—
30	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
31	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
32	#14579158	—	2	1	no	—
33	#14579158	—	2	1	no	—
34	#14579158	—	2	1	no	—
35	#14579158	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₇₉	—	2	1	no	—
37	C₂×C₄⋊Dic₇₉	—	2	1	no	—
38	C₂×C₄⋊Dic₇₉	—	2	1	no	—
39	#14579083	—	4	4	no	—
40	#14579081	—	4	4	no	—
41	C₄×C₇₉⋊C₈	—	2	1	no	—
42	C₄×C₇₉⋊C₈	—	2	1	no	—
43	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
45	C₂²×Q₈×C₇₉	—	2	1	no	—
46	C₂²×C₈×C₇₉	D₄	1	2	yes	—
47	C₄⋊Q₈×C₇₉	—	2	1	no	—
48	C₄⋊Q₈×C₇₉	—	2	1	no	—
49	C₄⋊Q₈×C₇₉	—	2	1	no	—
50	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
56	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₇₉	—	2	1	no	—
58	C₄×C₈×C₇₉	—	2	1	no	—
59	C₂³×D₇₉	—	4	1	no	—
60	C₂×D₃₁₆	—	2	2	no	—
61	C₂×D₃₁₆	—	2	2	no	—
62	C₂×D₃₁₆	—	2	2	no	—
63	C₂×D₃₁₆	—	4	1	no	—
64	C₂×D₃₁₆	—	4	1	no	—
65	C₂×D₃₁₆	—	4	1	no	—
66	C₂×D₃₁₆	—	4	2	no	—
67	C₂×D₃₁₆	—	4	2	no	—
68	C₂×D₃₁₆	—	4	2	no	—
69	C₂×D₃₁₆	—	4	2	no	—
70	C₂×D₃₁₆	—	4	2	no	—
71	C₂×D₃₁₆	—	4	2	no	—
72	C₂²×Dic₇₉	—	4	1	no	—
73	#11929611	—	4	16	no	—
74	C₂×C₇₉⋊C₈	—	2	2	no	—
75	C₂×C₇₉⋊C₈	—	2	2	no	—
76	C₂×C₇₉⋊C₈	—	2	2	no	—
77	C₂×C₇₉⋊C₈	—	2	2	no	—
78	C₂×C₇₉⋊C₈	—	2	2	no	—
79	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	4	yes	—
80	C₄⋊Dic₇₉	—	4	2	no	—
81	C₄⋊Dic₇₉	—	4	2	no	—
82	C₄⋊Dic₇₉	—	4	2	no	—
83	C₂×Q₈×C₇₉	—	2	2	no	—
84	C₂×Q₈×C₇₉	—	4	1	no	—
85	C₂×Q₈×C₇₉	—	4	2	no	—
86	C₂×Q₈×C₇₉	—	4	2	no	—
87	C₂²×C₄×C₇₉	—	4	1	no	—
88	C₂²×C₄×C₇₉	—	4	1	no	—
89	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	SD₁₆	1	4	yes	complemented
95	C₄⋊C₄×C₇₉	—	2	2	no	—
96	C₄⋊C₄×C₇₉	—	2	2	no	—
97	C₄⋊C₄×C₇₉	—	4	1	no	—
98	C₄⋊C₄×C₇₉	—	4	1	no	—
99	C₄⋊C₄×C₇₉	—	4	2	no	—
100	C₄⋊C₄×C₇₉	—	4	2	no	—
101	C₄⋊C₄×C₇₉	—	4	2	no	—
102	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇₉	#41	1	1	yes	characteristic
105	C₄²×C₇₉	#41	1	1	yes	characteristic
106	C₂³×C₇₉	#127	1	1	yes	characteristic
107	C₂²×D₇₉	—	4	2	no	—
108	C₂²×D₇₉	—	8	1	no	—
109	C₂²×D₇₉	—	8	1	no	—
110	C₂²×D₇₉	—	8	1	no	—
111	C₂²×D₇₉	—	8	1	no	—
112	C₂²×D₇₉	—	8	1	no	—
113	C₂²×D₇₉	—	8	1	no	—
114	Q₈×C₇₉	—	4	4	no	—
115	Q₈×C₇₉	—	8	2	no	—
116	C₂×C₄×C₇₉	—	4	2	no	—
117	C₂×C₄×C₇₉	—	8	1	no	—
118	C₂×C₄×C₇₉	—	8	1	no	—
119	C₂×C₄×C₇₉	—	8	1	no	—
120	C₂×C₄×C₇₉	—	8	1	no	—
121	C₂×C₄×C₇₉	—	8	1	no	—
122	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₇₉	#135	1	2	yes	—
130	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₇₉	#137	1	2	yes	—
134	C₂×Dic₇₉	—	8	1	no	—
135	C₂×Dic₇₉	—	8	1	no	—
136	C₂×Dic₇₉	—	8	1	no	—
137	D₃₁₆	—	4	4	no	—
138	D₃₁₆	—	4	4	no	—
139	D₃₁₆	—	4	4	no	—
140	D₃₁₆	—	8	2	no	—
141	D₃₁₆	—	8	2	no	—
142	D₃₁₆	—	8	2	no	—
143	C₈×C₇₉	—	2	4	no	—
144	C₇₉⋊C₈	—	2	4	no	—
145	C₇₉⋊C₈	—	4	2	no	—
146	C₂²×C₇₉	#390	1	1	yes	characteristic
147	C₂²×C₇₉	#492	1	1	yes	characteristic
148	C₂²×C₇₉	#495	1	1	yes	characteristic
149	C₂²×C₇₉	#496	1	1	yes	characteristic
150	C₂²×C₇₉	#496	1	1	yes	characteristic
151	C₂²×C₇₉	#496	1	1	yes	characteristic
152	C₂²×C₇₉	#497	1	1	yes	characteristic
153	C₂×D₇₉	—	8	2	no	—
154	C₂×D₇₉	—	8	2	no	—
155	C₂×D₇₉	—	8	2	no	—
156	C₂×D₇₉	—	16	1	no	—
157	C₂×D₇₉	—	16	1	no	—
158	C₂×D₇₉	—	16	1	no	—
159	C₂×D₇₉	—	16	1	no	—
160	C₄×C₇₉	—	8	2	no	—
161	C₄×C₇₉	—	16	1	no	—
162	C₄×C₇₉	#418	1	2	yes	—
163	C₄×C₇₉	C₄×SD₁₆	1	2	yes	—
164	C₄×C₇₉	#440	1	2	yes	—
165	C₄×C₇₉	#465	1	2	yes	—
166	C₄×C₇₉	#468	1	2	yes	—
167	C₄×C₇₉	#490	1	2	yes	—
168	Dic₇₉	—	16	1	no	—
169	#9829	—	79	1	no	maximal, Sylow
170	C₂×C₇₉	#1714	1	1	yes	characteristic
171	C₂×C₇₉	#1715	1	1	yes	characteristic
172	C₂×C₇₉	#1717	1	1	yes	characteristic
173	C₂×C₇₉	#1719	1	1	yes	characteristic
174	C₂×C₇₉	#1961	1	1	yes	characteristic
175	C₂×C₇₉	#1968	1	1	yes	characteristic
176	C₂×C₇₉	#1973	1	1	yes	characteristic
177	D₇₉	—	16	2	no	—
178	C₂×#492	—	79	1	no	—
179	#1990	—	79	1	no	—
180	#1990	D₇₉	1	1	yes	characteristic, complemented
181	#1957	—	79	1	no	—
182	#1957	—	79	1	no	—
183	#1834	—	79	1	no	—
184	C₂×#322	—	79	1	no	—
185	C₇₉	#9829	1	1	yes	characteristic, complemented, Sylow
186	C₂²×SD₁₆	—	158	2	no	—
187	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
188	C₂×#127	—	79	1	no	—
189	#492	—	79	4	no	—
190	#492	—	158	2	no	—
191	C₂×#102	—	158	2	no	—
192	C₂×#102	—	158	2	no	—
193	C₂×C₄×C₈	—	79	1	no	—
194	C₂×C₄×C₈	—	79	1	no	—
195	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
196	#384	—	79	1	no	—
197	#384	C₂×D₇₉	1	1	yes	characteristic
198	#340	—	2	2	no	—
199	#340	—	158	2	no	—
200	#340	—	158	2	no	—
201	#322	—	158	2	no	—
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×D₄	—	158	1	no	—
204	C₂²×D₄	—	158	1	no	—
205	C₂²×D₄	—	158	1	no	—
206	C₂×SD₁₆	—	316	4	no	—
207	C₂×SD₁₆	—	316	4	no	—
208	C₂×SD₁₆	—	158	8	no	—
209	C₂²×C₈	—	79	2	no	—
210	C₂²×C₈	—	79	2	no	—
211	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	C₄×D₇₉	1	2	yes	—
216	#127	—	79	2	no	—
217	#127	—	158	1	no	—
218	#127	—	158	1	no	—
219	#127	—	158	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	158	1	no	—
226	C₂×C₄⋊C₄	—	158	1	no	—
227	C₂×C₄⋊C₄	—	158	1	no	—
228	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
229	#102	—	316	4	no	—
230	#102	—	316	4	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	79	2	no	—
234	C₄×C₈	—	79	2	no	—
235	C₄×C₈	—	158	1	no	—
236	C₄×C₈	—	158	1	no	—
237	C₂⁴	—	316	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×D₄	—	316	1	no	—
243	C₂×D₄	—	316	1	no	—
244	C₂×D₄	—	316	1	no	—
245	C₂×D₄	—	316	2	no	—
246	C₂×D₄	—	316	2	no	—
247	C₂×D₄	—	316	2	no	—
248	C₂×D₄	—	316	2	no	—
249	C₂×D₄	—	316	2	no	—
250	C₂×D₄	—	316	2	no	—
251	C₂×D₄	—	158	2	no	—
252	C₂×D₄	—	158	2	no	—
253	C₂×D₄	—	158	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	316	1	no	—
257	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
260	SD₁₆	—	316	16	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	79	4	no	—
264	C₂×C₈	—	158	2	no	—
265	C₂×C₈	—	158	2	no	—
266	C₂×C₈	—	158	2	no	—
267	C₂×C₈	—	158	2	no	—
268	C₂×C₈	—	158	2	no	—
269	C₂×C₈	#11929611	1	4	yes	complemented
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	316	2	no	—
278	C₄⋊C₄	—	316	2	no	—
279	C₄⋊C₄	—	316	2	no	—
280	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
281	C₄²	D₄×D₇₉	1	1	yes	characteristic
282	C₄²	#11929623	1	1	yes	characteristic
283	C₄²	#11929624	1	1	yes	characteristic
284	C₂³	—	316	2	no	—
285	C₂³	—	632	1	no	—
286	C₂³	—	632	1	no	—
287	C₂³	—	632	1	no	—
288	C₂³	—	632	1	no	—
289	C₂³	—	632	1	no	—
290	C₂³	—	632	1	no	—
291	C₂³	#14579157	1	1	yes	characteristic
292	Q₈	—	4	4	no	—
293	Q₈	—	8	2	no	—
294	D₄	—	316	4	no	—
295	D₄	—	316	4	no	—
296	D₄	—	316	4	no	—
297	D₄	—	632	2	no	—
298	D₄	—	632	2	no	—
299	D₄	—	632	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	—	632	1	no	—
307	C₂×C₄	—	632	1	no	—
308	C₂×C₄	—	632	1	no	—
309	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
310	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
312	C₂×C₄	#14579133	1	1	yes	characteristic
313	C₂×C₄	#14579134	1	1	yes	characteristic
314	C₂×C₄	#14579147	1	1	yes	characteristic
315	C₂×C₄	C₂×#11929611	1	2	yes	—
316	C₂×C₄	#14579181	1	2	yes	—
317	C₂×C₄	#14579185	1	2	yes	—
318	C₂×C₄	#14579188	1	2	yes	—
319	C₂×C₄	#14579188	1	2	yes	—
320	C₂×C₄	#14579193	1	2	yes	—
321	C₈	—	2	4	no	—
322	C₈	—	316	2	no	—
323	C₈	—	158	4	no	—
324	C₂²	—	632	2	no	—
325	C₂²	—	632	2	no	—
326	C₂²	—	632	2	no	—
327	C₂²	—	1264	1	no	—
328	C₂²	—	1264	1	no	—
329	C₂²	—	1264	1	no	—
330	C₂²	—	1264	1	no	—
331	C₂²	#19098463	1	1	yes	characteristic
332	C₂²	#19098937	1	1	yes	characteristic
333	C₂²	#19098945	1	1	yes	characteristic
334	C₂²	#19098949	1	1	yes	characteristic
335	C₂²	#19098949	1	1	yes	characteristic
336	C₂²	#19098951	1	1	yes	characteristic
337	C₂²	#19098956	1	1	yes	characteristic
338	C₄	—	8	2	no	—
339	C₄	—	16	1	no	—
340	C₄	—	1264	1	no	—
341	C₄	#19098566	1	2	yes	—
342	C₄	C₄×#11929611	1	2	yes	—
343	C₄	#19098666	1	2	yes	—
344	C₄	#19098794	1	2	yes	—
345	C₄	#19098812	1	2	yes	—
346	C₄	#19098926	1	2	yes	—
347	C₂	—	1264	2	no	—
348	C₂	#23763023	1	1	yes	characteristic
349	C₂	#23763029	1	1	yes	characteristic
350	C₂	#23763040	1	1	yes	characteristic
351	C₂	#23763054	1	1	yes	characteristic
352	C₂	#23764222	1	1	yes	characteristic
353	C₂	#23764257	1	1	yes	characteristic
354	C₂	#23764278	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1990	C₂
socle	C₂³×C₇₉	#127
prime radical	C₂×#14579158	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1990	D₇₉
hypocenter	C₇₉	#9829
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
2. #23764362	C₂	semidirect
5. #23763527	C₂	semidirect
6. C₂×#19098284	C₂	semidirect
7. C₇₉×#1990	C₂	semidirect
11. #19098937	C₄	semidirect
94. C₂×C₈×C₇₉	SD₁₆	semidirect
180. #1990	D₇₉	semidirect
185. C₇₉	#9829	semidirect
269. C₂×C₈	#11929611	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098937	2
#23764362	2
#23764206	2
#23764205	2
#23763527	2
C₂×#19098284	2
C₇₉×#1990	2
#9829	79