Subgroups — #34189392

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098850	C₂	1	1	yes	maximal, characteristic
2	#23768574	C₂	1	1	yes	maximal, characteristic
3	#23764349	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098312	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098292	C₂	1	1	yes	maximal, characteristic
6	#23761240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#494×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
9	#19098850	—	2	2	no	—
10	#19098850	—	2	2	no	—
11	C₂×#14579092	—	2	1	no	—
12	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14579081	—	2	1	no	—
15	#19098474	—	2	2	no	—
16	#19098464	—	2	1	no	—
17	#19098428	—	2	1	no	—
18	#19098420	—	2	2	no	—
19	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
20	#19098325	—	2	2	no	—
21	#19098312	—	2	2	no	—
22	#19098292	—	2	2	no	—
23	C₂²×Q₁₆×C₇₉	—	2	2	no	—
24	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
26	#494×C₇₉	—	2	2	no	—
27	#494×C₇₉	—	2	2	no	—
28	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²×D₃₁₆	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₇₉	—	2	1	no	—
31	C₂²×C₇₉⋊C₈	—	2	1	no	—
32	C₂²×C₇₉⋊C₈	—	2	1	no	—
33	C₄×D₃₁₆	—	2	1	no	—
34	C₄×D₃₁₆	—	2	1	no	—
35	C₄×D₃₁₆	—	2	1	no	—
36	C₄×D₃₁₆	—	2	1	no	—
37	C₂×C₄⋊Dic₇₉	D₄	1	2	yes	—
38	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
39	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
40	C₂×(C₂×D₇₉)⋊C₄	—	2	4	no	—
41	C₂×C₄×Dic₇₉	—	2	1	no	—
42	#14579092	—	2	4	no	—
43	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
44	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
46	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
47	#14579081	—	2	4	no	—
48	C₇₉⋊#95	—	2	4	no	—
49	C₇₉⋊#95	—	2	1	no	—
50	C₂²×Q₈×C₇₉	—	2	1	no	—
51	C₂²×Q₈×C₇₉	D₄	1	2	yes	—
52	C₂×Q₁₆×C₇₉	—	2	8	no	—
53	C₂×Q₁₆×C₇₉	—	4	8	no	—
54	C₂²×C₈×C₇₉	—	2	1	no	—
55	C₄⋊Q₈×C₇₉	—	2	1	no	—
56	C₄⋊Q₈×C₇₉	—	2	1	no	—
57	C₄⋊Q₈×C₇₉	—	2	1	no	—
58	C₄⋊Q₈×C₇₉	—	2	1	no	—
59	C₄⋊Q₈×C₇₉	—	2	1	no	—
60	C₄⋊Q₈×C₇₉	—	2	1	no	—
61	C₄⋊Q₈×C₇₉	—	2	1	no	—
62	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
65	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
66	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
67	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₉	—	2	1	no	—
69	C₄×C₈×C₇₉	D₄	1	2	yes	—
70	C₂³×D₇₉	—	2	1	no	—
71	C₂×D₃₁₆	—	2	1	no	—
72	C₂×D₃₁₆	—	2	1	no	—
73	C₂×D₃₁₆	—	2	1	no	—
74	C₂×D₃₁₆	—	2	1	no	—
75	C₂×D₃₁₆	—	2	1	no	—
76	C₂×D₃₁₆	Q₁₆	1	2	yes	complemented
77	C₂²×Dic₇₉	—	2	1	no	—
78	C₂²×Dic₇₉	—	2	2	no	—
79	C₂×C₄×D₇₉	—	4	1	no	—
80	C₂×C₄×D₇₉	—	4	1	no	—
81	C₂×C₄×D₇₉	—	4	2	no	—
82	C₂×C₄×D₇₉	—	4	2	no	—
83	C₂×C₇₉⋊C₈	—	2	2	no	—
84	C₂×C₇₉⋊C₈	—	4	1	no	—
85	C₂×C₇₉⋊C₈	—	4	1	no	—
86	C₂×C₇₉⋊C₈	—	4	2	no	—
87	C₂×C₇₉⋊C₈	—	4	1	no	—
88	C₄⋊Dic₇₉	—	2	1	no	—
89	C₄⋊Dic₇₉	—	2	1	no	—
90	C₄⋊Dic₇₉	—	2	2	no	—
91	C₄⋊Dic₇₉	—	2	2	no	—
92	(C₂×D₇₉)⋊C₄	—	4	1	no	—
93	(C₂×D₇₉)⋊C₄	—	4	1	no	—
94	(C₂×D₇₉)⋊C₄	—	8	4	no	—
95	C₄×Dic₇₉	—	4	1	no	—
96	C₄×Dic₇₉	—	4	1	no	—
97	C₂×Q₈×C₇₉	—	2	2	no	—
98	C₂×Q₈×C₇₉	—	2	2	no	—
99	C₂×Q₈×C₇₉	—	2	2	no	—
100	C₂×Q₈×C₇₉	—	4	4	no	—
101	C₂×Q₈×C₇₉	—	4	4	no	—
102	C₂×Q₈×C₇₉	—	4	1	no	—
103	C₂²×C₄×C₇₉	—	2	2	no	—
104	C₂²×C₄×C₇₉	—	4	1	no	—
105	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₇₉	—	4	16	no	—
109	C₂×C₈×C₇₉	—	2	4	no	—
110	C₂×C₈×C₇₉	—	2	2	no	—
111	C₄⋊C₄×C₇₉	—	2	2	no	—
112	C₄⋊C₄×C₇₉	—	2	2	no	—
113	C₄⋊C₄×C₇₉	—	4	1	no	—
114	C₄⋊C₄×C₇₉	—	4	1	no	—
115	C₄⋊C₄×C₇₉	—	4	1	no	—
116	C₄⋊C₄×C₇₉	—	4	1	no	—
117	C₄⋊C₄×C₇₉	—	4	1	no	—
118	C₄⋊C₄×C₇₉	—	4	1	no	—
119	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₇₉	#41	1	1	yes	characteristic
123	C₂³×C₇₉	#120	1	1	yes	characteristic
124	C₂²×D₇₉	—	2	2	no	—
125	C₂²×D₇₉	—	4	1	no	—
126	C₂²×D₇₉	—	4	2	no	—
127	C₂²×D₇₉	—	4	1	no	—
128	C₂²×D₇₉	—	4	2	no	—
129	Q₈×C₇₉	—	4	4	no	—
130	Q₈×C₇₉	—	4	4	no	—
131	Q₈×C₇₉	—	8	2	no	—
132	Q₈×C₇₉	—	8	2	no	—
133	C₂×C₄×C₇₉	—	2	2	no	—
134	C₂×C₄×C₇₉	—	4	2	no	—
135	C₂×C₄×C₇₉	—	4	2	no	—
136	C₂×C₄×C₇₉	—	4	4	no	—
137	C₂×C₄×C₇₉	—	8	2	no	—
138	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
140	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
146	C₂×C₄×C₇₉	#135	1	2	yes	—
147	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
148	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
149	C₂×C₄×C₇₉	#137	1	2	yes	—
150	C₂×Dic₇₉	—	4	1	no	—
151	C₂×Dic₇₉	—	4	2	no	—
152	C₂×Dic₇₉	—	4	2	no	—
153	C₂×Dic₇₉	—	4	2	no	—
154	C₂×Dic₇₉	—	4	2	no	—
155	D₃₁₆	—	2	2	no	—
156	D₃₁₆	—	2	2	no	—
157	D₃₁₆	—	4	1	no	—
158	D₃₁₆	—	4	1	no	—
159	C₄×D₇₉	—	8	2	no	—
160	C₄×D₇₉	—	8	1	no	—
161	C₄×D₇₉	—	8	1	no	—
162	C₈×C₇₉	—	2	4	no	—
163	C₇₉⋊C₈	—	4	2	no	—
164	C₇₉⋊C₈	—	8	1	no	—
165	C₂²×C₇₉	#392	1	1	yes	characteristic
166	C₂²×C₇₉	#451	1	1	yes	characteristic
167	C₂²×C₇₉	#452	1	2	yes	—
168	C₂²×C₇₉	#452	1	1	yes	characteristic
169	C₂²×C₇₉	#453	1	2	yes	—
170	C₂×D₇₉	—	4	4	no	—
171	C₂×D₇₉	—	4	2	no	—
172	C₂×D₇₉	—	8	2	no	—
173	C₂×D₇₉	—	8	1	no	—
174	C₂×D₇₉	—	8	1	no	—
175	C₄×C₇₉	—	2	1	no	—
176	C₄×C₇₉	—	2	1	no	—
177	C₄×C₇₉	—	8	2	no	—
178	C₄×C₇₉	—	8	2	no	—
179	C₄×C₇₉	#451	1	1	yes	characteristic
180	C₄×C₇₉	#452	1	1	yes	characteristic
181	C₄×C₇₉	#464	1	2	yes	—
182	C₄×C₇₉	#467	1	1	yes	characteristic
183	C₄×C₇₉	#470	1	1	yes	characteristic
184	C₄×C₇₉	#495	1	1	yes	characteristic
185	C₄×C₇₉	#497	1	1	yes	characteristic
186	Dic₇₉	—	8	2	no	—
187	Dic₇₉	—	8	1	no	—
188	#10056	—	79	1	no	maximal, Sylow
189	C₂×C₇₉	#1632	1	2	yes	—
190	C₂×C₇₉	#1634	1	2	yes	—
191	C₂×C₇₉	#2001	1	1	yes	characteristic
192	C₂×C₇₉	#2002	1	1	yes	characteristic
193	C₂×C₇₉	#2015	1	1	yes	characteristic
194	D₇₉	—	8	2	no	—
195	C₂×#494	D₇₉	1	1	yes	characteristic, complemented
196	C₂×#476	—	79	1	no	—
197	#2563	—	79	1	no	—
198	#1987	—	79	1	no	—
199	C₂×#333	—	79	1	no	—
200	C₂×D₄⋊C₈	—	79	1	no	—
201	#1279	—	79	1	no	—
202	C₇₉	#10056	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₁₆	—	2	2	no	—
204	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
205	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
206	C₂×C₄×D₄	—	79	1	no	—
207	#494	—	2	2	no	—
208	#494	—	2	2	no	—
209	#476	—	158	2	no	—
210	#476	—	158	2	no	—
211	C₂×#106	—	158	1	no	—
212	C₂×C₄⋊C₈	—	79	1	no	—
213	C₂×C₄⋊C₈	—	79	1	no	—
214	C₂×#102	—	158	1	no	—
215	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
216	#393	—	158	2	no	—
217	#390	—	158	1	no	—
218	#382	—	158	1	no	—
219	#380	—	158	2	no	—
220	C₄×C₄⋊C₄	—	79	1	no	—
221	#340	—	158	2	no	—
222	#333	—	158	2	no	—
223	D₄⋊C₈	—	158	2	no	—
224	C₂²×Q₈	—	2	1	no	—
225	C₂²×Q₈	C₂²⋊D₇₉	1	2	yes	—
226	C₂²×D₄	—	79	1	no	—
227	C₂³×C₄	—	158	1	no	—
228	C₂×Q₁₆	—	2	8	no	—
229	C₂×Q₁₆	—	4	8	no	—
230	C₂²×C₈	—	2	1	no	—
231	C₂²×C₈	—	158	1	no	—
232	C₂²×C₈	—	158	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
241	C₄×D₄	—	158	1	no	—
242	C₄×D₄	—	158	1	no	—
243	C₄×D₄	—	158	1	no	—
244	C₄×D₄	—	158	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	79	1	no	—
250	C₂×C₄⋊C₄	—	79	2	no	—
251	C₂×C₂²⋊C₄	—	158	1	no	—
252	C₂×C₂²⋊C₄	—	158	4	no	—
253	C₂×C₄²	—	158	1	no	—
254	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
255	#106	—	158	4	no	—
256	C₄⋊C₈	—	79	2	no	—
257	C₄⋊C₈	—	158	1	no	—
258	C₄⋊C₈	—	158	1	no	—
259	C₄⋊C₈	—	158	1	no	—
260	#102	—	158	4	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
263	#95	—	158	4	no	—
264	#95	—	158	1	no	—
265	C₂⁴	—	158	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	4	4	no	—
270	C₂×Q₈	—	4	4	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×D₄	—	79	2	no	—
273	C₂×D₄	—	158	1	no	—
274	C₂×D₄	—	158	1	no	—
275	C₂×D₄	—	158	1	no	—
276	C₂×D₄	—	158	1	no	—
277	C₂×D₄	—	158	1	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	4	1	no	—
280	C₂²×C₄	—	316	1	no	—
281	C₂²×C₄	—	316	1	no	—
282	C₂²×C₄	—	316	2	no	—
283	C₂²×C₄	—	316	2	no	—
284	C₂²×C₄	—	158	1	no	—
285	C₂²×C₄	—	158	2	no	—
286	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
287	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
288	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
289	Q₁₆	—	4	16	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	316	1	no	—
293	C₂×C₈	—	316	1	no	—
294	C₂×C₈	—	316	1	no	—
295	C₂×C₈	—	316	2	no	—
296	C₂×C₈	—	158	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	158	1	no	—
306	C₄⋊C₄	—	158	1	no	—
307	C₄⋊C₄	—	158	2	no	—
308	C₄⋊C₄	—	158	2	no	—
309	C₂²⋊C₄	—	316	1	no	—
310	C₂²⋊C₄	—	316	1	no	—
311	C₂²⋊C₄	—	632	4	no	—
312	C₄²	—	316	1	no	—
313	C₄²	—	316	1	no	—
314	C₄²	D₄×D₇₉	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
317	C₄²	#11929623	1	1	yes	characteristic
318	C₂³	—	316	2	no	—
319	C₂³	—	316	1	no	—
320	C₂³	—	316	1	no	—
321	C₂³	—	316	2	no	—
322	C₂³	—	158	2	no	—
323	C₂³	#14579129	1	1	yes	characteristic
324	Q₈	—	4	4	no	—
325	Q₈	—	4	4	no	—
326	Q₈	—	8	2	no	—
327	Q₈	—	8	2	no	—
328	D₄	—	316	1	no	—
329	D₄	—	316	1	no	—
330	D₄	—	158	2	no	—
331	D₄	—	158	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	316	2	no	—
338	C₂×C₄	—	316	2	no	—
339	C₂×C₄	—	316	2	no	—
340	C₂×C₄	—	316	1	no	—
341	C₂×C₄	—	316	2	no	—
342	C₂×C₄	—	632	1	no	—
343	C₂×C₄	—	632	1	no	—
344	C₂×C₄	—	632	2	no	—
345	C₂×C₄	#14579129	1	1	yes	characteristic
346	C₂×C₄	#14579130	1	1	yes	characteristic
347	C₂×C₄	#14579132	1	1	yes	characteristic
348	C₂×C₄	#14579133	1	1	yes	characteristic
349	C₂×C₄	#14579133	1	1	yes	characteristic
350	C₂×C₄	#14579157	1	1	yes	characteristic
351	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
352	C₂×C₄	#14579184	1	2	yes	—
353	C₂×C₄	#14579184	1	1	yes	characteristic
354	C₂×C₄	#14579193	1	2	yes	—
355	C₂×C₄	#14579193	1	1	yes	characteristic
356	C₂×C₄	#14579193	1	1	yes	characteristic
357	C₈	—	2	4	no	—
358	C₈	—	316	2	no	—
359	C₈	—	632	1	no	—
360	C₂²	—	316	4	no	—
361	C₂²	—	316	2	no	—
362	C₂²	—	632	1	no	—
363	C₂²	—	632	2	no	—
364	C₂²	—	632	1	no	—
365	C₂²	#19098471	1	1	yes	characteristic
366	C₂²	#19098717	1	1	yes	characteristic
367	C₂²	#19098724	1	2	yes	—
368	C₂²	#19098724	1	1	yes	characteristic
369	C₂²	#19098730	1	2	yes	—
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	8	2	no	—
373	C₄	—	8	2	no	—
374	C₄	—	632	1	no	—
375	C₄	—	632	2	no	—
376	C₄	#19098717	1	1	yes	characteristic
377	C₄	#19098724	1	1	yes	characteristic
378	C₄	#19098787	1	2	yes	—
379	C₄	#19098801	1	1	yes	characteristic
380	C₄	#19098818	1	1	yes	characteristic
381	C₄	#19098948	1	1	yes	characteristic
382	C₄	#19098956	1	1	yes	characteristic
383	C₂	—	632	2	no	—
384	C₂	#23762531	1	2	yes	—
385	C₂	#23762543	1	2	yes	—
386	C₂	#23764407	1	1	yes	characteristic
387	C₂	#23764415	1	1	yes	characteristic
388	C₂	#23764491	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579129
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#494×C₇₉	C₂
socle	C₂³×C₇₉	#120
prime radical	C₂²×D₃₁₆	D₄
hypercenter	C₂×#494	D₇₉
hypocenter	C₇₉	#10056
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #23764349	C₂	semidirect
4. C₂×#19098312	C₂	semidirect
6. #23761240	C₂	semidirect
7. C₂×#494×C₇₉	C₂	semidirect
76. C₂×D₃₁₆	Q₁₆	semidirect
195. C₂×#494	D₇₉	semidirect
202. C₇₉	#10056	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098850	2
#23768574	2
#23764349	2
C₂×#19098312	2
C₂×#19098292	2
#23761240	2
C₂×#494×C₇₉	2
#10056	79