Subgroups — #34441806

Order 20224.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772377	C₂	1	1	yes	maximal, characteristic
2	#23765309	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765231	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765224	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₃₁₆	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765206	C₂	1	2	yes	maximal, complemented
8	#23765202	C₂	1	1	yes	maximal, characteristic, complemented
9	#23764995	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099100	C₄	1	1	yes	characteristic
13	C₂×#14579100	C₂²	1	2	yes	—
14	C₂×C₁₆×D₇₉	C₂²	1	2	yes	—
15	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
17	#19098694	C₄	1	1	yes	characteristic
18	#19098640	C₄	1	1	yes	characteristic
19	#19098633	C₄	1	2	yes	—
20	C₈×D₃₁₆	C₂²	1	2	yes	—
21	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
22	#19098623	C₄	1	2	yes	—
23	#19098619	C₄	1	1	yes	characteristic
24	#19098517	C₂²	1	2	yes	—
25	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
26	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
28	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
29	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
31	#19098339	C₂²	1	2	yes	—
32	#19098338	C₂²	1	2	yes	—
33	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
39	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×#11929607	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₇₉	C₂³	1	2	yes	—
45	#14579156	C₈	1	2	yes	—
46	#14579152	C₈	1	1	yes	characteristic
47	#14579150	C₈	1	1	yes	characteristic
48	#14579146	C₈	1	2	yes	—
49	#14579137	C₈	1	1	yes	characteristic
50	#14579136	C₈	1	1	yes	characteristic
51	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
52	C₄×D₃₁₆	C₂×C₄	1	1	yes	characteristic
53	#14579118	C₂×C₄	1	2	yes	—
54	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
55	#14579100	—	2	4	no	—
56	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₁₆×D₇₉	—	2	2	no	—
60	C₁₆×D₇₉	—	2	4	no	—
61	C₁₆×D₇₉	—	2	2	no	—
62	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
63	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
64	(C₂×D₇₉)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₇₉)⋊C₈	C₂×C₄	1	2	yes	—
66	(C₂×D₇₉)⋊C₈	C₂×C₄	1	2	yes	—
67	#14579072	C₂×C₄	1	2	yes	—
68	#14579071	C₂×C₄	1	2	yes	—
69	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
77	C₄⋊C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₉	C₂³	1	2	yes	—
80	C₂×D₃₁₆	C₂×C₈	1	2	yes	—
81	C₂×D₃₁₆	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₇₉	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₇₉	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
86	#11929607	—	2	4	no	—
87	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
90	C₈×D₇₉	—	2	1	no	—
91	C₈×D₇₉	—	2	1	no	—
92	C₈×D₇₉	—	2	2	no	—
93	C₈×D₇₉	—	2	2	no	—
94	C₈×D₇₉	—	2	2	no	—
95	C₄⋊Dic₇₉	C₂×C₈	1	1	yes	characteristic
96	C₄⋊Dic₇₉	C₂×C₈	1	2	yes	—
97	Dic₇₉⋊C₄	C₂×C₈	1	2	yes	—
98	Dic₇₉⋊C₄	C₂×C₈	1	2	yes	—
99	(C₂×D₇₉)⋊C₄	C₂×C₈	1	2	yes	—
100	(C₂×D₇₉)⋊C₄	C₂×C₈	1	2	yes	—
101	(C₂×D₇₉)⋊C₄	C₂×C₈	1	2	yes	—
102	(C₂×D₇₉)⋊C₄	C₂×C₈	1	2	yes	—
103	(C₂×D₇₉)⋊C₄	C₂×C₈	1	2	yes	—
104	C₄×Dic₇₉	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₇₉	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
107	C₇₉⋊C₁₆	—	2	1	no	—
108	C₇₉⋊C₁₆	—	2	2	no	—
109	C₇₉⋊C₁₆	#41	1	2	yes	—
110	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
115	C₄⋊C₄×C₇₉	C₂×C₈	1	2	yes	—
116	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
118	C₁₆×C₇₉	—	2	1	no	—
119	C₁₆×C₇₉	—	2	2	no	—
120	C₁₆×C₇₉	#41	1	2	yes	—
121	C₂²×D₇₉	C₂²×C₈	1	2	yes	—
122	C₂²×D₇₉	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₇₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₇₉	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₇₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₇₉	C₂²×C₈	1	2	yes	—
133	D₃₁₆	—	2	4	no	—
134	D₃₁₆	—	2	2	no	—
135	C₄×D₇₉	—	2	2	no	—
136	C₄×D₇₉	—	2	1	no	—
137	C₄×D₇₉	—	2	2	no	—
138	C₄×D₇₉	—	2	1	no	—
139	C₄×D₇₉	—	2	2	no	—
140	C₄×D₇₉	—	2	2	no	—
141	C₈×C₇₉	—	2	1	no	—
142	C₈×C₇₉	—	2	2	no	—
143	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
145	C₇₉⋊C₈	—	2	1	no	—
146	C₇₉⋊C₈	—	2	2	no	—
147	C₇₉⋊C₈	#131	1	2	yes	—
148	C₂²×C₇₉	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₇₉	—	2	1	no	—
150	C₂×D₇₉	—	2	1	no	—
151	C₂×D₇₉	—	2	2	no	—
152	C₂×D₇₉	—	2	2	no	—
153	C₂×D₇₉	—	2	2	no	—
154	C₂×D₇₉	#504	1	2	yes	—
155	C₄×C₇₉	—	2	2	no	—
156	C₄×C₇₉	—	2	1	no	—
157	C₄×C₇₉	—	2	2	no	—
158	C₄×C₇₉	#504	1	2	yes	—
159	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
160	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
161	Dic₇₉	—	2	1	no	—
162	Dic₇₉	—	2	1	no	—
163	Dic₇₉	—	2	1	no	—
164	Dic₇₉	—	2	2	no	—
165	#33480	—	79	1	no	maximal, Sylow
166	C₂×C₇₉	C₈×#41	1	1	yes	characteristic
167	C₂×C₇₉	C₂×#504	1	1	yes	characteristic
168	C₂×C₇₉	C₂×#504	1	1	yes	characteristic
169	D₇₉	—	2	2	no	—
170	D₇₉	—	4	2	no	—
171	#2966	—	79	1	no	—
172	#2183	—	79	1	no	—
173	Q₈×C₁₆	D₇₉	1	1	yes	characteristic, complemented
174	#2169	—	79	1	no	—
175	#2168	—	79	2	no	—
176	D₄×C₁₆	—	79	1	no	—
177	D₄×C₁₆	—	79	2	no	—
178	#2164	—	79	2	no	—
179	#2163	—	79	1	no	—
180	#2108	—	79	2	no	—
181	C₂×C₄×C₁₆	—	79	1	no	—
182	C₇₉	#33480	1	1	yes	characteristic, complemented, Sylow
183	#529	—	79	1	no	—
184	C₂×M₃₂	—	79	2	no	—
185	C₂²×C₁₆	—	79	1	no	—
186	C₂²×C₁₆	—	79	1	no	—
187	C₂²×C₁₆	—	79	2	no	—
188	#446	—	79	1	no	—
189	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
190	#436	—	79	1	no	—
191	#435	—	79	2	no	—
192	C₈×D₄	—	79	2	no	—
193	C₈×D₄	—	79	1	no	—
194	#433	—	79	2	no	—
195	#432	—	79	1	no	—
196	#405	—	79	2	no	—
197	C₂×C₄×C₈	—	79	1	no	—
198	C₄⋊C₁₆	—	79	1	no	—
199	C₄⋊C₁₆	—	79	2	no	—
200	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
202	C₂²⋊C₁₆	—	79	2	no	—
203	C₂²⋊C₁₆	—	79	2	no	—
204	C₂²⋊C₁₆	—	79	2	no	—
205	#346	—	79	2	no	—
206	#346	—	79	2	no	—
207	C₄×C₁₆	—	79	1	no	—
208	C₄×C₁₆	—	79	1	no	—
209	C₄×C₁₆	—	79	1	no	—
210	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
211	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
212	C₂×M₁₆	—	79	2	no	—
213	C₂²×C₈	—	79	1	no	—
214	C₂²×C₈	—	79	1	no	—
215	C₂²×C₈	—	79	2	no	—
216	#126	—	79	2	no	—
217	#125	—	79	1	no	—
218	#124	—	79	1	no	—
219	#123	—	79	2	no	—
220	C₂²⋊Q₈	—	79	1	no	—
221	#121	—	79	1	no	—
222	C₄×Q₈	C₄×D₇₉	1	1	yes	characteristic
223	C₄×D₄	—	79	2	no	—
224	C₄×D₄	—	79	1	no	—
225	#117	—	79	2	no	—
226	C₂×C₄²	—	79	1	no	—
227	M₃₂	—	158	4	no	—
228	C₂×C₁₆	—	79	1	no	—
229	C₂×C₁₆	—	79	1	no	—
230	C₂×C₁₆	—	79	2	no	—
231	C₂×C₁₆	—	158	2	no	—
232	C₂×C₁₆	—	158	2	no	—
233	C₂×C₁₆	—	158	4	no	—
234	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
236	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
237	C₄⋊C₈	—	79	1	no	—
238	C₄⋊C₈	—	79	2	no	—
239	C₄⋊C₈	C₄×D₇₉	1	1	yes	characteristic
240	C₄⋊C₈	C₄×D₇₉	1	2	yes	—
241	C₂²⋊C₈	—	79	2	no	—
242	C₂²⋊C₈	—	79	2	no	—
243	C₂²⋊C₈	—	79	2	no	—
244	#97	—	79	2	no	—
245	#97	—	79	2	no	—
246	C₄×C₈	—	79	1	no	—
247	C₄×C₈	—	79	1	no	—
248	C₄×C₈	—	79	1	no	—
249	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₇₉	1	2	yes	—
251	C₂×Q₈	C₈×D₇₉	1	1	yes	characteristic
252	C₂×D₄	—	79	2	no	—
253	C₂×D₄	—	79	1	no	—
254	C₂²×C₄	—	79	1	no	—
255	C₂²×C₄	—	79	1	no	—
256	C₂²×C₄	—	79	1	no	—
257	C₂²×C₄	—	79	2	no	—
258	M₁₆	—	158	4	no	—
259	C₂×C₈	—	79	1	no	—
260	C₂×C₈	—	79	1	no	—
261	C₂×C₈	—	79	2	no	—
262	C₂×C₈	—	158	2	no	—
263	C₂×C₈	—	158	1	no	—
264	C₂×C₈	—	158	2	no	—
265	C₂×C₈	—	158	1	no	—
266	C₂×C₈	—	158	2	no	—
267	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
269	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
270	C₄⋊C₄	—	79	2	no	—
271	C₄⋊C₄	—	79	2	no	—
272	C₄⋊C₄	—	79	1	no	—
273	C₄⋊C₄	—	79	2	no	—
274	C₄⋊C₄	C₈×D₇₉	1	1	yes	characteristic
275	C₄⋊C₄	C₈×D₇₉	1	2	yes	—
276	C₂²⋊C₄	—	79	2	no	—
277	C₂²⋊C₄	—	79	2	no	—
278	C₂²⋊C₄	—	79	2	no	—
279	C₂²⋊C₄	—	79	2	no	—
280	C₂²⋊C₄	—	79	2	no	—
281	C₄²	—	79	1	no	—
282	C₄²	—	79	1	no	—
283	C₄²	—	79	1	no	—
284	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₇₉	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	79	2	no	—
289	C₁₆	—	158	1	no	—
290	C₁₆	—	158	2	no	—
291	C₁₆	#11929625	1	2	yes	—
292	C₂³	—	79	1	no	—
293	C₂³	—	79	2	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	158	4	no	—
296	D₄	—	158	2	no	—
297	C₂×C₄	—	79	1	no	—
298	C₂×C₄	—	79	1	no	—
299	C₂×C₄	—	79	1	no	—
300	C₂×C₄	—	79	2	no	—
301	C₂×C₄	—	158	2	no	—
302	C₂×C₄	—	158	2	no	—
303	C₂×C₄	—	158	2	no	—
304	C₂×C₄	—	158	1	no	—
305	C₂×C₄	—	158	1	no	—
306	C₂×C₄	—	158	2	no	—
307	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
309	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	79	2	no	—
315	C₈	—	158	1	no	—
316	C₈	—	158	2	no	—
317	C₈	#41×D₇₉	1	1	yes	characteristic
318	C₈	C₂×#11929625	1	1	yes	characteristic
319	C₂²	—	79	2	no	—
320	C₂²	—	158	2	no	—
321	C₂²	—	158	2	no	—
322	C₂²	—	158	2	no	—
323	C₂²	—	158	1	no	—
324	C₂²	—	158	1	no	—
325	C₂²	C₂²×C₈×D₇₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	158	1	no	—
330	C₄	—	158	2	no	—
331	C₄	—	158	1	no	—
332	C₄	—	158	1	no	—
333	C₄	#19098979	1	2	yes	—
334	C₄	C₄×#11929625	1	1	yes	characteristic
335	C₄	#131×D₇₉	1	1	yes	characteristic
336	C₂	—	316	2	no	—
337	C₂	—	158	2	no	—
338	C₂	C₈×#11929625	1	1	yes	characteristic
339	C₂	#504×D₇₉	1	1	yes	characteristic
340	C₂	C₂×#19098979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₇₉
commutator, socle	C₂²×C₇₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	Q₈×C₁₆×C₇₉	C₂
prime radical	#14579136	C₈
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	Q₈×C₁₆	D₇₉
hypocenter	C₇₉	#33480
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #23765309	C₂	semidirect
3. #23765231	C₂	semidirect
4. #23765224	C₂	semidirect
5. C₁₆×D₃₁₆	C₂	semidirect
6. C₁₆×D₃₁₆	C₂	semidirect
7. #23765206	C₂	semidirect
8. #23765202	C₂	semidirect
9. #23764995	C₂	semidirect
10. C₄×C₁₆×D₇₉	C₂	semidirect
11. Q₈×C₁₆×C₇₉	C₂	semidirect
173. Q₈×C₁₆	D₇₉	semidirect
182. C₇₉	#33480	semidirect

Maximal subgroups

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

subgroup	index
#23772377	2
#23765309	2
#23765231	2
#23765224	2
C₁₆×D₃₁₆	2
#23765206	2
#23765202	2
#23764995	2
C₄×C₁₆×D₇₉	2
Q₈×C₁₆×C₇₉	2
#33480	79