Subgroups — #34445105

Order 20224.

622 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775794	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic
3	#23765624	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765587	C₂	1	2	yes	maximal, complemented
5	#23765547	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765499	C₂	1	2	yes	maximal, complemented
7	#23765440	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765327	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₇₉	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₇₉	C₂²	1	2	yes	—
13	#19099273	—	2	1	no	—
14	D₄×D₃₁₆	—	2	1	no	—
15	C₂×#14579104	C₂²	1	2	yes	complemented
16	C₂×C₁₆×D₇₉	C₂²	1	2	yes	complemented
17	#19098968	C₂²	1	1	yes	characteristic, complemented
18	#19098940	C₂²	1	1	yes	characteristic
19	#19098897	—	2	2	no	—
20	#19098858	—	2	1	no	—
21	#19098801	C₂²	1	2	yes	complemented
22	#19098800	C₂²	1	2	yes	—
23	#19098758	—	2	1	no	—
24	#19098698	—	2	2	no	—
25	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
26	#107×D₇₉	C₂²	1	2	yes	complemented
27	#19098390	C₂²	1	2	yes	complemented
28	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
29	#19098365	C₂²	1	2	yes	complemented
30	#19098365	C₂²	1	2	yes	complemented
31	#19098364	C₂²	1	2	yes	—
32	#19098363	C₂²	1	2	yes	complemented
33	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	complemented
34	C₂×SD₃₂×C₇₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
36	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
37	#367×C₇₉	C₂²	1	1	yes	characteristic, complemented
38	#358×C₇₉	C₂²	1	1	yes	characteristic, complemented
39	#357×C₇₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₃₁₆	—	4	1	no	—
42	C₂×D₄×D₇₉	—	2	2	no	—
43	C₂×D₆₃₂	—	2	1	no	—
44	C₂×D₆₃₂	C₂³	1	2	yes	—
45	C₂×#11929609	—	2	2	no	—
46	D₈×D₇₉	—	2	4	no	—
47	D₈×D₇₉	—	4	2	no	—
48	C₂×C₈×D₇₉	C₂³	1	2	yes	—
49	#14579161	—	2	1	no	—
50	#14579158	—	2	1	no	—
51	#14579153	—	4	1	no	—
52	#14579139	—	4	2	no	—
53	#14579137	—	2	2	no	—
54	#14579136	—	2	2	no	—
55	#14579135	—	4	1	no	—
56	#14579131	—	4	2	no	—
57	C₄×D₃₁₆	—	4	1	no	—
58	C₄×D₃₁₆	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₇₉	—	2	2	no	—
60	#14579104	—	4	2	no	—
61	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
62	C₁₆×D₇₉	—	4	2	no	—
63	#14579095	—	2	1	no	—
64	#14579095	C₂³	1	2	yes	—
65	#14579094	—	2	2	no	—
66	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
67	#14579085	—	2	2	no	—
68	#14579081	—	2	2	no	—
69	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
70	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₇₉	—	2	1	no	—
73	C₄×D₄×C₇₉	—	2	1	no	—
74	SD₃₂×C₇₉	—	2	2	no	—
75	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
77	#107×C₇₉	C₂³	1	1	yes	characteristic
78	#107×C₇₉	C₂³	1	1	yes	characteristic
79	#103×C₇₉	—	2	1	no	—
80	#102×C₇₉	—	2	1	no	—
81	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
82	C₂³×D₇₉	—	4	2	no	—
83	C₂×Dic₁₅₈	—	4	1	no	—
84	C₂×D₃₁₆	—	2	2	no	—
85	C₂×D₃₁₆	—	4	2	no	—
86	C₂×D₃₁₆	—	4	1	no	—
87	C₂×D₃₁₆	—	4	1	no	—
88	C₂×D₃₁₆	—	8	4	no	—
89	C₂×D₃₁₆	D₈	1	1	yes	characteristic
90	C₂×C₂²⋊D₇₉	—	4	2	no	—
91	D₄×D₇₉	—	4	4	no	—
92	D₄×D₇₉	—	8	2	no	—
93	C₂×C₄×D₇₉	—	4	2	no	—
94	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
95	D₆₃₂	—	2	2	no	—
96	D₆₃₂	—	2	2	no	—
97	D₆₃₂	—	8	1	no	—
98	#11929609	—	4	4	no	—
99	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
100	C₈×D₇₉	—	2	2	no	—
101	C₈×D₇₉	—	2	2	no	—
102	C₄⋊Dic₇₉	—	2	2	no	—
103	C₄⋊Dic₇₉	—	4	1	no	—
104	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
105	Dic₇₉⋊C₄	—	4	2	no	—
106	Dic₇₉⋊C₄	—	4	2	no	—
107	(C₂×D₇₉)⋊C₄	—	4	2	no	—
108	(C₂×D₇₉)⋊C₄	—	4	2	no	—
109	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
110	C₇₉⋊C₁₆	—	2	2	no	—
111	C₂×Q₈×C₇₉	—	2	1	no	—
112	C₂×D₄×C₇₉	—	2	1	no	—
113	C₂²×C₄×C₇₉	—	4	1	no	—
114	Q₁₆×C₇₉	—	2	1	no	—
115	Q₁₆×C₇₉	—	2	1	no	—
116	D₈×C₇₉	—	2	1	no	—
117	D₈×C₇₉	#41	1	2	yes	—
118	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₇₉	—	2	1	no	—
121	C₄⋊C₄×C₇₉	—	2	1	no	—
122	C₄⋊C₄×C₇₉	—	4	1	no	—
123	C₂²⋊C₄×C₇₉	—	4	1	no	—
124	C₄²×C₇₉	—	4	1	no	—
125	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₇₉	—	2	1	no	—
127	C₁₆×C₇₉	—	2	1	no	—
128	C₂³×C₇₉	—	4	1	no	—
129	C₂²×D₇₉	—	4	2	no	—
130	C₂²×D₇₉	—	8	4	no	—
131	C₂²×D₇₉	—	8	4	no	—
132	C₂²×D₇₉	—	16	2	no	—
133	C₂²×D₇₉	C₂×D₈	1	2	yes	—
134	Q₈×C₇₉	—	4	1	no	—
135	Q₈×C₇₉	—	4	1	no	—
136	D₄×C₇₉	—	2	2	no	—
137	D₄×C₇₉	—	4	1	no	—
138	C₂×C₄×C₇₉	—	4	1	no	—
139	C₂×C₄×C₇₉	—	4	1	no	—
140	C₂×C₄×C₇₉	—	4	1	no	—
141	C₂×C₄×C₇₉	—	8	2	no	—
142	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₇₉	—	8	4	no	—
146	C₂×Dic₇₉	—	4	2	no	—
147	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
148	D₃₁₆	—	4	2	no	—
149	D₃₁₆	—	4	2	no	—
150	D₃₁₆	—	8	4	no	—
151	D₃₁₆	—	8	2	no	—
152	D₃₁₆	—	16	1	no	—
153	D₃₁₆	SD₃₂	1	4	yes	complemented
154	C₄×D₇₉	—	2	2	no	—
155	C₄×D₇₉	—	2	2	no	—
156	C₄×D₇₉	—	16	2	no	—
157	Dic₁₅₈	—	16	1	no	—
158	C₈×C₇₉	—	2	1	no	—
159	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
160	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
161	C₇₉⋊C₈	—	2	2	no	—
162	C₂²×C₇₉	—	4	2	no	—
163	C₂²×C₇₉	—	8	1	no	—
164	C₂²×C₇₉	—	8	1	no	—
165	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
166	C₂×D₇₉	—	2	2	no	—
167	C₂×D₇₉	—	2	2	no	—
168	C₂×D₇₉	—	8	2	no	—
169	C₂×D₇₉	—	8	2	no	—
170	C₂×D₇₉	—	8	2	no	—
171	C₂×D₇₉	—	16	8	no	—
172	C₂×D₇₉	C₂×SD₃₂	1	4	yes	—
173	C₄×C₇₉	—	2	1	no	—
174	C₄×C₇₉	—	8	1	no	—
175	C₄×C₇₉	—	8	1	no	—
176	C₄×C₇₉	—	8	1	no	—
177	C₄×C₇₉	C₂×SD₃₂	1	2	yes	—
178	C₄×C₇₉	#546	1	1	yes	characteristic
179	C₄×C₇₉	#577	1	1	yes	characteristic
180	Dic₇₉	—	2	2	no	—
181	Dic₇₉	—	8	2	no	—
182	#33862	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	8	2	no	—
184	C₂×C₇₉	#3294	1	1	yes	characteristic
185	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
186	C₂×C₇₉	#3415	1	1	yes	characteristic
187	D₇₉	—	2	4	no	—
188	D₇₉	—	16	2	no	—
189	#3338	—	79	1	no	—
190	D₄×D₈	—	79	1	no	—
191	#2240	—	79	1	no	—
192	#2233	—	79	2	no	—
193	#2226	—	79	1	no	—
194	#2217	—	79	2	no	—
195	#2207	—	79	1	no	—
196	#2187	—	79	2	no	—
197	C₄×SD₃₂	D₇₉	1	1	yes	characteristic, complemented
198	D₄×C₁₆	—	79	1	no	—
199	C₂×#357	—	79	2	no	—
200	C₇₉	#33862	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₈	—	79	2	no	—
202	#554	—	158	1	no	—
203	D₄²	—	158	1	no	—
204	C₂×SD₃₂	—	79	2	no	—
205	C₂×SD₃₂	C₂×D₇₉	1	1	yes	characteristic
206	C₂²×C₁₆	—	79	2	no	—
207	#500	—	79	1	no	—
208	#493	—	79	1	no	—
209	#484	—	158	2	no	—
210	#477	—	158	1	no	—
211	#467	—	79	2	no	—
212	#466	—	79	2	no	—
213	#459	—	158	1	no	—
214	#447	—	158	2	no	—
215	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
216	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
217	C₈×D₄	—	79	1	no	—
218	C₂×#107	—	79	2	no	—
219	#367	—	79	2	no	—
220	#367	C₂×D₇₉	1	1	yes	characteristic, complemented
221	C₄⋊C₁₆	—	79	1	no	—
222	#358	C₂×D₇₉	1	1	yes	characteristic, complemented
223	#357	—	79	2	no	—
224	#357	—	79	2	no	—
225	#357	—	79	2	no	—
226	#357	—	79	2	no	—
227	#357	C₂×D₇₉	1	1	yes	characteristic
228	C₂²⋊C₁₆	—	79	2	no	—
229	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
230	C₂²×D₄	—	316	1	no	—
231	C₂²×D₄	—	158	2	no	—
232	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
233	C₂×D₈	—	316	2	no	—
234	C₂×D₈	—	79	2	no	—
235	C₂×D₈	—	158	2	no	—
236	C₂×D₈	—	158	4	no	—
237	C₂×D₈	—	158	1	no	—
238	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
239	C₂²×C₈	—	79	2	no	—
240	C₄⋊Q₈	—	158	1	no	—
241	#127	—	158	1	no	—
242	#125	—	316	1	no	—
243	C₂²⋊Q₈	—	316	2	no	—
244	C₂²⋊Q₈	—	158	2	no	—
245	#121	—	316	1	no	—
246	#121	—	158	2	no	—
247	#120	—	316	2	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×D₄	—	2	1	no	—
250	C₄×D₄	—	316	1	no	—
251	C₄×D₄	—	79	1	no	—
252	C₂×C₄⋊C₄	—	158	2	no	—
253	SD₃₂	—	2	2	no	—
254	SD₃₂	—	316	2	no	—
255	C₂×C₁₆	—	316	2	no	—
256	C₂×C₁₆	—	79	2	no	—
257	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
259	#107	—	79	2	no	—
260	#107	—	158	2	no	—
261	#107	—	158	1	no	—
262	#107	C₂²×D₇₉	1	1	yes	characteristic
263	#107	C₂²×D₇₉	1	1	yes	characteristic
264	C₄⋊C₈	—	79	1	no	—
265	#103	—	2	1	no	—
266	#103	—	158	2	no	—
267	#102	—	2	1	no	—
268	#102	—	158	2	no	—
269	C₂²⋊C₈	—	79	2	no	—
270	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
271	C₂⁴	—	316	2	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	316	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	316	2	no	—
276	C₂×D₄	—	316	2	no	—
277	C₂×D₄	—	316	4	no	—
278	C₂×D₄	—	316	1	no	—
279	C₂×D₄	—	316	1	no	—
280	C₂×D₄	—	79	1	no	—
281	C₂×D₄	—	632	4	no	—
282	C₂×D₄	—	632	2	no	—
283	C₂×D₄	—	158	2	no	—
284	C₂²×C₄	—	4	1	no	—
285	C₂²×C₄	—	316	2	no	—
286	C₂²×C₄	—	79	2	no	—
287	Q₁₆	—	2	1	no	—
288	Q₁₆	—	2	1	no	—
289	D₈	—	2	1	no	—
290	D₈	—	316	4	no	—
291	D₈	—	632	1	no	—
292	D₈	—	158	2	no	—
293	D₈	—	158	2	no	—
294	D₈	#11929624	1	2	yes	—
295	C₂×C₈	—	79	2	no	—
296	C₂×C₈	—	158	2	no	—
297	C₂×C₈	—	158	2	no	—
298	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	316	2	no	—
304	C₄⋊C₄	—	316	2	no	—
305	C₄⋊C₄	—	316	1	no	—
306	C₄⋊C₄	—	79	1	no	—
307	C₄⋊C₄	—	158	2	no	—
308	C₂²⋊C₄	—	4	1	no	—
309	C₂²⋊C₄	—	316	2	no	—
310	C₂²⋊C₄	—	316	2	no	—
311	C₂²⋊C₄	—	79	2	no	—
312	C₄²	—	4	1	no	—
313	C₄²	D₄×D₇₉	1	1	yes	characteristic
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	158	2	no	—
317	C₂³	—	4	1	no	—
318	C₂³	—	316	2	no	—
319	C₂³	—	79	2	no	—
320	C₂³	—	632	4	no	—
321	C₂³	—	632	4	no	—
322	C₂³	—	1264	2	no	—
323	Q₈	—	4	1	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	1264	1	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	316	2	no	—
329	D₄	—	316	2	no	—
330	D₄	—	79	4	no	—
331	D₄	—	632	4	no	—
332	D₄	—	632	4	no	—
333	D₄	—	632	2	no	—
334	D₄	—	1264	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	316	2	no	—
340	C₂×C₄	—	79	2	no	—
341	C₂×C₄	—	158	2	no	—
342	C₂×C₄	—	158	2	no	—
343	C₂×C₄	—	1264	2	no	—
344	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
345	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	158	2	no	—
349	C₈	C₂×#11929624	1	1	yes	characteristic
350	C₈	#14579207	1	1	yes	characteristic
351	C₂²	—	4	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	79	4	no	—
355	C₂²	—	632	2	no	—
356	C₂²	—	632	2	no	—
357	C₂²	—	632	2	no	—
358	C₂²	—	158	2	no	—
359	C₂²	—	158	2	no	—
360	C₂²	—	1264	8	no	—
361	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	632	2	no	—
367	C₄	—	158	2	no	—
368	C₄	SD₃₂×D₇₉	1	2	yes	—
369	C₄	#19099209	1	1	yes	characteristic
370	C₄	#19099370	1	1	yes	characteristic
371	C₂	—	8	2	no	—
372	C₂	—	158	4	no	—
373	C₂	—	1264	2	no	—
374	C₂	#23775215	1	1	yes	characteristic
375	C₂	C₂×SD₃₂×D₇₉	1	1	yes	characteristic
376	C₂	#23776538	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₄×SD₃₂×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	D₈×D₃₁₆	C₂
hypercenter	C₄×SD₃₂	D₇₉
hypocenter	C₇₉	#33862
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #23775794	C₂	semidirect
3. #23765624	C₂	semidirect
4. #23765587	C₂	semidirect
5. #23765547	C₂	semidirect
6. #23765499	C₂	semidirect
7. #23765440	C₂	semidirect
8. #23765327	C₂	semidirect
9. C₁₆×D₃₁₆	C₂	semidirect
10. #357×D₇₉	C₂	semidirect
11. C₄×SD₃₂×C₇₉	C₂	semidirect
15. C₂×#14579104	C₂²	semidirect
16. C₂×C₁₆×D₇₉	C₂²	semidirect
17. #19098968	C₂²	semidirect
21. #19098801	C₂²	semidirect
26. #107×D₇₉	C₂²	semidirect
27. #19098390	C₂²	semidirect
28. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
29. #19098365	C₂²	semidirect
30. #19098365	C₂²	semidirect
32. #19098363	C₂²	semidirect
33. (C₂×D₇₉)⋊C₁₆	C₂²	semidirect
35. C₄×Q₁₆×C₇₉	C₂²	semidirect
37. #367×C₇₉	C₂²	semidirect
38. #358×C₇₉	C₂²	semidirect
40. C₄×C₁₆×C₇₉	C₂²	semidirect
153. D₃₁₆	SD₃₂	semidirect
197. C₄×SD₃₂	D₇₉	semidirect
200. C₇₉	#33862	semidirect
215. C₄×Q₁₆	C₂×D₇₉	semidirect
220. #367	C₂×D₇₉	semidirect
222. #358	C₂×D₇₉	semidirect
229. C₄×C₁₆	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23775794	2
D₈×D₃₁₆	2
#23765624	2
#23765587	2
#23765547	2
#23765499	2
#23765440	2
#23765327	2
C₁₆×D₃₁₆	2
#357×D₇₉	2
C₄×SD₃₂×C₇₉	2
#33862	79