Subgroups — #34273551

Order 20224.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774325	C₂	1	1	yes	maximal, characteristic
2	#23772872	C₂	1	1	yes	maximal, characteristic
3	#23772481	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762828	C₂	1	4	yes	maximal, complemented
5	#23762750	C₂	1	4	yes	maximal, complemented
6	#23762170	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762085	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762050	C₂	1	1	yes	maximal, characteristic
9	C₂×#475×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579192	—	2	1	no	—
11	#19099113	C₂²	1	1	yes	characteristic
12	#19098937	C₂²	1	2	yes	—
13	#19098913	C₂²	1	2	yes	—
14	#19098785	—	2	2	no	—
15	#19098773	C₂²	1	2	yes	—
16	#19098765	—	2	2	no	—
17	#19098763	C₂²	1	2	yes	—
18	#19098702	—	2	2	no	—
19	Q₈×C₇₉⋊C₈	—	2	2	no	—
20	Q₈×C₇₉⋊C₈	C₂²	1	4	yes	—
21	#19098616	—	2	2	no	—
22	#19098616	C₂²	1	1	yes	characteristic
23	#19098616	C₂²	1	1	yes	characteristic
24	#19098587	C₂²	1	1	yes	characteristic
25	#19098562	—	2	1	no	—
26	#19098313	C₂²	1	4	yes	—
27	#19098310	C₂²	1	2	yes	—
28	#19098309	C₂²	1	4	yes	—
29	#19098306	C₂²	1	2	yes	—
30	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
32	#475×C₇₉	—	2	2	no	—
33	#475×C₇₉	C₂²	1	4	yes	—
34	C₂×#106×C₇₉	—	2	1	no	—
35	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×#103×C₇₉	—	2	1	no	—
37	C₂×#11929624	—	2	1	no	—
38	#14579192	—	4	4	no	—
39	C₂×#11929614	—	2	2	no	—
40	C₂×#11929611	—	2	2	no	—
41	C₂×#11929611	—	2	4	no	—
42	C₂×#11929608	—	2	1	no	—
43	#14579161	C₂³	1	1	yes	characteristic
44	#14579158	C₂³	1	1	yes	characteristic
45	#14579149	—	2	1	no	—
46	#14579135	—	2	1	no	—
47	#14579135	—	2	1	no	—
48	C₄×D₃₁₆	—	2	1	no	—
49	C₄×D₃₁₆	—	2	1	no	—
50	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
51	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
52	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
53	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
55	#14579085	—	2	4	no	—
56	#14579084	—	2	2	no	—
57	#14579081	—	2	2	no	—
58	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
59	C₂²⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
60	C₄×C₇₉⋊C₈	—	2	1	no	—
61	C₄×C₇₉⋊C₈	—	2	4	no	—
62	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
63	C₂²×Q₈×C₇₉	—	2	1	no	—
64	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₇₉	—	2	1	no	—
66	C₄⋊Q₈×C₇₉	—	2	1	no	—
67	C₄⋊Q₈×C₇₉	—	2	1	no	—
68	C₄⋊Q₈×C₇₉	—	2	1	no	—
69	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
70	C₄×Q₈×C₇₉	—	2	1	no	—
71	C₄×Q₈×C₇₉	—	2	1	no	—
72	C₄×Q₈×C₇₉	D₄	1	2	yes	—
73	C₄×Q₈×C₇₉	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
75	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
77	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₇₉	—	2	1	no	—
79	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
80	#106×C₇₉	—	2	4	no	—
81	C₄⋊C₈×C₇₉	—	2	1	no	—
82	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
83	#103×C₇₉	—	2	4	no	—
84	#11929624	—	4	2	no	—
85	#11929624	—	4	2	no	—
86	C₂×Dic₁₅₈	—	2	1	no	—
87	C₂×D₃₁₆	—	2	1	no	—
88	C₂×D₃₁₆	—	2	1	no	—
89	C₂×D₃₁₆	—	2	1	no	—
90	C₂×C₂²⋊D₇₉	—	4	1	no	—
91	C₂×C₄×D₇₉	—	4	1	no	—
92	#11929614	—	4	4	no	—
93	#11929611	—	4	4	no	—
94	#11929611	—	8	4	no	—
95	#11929608	—	4	2	no	—
96	C₂×C₇₉⋊C₈	—	2	2	no	—
97	C₂×C₇₉⋊C₈	—	2	2	no	—
98	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
99	C₄⋊Dic₇₉	—	2	1	no	—
100	C₄⋊Dic₇₉	—	2	1	no	—
101	(C₂×D₇₉)⋊C₄	—	4	1	no	—
102	(C₂×D₇₉)⋊C₄	—	4	1	no	—
103	C₂×Q₈×C₇₉	—	2	1	no	—
104	C₂×Q₈×C₇₉	—	2	2	no	—
105	C₂×Q₈×C₇₉	—	2	2	no	—
106	C₂×Q₈×C₇₉	—	4	1	no	—
107	C₂×Q₈×C₇₉	—	8	2	no	—
108	C₂×Q₈×C₇₉	SD₁₆	1	4	yes	complemented
109	C₂×Q₈×C₇₉	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₇₉	—	2	1	no	—
111	C₂²×C₄×C₇₉	—	2	1	no	—
112	C₂²×C₄×C₇₉	—	4	1	no	—
113	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
116	C₂×C₈×C₇₉	—	2	2	no	—
117	C₂×C₈×C₇₉	—	4	2	no	—
118	C₄⋊C₄×C₇₉	—	2	2	no	—
119	C₄⋊C₄×C₇₉	—	2	1	no	—
120	C₄⋊C₄×C₇₉	—	2	2	no	—
121	C₄⋊C₄×C₇₉	—	2	2	no	—
122	C₄⋊C₄×C₇₉	—	2	2	no	—
123	C₄⋊C₄×C₇₉	—	4	1	no	—
124	C₄⋊C₄×C₇₉	—	4	1	no	—
125	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
126	C₄²×C₇₉	—	2	2	no	—
127	C₄²×C₇₉	—	2	2	no	—
128	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₇₉	#41	1	1	yes	characteristic
131	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
132	C₂³×C₇₉	#121	1	1	yes	characteristic
133	C₂²×D₇₉	—	4	1	no	—
134	Q₈×C₇₉	—	2	2	no	—
135	Q₈×C₇₉	—	2	4	no	—
136	Q₈×C₇₉	—	4	4	no	—
137	Q₈×C₇₉	—	8	2	no	—
138	Q₈×C₇₉	C₂×SD₁₆	1	4	yes	—
139	C₂×C₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	—	2	4	no	—
141	C₂×C₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	—	4	1	no	—
144	C₂×C₄×C₇₉	—	4	2	no	—
145	C₂×C₄×C₇₉	—	8	1	no	—
146	C₂×C₄×C₇₉	—	8	2	no	—
147	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
151	C₂×C₄×C₇₉	#135	1	2	yes	—
152	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
155	C₂²⋊D₇₉	—	8	2	no	—
156	C₂×Dic₇₉	—	4	1	no	—
157	D₃₁₆	—	4	1	no	—
158	D₃₁₆	—	4	1	no	—
159	D₃₁₆	—	4	2	no	—
160	D₃₁₆	—	8	1	no	—
161	C₄×D₇₉	—	8	2	no	—
162	Dic₁₅₈	—	4	1	no	—
163	Dic₁₅₈	—	4	1	no	—
164	C₈×C₇₉	—	4	2	no	—
165	C₇₉⋊C₈	—	2	4	no	—
166	C₇₉⋊C₈	—	4	2	no	—
167	C₇₉⋊C₈	—	4	2	no	—
168	C₂²×C₇₉	—	2	1	no	—
169	C₂²×C₇₉	—	2	1	no	—
170	C₂²×C₇₉	#465	1	2	yes	—
171	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
172	C₂×D₇₉	—	8	1	no	—
173	C₂×D₇₉	—	8	1	no	—
174	C₂×D₇₉	—	8	1	no	—
175	C₄×C₇₉	—	2	1	no	—
176	C₄×C₇₉	—	2	4	no	—
177	C₄×C₇₉	—	2	1	no	—
178	C₄×C₇₉	—	2	1	no	—
179	C₄×C₇₉	—	4	2	no	—
180	C₄×C₇₉	—	8	2	no	—
181	C₄×C₇₉	#465	1	2	yes	—
182	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
183	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
184	C₄×C₇₉	#577	1	2	yes	—
185	Dic₇₉	—	8	1	no	—
186	#19397	—	79	1	no	maximal, Sylow
187	C₂×C₇₉	—	2	2	no	—
188	C₂×C₇₉	#3043	1	1	yes	characteristic
189	C₂×C₇₉	C₂×#465	1	1	yes	characteristic
190	C₂×C₇₉	#3069	1	1	yes	characteristic
191	D₇₉	—	16	1	no	—
192	#3206	—	79	1	no	—
193	C₂×#475	D₇₉	1	1	yes	characteristic, complemented
194	#3037	—	79	1	no	—
195	#2982	—	79	1	no	—
196	#1680	—	79	4	no	—
197	#1666	—	79	4	no	—
198	#1570	—	79	1	no	—
199	#1549	—	79	1	no	—
200	#1540	—	79	1	no	—
201	C₇₉	#19397	1	1	yes	characteristic, complemented, Sylow
202	C₂×#137	—	158	1	no	—
203	#532	—	79	1	no	—
204	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
205	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
206	#492	—	79	2	no	—
207	#487	—	79	2	no	—
208	#475	—	2	2	no	—
209	#475	C₂×D₇₉	1	4	yes	—
210	#464	—	158	2	no	—
211	#462	—	79	2	no	—
212	#461	—	158	2	no	—
213	#460	—	79	2	no	—
214	#448	—	158	2	no	—
215	C₈×Q₈	—	79	4	no	—
216	C₈×Q₈	—	158	2	no	—
217	#431	—	79	1	no	—
218	#431	—	79	1	no	—
219	#431	—	158	2	no	—
220	C₂×#106	—	2	1	no	—
221	#423	—	79	1	no	—
222	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
223	#417	—	158	1	no	—
224	C₂×#103	—	2	1	no	—
225	#334	—	79	4	no	—
226	#332	—	79	4	no	—
227	#332	—	79	2	no	—
228	#331	—	79	2	no	—
229	C₂×#41	—	158	1	no	—
230	C₂²×Q₈	—	2	1	no	—
231	C₂²×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
232	#137	—	316	4	no	—
233	C₂×Q₁₆	—	158	2	no	—
234	C₂×SD₁₆	—	158	4	no	—
235	C₂×SD₁₆	—	158	2	no	—
236	C₂×M₁₆	—	158	1	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	79	1	no	—
242	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
243	#127	—	79	1	no	—
244	#124	—	158	1	no	—
245	#121	—	158	1	no	—
246	#121	—	158	1	no	—
247	C₄×Q₈	—	2	1	no	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×Q₈	C₂²⋊D₇₉	1	2	yes	—
250	C₄×Q₈	C₂²×D₇₉	1	2	yes	—
251	C₄×D₄	—	158	1	no	—
252	C₄×D₄	—	158	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
259	#106	—	2	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	79	2	no	—
262	C₄⋊C₈	—	79	2	no	—
263	C₄⋊C₈	—	79	2	no	—
264	C₄⋊C₈	—	158	4	no	—
265	C₄⋊C₈	—	158	1	no	—
266	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
267	#103	—	2	4	no	—
268	#103	—	158	4	no	—
269	#103	—	158	2	no	—
270	#102	—	158	2	no	—
271	C₂²⋊C₈	—	79	2	no	—
272	C₂²⋊C₈	—	158	1	no	—
273	C₄×C₈	—	79	2	no	—
274	C₄×C₈	—	158	4	no	—
275	C₄×C₈	—	158	1	no	—
276	#41	—	316	2	no	—
277	#41	—	316	2	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	4	1	no	—
282	C₂×Q₈	—	8	2	no	—
283	C₂×Q₈	—	158	1	no	—
284	C₂×Q₈	#11929611	1	4	yes	complemented
285	C₂×Q₈	C₂×C₂²⋊D₇₉	1	2	yes	—
286	C₂×D₄	—	316	1	no	—
287	C₂×D₄	—	158	1	no	—
288	C₂×D₄	—	158	1	no	—
289	C₂×D₄	—	158	1	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	316	1	no	—
294	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
295	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
296	C₂²×C₄	#11929623	1	1	yes	characteristic
297	Q₁₆	—	316	4	no	—
298	SD₁₆	—	316	4	no	—
299	SD₁₆	—	632	4	no	—
300	M₁₆	—	316	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	2	no	—
303	C₂×C₈	—	79	4	no	—
304	C₂×C₈	—	158	2	no	—
305	C₂×C₈	—	158	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	158	1	no	—
314	C₄⋊C₄	—	158	1	no	—
315	C₄⋊C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
316	C₂²⋊C₄	—	316	1	no	—
317	C₂²⋊C₄	—	316	1	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	D₄×D₇₉	1	1	yes	characteristic
321	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
322	C₄²	#11929623	1	1	yes	characteristic
323	C₄²	C₂³×D₇₉	1	1	yes	characteristic
324	C₂³	—	316	1	no	—
325	C₂³	#14579133	1	1	yes	characteristic
326	Q₈	—	2	4	no	—
327	Q₈	—	2	2	no	—
328	Q₈	—	4	4	no	—
329	Q₈	—	8	2	no	—
330	Q₈	—	316	1	no	—
331	Q₈	—	316	1	no	—
332	Q₈	C₂×#11929611	1	4	yes	—
333	D₄	—	316	1	no	—
334	D₄	—	316	1	no	—
335	D₄	—	316	2	no	—
336	D₄	—	632	1	no	—
337	D₄	—	632	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	8	1	no	—
345	C₂×C₄	—	8	2	no	—
346	C₂×C₄	—	316	1	no	—
347	C₂×C₄	—	632	2	no	—
348	C₂×C₄	#14579133	1	1	yes	characteristic
349	C₂×C₄	#14579133	1	1	yes	characteristic
350	C₂×C₄	#14579133	1	1	yes	characteristic
351	C₂×C₄	C₂×#11929611	1	2	yes	—
352	C₂×C₄	#14579181	1	2	yes	—
353	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
355	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
356	C₈	—	4	2	no	—
357	C₈	—	316	2	no	—
358	C₈	—	316	2	no	—
359	C₈	—	158	4	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	632	1	no	—
363	C₂²	—	632	1	no	—
364	C₂²	—	632	1	no	—
365	C₂²	#19098794	1	2	yes	—
366	C₂²	C₂×#14579133	1	1	yes	characteristic
367	C₄	—	2	4	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	2	no	—
373	C₄	—	632	1	no	—
374	C₄	#19098794	1	2	yes	—
375	C₄	C₂²×#11929611	1	1	yes	characteristic
376	C₄	C₂×#14579181	1	1	yes	characteristic
377	C₄	#19099369	1	2	yes	—
378	C₂	—	2	2	no	—
379	C₂	—	1264	1	no	—
380	C₂	#23772918	1	1	yes	characteristic
381	C₂	C₂×#19098794	1	1	yes	characteristic
382	C₂	#23773134	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579133
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#475×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#121
prime radical	#19099113	C₂²
hypercenter	C₂×#475	D₇₉
hypocenter	C₇₉	#19397
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#475

Complements

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

N	H ≅ G/N	type
3. #23772481	C₂	semidirect
4. #23762828	C₂	semidirect
5. #23762750	C₂	semidirect
6. #23762170	C₂	semidirect
7. #23762085	C₂	semidirect
9. C₂×#475×C₇₉	C₂	semidirect
108. C₂×Q₈×C₇₉	SD₁₆	semidirect
193. C₂×#475	D₇₉	semidirect
201. C₇₉	#19397	semidirect
284. C₂×Q₈	#11929611	semidirect

Maximal subgroups

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

subgroup	index
#23774325	2
#23772872	2
#23772481	2
#23762828	2
#23762750	2
#23762170	2
#23762085	2
#23762050	2
C₂×#475×C₇₉	2
#19397	79