Subgroups — #34183677

Order 20224.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772423	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098661	C₂	1	1	yes	maximal, characteristic
3	#23761792	C₂	1	1	yes	maximal, characteristic
4	#23761724	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761244	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761230	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11929611	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
10	#19098661	—	2	2	no	—
11	#19098661	—	2	2	no	—
12	C₈×D₃₁₆	—	2	2	no	—
13	#19098628	—	2	2	no	—
14	#19098616	C₂²	1	1	yes	characteristic
15	#19098616	C₂²	1	1	yes	characteristic
16	C₂×#14579094	C₄	1	1	yes	characteristic
17	C₂×#14579085	C₄	1	1	yes	characteristic
18	C₂×#14579081	C₄	1	1	yes	characteristic
19	C₂²⋊C₈×D₇₉	—	2	2	no	—
20	C₂×#14579072	C₂²	1	1	yes	characteristic
21	#19098292	—	2	2	no	—
22	#19098289	—	2	2	no	—
23	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
24	#431×C₇₉	C₂²	1	1	yes	characteristic
25	#431×C₇₉	C₂²	1	1	yes	characteristic
26	Q₈⋊C₈×C₇₉	—	2	2	no	—
27	C₂²×D₃₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₇₉	—	2	1	no	—
29	C₂×#11929611	—	2	2	no	—
30	C₂×#11929611	—	2	2	no	—
31	C₂×#11929611	C₈	1	4	yes	complemented
32	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₇₉	—	2	4	no	—
35	C₄×D₃₁₆	—	2	1	no	—
36	C₄×D₃₁₆	—	2	1	no	—
37	C₄×D₃₁₆	D₄	1	2	yes	—
38	C₄×D₃₁₆	D₄	1	2	yes	—
39	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
41	#14579094	—	2	2	no	—
42	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
43	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
44	#14579085	—	2	2	no	—
45	#14579081	—	2	2	no	—
46	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
48	(C₂×D₇₉)⋊C₈	—	2	4	no	—
49	#14579072	—	2	1	no	—
50	#14579072	—	2	1	no	—
51	C₄×C₇₉⋊C₈	—	2	1	no	—
52	C₄×C₇₉⋊C₈	—	2	1	no	—
53	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×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	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₇₉	—	2	1	no	—
61	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₇₉	—	2	1	no	—
63	C₄⋊C₈×C₇₉	—	2	1	no	—
64	C₂²⋊C₈×C₇₉	—	2	1	no	—
65	C₂²⋊C₈×C₇₉	—	2	1	no	—
66	C₄×C₈×C₇₉	—	2	1	no	—
67	C₄×C₈×C₇₉	D₄	1	2	yes	—
68	C₂³×D₇₉	—	2	1	no	—
69	C₂×D₃₁₆	—	2	1	no	—
70	C₂×D₃₁₆	—	2	1	no	—
71	C₂×D₃₁₆	—	2	1	no	—
72	C₂×D₃₁₆	C₂²⋊C₄	1	2	yes	—
73	C₂×D₃₁₆	C₂×C₈	1	2	yes	—
74	C₂×D₃₁₆	M₁₆	1	2	yes	—
75	C₂²×Dic₇₉	—	2	1	no	—
76	C₂×C₄×D₇₉	—	2	2	no	—
77	C₂×C₄×D₇₉	—	2	2	no	—
78	C₂×C₄×D₇₉	—	8	2	no	—
79	#11929611	—	2	8	no	—
80	C₂×C₇₉⋊C₈	—	2	1	no	—
81	C₂×C₇₉⋊C₈	—	2	1	no	—
82	C₂×C₇₉⋊C₈	—	2	2	no	—
83	C₂×C₇₉⋊C₈	—	2	1	no	—
84	C₂×C₇₉⋊C₈	—	2	1	no	—
85	C₂×C₇₉⋊C₈	—	2	1	no	—
86	C₂×C₇₉⋊C₈	—	4	1	no	—
87	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	complemented
88	C₈×D₇₉	—	8	4	no	—
89	C₄⋊Dic₇₉	—	2	1	no	—
90	C₄⋊Dic₇₉	C₂²⋊C₄	1	2	yes	—
91	(C₂×D₇₉)⋊C₄	—	2	2	no	—
92	(C₂×D₇₉)⋊C₄	—	2	2	no	—
93	C₂×Q₈×C₇₉	—	2	1	no	—
94	C₂×Q₈×C₇₉	—	2	1	no	—
95	C₂×Q₈×C₇₉	—	2	1	no	—
96	C₂×Q₈×C₇₉	—	2	1	no	—
97	C₂×Q₈×C₇₉	—	2	1	no	—
98	C₂×Q₈×C₇₉	C₂×C₈	1	2	yes	complemented
99	C₂²×C₄×C₇₉	—	2	1	no	—
100	C₂²×C₄×C₇₉	—	2	1	no	—
101	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
104	C₂×C₈×C₇₉	—	2	2	no	—
105	C₂×C₈×C₇₉	—	4	1	no	—
106	C₄⋊C₄×C₇₉	—	2	1	no	—
107	C₄⋊C₄×C₇₉	—	2	1	no	—
108	C₄⋊C₄×C₇₉	—	4	1	no	—
109	C₄⋊C₄×C₇₉	—	4	1	no	—
110	C₄²×C₇₉	—	4	1	no	—
111	C₄²×C₇₉	—	4	1	no	—
112	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₉	#41	1	1	yes	characteristic
116	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₇₉	—	2	2	no	—
118	C₂²×D₇₉	—	2	2	no	—
119	C₂²×D₇₉	—	2	2	no	—
120	C₂²×D₇₉	—	8	1	no	—
121	C₂²×D₇₉	—	8	1	no	—
122	Q₈×C₇₉	—	2	2	no	—
123	Q₈×C₇₉	—	2	2	no	—
124	Q₈×C₇₉	—	4	1	no	—
125	Q₈×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	2	2	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	4	1	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	8	1	no	—
133	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
136	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	8	1	no	—
145	D₃₁₆	—	2	2	no	—
146	D₃₁₆	—	2	2	no	—
147	D₃₁₆	—	2	2	no	—
148	D₃₁₆	C₂²⋊C₈	1	4	yes	—
149	C₄×D₇₉	—	4	4	no	—
150	C₄×D₇₉	—	8	2	no	—
151	C₈×C₇₉	—	4	2	no	—
152	C₈×C₇₉	—	8	1	no	—
153	C₇₉⋊C₈	—	2	2	no	—
154	C₇₉⋊C₈	—	4	2	no	—
155	C₇₉⋊C₈	—	4	1	no	—
156	C₇₉⋊C₈	—	8	1	no	—
157	C₂²×C₇₉	—	2	1	no	—
158	C₂²×C₇₉	—	2	1	no	—
159	C₂²×C₇₉	#386	1	1	yes	characteristic
160	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
161	C₂²×C₇₉	#435	1	1	yes	characteristic
162	C₂×D₇₉	—	2	4	no	—
163	C₂×D₇₉	—	4	2	no	—
164	C₂×D₇₉	—	4	2	no	—
165	C₂×D₇₉	—	8	2	no	—
166	C₂×D₇₉	—	8	2	no	—
167	C₄×C₇₉	—	2	1	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	—	2	1	no	—
171	C₄×C₇₉	—	4	2	no	—
172	C₄×C₇₉	—	8	1	no	—
173	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
174	C₄×C₇₉	#408	1	1	yes	characteristic
175	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
176	C₄×C₇₉	#435	1	1	yes	characteristic
177	Dic₇₉	—	4	2	no	—
178	#9087	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	2	1	no	—
180	C₂×C₇₉	—	2	1	no	—
181	C₂×C₇₉	#1869	1	1	yes	characteristic
182	C₂×C₇₉	#1902	1	1	yes	characteristic
183	C₂×C₇₉	#1904	1	1	yes	characteristic
184	D₇₉	—	4	4	no	—
185	#2972	—	79	1	no	—
186	C₂×#440	—	79	1	no	—
187	#1479	D₇₉	1	1	yes	characteristic, complemented
188	#1478	—	79	1	no	—
189	#1461	—	79	1	no	—
190	#1281	—	79	1	no	—
191	#1275	—	79	1	no	—
192	C₇₉	#9087	1	1	yes	characteristic, complemented, Sylow
193	C₂²×SD₁₆	—	79	1	no	—
194	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
195	C₂×C₄×D₄	—	79	1	no	—
196	#440	—	158	2	no	—
197	#440	—	158	2	no	—
198	C₈×D₄	—	158	2	no	—
199	C₈×D₄	—	158	2	no	—
200	#431	—	79	1	no	—
201	#431	—	79	1	no	—
202	#431	C₂×D₇₉	1	1	yes	characteristic
203	#431	C₂×D₇₉	1	1	yes	characteristic
204	C₂×#107	—	79	1	no	—
205	C₂×#103	—	79	1	no	—
206	C₂×#102	—	79	1	no	—
207	C₂×C₂²⋊C₈	—	158	2	no	—
208	C₂×#97	—	79	1	no	—
209	Q₈⋊C₈	—	2	2	no	—
210	D₄⋊C₈	—	158	2	no	—
211	#324	—	158	2	no	—
212	C₂²×Q₈	C₄×D₇₉	1	1	yes	characteristic
213	C₂²×D₄	—	79	1	no	—
214	C₂³×C₄	—	158	1	no	—
215	C₂×SD₁₆	—	79	4	no	—
216	C₂×SD₁₆	—	158	2	no	—
217	C₂×SD₁₆	—	158	2	no	—
218	C₂²×C₈	—	79	1	no	—
219	C₂²×C₈	—	79	1	no	—
220	C₂²×C₈	—	158	4	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×D₄	—	79	2	no	—
226	C₄×D₄	—	79	2	no	—
227	C₄×D₄	—	158	1	no	—
228	C₄×D₄	—	158	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	79	1	no	—
231	C₂×C₄⋊C₄	C₄×D₇₉	1	1	yes	characteristic
232	C₂×C₂²⋊C₄	—	158	1	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
235	#107	—	158	2	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	158	1	no	—
239	C₄⋊C₈	—	158	1	no	—
240	#103	—	158	2	no	—
241	#102	—	158	2	no	—
242	C₂²⋊C₈	—	2	1	no	—
243	C₂²⋊C₈	—	2	1	no	—
244	C₂²⋊C₈	—	79	2	no	—
245	C₂²⋊C₈	—	158	4	no	—
246	C₂²⋊C₈	—	158	1	no	—
247	#97	—	158	1	no	—
248	#97	—	158	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	158	1	no	—
251	C₄×C₈	—	158	1	no	—
252	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
253	C₂⁴	—	158	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	C₈×D₇₉	1	2	yes	complemented
260	C₂×D₄	—	79	2	no	—
261	C₂×D₄	—	79	2	no	—
262	C₂×D₄	—	79	2	no	—
263	C₂×D₄	—	158	1	no	—
264	C₂×D₄	—	158	1	no	—
265	C₂×D₄	—	158	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	632	2	no	—
269	C₂²×C₄	—	158	2	no	—
270	C₂²×C₄	—	158	1	no	—
271	C₂²×C₄	—	158	2	no	—
272	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
274	C₂²×C₄	#11929623	1	1	yes	characteristic
275	SD₁₆	—	158	8	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	316	1	no	—
279	C₂×C₈	—	79	2	no	—
280	C₂×C₈	—	632	4	no	—
281	C₂×C₈	—	158	1	no	—
282	C₂×C₈	—	158	2	no	—
283	C₂×C₈	—	158	1	no	—
284	C₂×C₈	—	158	1	no	—
285	C₂×C₈	—	158	1	no	—
286	C₂×C₈	—	158	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	79	2	no	—
292	C₄⋊C₄	—	158	1	no	—
293	C₂²⋊C₄	—	158	2	no	—
294	C₂²⋊C₄	—	158	2	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	4	1	no	—
297	C₄²	D₄×D₇₉	1	1	yes	characteristic
298	C₄²	D₄×D₇₉	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
300	C₄²	#11929624	1	1	yes	characteristic
301	C₂³	—	632	1	no	—
302	C₂³	—	632	1	no	—
303	C₂³	—	158	2	no	—
304	C₂³	—	158	2	no	—
305	C₂³	—	158	2	no	—
306	C₂³	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	D₄	—	79	4	no	—
312	D₄	—	158	2	no	—
313	D₄	—	158	2	no	—
314	D₄	—	158	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	316	4	no	—
323	C₂×C₄	—	632	2	no	—
324	C₂×C₄	—	632	1	no	—
325	C₂×C₄	—	158	2	no	—
326	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
327	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
328	C₂×C₄	#14579129	1	1	yes	characteristic
329	C₂×C₄	#14579132	1	1	yes	characteristic
330	C₂×C₄	#14579134	1	1	yes	characteristic
331	C₂×C₄	#14579143	1	1	yes	characteristic
332	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
333	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
334	C₂×C₄	#14579167	1	1	yes	characteristic
335	C₂×C₄	#14579169	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	8	1	no	—
338	C₈	—	316	2	no	—
339	C₈	—	316	1	no	—
340	C₈	—	632	1	no	—
341	C₈	—	158	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	316	2	no	—
345	C₂²	—	316	2	no	—
346	C₂²	—	632	2	no	—
347	C₂²	—	632	2	no	—
348	C₂²	—	158	4	no	—
349	C₂²	#19098439	1	1	yes	characteristic
350	C₂²	(C₂²⋊D₇₉)⋊C₈	1	1	yes	characteristic
351	C₂²	#19098637	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	8	1	no	—
358	C₄	—	316	2	no	—
359	C₄	C₂²⋊C₈×D₇₉	1	1	yes	characteristic
360	C₄	#19098528	1	1	yes	characteristic
361	C₄	C₈×C₂²⋊D₇₉	1	1	yes	characteristic
362	C₄	#19098632	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	316	4	no	—
366	C₂	#23763689	1	1	yes	characteristic
367	C₂	#23763870	1	1	yes	characteristic
368	C₂	#23763884	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098439
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1479	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂²×D₃₁₆	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1479	D₇₉
hypocenter	C₇₉	#9087
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1479

Complements

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

N	H ≅ G/N	type
4. #23761724	C₂	semidirect
5. #23761244	C₂	semidirect
6. #23761230	C₂	semidirect
7. C₇₉×#1479	C₂	semidirect
31. C₂×#11929611	C₈	semidirect
87. C₂×C₇₉⋊C₈	C₂×C₈	semidirect
98. C₂×Q₈×C₇₉	C₂×C₈	semidirect
187. #1479	D₇₉	semidirect
192. C₇₉	#9087	semidirect
259. C₂×Q₈	C₈×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23772423	2
C₂×#19098661	2
#23761792	2
#23761724	2
#23761244	2
#23761230	2
C₇₉×#1479	2
#9087	79