Subgroups — #34287349

Order 20224.

806 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098693	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
3	#23765877	C₂	1	1	yes	maximal, characteristic
4	#23763273	C₂	1	2	yes	maximal
5	#23763267	C₂	1	1	yes	maximal, characteristic
6	#23763266	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
8	#23763256	C₂	1	2	yes	maximal
9	#23763253	C₂	1	1	yes	maximal, characteristic
10	#23763150	C₂	1	2	yes	maximal
11	C₇₉×#1744	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
13	#19098693	C₄	1	8	yes	complemented
14	C₈×Dic₁₅₈	C₄	1	8	yes	complemented
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₇₉⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₇₉⋊C₈	C₂²	1	2	yes	—
19	C₂×#14579072	C₂²	1	2	yes	—
20	C₂×#14579071	C₂²	1	2	yes	—
21	C₂×#14579071	C₂²	1	2	yes	—
22	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
23	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
24	#19098469	C₄	1	2	yes	—
25	#19098461	C₄	1	2	yes	—
26	#19098447	C₄	1	4	yes	—
27	C₄×Dic₇₉⋊C₄	C₂²	1	2	yes	—
28	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
29	#19098409	C₂²	1	2	yes	—
30	#19098318	C₂²	1	2	yes	—
31	#19098317	C₂²	1	2	yes	—
32	#19098317	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	C₄³×C₇₉	C₂²	1	1	yes	characteristic
36	#336×C₇₉	C₂²	1	1	yes	characteristic
37	#336×C₇₉	C₂²	1	1	yes	characteristic
38	#336×C₇₉	C₂²	1	1	yes	characteristic
39	#336×C₇₉	C₂²	1	1	yes	characteristic
40	C₂²×Dic₁₅₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
43	C₄×Dic₁₅₈	C₂×C₄	1	4	yes	—
44	C₄×Dic₁₅₈	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₇₉⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×Dic₇₉⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
50	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
51	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
52	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
53	Dic₇₉⋊C₈	C₂×C₄	1	8	yes	—
54	Dic₇₉⋊C₈	C₂×C₄	1	8	yes	—
55	#14579072	—	4	2	no	—
56	#14579071	—	4	2	no	—
57	#14579071	C₂×C₄	1	8	yes	—
58	C₈×Dic₇₉	—	4	2	no	—
59	C₈×Dic₇₉	C₂×C₄	1	8	yes	—
60	C₇₉⋊#95	C₂×C₄	1	4	yes	—
61	C₇₉⋊#95	C₂×C₄	1	4	yes	—
62	C₇₉⋊#95	C₂×C₄	1	2	yes	—
63	C₇₉⋊#95	C₂×C₄	1	4	yes	—
64	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
74	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
75	C₂×Dic₁₅₈	—	2	4	no	—
76	C₂×Dic₁₅₈	C₄²	1	4	yes	—
77	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
78	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
80	C₂×C₇₉⋊C₈	—	2	2	no	—
81	C₂×C₇₉⋊C₈	—	2	2	no	—
82	C₂×C₇₉⋊C₈	—	2	2	no	—
83	C₂×C₇₉⋊C₈	—	2	2	no	—
84	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
85	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
86	C₄⋊Dic₇₉	—	2	4	no	—
87	C₄⋊Dic₇₉	C₄²	1	4	yes	—
88	Dic₇₉⋊C₄	—	4	4	no	—
89	Dic₇₉⋊C₄	C₄²	1	8	yes	—
90	C₄×Dic₇₉	—	2	2	no	—
91	C₄×Dic₇₉	—	2	2	no	—
92	C₄×Dic₇₉	C₂²×C₄	1	4	yes	—
93	C₄×Dic₇₉	C₂²×C₄	1	4	yes	—
94	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₇₉	—	2	1	no	—
100	C₂×C₈×C₇₉	—	2	1	no	—
101	C₂×C₈×C₇₉	—	2	1	no	—
102	C₂×C₈×C₇₉	—	2	1	no	—
103	C₂×C₈×C₇₉	—	2	1	no	—
104	C₂×C₈×C₇₉	—	2	1	no	—
105	C₂×C₈×C₇₉	—	2	1	no	—
106	C₂×C₈×C₇₉	—	2	1	no	—
107	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
111	C₄²×C₇₉	—	2	4	no	—
112	C₄²×C₇₉	—	2	1	no	—
113	C₄²×C₇₉	—	2	1	no	—
114	C₄²×C₇₉	—	2	1	no	—
115	C₄²×C₇₉	—	2	1	no	—
116	C₄²×C₇₉	—	4	2	no	—
117	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
118	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
119	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	—	2	2	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	—	2	2	no	—
123	C₂×C₄×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	—	2	2	no	—
125	C₂×C₄×C₇₉	—	2	2	no	—
126	C₂×C₄×C₇₉	—	2	2	no	—
127	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₇₉	#131	1	4	yes	—
130	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×Dic₇₉	—	2	4	no	—
137	C₂×Dic₇₉	—	2	4	no	—
138	C₂×Dic₇₉	—	2	4	no	—
139	C₂×Dic₇₉	—	2	4	no	—
140	C₂×Dic₇₉	—	2	4	no	—
141	C₂×Dic₇₉	C₂×C₄²	1	4	yes	—
142	C₂×Dic₇₉	C₂×C₄²	1	4	yes	—
143	Dic₁₅₈	—	2	8	no	—
144	C₈×C₇₉	—	2	2	no	—
145	C₈×C₇₉	—	2	2	no	—
146	C₈×C₇₉	—	2	2	no	—
147	C₈×C₇₉	—	2	2	no	—
148	C₇₉⋊C₈	—	2	4	no	—
149	C₇₉⋊C₈	—	2	4	no	—
150	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
151	C₂²×C₇₉	#519	1	1	yes	characteristic
152	C₂²×C₇₉	#520	1	1	yes	characteristic
153	C₂²×C₇₉	C₂×#131	1	2	yes	—
154	C₂²×C₇₉	#568	1	2	yes	—
155	C₄×C₇₉	—	2	2	no	—
156	C₄×C₇₉	—	4	2	no	—
157	C₄×C₇₉	—	4	2	no	—
158	C₄×C₇₉	#405	1	4	yes	—
159	C₄×C₇₉	#519	1	1	yes	characteristic
160	C₄×C₇₉	#519	1	1	yes	characteristic
161	C₄×C₇₉	#520	1	1	yes	characteristic
162	C₄×C₇₉	#520	1	1	yes	characteristic
163	Dic₇₉	—	2	4	no	—
164	Dic₇₉	—	2	4	no	—
165	Dic₇₉	—	4	4	no	—
166	#20574	—	79	1	no	maximal, Sylow
167	C₂×C₇₉	#2276	1	1	yes	characteristic
168	C₂×C₇₉	C₂×#405	1	1	yes	characteristic
169	C₂×C₇₉	#2875	1	1	yes	characteristic
170	C₂×C₇₉	#2985	1	2	yes	—
171	C₂×C₇₉	#2993	1	2	yes	—
172	C₂×#446	—	79	1	no	—
173	C₂×C₈×Q₈	—	79	1	no	—
174	#2295	—	79	1	no	—
175	#1771	—	79	2	no	—
176	#1770	—	79	1	no	—
177	#1770	—	79	2	no	—
178	C₈×C₄⋊C₄	—	79	1	no	—
179	#1768	—	79	2	no	—
180	#1767	—	79	1	no	—
181	#1746	—	79	2	no	—
182	#1744	D₇₉	1	1	yes	characteristic
183	C₇₉	#20574	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄×Q₈	—	79	1	no	—
185	#446	—	79	8	no	—
186	C₈×Q₈	—	79	8	no	—
187	C₂×C₄⋊C₈	—	79	2	no	—
188	C₂×C₄⋊C₈	—	79	2	no	—
189	C₂×C₄⋊C₈	—	79	1	no	—
190	C₂×C₄⋊C₈	—	79	1	no	—
191	C₂×#97	—	79	2	no	—
192	C₂×#97	—	79	2	no	—
193	C₂×#97	—	79	2	no	—
194	C₂×C₄×C₈	—	79	2	no	—
195	C₂×C₄×C₈	—	79	2	no	—
196	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
198	#391	—	79	2	no	—
199	#389	—	79	2	no	—
200	#387	—	79	4	no	—
201	C₄×C₄⋊C₄	—	79	1	no	—
202	C₄×C₄⋊C₄	—	79	2	no	—
203	#376	—	79	2	no	—
204	C₄³	C₂×D₇₉	1	1	yes	characteristic
205	#336	—	79	2	no	—
206	#336	—	79	2	no	—
207	#336	—	79	2	no	—
208	#336	C₂×D₇₉	1	1	yes	characteristic
209	#336	C₂×D₇₉	1	1	yes	characteristic
210	#336	C₂×D₇₉	1	1	yes	characteristic
211	#336	C₂×D₇₉	1	1	yes	characteristic
212	C₂²×Q₈	—	79	1	no	—
213	C₂²×C₈	—	79	2	no	—
214	C₂²×C₈	—	79	2	no	—
215	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
219	C₄×Q₈	—	79	4	no	—
220	C₄×Q₈	—	79	4	no	—
221	C₂×C₄⋊C₄	—	79	4	no	—
222	C₂×C₄⋊C₄	—	79	2	no	—
223	C₂×C₄⋊C₄	—	79	1	no	—
224	C₂×C₄⋊C₄	—	79	2	no	—
225	C₂×C₄²	—	79	2	no	—
226	C₂×C₄²	—	79	2	no	—
227	C₂×C₄²	C₄×D₇₉	1	2	yes	—
228	C₂×C₄²	C₄×D₇₉	1	2	yes	—
229	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
232	C₄⋊C₈	—	79	8	no	—
233	C₄⋊C₈	—	79	8	no	—
234	C₄⋊C₈	—	79	4	no	—
235	C₄⋊C₈	—	79	4	no	—
236	#97	—	316	2	no	—
237	#97	—	316	2	no	—
238	#97	—	79	8	no	—
239	C₄×C₈	—	316	2	no	—
240	C₄×C₈	—	79	8	no	—
241	C₄×C₈	C₄×D₇₉	1	4	yes	—
242	C₄×C₈	C₄×D₇₉	1	4	yes	—
243	#95	—	79	4	no	—
244	#95	—	79	4	no	—
245	#95	—	79	2	no	—
246	#95	—	79	4	no	—
247	C₂×Q₈	—	79	4	no	—
248	C₂×Q₈	—	158	4	no	—
249	C₂²×C₄	—	79	2	no	—
250	C₂²×C₄	—	79	2	no	—
251	C₂²×C₄	—	79	4	no	—
252	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
253	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
254	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	79	4	no	—
266	C₂×C₈	—	79	4	no	—
267	C₂×C₈	—	158	2	no	—
268	C₂×C₈	—	158	2	no	—
269	C₂×C₈	—	158	2	no	—
270	C₂×C₈	—	158	2	no	—
271	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
273	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
274	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
275	C₄⋊C₄	—	316	4	no	—
276	C₄⋊C₄	—	79	8	no	—
277	C₄⋊C₄	—	79	4	no	—
278	C₄⋊C₄	—	158	4	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	4	2	no	—
285	C₄²	—	79	4	no	—
286	C₄²	—	79	4	no	—
287	C₄²	—	158	2	no	—
288	C₄²	—	158	2	no	—
289	C₄²	C₂×C₄×D₇₉	1	2	yes	—
290	C₄²	C₂×C₄×D₇₉	1	2	yes	—
291	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
292	Q₈	—	158	8	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	79	4	no	—
301	C₂×C₄	—	79	4	no	—
302	C₂×C₄	—	158	4	no	—
303	C₂×C₄	—	158	4	no	—
304	C₂×C₄	—	158	4	no	—
305	C₂×C₄	—	158	4	no	—
306	C₂×C₄	—	158	4	no	—
307	C₂×C₄	C₄²×D₇₉	1	2	yes	—
308	C₂×C₄	C₄²×D₇₉	1	2	yes	—
309	C₂×C₄	#14579167	1	4	yes	—
310	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
312	C₂×C₄	#14579207	1	1	yes	characteristic
313	C₂×C₄	#14579208	1	1	yes	characteristic
314	C₂×C₄	#14579208	1	1	yes	characteristic
315	C₂×C₄	#14579208	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	158	4	no	—
321	C₈	—	158	4	no	—
322	C₂²	C₂×C₄²×D₇₉	1	1	yes	characteristic
323	C₂²	#19099040	1	1	yes	characteristic
324	C₂²	#19099043	1	1	yes	characteristic
325	C₂²	C₂×#14579167	1	2	yes	—
326	C₂²	#19099329	1	2	yes	—
327	C₄	—	2	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	316	4	no	—
331	C₄	—	158	4	no	—
332	C₄	—	158	4	no	—
333	C₄	#19098517	1	4	yes	—
334	C₄	#19099040	1	1	yes	characteristic
335	C₄	#19099040	1	1	yes	characteristic
336	C₄	#19099043	1	1	yes	characteristic
337	C₄	#19099043	1	1	yes	characteristic
338	C₂	#23765763	1	1	yes	characteristic
339	C₂	C₂×#19098517	1	1	yes	characteristic
340	C₂	#23771825	1	1	yes	characteristic
341	C₂	#23772494	1	2	yes	—
342	C₂	#23772551	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₇₉×#1744	C₂
socle, prime radical	C₂³×C₇₉	C₂³×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1744	D₇₉
hypocenter	C₇₉	#20574
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1744

Complements

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

N	H ≅ G/N	type
13. #19098693	C₄	semidirect
14. C₈×Dic₁₅₈	C₄	semidirect
183. C₇₉	#20574	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098693	2
C₂×C₈×Dic₁₅₈	2
#23765877	2
#23763273	2
#23763267	2
#23763266	2
C₈×C₄⋊Dic₇₉	2
#23763256	2
#23763253	2
#23763150	2
C₇₉×#1744	2
#20574	79