Subgroups — #34287352

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×Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#23765878	C₂	1	1	yes	maximal, characteristic
3	#23763272	C₂	1	2	yes	maximal
4	#23763270	C₂	1	2	yes	maximal
5	#23763267	C₂	1	1	yes	maximal, characteristic
6	#23763265	C₂	1	1	yes	maximal, characteristic
7	#23763259	C₂	1	2	yes	maximal
8	#23763255	C₂	1	1	yes	maximal, characteristic
9	#23763153	C₂	1	2	yes	maximal
10	#23763145	C₂	1	1	yes	maximal, characteristic
11	C₂×#446×C₇₉	C₂	1	1	yes	maximal, characteristic
12	Q₈×C₇₉⋊C₈	C₄	1	8	yes	complemented
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
15	C₂×#14579071	C₂²	1	2	yes	—
16	C₂×#14579071	C₂²	1	2	yes	—
17	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
19	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
20	#19098470	C₄	1	2	yes	—
21	#19098462	C₄	1	2	yes	—
22	#19098449	C₄	1	4	yes	—
23	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
25	#19098409	C₂²	1	2	yes	—
26	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
27	#19098317	C₂²	1	2	yes	—
28	#19098317	C₂²	1	1	yes	characteristic
29	#19098317	C₂²	1	2	yes	—
30	#19098317	C₂²	1	2	yes	—
31	#19098317	C₂²	1	1	yes	characteristic
32	#19098317	C₂²	1	1	yes	characteristic
33	#19098317	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
35	#446×C₇₉	C₄	1	8	yes	complemented
36	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
38	C₂×#97×C₇₉	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
40	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
50	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
51	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
52	#14579071	—	4	2	no	—
53	#14579071	—	4	2	no	—
54	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
55	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
56	C₈×Dic₇₉	—	4	2	no	—
57	C₇₉⋊#95	C₂×C₄	1	4	yes	—
58	C₇₉⋊#95	C₂×C₄	1	4	yes	—
59	C₇₉⋊#95	C₂×C₄	1	2	yes	—
60	C₇₉⋊#95	C₂×C₄	1	4	yes	—
61	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
66	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
67	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
71	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
72	C₄⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
73	#97×C₇₉	C₂×C₄	1	8	yes	—
74	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
75	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
77	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
78	C₂×C₇₉⋊C₈	—	2	1	no	—
79	C₂×C₇₉⋊C₈	—	2	2	no	—
80	C₂×C₇₉⋊C₈	—	2	1	no	—
81	C₂×C₇₉⋊C₈	—	2	1	no	—
82	C₂×C₇₉⋊C₈	—	2	1	no	—
83	C₂×C₇₉⋊C₈	—	2	2	no	—
84	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
87	C₄⋊Dic₇₉	—	2	4	no	—
88	C₄⋊Dic₇₉	—	4	4	no	—
89	C₄×Dic₇₉	—	2	4	no	—
90	C₄×Dic₇₉	—	2	2	no	—
91	C₄×Dic₇₉	—	2	1	no	—
92	C₄×Dic₇₉	—	2	1	no	—
93	C₄×Dic₇₉	—	2	1	no	—
94	C₄×Dic₇₉	—	2	2	no	—
95	C₄×Dic₇₉	—	2	1	no	—
96	C₄×Dic₇₉	—	4	2	no	—
97	C₂×Q₈×C₇₉	—	2	4	no	—
98	C₂×Q₈×C₇₉	C₄²	1	4	yes	—
99	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
104	C₂×C₈×C₇₉	—	2	2	no	—
105	C₂×C₈×C₇₉	—	2	1	no	—
106	C₂×C₈×C₇₉	—	2	2	no	—
107	C₂×C₈×C₇₉	—	2	1	no	—
108	C₂×C₈×C₇₉	—	2	1	no	—
109	C₂×C₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄×C₇₉	C₄²	1	8	yes	—
114	C₄⋊C₄×C₇₉	C₄²	1	4	yes	—
115	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
116	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
117	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
118	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
119	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
120	Q₈×C₇₉	—	2	8	no	—
121	C₂×C₄×C₇₉	—	2	4	no	—
122	C₂×C₄×C₇₉	—	2	2	no	—
123	C₂×C₄×C₇₉	—	2	4	no	—
124	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
126	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
127	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₇₉	#131	1	4	yes	—
129	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₇₉	2₋¹⁺⁴	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×Dic₇₉	—	2	2	no	—
136	C₂×Dic₇₉	—	2	4	no	—
137	C₂×Dic₇₉	—	2	2	no	—
138	C₂×Dic₇₉	—	2	2	no	—
139	C₂×Dic₇₉	—	2	2	no	—
140	C₂×Dic₇₉	—	2	4	no	—
141	C₂×Dic₇₉	—	2	2	no	—
142	C₂×Dic₇₉	—	2	2	no	—
143	C₂×Dic₇₉	—	2	4	no	—
144	C₈×C₇₉	—	2	2	no	—
145	C₈×C₇₉	—	2	4	no	—
146	C₈×C₇₉	—	2	2	no	—
147	C₇₉⋊C₈	—	2	2	no	—
148	C₇₉⋊C₈	—	2	2	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	4	no	—
156	C₄×C₇₉	—	2	2	no	—
157	C₄×C₇₉	—	2	4	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	—	4	2	no	—
164	Dic₇₉	—	4	2	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	D₇₉	1	1	yes	characteristic
173	C₂×C₈×Q₈	—	79	1	no	—
174	#2295	—	79	1	no	—
175	#1771	—	79	2	no	—
176	#1770	—	79	2	no	—
177	#1770	—	79	1	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	—	79	1	no	—
183	C₇₉	#20574	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
185	#446	Dic₇₉	1	8	yes	complemented
186	C₈×Q₈	—	79	8	no	—
187	C₂×C₄⋊C₈	—	79	1	no	—
188	C₂×C₄⋊C₈	—	79	2	no	—
189	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
191	C₂×#97	—	79	2	no	—
192	C₂×#97	—	79	2	no	—
193	C₂×#97	C₂×D₇₉	1	2	yes	—
194	C₂×C₄×C₈	—	79	2	no	—
195	C₂×C₄×C₈	—	79	1	no	—
196	C₂×C₄×C₈	—	79	2	no	—
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	—	79	1	no	—
205	#336	—	79	1	no	—
206	#336	—	79	1	no	—
207	#336	—	79	1	no	—
208	#336	—	79	1	no	—
209	#336	—	79	2	no	—
210	#336	—	79	2	no	—
211	#336	—	79	2	no	—
212	C₂²×Q₈	C₄×D₇₉	1	1	yes	characteristic
213	C₂²×C₈	—	79	1	no	—
214	C₂²×C₈	—	79	1	no	—
215	C₂²×C₈	—	79	2	no	—
216	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
218	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
219	C₄×Q₈	C₄×D₇₉	1	4	yes	—
220	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
221	C₂×C₄⋊C₄	—	79	4	no	—
222	C₂×C₄⋊C₄	—	79	2	no	—
223	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₄×D₇₉	1	1	yes	characteristic
225	C₂×C₄²	—	79	2	no	—
226	C₂×C₄²	—	79	2	no	—
227	C₂×C₄²	—	79	1	no	—
228	C₂×C₄²	—	79	1	no	—
229	C₂×C₄²	—	79	2	no	—
230	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
232	C₄⋊C₈	—	79	4	no	—
233	C₄⋊C₈	—	79	8	no	—
234	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
235	C₄⋊C₈	C₂×Dic₇₉	1	8	yes	—
236	#97	—	316	2	no	—
237	#97	—	316	2	no	—
238	#97	C₂×Dic₇₉	1	8	yes	—
239	C₄×C₈	—	316	2	no	—
240	C₄×C₈	—	79	4	no	—
241	C₄×C₈	—	79	8	no	—
242	C₄×C₈	C₂×Dic₇₉	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	—	2	4	no	—
248	C₂×Q₈	C₄×Dic₇₉	1	4	yes	—
249	C₂²×C₄	—	79	2	no	—
250	C₂²×C₄	—	79	4	no	—
251	C₂²×C₄	—	79	2	no	—
252	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
253	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
255	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	79	2	no	—
264	C₂×C₈	—	79	2	no	—
265	C₂×C₈	—	79	4	no	—
266	C₂×C₈	—	158	1	no	—
267	C₂×C₈	—	158	2	no	—
268	C₂×C₈	—	158	1	no	—
269	C₂×C₈	—	158	1	no	—
270	C₂×C₈	—	158	1	no	—
271	C₂×C₈	—	158	2	no	—
272	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
273	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
274	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
275	C₄⋊C₄	—	316	4	no	—
276	C₄⋊C₄	—	158	4	no	—
277	C₄⋊C₄	C₄×Dic₇₉	1	8	yes	—
278	C₄⋊C₄	C₄×Dic₇₉	1	4	yes	—
279	C₄²	—	316	2	no	—
280	C₄²	—	158	2	no	—
281	C₄²	—	158	1	no	—
282	C₄²	—	158	1	no	—
283	C₄²	—	158	1	no	—
284	C₄²	—	158	1	no	—
285	C₄²	—	158	2	no	—
286	C₄²	—	158	4	no	—
287	C₄²	C₂×C₄×D₇₉	1	4	yes	—
288	C₄²	C₂×C₄×D₇₉	1	2	yes	—
289	C₄²	C₂²×Dic₇₉	1	2	yes	—
290	C₄²	C₂²×Dic₇₉	1	4	yes	—
291	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
292	Q₈	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	158	2	no	—
297	C₂×C₄	—	158	4	no	—
298	C₂×C₄	—	158	2	no	—
299	C₂×C₄	—	158	2	no	—
300	C₂×C₄	—	158	2	no	—
301	C₂×C₄	—	158	4	no	—
302	C₂×C₄	—	158	2	no	—
303	C₂×C₄	—	158	2	no	—
304	C₂×C₄	—	158	4	no	—
305	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
306	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
307	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
308	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
309	C₂×C₄	#14579169	1	4	yes	—
310	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
312	C₂×C₄	#14579207	1	1	yes	characteristic
313	C₂×C₄	#14579208	1	1	yes	characteristic
314	C₂×C₄	#14579209	1	1	yes	characteristic
315	C₂×C₄	#14579209	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	4	no	—
318	C₈	—	2	2	no	—
319	C₈	—	158	2	no	—
320	C₈	—	158	2	no	—
321	C₈	—	158	4	no	—
322	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
323	C₂²	#19099039	1	1	yes	characteristic
324	C₂²	#19099045	1	1	yes	characteristic
325	C₂²	C₂×#14579169	1	2	yes	—
326	C₂²	#19099328	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	4	no	—
330	C₄	—	316	2	no	—
331	C₄	—	316	2	no	—
332	C₄	—	316	4	no	—
333	C₄	#19098521	1	4	yes	—
334	C₄	#19099039	1	1	yes	characteristic
335	C₄	#19099041	1	1	yes	characteristic
336	C₄	#19099045	1	1	yes	characteristic
337	C₄	#19099046	1	1	yes	characteristic
338	C₂	#23765765	1	1	yes	characteristic
339	C₂	C₂×#19098521	1	1	yes	characteristic
340	C₂	#23771826	1	1	yes	characteristic
341	C₂	#23772496	1	2	yes	—
342	C₂	#23772552	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₂×#446×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂³×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#446	D₇₉
hypocenter	C₇₉	#20574
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
12. Q₈×C₇₉⋊C₈	C₄	semidirect
35. #446×C₇₉	C₄	semidirect
183. C₇₉	#20574	semidirect
185. #446	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₇₉⋊C₈	2
#23765878	2
#23763272	2
#23763270	2
#23763267	2
#23763265	2
#23763259	2
#23763255	2
#23763153	2
#23763145	2
C₂×#446×C₇₉	2
#20574	79