Subgroups — #34287326

Order 20224.

826 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×#19098694	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23763275	C₂	1	2	yes	maximal
4	#23763269	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×C₇₉⋊C₈	C₂	1	2	yes	maximal
6	C₄⋊C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#23763257	C₂	1	2	yes	maximal
8	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
9	#23763152	C₂	1	2	yes	maximal
10	#23763146	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#2295	C₂	1	1	yes	maximal, characteristic
12	#19098694	C₄	1	8	yes	complemented
13	Q₈×C₇₉⋊C₈	C₄	1	8	yes	complemented
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
18	C₂×#14579072	C₂²	1	2	yes	—
19	C₂×#14579072	C₂²	1	2	yes	—
20	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
25	#19098318	C₂²	1	2	yes	—
26	#19098318	C₂²	1	1	yes	characteristic
27	#19098318	C₂²	1	1	yes	characteristic
28	#19098318	C₂²	1	2	yes	—
29	#19098318	C₂²	1	2	yes	—
30	#19098318	C₂²	1	1	yes	characteristic
31	#19098318	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
33	#391×C₇₉	C₄	1	2	yes	—
34	#389×C₇₉	C₄	1	2	yes	—
35	#387×C₇₉	C₄	1	4	yes	—
36	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
38	#376×C₇₉	C₂²	1	2	yes	—
39	C₄³×C₇₉	C₂²	1	1	yes	characteristic
40	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
45	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
46	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
47	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
48	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
49	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
50	#14579072	—	4	2	no	—
51	#14579072	C₂×C₄	1	8	yes	—
52	C₄×C₇₉⋊C₈	—	4	2	no	—
53	C₄×C₇₉⋊C₈	—	4	2	no	—
54	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
55	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
56	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
57	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
58	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
59	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
68	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
71	#95×C₇₉	C₂×C₄	1	4	yes	—
72	#95×C₇₉	C₂×C₄	1	4	yes	—
73	#95×C₇₉	C₂×C₄	1	2	yes	—
74	#95×C₇₉	C₂×C₄	1	4	yes	—
75	C₂×C₇₉⋊C₈	—	2	1	no	—
76	C₂×C₇₉⋊C₈	—	2	2	no	—
77	C₂×C₇₉⋊C₈	—	2	2	no	—
78	C₂×C₇₉⋊C₈	—	2	1	no	—
79	C₂×C₇₉⋊C₈	—	2	1	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	—	2	1	no	—
85	C₂×C₇₉⋊C₈	—	2	2	no	—
86	C₂×C₇₉⋊C₈	—	2	1	no	—
87	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
90	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
93	C₂×Q₈×C₇₉	—	2	4	no	—
94	C₂×Q₈×C₇₉	C₄²	1	4	yes	—
95	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
97	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₇₉	—	2	8	no	—
104	C₄⋊C₄×C₇₉	—	2	4	no	—
105	C₄⋊C₄×C₇₉	C₄²	1	8	yes	—
106	C₄⋊C₄×C₇₉	C₄²	1	4	yes	—
107	C₄²×C₇₉	—	2	4	no	—
108	C₄²×C₇₉	—	2	4	no	—
109	C₄²×C₇₉	—	2	1	no	—
110	C₄²×C₇₉	—	2	2	no	—
111	C₄²×C₇₉	—	2	1	no	—
112	C₄²×C₇₉	—	2	1	no	—
113	C₄²×C₇₉	—	2	1	no	—
114	C₄²×C₇₉	—	2	2	no	—
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	2	no	—
122	C₂×C₄×C₇₉	—	2	2	no	—
123	C₂×C₄×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	—	2	4	no	—
125	C₂×C₄×C₇₉	—	2	4	no	—
126	C₂×C₄×C₇₉	—	2	2	no	—
127	C₂×C₄×C₇₉	—	2	4	no	—
128	C₂×C₄×C₇₉	—	2	2	no	—
129	C₂×C₄×C₇₉	—	2	2	no	—
130	C₂×C₄×C₇₉	—	2	4	no	—
131	C₂×C₄×C₇₉	—	2	4	no	—
132	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
134	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
135	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₇₉	#131	1	4	yes	—
137	C₂×C₄×C₇₉	#131	1	4	yes	—
138	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₇₉⋊C₈	—	2	2	no	—
145	C₇₉⋊C₈	—	2	2	no	—
146	C₇₉⋊C₈	—	2	4	no	—
147	C₇₉⋊C₈	—	2	2	no	—
148	C₇₉⋊C₈	—	2	4	no	—
149	C₇₉⋊C₈	—	2	2	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	C₂×#131	1	2	yes	—
155	C₄×C₇₉	—	2	4	no	—
156	C₄×C₇₉	—	2	2	no	—
157	C₄×C₇₉	—	2	4	no	—
158	C₄×C₇₉	—	2	4	no	—
159	C₄×C₇₉	—	4	2	no	—
160	C₄×C₇₉	—	4	4	no	—
161	C₄×C₇₉	#405	1	4	yes	—
162	C₄×C₇₉	#519	1	1	yes	characteristic
163	C₄×C₇₉	#519	1	1	yes	characteristic
164	C₄×C₇₉	#520	1	1	yes	characteristic
165	C₄×C₇₉	#520	1	1	yes	characteristic
166	#20571	—	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	C₄×#131	1	1	yes	characteristic
170	C₂×C₇₉	#2986	1	2	yes	—
171	C₂×C₇₉	#2994	1	2	yes	—
172	C₂×#446	—	79	1	no	—
173	C₂×C₈×Q₈	—	79	1	no	—
174	#2295	D₇₉	1	1	yes	characteristic
175	#1771	—	79	2	no	—
176	#1770	—	79	1	no	—
177	C₈×C₄⋊C₄	—	79	2	no	—
178	C₈×C₄⋊C₄	—	79	1	no	—
179	#1768	—	79	2	no	—
180	C₄×C₄⋊C₈	—	79	1	no	—
181	#1746	—	79	2	no	—
182	#1744	—	79	1	no	—
183	C₇₉	#20571	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
185	#446	—	79	8	no	—
186	C₈×Q₈	—	79	8	no	—
187	C₂×C₄⋊C₈	—	79	1	no	—
188	C₂×C₄⋊C₈	—	79	2	no	—
189	C₂×C₄⋊C₈	—	79	1	no	—
190	C₂×C₄⋊C₈	—	79	2	no	—
191	C₂×#97	—	79	2	no	—
192	C₂×#97	—	79	2	no	—
193	C₂×C₄×C₈	—	79	2	no	—
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	—	79	1	no	—
198	#391	Dic₇₉	1	2	yes	—
199	#389	Dic₇₉	1	2	yes	—
200	#387	Dic₇₉	1	4	yes	—
201	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
202	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
203	#376	C₂×D₇₉	1	2	yes	—
204	C₄³	C₂×D₇₉	1	1	yes	characteristic
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	C₂×Dic₇₉	1	1	yes	characteristic
213	C₂²×C₈	—	79	1	no	—
214	C₂²×C₈	—	79	1	no	—
215	C₂²×C₈	—	79	1	no	—
216	C₂²×C₈	—	79	2	no	—
217	C₂²×C₈	—	79	1	no	—
218	C₂²×C₈	—	79	2	no	—
219	C₄×Q₈	C₄×D₇₉	1	4	yes	—
220	C₄×Q₈	C₄×D₇₉	1	4	yes	—
221	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
222	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
223	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
226	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
227	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
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	—	79	4	no	—
235	C₄⋊C₈	—	79	8	no	—
236	#97	—	316	2	no	—
237	#97	—	79	8	no	—
238	C₄×C₈	—	316	2	no	—
239	C₄×C₈	—	316	2	no	—
240	C₄×C₈	—	79	4	no	—
241	C₄×C₈	—	79	8	no	—
242	C₄×C₈	—	79	4	no	—
243	#95	C₂×Dic₇₉	1	4	yes	—
244	#95	C₂×Dic₇₉	1	4	yes	—
245	#95	C₂×Dic₇₉	1	2	yes	—
246	#95	C₂×Dic₇₉	1	4	yes	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	C₄×Dic₇₉	1	4	yes	—
249	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
251	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
254	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
255	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
257	C₂×C₈	—	79	2	no	—
258	C₂×C₈	—	79	2	no	—
259	C₂×C₈	—	79	4	no	—
260	C₂×C₈	—	79	2	no	—
261	C₂×C₈	—	79	2	no	—
262	C₂×C₈	—	79	4	no	—
263	C₂×C₈	—	158	2	no	—
264	C₂×C₈	—	158	1	no	—
265	C₂×C₈	—	158	2	no	—
266	C₂×C₈	—	158	1	no	—
267	C₂×C₈	—	158	1	no	—
268	C₂×C₈	—	158	1	no	—
269	C₂×C₈	—	158	1	no	—
270	C₂×C₈	—	158	1	no	—
271	C₂×C₈	—	158	1	no	—
272	C₂×C₈	—	158	2	no	—
273	C₂×C₈	—	158	1	no	—
274	C₂×C₈	—	158	2	no	—
275	C₄⋊C₄	—	2	8	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	C₄×Dic₇₉	1	8	yes	—
278	C₄⋊C₄	C₄×Dic₇₉	1	4	yes	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	4	no	—
287	C₄²	C₂×C₄×D₇₉	1	4	yes	—
288	C₄²	C₂×C₄×D₇₉	1	2	yes	—
289	C₄²	C₂×C₄×D₇₉	1	2	yes	—
290	C₄²	C₂×C₄×D₇₉	1	4	yes	—
291	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
292	Q₈	—	2	8	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
305	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
306	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
307	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
308	C₂×C₄	#14579168	1	4	yes	—
309	C₂×C₄	#14579168	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	#14579209	1	1	yes	characteristic
315	C₂×C₄	#14579209	1	1	yes	characteristic
316	C₈	—	158	2	no	—
317	C₈	—	158	2	no	—
318	C₈	—	158	4	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	#19099041	1	1	yes	characteristic
324	C₂²	#19099046	1	1	yes	characteristic
325	C₂²	C₂×#14579168	1	2	yes	—
326	C₂²	C₂×#14579168	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	4	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	4	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	4	no	—
333	C₄	#19098519	1	4	yes	—
334	C₄	#19099039	1	1	yes	characteristic
335	C₄	#19099039	1	1	yes	characteristic
336	C₄	#19099045	1	1	yes	characteristic
337	C₄	#19099045	1	1	yes	characteristic
338	C₂	#23765765	1	1	yes	characteristic
339	C₂	C₂×#19098519	1	1	yes	characteristic
340	C₂	C₄×#14579168	1	1	yes	characteristic
341	C₂	#23772502	1	2	yes	—
342	C₂	#23772560	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Fitting	C₇₉×#2295	C₂
socle, prime radical	C₂³×C₇₉	C₂³×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2295	D₇₉
hypocenter	C₇₉	#20571
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
12. #19098694	C₄	semidirect
13. Q₈×C₇₉⋊C₈	C₄	semidirect
183. C₇₉	#20571	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098694	2
C₂×Q₈×C₇₉⋊C₈	2
#23763275	2
#23763269	2
C₄⋊C₄×C₇₉⋊C₈	2
#23763257	2
C₄×C₄⋊(C₇₉⋊C₈)	2
#23763152	2
#23763146	2
C₇₉×#2295	2
#20571	79