Subgroups — #34287337

Order 20224.

898 conjugacy classes of subgroups.

The subgroup lattice has 350 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	C₄×Q₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23763277	C₂	1	2	yes	maximal
4	#23763272	C₂	1	2	yes	maximal
5	#23763265	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
7	#23763255	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₈×Dic₇₉	C₂	1	2	yes	maximal
9	#23763148	C₂	1	2	yes	maximal
10	#23763145	C₂	1	1	yes	maximal, characteristic
11	C₂×#446×C₇₉	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×Dic₇₉	C₄	1	2	yes	—
13	#19098694	C₄	1	8	yes	complemented
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#14579072	C₂²	1	2	yes	—
17	C₂×#14579071	C₂²	1	2	yes	—
18	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
20	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
21	C₄⋊C₄×Dic₇₉	C₄	1	4	yes	—
22	C₄⋊C₄×Dic₇₉	C₄	1	2	yes	—
23	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
25	C₄²×Dic₇₉	C₂²	1	2	yes	—
26	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
27	#19098317	C₂²	1	1	yes	characteristic
28	#19098317	C₂²	1	2	yes	—
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	2	yes	—
37	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
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	2	yes	—
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	Q₈×Dic₇₉	—	2	8	no	—
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	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
49	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
52	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
53	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
54	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
55	#14579072	C₂×C₄	1	8	yes	—
56	#14579071	—	4	2	no	—
57	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
58	C₈×Dic₇₉	—	4	2	no	—
59	C₈×Dic₇₉	—	4	2	no	—
60	C₇₉⋊#95	C₂×C₄	1	4	yes	—
61	C₇₉⋊#95	C₂×C₄	1	2	yes	—
62	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
65	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
67	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
72	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
73	C₄⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
74	#97×C₇₉	C₂×C₄	1	8	yes	—
75	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
76	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
78	C₂²×Dic₇₉	C₂²×C₄	1	2	yes	—
79	C₂×C₇₉⋊C₈	—	2	1	no	—
80	C₂×C₇₉⋊C₈	—	2	2	no	—
81	C₂×C₇₉⋊C₈	—	2	1	no	—
82	C₂×C₇₉⋊C₈	—	2	1	no	—
83	C₂×C₇₉⋊C₈	—	2	1	no	—
84	C₂×C₇₉⋊C₈	—	2	2	no	—
85	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
88	C₄⋊Dic₇₉	—	2	8	no	—
89	C₄⋊Dic₇₉	M₁₆	1	8	yes	—
90	C₄×Dic₇₉	—	2	8	no	—
91	C₄×Dic₇₉	—	2	8	no	—
92	C₄×Dic₇₉	—	2	4	no	—
93	C₄×Dic₇₉	—	2	4	no	—
94	C₄×Dic₇₉	—	2	2	no	—
95	C₄×Dic₇₉	—	2	1	no	—
96	C₄×Dic₇₉	—	2	1	no	—
97	C₄×Dic₇₉	—	2	2	no	—
98	C₄×Dic₇₉	—	2	1	no	—
99	C₄×Dic₇₉	—	2	1	no	—
100	C₂×Q₈×C₇₉	—	2	4	no	—
101	C₂×Q₈×C₇₉	C₄²	1	4	yes	—
102	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₇₉	—	2	1	no	—
108	C₂×C₈×C₇₉	—	2	2	no	—
109	C₂×C₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	—	2	2	no	—
111	C₂×C₈×C₇₉	—	2	1	no	—
112	C₂×C₈×C₇₉	—	2	1	no	—
113	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
116	C₄⋊C₄×C₇₉	C₄²	1	8	yes	—
117	C₄⋊C₄×C₇₉	C₄²	1	4	yes	—
118	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
119	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
120	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
121	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
122	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
123	Q₈×C₇₉	—	2	8	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	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
129	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
130	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
132	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
138	C₂×Dic₇₉	—	2	2	no	—
139	C₂×Dic₇₉	—	2	4	no	—
140	C₂×Dic₇₉	—	2	2	no	—
141	C₂×Dic₇₉	—	2	4	no	—
142	C₂×Dic₇₉	—	2	2	no	—
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	2	2	no	—
145	C₂×Dic₇₉	C₂×M₁₆	1	8	yes	—
146	C₂×Dic₇₉	#131	1	4	yes	—
147	C₈×C₇₉	—	2	2	no	—
148	C₈×C₇₉	—	2	2	no	—
149	C₈×C₇₉	—	2	4	no	—
150	C₇₉⋊C₈	—	2	2	no	—
151	C₇₉⋊C₈	—	2	2	no	—
152	C₇₉⋊C₈	—	2	4	no	—
153	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
154	C₂²×C₇₉	#519	1	1	yes	characteristic
155	C₂²×C₇₉	#520	1	1	yes	characteristic
156	C₂²×C₇₉	C₂²×M₁₆	1	2	yes	—
157	C₂²×C₇₉	C₂×#131	1	2	yes	—
158	C₄×C₇₉	—	2	4	no	—
159	C₄×C₇₉	—	2	2	no	—
160	C₄×C₇₉	—	2	4	no	—
161	C₄×C₇₉	C₄×M₁₆	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	Dic₇₉	—	2	4	no	—
167	Dic₇₉	—	2	8	no	—
168	Dic₇₉	—	4	2	no	—
169	#20573	—	79	1	no	maximal, Sylow
170	C₂×C₇₉	#2276	1	1	yes	characteristic
171	C₂×C₇₉	C₂×C₄×M₁₆	1	1	yes	characteristic
172	C₂×C₇₉	C₄×#131	1	1	yes	characteristic
173	C₂×C₇₉	#2984	1	2	yes	—
174	C₂×C₇₉	#2990	1	2	yes	—
175	C₂×#446	—	79	1	no	—
176	C₂×#446	D₇₉	1	1	yes	characteristic
177	C₄²×Q₈	—	79	1	no	—
178	#1771	—	79	2	no	—
179	#1771	—	79	2	no	—
180	C₈×C₄⋊C₄	—	79	1	no	—
181	C₈×C₄⋊C₄	—	79	1	no	—
182	#1767	—	79	1	no	—
183	C₄×C₄⋊C₈	—	79	2	no	—
184	#1745	—	79	2	no	—
185	#1744	—	79	1	no	—
186	C₇₉	#20573	1	1	yes	characteristic, complemented, Sylow
187	C₂×C₄×Q₈	—	79	2	no	—
188	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
189	#446	—	79	8	no	—
190	#446	Dic₇₉	1	8	yes	complemented
191	C₂×C₄⋊C₈	—	79	2	no	—
192	C₂×C₄⋊C₈	—	79	1	no	—
193	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
194	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
195	C₂×#97	—	79	2	no	—
196	C₂×#97	—	79	2	no	—
197	C₂×#97	C₂×D₇₉	1	2	yes	—
198	C₂×C₄×C₈	—	79	2	no	—
199	C₂×C₄×C₈	—	79	2	no	—
200	C₂×C₄×C₈	—	79	1	no	—
201	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
202	C₄×C₄⋊C₄	—	79	2	no	—
203	C₄×C₄⋊C₄	—	79	4	no	—
204	C₄×C₄⋊C₄	—	79	1	no	—
205	C₄×C₄⋊C₄	—	79	2	no	—
206	C₄³	—	79	1	no	—
207	C₄³	—	79	2	no	—
208	#336	—	79	1	no	—
209	#336	—	79	1	no	—
210	#336	—	79	1	no	—
211	#336	—	79	1	no	—
212	#336	—	79	2	no	—
213	#336	—	79	2	no	—
214	#336	—	79	2	no	—
215	C₂²×Q₈	C₄×D₇₉	1	1	yes	characteristic
216	C₂²×C₈	—	79	1	no	—
217	C₂²×C₈	—	79	1	no	—
218	C₂²×C₈	—	79	2	no	—
219	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
222	C₄×Q₈	—	158	8	no	—
223	C₄×Q₈	C₄×D₇₉	1	4	yes	—
224	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
225	C₂×C₄⋊C₄	—	79	2	no	—
226	C₂×C₄⋊C₄	—	79	4	no	—
227	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
228	C₂×C₄⋊C₄	C₄×D₇₉	1	1	yes	characteristic
229	C₂×C₄²	—	79	4	no	—
230	C₂×C₄²	—	79	2	no	—
231	C₂×C₄²	—	79	4	no	—
232	C₂×C₄²	—	79	2	no	—
233	C₂×C₄²	—	79	1	no	—
234	C₂×C₄²	—	79	2	no	—
235	C₂×C₄²	—	79	1	no	—
236	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
238	C₄⋊C₈	—	79	4	no	—
239	C₄⋊C₈	—	79	8	no	—
240	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
241	C₄⋊C₈	C₂×Dic₇₉	1	8	yes	—
242	#97	—	316	2	no	—
243	#97	—	79	8	no	—
244	#97	C₂×Dic₇₉	1	8	yes	—
245	C₄×C₈	—	316	2	no	—
246	C₄×C₈	—	316	2	no	—
247	C₄×C₈	—	79	4	no	—
248	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
249	#95	—	79	2	no	—
250	#95	—	79	4	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	C₄×Dic₇₉	1	4	yes	—
253	C₂²×C₄	—	79	4	no	—
254	C₂²×C₄	—	79	2	no	—
255	C₂²×C₄	—	79	2	no	—
256	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
258	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
259	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
260	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	79	2	no	—
268	C₂×C₈	—	79	2	no	—
269	C₂×C₈	—	79	4	no	—
270	C₂×C₈	—	158	1	no	—
271	C₂×C₈	—	158	2	no	—
272	C₂×C₈	—	158	1	no	—
273	C₂×C₈	—	158	1	no	—
274	C₂×C₈	—	158	1	no	—
275	C₂×C₈	—	158	2	no	—
276	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
277	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
278	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
279	C₄⋊C₄	—	79	8	no	—
280	C₄⋊C₄	—	158	8	no	—
281	C₄⋊C₄	C₄×Dic₇₉	1	8	yes	—
282	C₄⋊C₄	C₄×Dic₇₉	1	4	yes	—
283	C₄²	—	158	4	no	—
284	C₄²	—	158	8	no	—
285	C₄²	—	158	8	no	—
286	C₄²	—	158	2	no	—
287	C₄²	—	158	1	no	—
288	C₄²	—	158	1	no	—
289	C₄²	—	158	1	no	—
290	C₄²	—	158	1	no	—
291	C₄²	—	158	2	no	—
292	C₄²	—	158	4	no	—
293	C₄²	C₂×C₄×D₇₉	1	2	yes	—
294	C₄²	C₂×C₄×D₇₉	1	4	yes	—
295	C₄²	C₂²×Dic₇₉	1	2	yes	—
296	C₄²	C₂²×Dic₇₉	1	4	yes	—
297	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
298	Q₈	—	2	8	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	79	8	no	—
303	C₂×C₄	—	79	4	no	—
304	C₂×C₄	—	158	2	no	—
305	C₂×C₄	—	158	2	no	—
306	C₂×C₄	—	158	2	no	—
307	C₂×C₄	—	158	2	no	—
308	C₂×C₄	—	158	4	no	—
309	C₂×C₄	—	158	2	no	—
310	C₂×C₄	—	158	4	no	—
311	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
312	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
313	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
314	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
315	C₂×C₄	M₁₆×D₇₉	1	4	yes	—
316	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
317	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
318	C₂×C₄	#14579207	1	1	yes	characteristic
319	C₂×C₄	#14579208	1	1	yes	characteristic
320	C₂×C₄	#14579209	1	1	yes	characteristic
321	C₂×C₄	#14579209	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	4	no	—
325	C₈	—	158	2	no	—
326	C₈	—	158	2	no	—
327	C₈	—	158	4	no	—
328	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
329	C₂²	#19099039	1	1	yes	characteristic
330	C₂²	#19099045	1	1	yes	characteristic
331	C₂²	C₂×M₁₆×D₇₉	1	2	yes	—
332	C₂²	#131×D₇₉	1	2	yes	—
333	C₄	—	2	4	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	4	no	—
336	C₄	—	316	2	no	—
337	C₄	—	158	4	no	—
338	C₄	—	158	8	no	—
339	C₄	M₁₆×Dic₇₉	1	4	yes	—
340	C₄	#19099039	1	1	yes	characteristic
341	C₄	#19099041	1	1	yes	characteristic
342	C₄	#19099045	1	1	yes	characteristic
343	C₄	#19099046	1	1	yes	characteristic
344	C₂	#23765765	1	1	yes	characteristic
345	C₂	C₂×M₁₆×Dic₇₉	1	1	yes	characteristic
346	C₂	#131×Dic₇₉	1	1	yes	characteristic
347	C₂	#23772486	1	2	yes	—
348	C₂	#23772524	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	#20573
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
13. #19098694	C₄	semidirect
35. #446×C₇₉	C₄	semidirect
186. C₇₉	#20573	semidirect
190. #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×#19098694	2
C₄×Q₈×Dic₇₉	2
#23763277	2
#23763272	2
#23763265	2
C₈×C₄⋊Dic₇₉	2
#23763255	2
C₄⋊C₈×Dic₇₉	2
#23763148	2
#23763145	2
C₂×#446×C₇₉	2
#20573	79