Subgroups — #34178369

Order 20224.

654 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098620	C₂	1	1	yes	maximal, characteristic
2	#23765807	C₂	1	1	yes	maximal, characteristic
3	#23761284	C₂	1	4	yes	maximal, complemented
4	C₂×#423×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579120	—	2	1	no	—
6	#19098620	—	2	2	no	—
7	#19098620	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
10	C₂×#14579072	—	2	1	no	—
11	#19098433	C₂²	1	2	yes	—
12	#19098429	—	2	1	no	—
13	#19098423	—	2	1	no	—
14	#19098423	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₇₉	—	2	1	no	—
16	C₄×C₄⋊Dic₇₉	C₄	1	2	yes	—
17	C₂²×M₁₆×C₇₉	—	2	1	no	—
18	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
19	#423×C₇₉	—	2	2	no	—
20	#423×C₇₉	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	#14579120	—	4	4	no	—
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
29	C₂×C₄×Dic₇₉	—	2	2	no	—
30	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
32	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
33	#14579072	—	2	4	no	—
34	C₇₉⋊#95	—	2	2	no	—
35	C₇₉⋊#95	—	2	2	no	—
36	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
39	C₂×M₁₆×C₇₉	—	2	4	no	—
40	C₂×M₁₆×C₇₉	—	4	4	no	—
41	C₂²×C₈×C₇₉	—	2	2	no	—
42	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	D₄	1	2	yes	—
47	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	Q₈	1	2	yes	—
49	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₇₉	—	2	4	no	—
51	C₄⋊C₈×C₇₉	—	2	2	no	—
52	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
53	C₂²×Dic₇₉	—	2	4	no	—
54	C₂×C₇₉⋊C₈	—	2	4	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	4	2	no	—
57	C₄⋊Dic₇₉	—	4	4	no	—
58	C₄⋊Dic₇₉	—	4	2	no	—
59	C₄⋊Dic₇₉	—	4	4	no	—
60	C₄⋊Dic₇₉	—	4	2	no	—
61	C₂²⋊Dic₇₉	—	4	4	no	—
62	C₄×Dic₇₉	—	4	4	no	—
63	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	—	2	2	no	—
65	C₂²×C₄×C₇₉	—	2	2	no	—
66	C₂²×C₄×C₇₉	—	2	2	no	—
67	C₂²×C₄×C₇₉	—	2	2	no	—
68	C₂²×C₄×C₇₉	—	4	2	no	—
69	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
70	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	M₁₆×C₇₉	—	4	8	no	—
82	C₂×C₈×C₇₉	—	2	4	no	—
83	C₂×C₈×C₇₉	—	4	4	no	—
84	C₄²×C₇₉	C₄²	1	2	yes	—
85	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
88	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
89	C₄²×C₇₉	C₄⋊C₄	1	2	yes	—
90	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
94	C₂³×C₇₉	—	2	2	no	—
95	C₂³×C₇₉	—	2	2	no	—
96	C₂³×C₇₉	—	2	1	no	—
97	C₂³×C₇₉	—	2	1	no	—
98	C₂³×C₇₉	#95	1	2	yes	—
99	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	—	2	2	no	—
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	1	no	—
104	C₂×C₄×C₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	2	4	no	—
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	—	2	2	no	—
108	C₂×C₄×C₇₉	—	2	4	no	—
109	C₂×C₄×C₇₉	—	2	4	no	—
110	C₂×C₄×C₇₉	—	4	2	no	—
111	C₂×C₄×C₇₉	—	4	2	no	—
112	C₂×C₄×C₇₉	#95	1	2	yes	—
113	C₂×C₄×C₇₉	#95	1	2	yes	—
114	C₂×C₄×C₇₉	#95	1	2	yes	—
115	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₇₉	#137	1	2	yes	—
124	C₂×C₄×C₇₉	#137	1	2	yes	—
125	C₂×Dic₇₉	—	4	4	no	—
126	C₂×Dic₇₉	—	4	4	no	—
127	C₂×Dic₇₉	—	4	4	no	—
128	C₈×C₇₉	—	4	4	no	—
129	C₇₉⋊C₈	—	4	4	no	—
130	C₂²×C₇₉	—	2	2	no	—
131	C₂²×C₇₉	—	2	2	no	—
132	C₂²×C₇₉	—	2	2	no	—
133	C₂²×C₇₉	—	4	2	no	—
134	C₂²×C₇₉	—	4	2	no	—
135	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
136	C₂²×C₇₉	#413	1	2	yes	—
137	C₂²×C₇₉	#417	1	2	yes	—
138	C₂²×C₇₉	#418	1	1	yes	characteristic
139	C₂²×C₇₉	#419	1	1	yes	characteristic
140	C₄×C₇₉	—	2	2	no	—
141	C₄×C₇₉	—	2	2	no	—
142	C₄×C₇₉	—	2	2	no	—
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	—	2	1	no	—
145	C₄×C₇₉	#417	1	2	yes	—
146	C₄×C₇₉	#418	1	1	yes	characteristic
147	C₄×C₇₉	#419	1	1	yes	characteristic
148	C₄×C₇₉	#428	1	2	yes	—
149	C₄×C₇₉	#428	1	2	yes	—
150	Dic₇₉	—	8	4	no	—
151	#8042	—	79	1	no	maximal, Sylow
152	C₂×C₇₉	—	4	2	no	—
153	C₂×C₇₉	#1557	1	2	yes	—
154	C₂×C₇₉	#1559	1	2	yes	—
155	C₂×C₇₉	#1736	1	1	yes	characteristic
156	C₂×C₇₉	#1736	1	1	yes	characteristic
157	C₂×C₇₉	#1742	1	1	yes	characteristic
158	C₂×#432	—	79	1	no	—
159	C₂×#423	D₇₉	1	1	yes	characteristic
160	#2285	—	79	1	no	—
161	#1296	—	79	4	no	—
162	C₇₉	#8042	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	2	1	no	—
164	C₂×#117	—	158	1	no	—
165	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
166	#432	—	79	4	no	—
167	#432	—	158	2	no	—
168	#423	—	2	2	no	—
169	#423	Dic₇₉	1	4	yes	complemented
170	C₂×C₄⋊C₈	—	2	1	no	—
171	C₂×C₄⋊C₈	—	79	2	no	—
172	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
173	C₂×C₂²⋊C₈	—	158	1	no	—
174	C₂×#97	—	158	1	no	—
175	#384	—	79	2	no	—
176	#382	—	158	1	no	—
177	#380	—	79	2	no	—
178	#380	—	158	1	no	—
179	C₄×C₄⋊C₄	—	79	2	no	—
180	C₄×C₄⋊C₄	—	158	1	no	—
181	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
182	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
183	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	4	4	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	158	2	no	—
188	#117	—	316	4	no	—
189	C₂×C₄⋊C₄	—	79	4	no	—
190	C₂×C₄⋊C₄	—	158	2	no	—
191	C₂×C₄⋊C₄	—	158	2	no	—
192	C₂×C₂²⋊C₄	—	158	2	no	—
193	C₂×C₄²	—	158	2	no	—
194	C₂×C₄²	Dic₁₅₈	1	2	yes	—
195	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₇₉	1	2	yes	—
197	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
199	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
200	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	79	4	no	—
205	C₄⋊C₈	—	158	2	no	—
206	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
207	C₂²⋊C₈	—	158	4	no	—
208	#97	—	158	4	no	—
209	#95	—	158	2	no	—
210	#95	—	158	2	no	—
211	C₂⁴	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	158	4	no	—
218	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
219	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	2	yes	—
223	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
224	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
225	C₂²×C₄	Dic₇₉⋊C₄	1	2	yes	—
226	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
230	M₁₆	—	4	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	4	no	—
233	C₂×C₈	—	316	2	no	—
234	C₂×C₈	—	316	2	no	—
235	C₂×C₈	—	158	4	no	—
236	C₄⋊C₄	—	316	4	no	—
237	C₄⋊C₄	—	316	2	no	—
238	C₄⋊C₄	—	316	4	no	—
239	C₄⋊C₄	—	316	2	no	—
240	C₂²⋊C₄	—	316	4	no	—
241	C₄²	—	316	4	no	—
242	C₄²	C₄×Dic₇₉	1	2	yes	—
243	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
244	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
245	C₄²	Dic₇₉⋊C₄	1	2	yes	—
246	C₄²	Dic₇₉⋊C₄	1	2	yes	—
247	C₄²	C₄⋊Dic₇₉	1	2	yes	—
248	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
249	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
251	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	C₇₉⋊#95	1	2	yes	—
257	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	316	4	no	—
271	C₂×C₄	—	316	4	no	—
272	C₂×C₄	—	316	4	no	—
273	C₂×C₄	C₇₉⋊#95	1	2	yes	—
274	C₂×C₄	C₇₉⋊#95	1	2	yes	—
275	C₂×C₄	C₇₉⋊#95	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
277	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
279	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
282	C₂×C₄	#14579187	1	2	yes	—
283	C₂×C₄	#14579189	1	2	yes	—
284	C₂×C₄	#14579192	1	2	yes	—
285	C₂×C₄	#14579194	1	2	yes	—
286	C₈	—	4	4	no	—
287	C₈	—	316	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×C₇₉⋊#95	1	1	yes	characteristic
294	C₂²	#19098545	1	2	yes	—
295	C₂²	#19098562	1	2	yes	—
296	C₂²	#19098569	1	1	yes	characteristic
297	C₂²	#19098573	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	632	4	no	—
304	C₄	#19098565	1	2	yes	—
305	C₄	#19098568	1	1	yes	characteristic
306	C₄	#19098572	1	1	yes	characteristic
307	C₄	#19098603	1	2	yes	—
308	C₄	#19098605	1	2	yes	—
309	C₂	—	4	2	no	—
310	C₂	#23762120	1	2	yes	—
311	C₂	#23762128	1	2	yes	—
312	C₂	#23763119	1	1	yes	characteristic
313	C₂	#23763120	1	1	yes	characteristic
314	C₂	#23763138	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#423×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#423	D₇₉
hypocenter	C₇₉	#8042
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #23761284	C₂	semidirect
7. #19098620	C₄	semidirect
20. #423×C₇₉	C₄	semidirect
162. C₇₉	#8042	semidirect
169. #423	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098620	2
#23765807	2
#23761284	2
C₂×#423×C₇₉	2
#8042	79