Subgroups — #34178762

Order 20224.

578 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098616	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19098616	—	2	2	no	—
9	#19098616	—	2	2	no	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
11	C₂×C₂²⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
12	C₂×C₂²⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
13	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
17	#19098318	C₄	1	2	yes	—
18	#19098285	—	2	2	no	—
19	C₈×C₇₉⋊C₈	—	2	2	no	—
20	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₇₉	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#336×C₇₉	C₄	1	2	yes	—
25	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
30	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
31	C₂²⋊(C₇₉⋊C₈)	C₈	1	8	yes	complemented
32	C₄×C₇₉⋊C₈	—	2	1	no	—
33	C₄×C₇₉⋊C₈	—	2	1	no	—
34	C₄×C₇₉⋊C₈	—	2	1	no	—
35	C₄×C₇₉⋊C₈	—	2	1	no	—
36	C₄×C₇₉⋊C₈	—	2	1	no	—
37	C₄×C₇₉⋊C₈	—	2	1	no	—
38	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
39	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
40	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₇₉	—	2	1	no	—
46	C₂×C₄²×C₇₉	—	2	1	no	—
47	C₂×C₄²×C₇₉	—	2	1	no	—
48	C₂×C₄²×C₇₉	—	2	1	no	—
49	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
53	C₂²⋊C₈×C₇₉	—	2	4	no	—
54	C₄×C₈×C₇₉	—	2	1	no	—
55	C₄×C₈×C₇₉	—	2	1	no	—
56	C₄×C₈×C₇₉	—	2	1	no	—
57	C₄×C₈×C₇₉	—	2	1	no	—
58	C₂×C₇₉⋊C₈	—	2	2	no	—
59	C₂×C₇₉⋊C₈	—	2	2	no	—
60	C₂×C₇₉⋊C₈	—	2	2	no	—
61	C₂×C₇₉⋊C₈	—	2	2	no	—
62	C₂×C₇₉⋊C₈	—	2	2	no	—
63	C₂×C₇₉⋊C₈	—	2	2	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	2	2	no	—
66	C₂×C₇₉⋊C₈	—	2	2	no	—
67	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	4	yes	—
68	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
69	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
70	C₂⁴×C₇₉	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	—	2	1	no	—
72	C₂²×C₄×C₇₉	—	2	1	no	—
73	C₂²×C₄×C₇₉	—	2	2	no	—
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	—	2	1	no	—
77	C₂²×C₄×C₇₉	—	2	1	no	—
78	C₂²×C₄×C₇₉	—	2	2	no	—
79	C₂²×C₄×C₇₉	—	2	2	no	—
80	C₂²×C₄×C₇₉	—	2	2	no	—
81	C₂²×C₄×C₇₉	—	2	2	no	—
82	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×C₈	1	4	yes	—
86	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₉	—	2	2	no	—
90	C₂×C₈×C₇₉	—	2	2	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
95	C₄²×C₇₉	—	2	1	no	—
96	C₄²×C₇₉	—	2	1	no	—
97	C₄²×C₇₉	—	2	1	no	—
98	C₄²×C₇₉	—	2	1	no	—
99	C₄²×C₇₉	—	2	1	no	—
100	C₄²×C₇₉	—	2	1	no	—
101	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₇₉	#41	1	1	yes	characteristic
104	C₄²×C₇₉	#41	1	1	yes	characteristic
105	C₂³×C₇₉	—	2	1	no	—
106	C₂³×C₇₉	—	2	1	no	—
107	C₂³×C₇₉	—	2	1	no	—
108	C₂³×C₇₉	—	2	1	no	—
109	C₂³×C₇₉	—	2	1	no	—
110	C₂³×C₇₉	—	2	1	no	—
111	C₂³×C₇₉	C₄×C₈	1	2	yes	—
112	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	4	no	—
116	C₂×C₄×C₇₉	—	2	4	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	4	2	no	—
124	C₂×C₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	—	4	2	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
128	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
136	C₂×C₄×C₇₉	#131	1	2	yes	—
137	C₂×C₄×C₇₉	#131	1	2	yes	—
138	C₂×C₄×C₇₉	#131	1	2	yes	—
139	C₂×C₄×C₇₉	#131	1	2	yes	—
140	C₈×C₇₉	—	2	4	no	—
141	C₈×C₇₉	—	4	2	no	—
142	C₇₉⋊C₈	—	2	4	no	—
143	C₇₉⋊C₈	—	2	4	no	—
144	C₇₉⋊C₈	—	2	4	no	—
145	C₇₉⋊C₈	—	4	2	no	—
146	C₂²×C₇₉	—	2	2	no	—
147	C₂²×C₇₉	—	2	2	no	—
148	C₂²×C₇₉	—	2	2	no	—
149	C₂²×C₇₉	—	4	1	no	—
150	C₂²×C₇₉	—	4	1	no	—
151	C₂²×C₇₉	—	4	1	no	—
152	C₂²×C₇₉	—	4	1	no	—
153	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
155	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
156	C₂²×C₇₉	#405	1	1	yes	characteristic
157	C₂²×C₇₉	#405	1	1	yes	characteristic
158	C₂²×C₇₉	#405	1	1	yes	characteristic
159	C₂²×C₇₉	#405	1	1	yes	characteristic
160	C₄×C₇₉	—	2	2	no	—
161	C₄×C₇₉	—	4	1	no	—
162	C₄×C₇₉	—	4	1	no	—
163	C₄×C₇₉	#408	1	2	yes	—
164	C₄×C₇₉	#433	1	2	yes	—
165	C₄×C₇₉	C₈×D₄	1	2	yes	—
166	C₄×C₇₉	C₈×D₄	1	2	yes	—
167	C₄×C₇₉	#435	1	2	yes	—
168	C₄×C₇₉	#436	1	2	yes	—
169	#8148	—	79	1	no	maximal, Sylow
170	C₂×C₇₉	—	4	2	no	—
171	C₂×C₇₉	C₈×M₁₆	1	1	yes	characteristic
172	C₂×C₇₉	#1450	1	1	yes	characteristic
173	C₂×C₇₉	#1453	1	1	yes	characteristic
174	C₂×C₇₉	#1454	1	1	yes	characteristic
175	C₂×C₇₉	#1750	1	1	yes	characteristic
176	C₂×C₇₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₇₉	#1753	1	1	yes	characteristic
178	C₂×#431	—	79	1	no	—
179	C₄×C₂²⋊C₈	—	79	1	no	—
180	C₄×C₂²⋊C₈	D₇₉	1	1	yes	characteristic
181	C₂×#322	—	79	1	no	—
182	C₂×C₈²	—	79	1	no	—
183	#1273	—	79	1	no	—
184	#1273	—	79	1	no	—
185	C₇₉	#8148	1	1	yes	characteristic, complemented, Sylow
186	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
187	#431	—	158	2	no	—
188	#431	—	158	2	no	—
189	C₂×C₄⋊C₈	—	79	2	no	—
190	C₂×C₂²⋊C₈	—	79	2	no	—
191	C₂×C₂²⋊C₈	—	79	2	no	—
192	C₂×C₂²⋊C₈	Dic₇₉	1	2	yes	—
193	C₂×C₄×C₈	—	79	1	no	—
194	C₂×C₄×C₈	—	79	1	no	—
195	C₂×C₄×C₈	—	79	1	no	—
196	C₂×C₄×C₈	—	79	1	no	—
197	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
199	#336	—	79	2	no	—
200	#336	Dic₇₉	1	2	yes	—
201	#322	—	158	2	no	—
202	C₈²	—	158	2	no	—
203	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
204	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
205	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
206	C₂²×C₈	—	79	2	no	—
207	C₂²×C₈	—	79	2	no	—
208	C₂²×C₈	—	79	2	no	—
209	C₂²×C₈	—	79	2	no	—
210	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
211	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
220	C₄⋊C₈	—	158	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₂²⋊C₈	—	79	8	no	—
223	C₂²⋊C₈	—	158	4	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	79	2	no	—
229	C₄×C₈	—	79	2	no	—
230	C₄×C₈	—	158	1	no	—
231	C₄×C₈	—	158	1	no	—
232	C₄×C₈	—	158	1	no	—
233	C₄×C₈	—	158	1	no	—
234	C₄×C₈	—	158	1	no	—
235	C₄×C₈	—	158	1	no	—
236	C₂⁴	C₄×Dic₇₉	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
249	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
250	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
251	C₂²×C₄	C₈×D₇₉	1	4	yes	—
252	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	79	4	no	—
261	C₂×C₈	—	79	4	no	—
262	C₂×C₈	—	79	4	no	—
263	C₂×C₈	—	158	2	no	—
264	C₂×C₈	—	158	2	no	—
265	C₂×C₈	—	158	2	no	—
266	C₂×C₈	—	158	2	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	—	158	2	no	—
272	C₂×C₈	#11929608	1	4	yes	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	D₄×D₇₉	1	1	yes	characteristic
280	C₄²	D₄×D₇₉	1	1	yes	characteristic
281	C₄²	#11929623	1	1	yes	characteristic
282	C₄²	#11929623	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	C₈×Dic₇₉	1	2	yes	—
290	C₂³	C₂×C₄×Dic₇₉	1	1	yes	characteristic
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₈×Dic₇₉	1	2	yes	—
306	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
307	C₂×C₄	#14579117	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
310	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
311	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
312	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
313	C₂×C₄	C₂×#11929608	1	2	yes	—
314	C₂×C₄	#14579167	1	2	yes	—
315	C₂×C₄	#14579168	1	2	yes	—
316	C₂×C₄	#14579169	1	2	yes	—
317	C₂×C₄	#14579169	1	2	yes	—
318	C₈	—	2	4	no	—
319	C₈	—	4	2	no	—
320	C₈	—	316	2	no	—
321	C₈	—	158	4	no	—
322	C₈	—	158	4	no	—
323	C₈	—	158	4	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	C₂²⋊C₄×Dic₇₉	1	1	yes	characteristic
332	C₂²	C₂×C₈×Dic₇₉	1	1	yes	characteristic
333	C₂²	C₄×#11929608	1	1	yes	characteristic
334	C₂²	#19098518	1	1	yes	characteristic
335	C₂²	#19098519	1	1	yes	characteristic
336	C₂²	#19098521	1	1	yes	characteristic
337	C₂²	#19098521	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	#19098528	1	2	yes	—
342	C₄	#19098621	1	2	yes	—
343	C₄	(C₂²⋊D₇₉)⋊C₈	1	2	yes	—
344	C₄	(C₂²⋊D₇₉)⋊C₈	1	2	yes	—
345	C₄	#19098635	1	2	yes	—
346	C₄	#19098641	1	2	yes	—
347	C₂	—	4	2	no	—
348	C₂	C₈×#11929608	1	1	yes	characteristic
349	C₂	#23761684	1	1	yes	characteristic
350	C₂	#23761697	1	1	yes	characteristic
351	C₂	#23761700	1	1	yes	characteristic
352	C₂	#23763168	1	1	yes	characteristic
353	C₂	#23763174	1	1	yes	characteristic
354	C₂	#23763184	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂×C₄×C₈
Fitting	C₄×C₂²⋊C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₄²
prime radical	C₂⁴×C₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×C₂²⋊C₈	D₇₉
hypocenter	C₇₉	#8148
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#19098285	C₂	semidirect
4. C₂×C₈×C₇₉⋊C₈	C₂	semidirect
5. C₇₉⋊#1273	C₂	semidirect
6. C₇₉⋊#1273	C₂	semidirect
31. C₂²⋊(C₇₉⋊C₈)	C₈	semidirect
185. C₇₉	#8148	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098616	2
C₄×C₂²⋊(C₇₉⋊C₈)	2
C₂×#19098285	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉⋊#1273	2
C₄×C₂²⋊C₈×C₇₉	2
#8148	79