Subgroups — #34178754

Order 20224.

618 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11929607	C₂	1	1	yes	maximal, characteristic
2	C₄×(C₂×D₇₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×(C₂×D₇₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098285	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	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11929607	C₄	1	2	yes	—
9	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₇₉)⋊C₈	C₄	1	2	yes	—
11	C₂×(C₂×D₇₉)⋊C₈	C₄	1	2	yes	—
12	C₄×#11929607	—	2	2	no	—
13	C₄×#11929607	—	2	2	no	—
14	C₂×#14579071	C₄	1	2	yes	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
18	#19098317	C₄	1	2	yes	—
19	#19098317	C₄	1	2	yes	—
20	#19098285	—	2	2	no	—
21	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#322×C₇₉	—	2	2	no	—
25	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂×#11929607	—	2	4	no	—
29	C₂×#11929607	—	2	4	no	—
30	C₂×#11929607	—	2	4	no	—
31	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₇₉	—	2	1	no	—
38	C₄²×D₇₉	—	2	1	no	—
39	C₄²×D₇₉	—	2	1	no	—
40	C₄²×D₇₉	—	2	1	no	—
41	(C₂×D₇₉)⋊C₈	—	2	4	no	—
42	(C₂×D₇₉)⋊C₈	—	2	4	no	—
43	#14579071	—	2	4	no	—
44	C₄×C₇₉⋊C₈	—	2	1	no	—
45	C₄×C₇₉⋊C₈	—	2	1	no	—
46	C₄×C₇₉⋊C₈	—	2	1	no	—
47	C₄×C₇₉⋊C₈	—	2	1	no	—
48	C₄×C₇₉⋊C₈	—	2	1	no	—
49	C₄×C₇₉⋊C₈	—	2	1	no	—
50	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
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	C₄×C₈×C₇₉	D₄	1	2	yes	—
59	C₄×C₈×C₇₉	D₄	1	2	yes	—
60	C₂³×D₇₉	C₄²	1	1	yes	characteristic
61	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
62	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
63	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
64	C₂×C₄×D₇₉	—	2	1	no	—
65	C₂×C₄×D₇₉	—	2	1	no	—
66	C₂×C₄×D₇₉	—	2	1	no	—
67	C₂×C₄×D₇₉	—	2	1	no	—
68	C₂×C₄×D₇₉	—	2	1	no	—
69	C₂×C₄×D₇₉	—	2	1	no	—
70	C₂×C₄×D₇₉	—	2	2	no	—
71	C₂×C₄×D₇₉	—	2	2	no	—
72	C₂×C₄×D₇₉	—	2	2	no	—
73	C₂×C₄×D₇₉	—	2	2	no	—
74	C₂×C₄×D₇₉	—	2	2	no	—
75	C₂×C₄×D₇₉	M₁₆	1	4	yes	—
76	#11929607	—	2	16	no	—
77	C₂×C₇₉⋊C₈	—	2	2	no	—
78	C₂×C₇₉⋊C₈	—	2	2	no	—
79	C₂×C₇₉⋊C₈	—	2	2	no	—
80	C₂×C₇₉⋊C₈	—	2	2	no	—
81	C₂×C₇₉⋊C₈	—	2	2	no	—
82	C₂×C₇₉⋊C₈	—	2	2	no	—
83	C₂×C₇₉⋊C₈	—	2	2	no	—
84	C₂×C₇₉⋊C₈	M₁₆	1	4	yes	—
85	C₂×C₇₉⋊C₈	M₁₆	1	4	yes	complemented
86	C₄×Dic₇₉	—	2	1	no	—
87	C₄×Dic₇₉	—	2	1	no	—
88	C₄×Dic₇₉	—	2	1	no	—
89	C₄×Dic₇₉	—	2	1	no	—
90	C₄×Dic₇₉	—	2	1	no	—
91	C₄×Dic₇₉	—	2	1	no	—
92	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	—	2	2	no	—
97	C₂×C₈×C₇₉	—	2	2	no	—
98	C₂×C₈×C₇₉	—	2	2	no	—
99	C₂×C₈×C₇₉	—	2	2	no	—
100	C₂×C₈×C₇₉	—	2	2	no	—
101	C₂×C₈×C₇₉	—	2	2	no	—
102	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
103	C₂×C₈×C₇₉	M₁₆	1	4	yes	complemented
104	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇₉	#41	1	1	yes	characteristic
107	C₄²×C₇₉	#41	1	1	yes	characteristic
108	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
109	C₂²×D₇₉	—	2	1	no	—
110	C₂²×D₇₉	—	2	1	no	—
111	C₂²×D₇₉	—	2	1	no	—
112	C₂²×D₇₉	—	2	1	no	—
113	C₂²×D₇₉	—	2	1	no	—
114	C₂²×D₇₉	—	2	1	no	—
115	C₂²×D₇₉	#97	1	2	yes	—
116	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₇₉	#131	1	2	yes	—
127	C₂×C₄×C₇₉	#131	1	2	yes	—
128	C₂×Dic₇₉	—	2	1	no	—
129	C₂×Dic₇₉	—	2	1	no	—
130	C₂×Dic₇₉	—	2	1	no	—
131	C₂×Dic₇₉	—	2	1	no	—
132	C₂×Dic₇₉	—	2	1	no	—
133	C₂×Dic₇₉	—	2	1	no	—
134	C₂×Dic₇₉	—	2	1	no	—
135	C₂×Dic₇₉	—	2	1	no	—
136	C₂×Dic₇₉	#97	1	2	yes	—
137	C₄×D₇₉	—	2	4	no	—
138	C₄×D₇₉	—	2	4	no	—
139	C₄×D₇₉	—	4	2	no	—
140	C₄×D₇₉	—	4	2	no	—
141	C₄×D₇₉	—	4	2	no	—
142	C₄×D₇₉	—	4	2	no	—
143	C₈×C₇₉	—	2	4	no	—
144	C₈×C₇₉	—	2	4	no	—
145	C₈×C₇₉	—	4	2	no	—
146	C₇₉⋊C₈	—	2	4	no	—
147	C₇₉⋊C₈	—	2	4	no	—
148	C₇₉⋊C₈	—	4	2	no	—
149	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
151	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
152	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
153	C₂²×C₇₉	C₄×M₁₆	1	1	yes	characteristic
154	C₂²×C₇₉	#405	1	1	yes	characteristic
155	C₂²×C₇₉	#405	1	1	yes	characteristic
156	C₂×D₇₉	—	2	2	no	—
157	C₂×D₇₉	—	2	2	no	—
158	C₂×D₇₉	—	2	2	no	—
159	C₂×D₇₉	—	4	1	no	—
160	C₂×D₇₉	—	4	1	no	—
161	C₂×D₇₉	—	4	1	no	—
162	C₂×D₇₉	—	4	1	no	—
163	C₄×C₇₉	#407	1	2	yes	—
164	C₄×C₇₉	#432	1	2	yes	—
165	C₄×C₇₉	#435	1	2	yes	—
166	C₄×C₇₉	#435	1	2	yes	—
167	C₄×C₇₉	#436	1	2	yes	—
168	C₄×C₇₉	#436	1	2	yes	—
169	Dic₇₉	—	2	2	no	—
170	Dic₇₉	—	4	1	no	—
171	Dic₇₉	—	4	1	no	—
172	#8147	—	79	1	no	maximal, Sylow
173	C₂×C₇₉	#1451	1	1	yes	characteristic
174	C₂×C₇₉	#1451	1	1	yes	characteristic
175	C₂×C₇₉	#1453	1	1	yes	characteristic
176	C₂×C₇₉	#1455	1	1	yes	characteristic
177	C₂×C₇₉	#1749	1	1	yes	characteristic
178	C₂×C₇₉	#1753	1	1	yes	characteristic
179	C₂×C₇₉	#1753	1	1	yes	characteristic
180	D₇₉	—	4	2	no	—
181	C₂×C₄×M₁₆	—	79	1	no	—
182	C₄×C₂²⋊C₈	—	79	1	no	—
183	C₄×C₂²⋊C₈	—	79	1	no	—
184	C₂×#322	—	79	1	no	—
185	C₂×#322	D₇₉	1	1	yes	characteristic, complemented
186	#1273	—	79	1	no	—
187	#1273	—	79	1	no	—
188	C₇₉	#8147	1	1	yes	characteristic, complemented, Sylow
189	C₂²×M₁₆	—	79	2	no	—
190	C₂²×C₄²	—	79	1	no	—
191	C₂×C₂²⋊C₈	—	79	2	no	—
192	C₂×C₂²⋊C₈	—	79	2	no	—
193	C₄×M₁₆	—	158	2	no	—
194	C₄×M₁₆	—	158	2	no	—
195	C₂×#97	—	79	2	no	—
196	C₂×C₄×C₈	—	79	1	no	—
197	C₂×C₄×C₈	—	79	1	no	—
198	C₂×C₄×C₈	—	79	1	no	—
199	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
202	#336	—	79	2	no	—
203	#336	—	79	2	no	—
204	#322	—	2	2	no	—
205	#322	—	158	2	no	—
206	C₂³×C₄	—	79	1	no	—
207	C₂³×C₄	—	79	1	no	—
208	C₂³×C₄	—	79	1	no	—
209	C₂×M₁₆	—	158	4	no	—
210	C₂×M₁₆	—	158	4	no	—
211	C₂×M₁₆	—	158	4	no	—
212	C₂²×C₈	—	79	2	no	—
213	C₂²×C₈	—	79	2	no	—
214	C₂²×C₈	—	79	2	no	—
215	C₂²×C₈	C₄×D₇₉	1	2	yes	—
216	C₂²×C₈	C₄×D₇₉	1	2	yes	—
217	C₂²×C₈	C₄×D₇₉	1	2	yes	—
218	C₂×C₄²	—	79	1	no	—
219	C₂×C₄²	—	79	1	no	—
220	C₂×C₄²	—	79	1	no	—
221	C₂×C₄²	—	158	1	no	—
222	C₂×C₄²	—	158	1	no	—
223	C₂×C₄²	—	158	1	no	—
224	C₂×C₄²	—	158	1	no	—
225	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
226	C₂²⋊C₈	—	158	4	no	—
227	C₂²⋊C₈	—	158	4	no	—
228	#97	—	158	4	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	158	1	no	—
234	C₄×C₈	—	158	1	no	—
235	C₄×C₈	—	158	1	no	—
236	C₄×C₈	—	158	1	no	—
237	C₄×C₈	—	158	1	no	—
238	C₄×C₈	—	158	1	no	—
239	C₄×C₈	D₃₁₆	1	2	yes	—
240	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
241	C₂⁴	—	79	1	no	—
242	C₂²×C₄	—	79	1	no	—
243	C₂²×C₄	—	79	4	no	—
244	C₂²×C₄	—	79	1	no	—
245	C₂²×C₄	—	79	1	no	—
246	C₂²×C₄	—	158	1	no	—
247	C₂²×C₄	—	158	1	no	—
248	C₂²×C₄	—	158	1	no	—
249	C₂²×C₄	—	158	1	no	—
250	C₂²×C₄	—	158	2	no	—
251	C₂²×C₄	—	158	2	no	—
252	C₂²×C₄	—	158	2	no	—
253	C₂²×C₄	—	158	2	no	—
254	C₂²×C₄	—	158	1	no	—
255	C₂²×C₄	—	158	1	no	—
256	C₂²×C₄	—	158	2	no	—
257	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
260	M₁₆	—	158	16	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	79	4	no	—
269	C₂×C₈	—	79	4	no	—
270	C₂×C₈	—	158	2	no	—
271	C₂×C₈	—	158	2	no	—
272	C₂×C₈	—	158	2	no	—
273	C₂×C₈	—	158	2	no	—
274	C₂×C₈	—	158	2	no	—
275	C₂×C₈	—	158	2	no	—
276	C₂×C₈	—	158	2	no	—
277	C₂×C₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
278	C₂×C₈	#11929607	1	4	yes	complemented
279	C₄²	—	158	1	no	—
280	C₄²	—	158	1	no	—
281	C₄²	—	158	1	no	—
282	C₄²	—	158	1	no	—
283	C₄²	—	158	1	no	—
284	C₄²	—	158	1	no	—
285	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
286	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
287	C₄²	#11929624	1	1	yes	characteristic
288	C₄²	#11929624	1	1	yes	characteristic
289	C₂³	—	79	2	no	—
290	C₂³	—	158	1	no	—
291	C₂³	—	158	1	no	—
292	C₂³	—	158	1	no	—
293	C₂³	—	158	1	no	—
294	C₂³	—	158	1	no	—
295	C₂³	—	158	1	no	—
296	C₂³	C₄²×D₇₉	1	1	yes	characteristic
297	C₂×C₄	—	316	2	no	—
298	C₂×C₄	—	316	2	no	—
299	C₂×C₄	—	316	2	no	—
300	C₂×C₄	—	316	2	no	—
301	C₂×C₄	—	79	2	no	—
302	C₂×C₄	—	158	1	no	—
303	C₂×C₄	—	158	1	no	—
304	C₂×C₄	—	158	1	no	—
305	C₂×C₄	—	158	1	no	—
306	C₂×C₄	—	158	1	no	—
307	C₂×C₄	—	158	1	no	—
308	C₂×C₄	—	158	1	no	—
309	C₂×C₄	—	158	1	no	—
310	C₂×C₄	—	158	4	no	—
311	C₂×C₄	—	158	4	no	—
312	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
313	C₂×C₄	#14579118	1	1	yes	characteristic
314	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
315	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
316	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
317	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
318	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
319	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
320	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
321	C₂×C₄	C₂×#11929607	1	2	yes	—
322	C₂×C₄	#14579167	1	2	yes	—
323	C₂×C₄	#14579169	1	2	yes	—
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₈	—	4	2	no	—
327	C₈	—	316	2	no	—
328	C₈	—	158	4	no	—
329	C₈	—	158	4	no	—
330	C₂²	—	316	1	no	—
331	C₂²	—	316	1	no	—
332	C₂²	—	316	1	no	—
333	C₂²	—	316	1	no	—
334	C₂²	—	158	2	no	—
335	C₂²	—	158	2	no	—
336	C₂²	—	158	2	no	—
337	C₂²	C₄×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
338	C₂²	#97×D₇₉	1	1	yes	characteristic
339	C₂²	C₄×#11929607	1	1	yes	characteristic
340	C₂²	#19098515	1	1	yes	characteristic
341	C₂²	#19098515	1	1	yes	characteristic
342	C₂²	#19098517	1	1	yes	characteristic
343	C₂²	#19098520	1	1	yes	characteristic
344	C₄	—	316	1	no	—
345	C₄	—	316	1	no	—
346	C₄	—	158	2	no	—
347	C₄	#19098526	1	2	yes	—
348	C₄	#19098618	1	2	yes	—
349	C₄	#19098634	1	2	yes	—
350	C₄	#19098636	1	2	yes	—
351	C₄	#19098639	1	2	yes	—
352	C₄	#19098642	1	2	yes	—
353	C₂	—	316	2	no	—
354	C₂	#23761686	1	1	yes	characteristic
355	C₂	#23761688	1	1	yes	characteristic
356	C₂	#23761696	1	1	yes	characteristic
357	C₂	#23761704	1	1	yes	characteristic
358	C₂	#23763163	1	1	yes	characteristic
359	C₂	#23763183	1	1	yes	characteristic
360	C₂	#23763185	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₉
commutator, socle	C₂³×C₇₉	C₂×C₄²
Fitting	C₂×#322×C₇₉	C₂
prime radical	C₂³×D₇₉	C₄²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#8147
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. C₂×#19098285	C₂	semidirect
5. C₇₉⋊#1273	C₂	semidirect
6. C₇₉⋊#1273	C₂	semidirect
7. C₂×#322×C₇₉	C₂	semidirect
85. C₂×C₇₉⋊C₈	M₁₆	semidirect
103. C₂×C₈×C₇₉	M₁₆	semidirect
185. C₂×#322	D₇₉	semidirect
188. C₇₉	#8147	semidirect
278. C₂×C₈	#11929607	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11929607	2
C₄×(C₂×D₇₉)⋊C₈	2
C₂×#19098285	2
C₇₉⋊#1273	2
C₂×#322×C₇₉	2
#8147	79