Subgroups — C₄×#19098962 #34297007

Order 20224.

774 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×#19098962	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579153	C₂	1	1	yes	maximal, characteristic
3	#23764366	C₂	1	1	yes	maximal, characteristic
4	#23764364	C₂	1	1	yes	maximal, characteristic
5	#23764130	C₂	1	2	yes	maximal
6	#23764125	C₂	1	2	yes	maximal
7	#23763562	C₂	1	1	yes	maximal, characteristic
8	C₄×#14579094	C₂	1	2	yes	maximal
9	C₄×#14579092	C₂	1	2	yes	maximal
10	C₄²×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
11	C₄×#125×C₇₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#14579153	C₂²	1	1	yes	characteristic
13	#19098962	C₄	1	8	yes	direct product factor
14	C₂×#14579094	C₂²	1	2	yes	—
15	C₂×#14579094	C₂²	1	2	yes	—
16	C₂×#14579092	C₂²	1	2	yes	—
17	C₂×#14579092	C₂²	1	2	yes	—
18	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
24	#19098461	C₂²	1	2	yes	—
25	#19098447	—	2	2	no	—
26	#19098429	C₂²	1	1	yes	characteristic
27	C₄×Dic₇₉⋊C₄	—	2	2	no	—
28	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
29	#19098325	C₂²	1	2	yes	—
30	#19098325	C₂²	1	2	yes	—
31	#19098325	C₂²	1	2	yes	—
32	#19098325	C₂²	1	2	yes	—
33	C₂×#125×C₇₉	C₂²	1	1	yes	characteristic
34	#389×C₇₉	C₂²	1	2	yes	—
35	#387×C₇₉	—	2	2	no	—
36	#382×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₇₉	—	2	2	no	—
38	C₄×C₄⋊C₄×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	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
44	#14579153	—	2	2	no	—
45	#14579153	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
48	C₂×Dic₇₉⋊C₄	—	2	2	no	—
49	C₂×Dic₇₉⋊C₄	—	2	2	no	—
50	C₂×C₄×Dic₇₉	—	2	2	no	—
51	#14579094	—	2	4	no	—
52	#14579094	C₂×C₄	1	8	yes	—
53	#14579092	—	2	4	no	—
54	#14579092	C₂×C₄	1	8	yes	—
55	C₄×C₇₉⋊C₈	—	2	2	no	—
56	C₄×C₇₉⋊C₈	—	2	2	no	—
57	C₄×C₇₉⋊C₈	—	2	2	no	—
58	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
59	C₄×C₇₉⋊C₈	Q₈	1	4	yes	—
60	C₄×C₇₉⋊C₈	Q₈	1	4	yes	—
61	C₇₉⋊#95	—	2	2	no	—
62	C₇₉⋊#95	—	2	2	no	—
63	C₇₉⋊#95	—	2	2	no	—
64	#125×C₇₉	—	2	2	no	—
65	#125×C₇₉	C₂×C₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
67	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
68	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
70	C₂×C₄²×C₇₉	—	2	2	no	—
71	C₂×C₄²×C₇₉	D₄	1	2	yes	—
72	C₂×C₄²×C₇₉	D₄	1	2	yes	—
73	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
76	#95×C₇₉	—	2	2	no	—
77	#95×C₇₉	—	2	2	no	—
78	#95×C₇₉	—	2	2	no	—
79	C₂²×Dic₇₉	—	2	2	no	—
80	C₂²×Dic₇₉	—	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	1	no	—
85	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
88	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
90	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
91	C₂×C₇₉⋊C₈	#41	1	2	yes	—
92	C₂×C₇₉⋊C₈	#41	1	2	yes	—
93	C₄⋊Dic₇₉	—	2	2	no	—
94	C₄⋊Dic₇₉	—	2	2	no	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
97	Dic₇₉⋊C₄	—	2	8	no	—
98	Dic₇₉⋊C₄	—	4	4	no	—
99	C₄×Dic₇₉	—	4	4	no	—
100	C₂²×C₄×C₇₉	—	2	2	no	—
101	C₂²×C₄×C₇₉	—	2	2	no	—
102	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₇₉	—	2	8	no	—
108	C₄⋊C₄×C₇₉	—	2	2	no	—
109	C₄⋊C₄×C₇₉	—	2	2	no	—
110	C₄⋊C₄×C₇₉	—	2	2	no	—
111	C₄⋊C₄×C₇₉	—	4	4	no	—
112	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
113	C₄²×C₇₉	—	2	4	no	—
114	C₄²×C₇₉	—	2	4	no	—
115	C₄²×C₇₉	—	2	1	no	—
116	C₄²×C₇₉	—	2	1	no	—
117	C₄²×C₇₉	—	4	4	no	—
118	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
119	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
120	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
121	C₄²×C₇₉	#41	1	2	yes	—
122	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	—	2	2	no	—
125	C₂×C₄×C₇₉	—	2	2	no	—
126	C₂×C₄×C₇₉	—	2	2	no	—
127	C₂×C₄×C₇₉	—	2	2	no	—
128	C₂×C₄×C₇₉	—	2	4	no	—
129	C₂×C₄×C₇₉	—	4	2	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	—	4	2	no	—
132	C₂×C₄×C₇₉	—	4	2	no	—
133	C₂×C₄×C₇₉	—	4	2	no	—
134	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
135	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₇₉	#135	1	4	yes	—
138	C₂×C₄×C₇₉	#135	1	4	yes	—
139	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₇₉	—	2	4	no	—
146	C₂×Dic₇₉	—	4	2	no	—
147	C₂×Dic₇₉	—	4	2	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	—	4	2	no	—
150	C₂×Dic₇₉	—	4	2	no	—
151	C₇₉⋊C₈	—	2	2	no	—
152	C₇₉⋊C₈	—	2	2	no	—
153	C₇₉⋊C₈	C₄×Q₈	1	4	yes	—
154	C₇₉⋊C₈	C₄×Q₈	1	4	yes	—
155	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₂²×C₇₉	#532	1	1	yes	characteristic
158	C₂²×C₇₉	C₂×#135	1	2	yes	—
159	C₂²×C₇₉	C₂×#135	1	2	yes	—
160	C₄×C₇₉	—	2	2	no	—
161	C₄×C₇₉	—	2	4	no	—
162	C₄×C₇₉	—	2	4	no	—
163	C₄×C₇₉	—	4	4	no	—
164	C₄×C₇₉	—	8	2	no	—
165	C₄×C₇₉	#499	1	4	yes	complemented
166	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₇₉	#533	1	1	yes	characteristic
169	C₄×C₇₉	#533	1	1	yes	characteristic
170	Dic₇₉	—	4	4	no	—
171	Dic₇₉	—	8	2	no	—
172	C₄×#499	—	79	1	no	maximal, Sylow
173	C₂×C₇₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₇₉	C₄×#135	1	2	yes	—
175	C₂×C₇₉	C₂×#499	1	1	yes	characteristic
176	C₂×C₇₉	#3160	1	1	yes	characteristic
177	C₂×C₇₉	#3168	1	2	yes	—
178	C₂×#499	—	79	1	no	—
179	C₄×#125	—	79	1	no	—
180	C₄×#125	D₇₉	1	1	yes	characteristic
181	#1991	—	79	1	no	—
182	#1991	—	79	1	no	—
183	#1942	—	79	2	no	—
184	#1941	—	79	2	no	—
185	#1846	—	79	1	no	—
186	C₄×#107	—	79	2	no	—
187	C₄×#106	—	79	2	no	—
188	C₄²×C₈	—	79	1	no	—
189	C₇₉	C₄×#499	1	1	yes	characteristic, complemented, Sylow
190	C₂×#125	—	79	1	no	—
191	C₂×#125	C₂×D₇₉	1	1	yes	characteristic
192	#499	—	79	8	no	—
193	C₂×#107	—	79	2	no	—
194	C₂×#107	—	79	2	no	—
195	C₂×#106	—	79	2	no	—
196	C₂×#106	—	79	2	no	—
197	C₂×C₄×C₈	—	79	1	no	—
198	C₂×C₄×C₈	—	79	1	no	—
199	C₂×C₄×C₈	—	79	1	no	—
200	C₂×C₄×C₈	—	79	1	no	—
201	C₂×C₄×C₈	—	79	1	no	—
202	C₂×C₄×C₈	—	79	1	no	—
203	#389	—	79	2	no	—
204	#389	C₂×D₇₉	1	2	yes	—
205	#387	—	2	2	no	—
206	#387	—	158	2	no	—
207	#382	—	79	1	no	—
208	#382	C₂×D₇₉	1	1	yes	characteristic
209	C₄×C₄⋊C₄	—	2	2	no	—
210	C₄×C₄⋊C₄	—	79	2	no	—
211	C₄×C₄⋊C₄	—	158	2	no	—
212	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
213	C₄³	C₂×D₇₉	1	1	yes	characteristic
214	#340	—	79	2	no	—
215	#340	—	79	2	no	—
216	#340	—	79	2	no	—
217	#340	—	79	2	no	—
218	C₂²×C₈	—	79	1	no	—
219	C₂²×C₈	—	79	1	no	—
220	C₂²×C₈	—	79	1	no	—
221	C₂²×C₈	—	79	1	no	—
222	#125	—	2	2	no	—
223	#125	—	79	4	no	—
224	#125	—	158	2	no	—
225	#125	C₄×D₇₉	1	4	yes	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	79	2	no	—
229	C₂×C₄⋊C₄	—	79	2	no	—
230	C₂×C₄⋊C₄	—	158	2	no	—
231	C₂×C₄⋊C₄	—	158	2	no	—
232	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
233	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	—	158	2	no	—
236	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
237	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
238	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
241	#107	—	79	8	no	—
242	#107	—	158	4	no	—
243	#106	—	79	8	no	—
244	#106	—	158	4	no	—
245	C₄×C₈	—	79	4	no	—
246	C₄×C₈	—	79	4	no	—
247	C₄×C₈	—	79	4	no	—
248	C₄×C₈	—	158	2	no	—
249	C₄×C₈	—	158	2	no	—
250	C₄×C₈	—	158	2	no	—
251	#95	—	2	2	no	—
252	#95	—	2	2	no	—
253	#95	—	2	2	no	—
254	#95	—	158	2	no	—
255	#95	—	158	2	no	—
256	#95	—	158	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	158	2	no	—
260	C₂²×C₄	—	158	2	no	—
261	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
262	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
266	C₂×C₈	—	79	2	no	—
267	C₂×C₈	—	79	2	no	—
268	C₂×C₈	—	79	2	no	—
269	C₂×C₈	—	79	2	no	—
270	C₂×C₈	—	79	2	no	—
271	C₂×C₈	—	79	2	no	—
272	C₂×C₈	—	79	2	no	—
273	C₂×C₈	—	79	2	no	—
274	C₂×C₈	—	158	1	no	—
275	C₂×C₈	—	158	1	no	—
276	C₂×C₈	—	158	1	no	—
277	C₂×C₈	—	158	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	8	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	4	no	—
283	C₄⋊C₄	—	316	4	no	—
284	C₄⋊C₄	—	79	4	no	—
285	C₄⋊C₄	—	158	2	no	—
286	C₄⋊C₄	—	158	2	no	—
287	C₄⋊C₄	—	158	2	no	—
288	C₄⋊C₄	—	158	8	no	—
289	C₄⋊C₄	C₂×C₄×D₇₉	1	4	yes	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	4	no	—
293	C₄²	—	2	4	no	—
294	C₄²	—	4	4	no	—
295	C₄²	—	316	4	no	—
296	C₄²	C₂×C₄×D₇₉	1	2	yes	—
297	C₄²	Q₈×D₇₉	1	2	yes	—
298	C₄²	Q₈×D₇₉	1	2	yes	—
299	C₄²	#11929624	1	2	yes	—
300	C₂³	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	316	2	no	—
313	C₂×C₄	—	316	2	no	—
314	C₂×C₄	—	316	2	no	—
315	C₂×C₄	—	316	2	no	—
316	C₂×C₄	—	316	2	no	—
317	C₂×C₄	—	158	4	no	—
318	C₂×C₄	C₄×C₂²⋊D₇₉	1	4	yes	—
319	C₂×C₄	#14579159	1	2	yes	—
320	C₂×C₄	#14579159	1	2	yes	—
321	C₂×C₄	#14579181	1	4	yes	—
322	C₂×C₄	#14579181	1	4	yes	—
323	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
325	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
326	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
327	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
328	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
329	C₈	—	79	4	no	—
330	C₈	—	79	4	no	—
331	C₈	—	158	2	no	—
332	C₈	—	158	2	no	—
333	C₂²	C₂×C₄×C₂²⋊D₇₉	1	1	yes	characteristic
334	C₂²	C₂×#14579159	1	1	yes	characteristic
335	C₂²	#19099109	1	1	yes	characteristic
336	C₂²	C₂×#14579181	1	2	yes	—
337	C₂²	C₂×#14579181	1	2	yes	—
338	C₄	—	2	4	no	—
339	C₄	—	2	4	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	4	no	—
342	C₄	—	8	2	no	—
343	C₄	—	316	4	no	—
344	C₄	—	632	2	no	—
345	C₄	#19098962	1	4	yes	direct product factor
346	C₄	C₄×Q₈×D₇₉	1	1	yes	characteristic
347	C₄	C₄×Q₈×D₇₉	1	1	yes	characteristic
348	C₄	#19099117	1	1	yes	characteristic
349	C₄	#19099117	1	1	yes	characteristic
350	C₂	C₄×#14579159	1	1	yes	characteristic
351	C₂	C₄×#14579181	1	2	yes	—
352	C₂	C₂×#19098962	1	1	yes	characteristic
353	C₂	#23773910	1	1	yes	characteristic
354	C₂	#23773964	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₄×#125×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄×#125	D₇₉
hypocenter	C₇₉	C₄×#499
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
13. #19098962	C₄	direct (N × H)
165. C₄×C₇₉	#499	semidirect
189. C₇₉	C₄×#499	semidirect
345. C₄	#19098962	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098962	2
C₄×#14579153	2
#23764366	2
#23764364	2
#23764130	2
#23764125	2
#23763562	2
C₄×#14579094	2
C₄×#14579092	2
C₄²×C₇₉⋊C₈	2
C₄×#125×C₇₉	2
C₄×#499	79

Subgroups — C4×#19098962 #34297007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098962 #34297007