Subgroups — #34257500

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772066	C₂	1	2	yes	maximal
2	#23762180	C₂	1	4	yes	maximal, complemented
3	#23762175	C₂	1	2	yes	maximal, complemented
4	#23762163	C₂	1	2	yes	maximal, complemented
5	#23762156	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762148	C₂	1	2	yes	maximal, complemented
7	C₂×#19098314	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#41×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#19098624	—	2	4	no	—
10	#19098620	C₂²	1	4	yes	—
11	#19098616	C₂²	1	4	yes	—
12	#19098587	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
14	#19098530	—	2	2	no	—
15	#19098519	—	2	2	no	—
16	#19098316	C₂²	1	4	yes	—
17	#19098316	C₂²	1	4	yes	—
18	#19098314	C₂²	1	2	yes	—
19	#19098314	C₂²	1	2	yes	—
20	#19098314	C₂²	1	1	yes	characteristic
21	#19098314	C₂²	1	1	yes	characteristic
22	#19098314	C₂²	1	2	yes	—
23	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₇₉	C₄	1	1	yes	characteristic
25	C₈×Q₈×C₇₉	C₄	1	1	yes	characteristic
26	C₈×D₄×C₇₉	C₄	1	1	yes	characteristic
27	C₈×D₄×C₇₉	C₄	1	2	yes	—
28	C₈×D₄×C₇₉	C₄	1	1	yes	characteristic
29	C₈×D₄×C₇₉	C₄	1	2	yes	—
30	#431×C₇₉	C₂²	1	2	yes	—
31	#431×C₇₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
34	C₂×#11929608	—	2	2	no	—
35	C₂²×C₇₉⋊C₈	—	2	2	no	—
36	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
39	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
40	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
41	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
42	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
43	#14579072	—	2	4	no	—
44	C₄×C₇₉⋊C₈	—	2	4	no	—
45	C₂×#41×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₈×C₇₉	Q₈	1	1	yes	characteristic
52	C₄×D₄×C₇₉	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄×C₇₉	Q₈	1	2	yes	—
55	C₄×D₄×C₇₉	Q₈	1	1	yes	characteristic
56	#117×C₇₉	D₄	1	2	yes	—
57	#117×C₇₉	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
60	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
61	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₇₉	C₂³	1	2	yes	—
73	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	#11929608	—	4	4	no	—
75	C₂×C₇₉⋊C₈	—	2	2	no	—
76	C₂×C₇₉⋊C₈	—	2	4	no	—
77	C₂×C₇₉⋊C₈	—	2	2	no	—
78	C₂×C₇₉⋊C₈	—	4	4	no	—
79	#41×C₇₉	—	2	4	no	—
80	C₂×Q₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
81	C₂×D₄×C₇₉	C₄⋊C₄	1	2	yes	—
82	C₂×D₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
88	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₇₉	—	2	2	no	—
90	C₂×C₈×C₇₉	—	2	4	no	—
91	C₂×C₈×C₇₉	—	4	1	no	—
92	C₂×C₈×C₇₉	—	4	2	no	—
93	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
100	C₄⋊C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
102	C₄⋊C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
103	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
105	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
106	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
107	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₇₉	C₂×D₄	1	2	yes	—
109	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
111	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
113	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
114	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	Q₈×C₇₉	—	2	2	no	—
116	D₄×C₇₉	—	2	4	no	—
117	D₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	2	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	—	2	4	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#135	1	4	yes	—
132	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
135	C₈×C₇₉	—	2	1	no	—
136	C₈×C₇₉	—	2	2	no	—
137	C₈×C₇₉	—	2	1	no	—
138	C₈×C₇₉	—	2	1	no	—
139	C₈×C₇₉	—	2	2	no	—
140	C₈×C₇₉	#131	1	2	yes	—
141	C₇₉⋊C₈	—	4	2	no	—
142	C₇₉⋊C₈	—	4	2	no	—
143	C₇₉⋊C₈	—	4	4	no	—
144	C₂²×C₇₉	—	2	1	no	—
145	C₂²×C₇₉	—	2	1	no	—
146	C₂²×C₇₉	—	2	2	no	—
147	C₂²×C₇₉	—	2	2	no	—
148	C₂²×C₇₉	—	2	2	no	—
149	C₂²×C₇₉	#427	1	2	yes	—
150	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₇₉	—	2	1	no	—
152	C₄×C₇₉	—	2	2	no	—
153	C₄×C₇₉	—	2	2	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	—	2	1	no	—
156	C₄×C₇₉	—	2	2	no	—
157	C₄×C₇₉	#427	1	2	yes	—
158	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
159	C₄×C₇₉	#568	1	1	yes	characteristic
160	C₄×C₇₉	C₂×#135	1	2	yes	—
161	#17827	—	79	1	no	maximal, Sylow
162	C₂×C₇₉	—	2	2	no	—
163	C₂×C₇₉	—	4	2	no	—
164	C₂×C₇₉	#2906	1	1	yes	characteristic
165	C₂×C₇₉	C₂×#427	1	1	yes	characteristic
166	C₂×C₇₉	#2916	1	1	yes	characteristic
167	C₈×#41	D₇₉	1	1	yes	characteristic
168	#2925	—	79	2	no	—
169	#1572	—	79	4	no	—
170	#1571	—	79	2	no	—
171	#1568	—	79	2	no	—
172	#1566	—	79	1	no	—
173	#1564	—	79	2	no	—
174	C₂×#334	—	79	1	no	—
175	C₇₉	#17827	1	1	yes	characteristic, complemented, Sylow
176	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
177	C₈×Q₈	Dic₇₉	1	1	yes	characteristic
178	C₈×Q₈	Dic₇₉	1	1	yes	characteristic
179	C₈×D₄	Dic₇₉	1	1	yes	characteristic
180	C₈×D₄	Dic₇₉	1	2	yes	—
181	C₈×D₄	Dic₇₉	1	1	yes	characteristic
182	C₈×D₄	Dic₇₉	1	2	yes	—
183	#433	—	158	4	no	—
184	#432	—	79	4	no	—
185	#431	—	79	4	no	—
186	#431	C₂×D₇₉	1	1	yes	characteristic
187	#431	C₂×D₇₉	1	2	yes	—
188	#423	—	79	2	no	—
189	C₂×C₄⋊C₈	—	79	2	no	—
190	#408	—	158	2	no	—
191	#405	—	158	2	no	—
192	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
194	#335	—	79	4	no	—
195	#335	—	79	4	no	—
196	#334	—	79	1	no	—
197	#334	—	79	1	no	—
198	#334	—	79	2	no	—
199	#334	—	79	2	no	—
200	#334	—	79	2	no	—
201	C₂×#41	D₃₁₆	1	1	yes	characteristic
202	C₂×M₁₆	—	158	2	no	—
203	C₂²×C₈	—	158	2	no	—
204	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
205	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
206	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
207	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
208	C₄×Q₈	Dic₁₅₈	1	1	yes	characteristic
209	C₄×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
210	C₄×D₄	Dic₁₅₈	1	2	yes	—
211	C₄×D₄	Dic₁₅₈	1	1	yes	characteristic
212	C₄×D₄	C₂×Dic₇₉	1	2	yes	—
213	C₄×D₄	C₂×Dic₇₉	1	1	yes	characteristic
214	#117	D₃₁₆	1	1	yes	characteristic
215	#117	D₃₁₆	1	2	yes	—
216	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
218	C₄⋊C₈	—	79	2	no	—
219	C₄⋊C₈	—	79	2	no	—
220	C₄⋊C₈	—	79	4	no	—
221	C₄⋊C₈	—	158	2	no	—
222	C₄⋊C₈	—	158	2	no	—
223	C₄⋊C₈	C₂×Dic₇₉	1	1	yes	characteristic
224	C₄⋊C₈	C₂×Dic₇₉	1	1	yes	characteristic
225	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
226	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
227	C₂²⋊C₈	—	158	4	no	—
228	C₂²⋊C₈	—	158	4	no	—
229	C₂²⋊C₈	C₂×Dic₇₉	1	2	yes	—
230	C₂²⋊C₈	C₂×Dic₇₉	1	2	yes	—
231	C₂²⋊C₈	C₂×Dic₇₉	1	2	yes	—
232	#97	—	158	4	no	—
233	C₄×C₈	—	158	4	no	—
234	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
235	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
236	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
237	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
238	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₇₉	1	2	yes	—
241	#41	—	2	4	no	—
242	C₂×Q₈	C₄⋊Dic₇₉	1	1	yes	characteristic
243	C₂×D₄	C₄⋊Dic₇₉	1	2	yes	—
244	C₂×D₄	C₄⋊Dic₇₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
248	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
249	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
250	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
251	M₁₆	—	316	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	316	4	no	—
257	C₂×C₈	—	158	2	no	—
258	C₂×C₈	—	158	4	no	—
259	C₂×C₈	—	158	2	no	—
260	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
263	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
265	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
266	C₄⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
267	C₄⋊C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
268	C₄⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
269	C₄⋊C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
270	C₂²⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
271	C₂²⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
272	C₂²⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
273	C₄²	C₂²×Dic₇₉	1	2	yes	—
274	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
275	C₄²	C₂×D₃₁₆	1	2	yes	—
276	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
277	C₄²	C₂×Dic₁₅₈	1	2	yes	—
278	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
279	C₄²	C₂³×D₇₉	1	1	yes	characteristic
280	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
281	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	D₄	—	2	4	no	—
284	D₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
296	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
298	C₂×C₄	#14579183	1	4	yes	—
299	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
301	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	2	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	316	2	no	—
308	C₈	—	316	2	no	—
309	C₈	—	316	4	no	—
310	C₈	#14579168	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	#19098601	1	2	yes	—
317	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	#19098601	1	2	yes	—
325	C₄	C₂×#14579168	1	1	yes	characteristic
326	C₄	#19099330	1	1	yes	characteristic
327	C₄	C₂×#14579183	1	2	yes	—
328	C₂	—	2	2	no	—
329	C₂	—	4	2	no	—
330	C₂	#23771968	1	1	yes	characteristic
331	C₂	C₂×#19098601	1	1	yes	characteristic
332	C₂	#23772011	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₈×#41×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₇₉	D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₈×#41	D₇₉
hypocenter	C₇₉	#17827
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #23762180	C₂	semidirect
3. #23762175	C₂	semidirect
4. #23762163	C₂	semidirect
5. #23762156	C₂	semidirect
6. #23762148	C₂	semidirect
7. C₂×#19098314	C₂	semidirect
175. C₇₉	#17827	semidirect

Maximal subgroups

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

subgroup	index
#23772066	2
#23762180	2
#23762175	2
#23762163	2
#23762156	2
#23762148	2
C₂×#19098314	2
C₈×#41×C₇₉	2
#17827	79