Subgroups — #34257478

Order 20224.

572 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772517	C₂	1	2	yes	maximal
2	#23762189	C₂	1	2	yes	maximal, complemented
3	#23762185	C₂	1	2	yes	maximal, complemented
4	#23762180	C₂	1	2	yes	maximal, complemented
5	#23762171	C₂	1	2	yes	maximal, complemented
6	#23762167	C₂	1	2	yes	maximal, complemented
7	#23762163	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762156	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#2922	C₂	1	1	yes	maximal, characteristic
10	#19098694	—	2	2	no	—
11	#19098641	—	2	2	no	—
12	#19098635	—	2	2	no	—
13	#19098635	—	2	2	no	—
14	#19098620	C₂²	1	2	yes	—
15	#19098620	C₂²	1	2	yes	—
16	#19098616	C₂²	1	2	yes	—
17	#19098587	C₂²	1	2	yes	—
18	#19098587	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
20	#19098316	C₂²	1	2	yes	—
21	#19098316	C₂²	1	2	yes	—
22	#19098316	C₂²	1	2	yes	—
23	#19098314	C₂²	1	1	yes	characteristic
24	#19098314	C₂²	1	2	yes	—
25	#19098314	C₂²	1	1	yes	characteristic
26	#19098314	C₂²	1	2	yes	—
27	#19098314	C₂²	1	1	yes	characteristic
28	#19098314	C₂²	1	2	yes	—
29	#19098314	C₂²	1	1	yes	characteristic
30	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
31	#433×C₇₉	C₄	1	4	yes	—
32	#431×C₇₉	C₂²	1	2	yes	—
33	#431×C₇₉	C₂²	1	2	yes	—
34	#424×C₇₉	C₄	1	2	yes	—
35	#408×C₇₉	C₄	1	2	yes	—
36	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×#11929608	—	2	2	no	—
39	C₂×#11929608	—	2	2	no	—
40	C₂²×C₇₉⋊C₈	—	2	2	no	—
41	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
44	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
45	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
46	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
47	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
48	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
49	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
50	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
51	#14579072	—	2	2	no	—
52	#14579072	—	2	2	no	—
53	C₄×C₇₉⋊C₈	—	2	2	no	—
54	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
58	C₄×Q₈×C₇₉	Q₈	1	1	yes	characteristic
59	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
60	C₄×D₄×C₇₉	D₄	1	2	yes	—
61	C₄×D₄×C₇₉	Q₈	1	2	yes	—
62	C₄×D₄×C₇₉	Q₈	1	1	yes	characteristic
63	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
64	#117×C₇₉	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
66	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
70	C₂²⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
71	#97×C₇₉	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₉	C₂³	1	2	yes	—
76	#11929608	—	8	2	no	—
77	#11929608	—	8	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	—	8	2	no	—
83	#41×C₇₉	—	4	2	no	—
84	C₂×Q₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
85	C₂×D₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
86	C₂×D₄×C₇₉	C₄⋊C₄	1	2	yes	—
87	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
93	M₁₆×C₇₉	—	4	2	no	—
94	C₂×C₈×C₇₉	—	4	2	no	—
95	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
98	C₄⋊C₄×C₇₉	C₄⋊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	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
103	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
104	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
105	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
107	C₄²×C₇₉	C₂×D₄	1	2	yes	—
108	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
110	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
112	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	Q₈×C₇₉	—	4	1	no	—
115	D₄×C₇₉	—	4	1	no	—
116	D₄×C₇₉	—	4	2	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	4	no	—
119	C₂×C₄×C₇₉	—	2	2	no	—
120	C₂×C₄×C₇₉	—	2	2	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	—	4	2	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
133	C₈×C₇₉	—	2	2	no	—
134	C₈×C₇₉	—	2	2	no	—
135	C₈×C₇₉	—	2	4	no	—
136	C₇₉⋊C₈	—	4	2	no	—
137	C₇₉⋊C₈	—	4	2	no	—
138	C₇₉⋊C₈	—	4	2	no	—
139	C₇₉⋊C₈	—	4	2	no	—
140	C₂²×C₇₉	—	2	1	no	—
141	C₂²×C₇₉	—	2	2	no	—
142	C₂²×C₇₉	—	2	2	no	—
143	C₂²×C₇₉	—	2	1	no	—
144	C₂²×C₇₉	—	2	2	no	—
145	C₂²×C₇₉	—	2	1	no	—
146	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₇₉	—	2	1	no	—
148	C₄×C₇₉	—	2	2	no	—
149	C₄×C₇₉	—	2	2	no	—
150	C₄×C₇₉	—	2	1	no	—
151	C₄×C₇₉	—	2	1	no	—
152	C₄×C₇₉	—	2	1	no	—
153	C₄×C₇₉	—	2	2	no	—
154	C₄×C₇₉	#568	1	2	yes	—
155	C₄×C₇₉	#577	1	2	yes	—
156	#17824	—	79	1	no	maximal, Sylow
157	C₂×C₇₉	—	4	1	no	—
158	C₂×C₇₉	—	4	2	no	—
159	C₂×C₇₉	#2905	1	1	yes	characteristic
160	C₂×C₇₉	#2912	1	1	yes	characteristic
161	C₂×C₇₉	#2916	1	1	yes	characteristic
162	#2989	—	79	2	no	—
163	#2922	D₇₉	1	1	yes	characteristic
164	#1574	—	79	2	no	—
165	#1573	—	79	2	no	—
166	#1572	—	79	2	no	—
167	#1570	—	79	2	no	—
168	#1569	—	79	2	no	—
169	#1568	—	79	1	no	—
170	#1566	—	79	1	no	—
171	C₇₉	#17824	1	1	yes	characteristic, complemented, Sylow
172	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
173	#446	—	158	2	no	—
174	#436	—	158	2	no	—
175	#435	—	158	2	no	—
176	#435	—	158	2	no	—
177	#433	Dic₇₉	1	4	yes	—
178	#432	—	79	2	no	—
179	#432	—	79	2	no	—
180	#431	—	79	2	no	—
181	#431	C₂×D₇₉	1	2	yes	—
182	#431	C₂×D₇₉	1	2	yes	—
183	#424	Dic₇₉	1	2	yes	—
184	#423	—	79	2	no	—
185	#423	—	79	2	no	—
186	C₂×C₄⋊C₈	—	79	2	no	—
187	#408	Dic₇₉	1	2	yes	—
188	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
190	#335	—	79	2	no	—
191	#335	—	79	2	no	—
192	#335	—	79	2	no	—
193	#334	—	79	2	no	—
194	#334	—	79	1	no	—
195	#334	—	79	1	no	—
196	#334	—	79	1	no	—
197	#334	—	79	1	no	—
198	#334	—	79	2	no	—
199	#334	—	79	2	no	—
200	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
201	C₂×M₁₆	—	158	2	no	—
202	C₂×M₁₆	—	158	2	no	—
203	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
204	C₂²×C₈	—	158	2	no	—
205	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
206	C₄×Q₈	Dic₁₅₈	1	1	yes	characteristic
207	C₄×Q₈	D₃₁₆	1	1	yes	characteristic
208	C₄×D₄	Dic₁₅₈	1	1	yes	characteristic
209	C₄×D₄	Dic₁₅₈	1	2	yes	—
210	C₄×D₄	D₃₁₆	1	1	yes	characteristic
211	C₄×D₄	D₃₁₆	1	2	yes	—
212	#117	C₂×Dic₇₉	1	1	yes	characteristic
213	#117	C₂×Dic₇₉	1	2	yes	—
214	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
216	C₄⋊C₈	—	79	2	no	—
217	C₄⋊C₈	—	79	2	no	—
218	C₄⋊C₈	—	79	2	no	—
219	C₄⋊C₈	—	79	2	no	—
220	C₄⋊C₈	—	158	2	no	—
221	C₄⋊C₈	—	158	2	no	—
222	C₄⋊C₈	—	158	2	no	—
223	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
224	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
225	C₂²⋊C₈	—	158	2	no	—
226	C₂²⋊C₈	—	158	2	no	—
227	C₂²⋊C₈	—	158	2	no	—
228	C₂²⋊C₈	C₂×Dic₇₉	1	4	yes	—
229	C₂²⋊C₈	C₂×Dic₇₉	1	4	yes	—
230	#97	—	158	2	no	—
231	#97	—	158	2	no	—
232	#97	C₂×Dic₇₉	1	2	yes	—
233	C₄×C₈	—	158	2	no	—
234	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
235	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₇₉	1	2	yes	—
238	#41	—	4	2	no	—
239	C₂×Q₈	C₄⋊Dic₇₉	1	1	yes	characteristic
240	C₂×D₄	C₄⋊Dic₇₉	1	1	yes	characteristic
241	C₂×D₄	C₄⋊Dic₇₉	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
245	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
246	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
247	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	632	2	no	—
250	M₁₆	—	632	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	632	2	no	—
253	C₂×C₈	—	158	2	no	—
254	C₂×C₈	—	158	2	no	—
255	C₂×C₈	—	158	2	no	—
256	C₂×C₈	—	158	2	no	—
257	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
258	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
259	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
260	C₄⋊C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
261	C₄⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
262	C₄⋊C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
263	C₄⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
264	C₂²⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
265	C₂²⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
266	C₂²⋊C₄	C₄⋊Dic₇₉	1	2	yes	—
267	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
268	C₄²	C₂²×Dic₇₉	1	2	yes	—
269	C₄²	C₂×D₃₁₆	1	2	yes	—
270	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
271	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
272	C₄²	C₂×Dic₁₅₈	1	2	yes	—
273	C₄²	C₂³×D₇₉	1	1	yes	characteristic
274	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
275	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
276	Q₈	—	4	1	no	—
277	D₄	—	4	1	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₈	—	316	2	no	—
299	C₈	—	316	2	no	—
300	C₈	—	316	2	no	—
301	C₈	—	316	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	#19099330	1	2	yes	—
317	C₄	#19099375	1	2	yes	—
318	C₂	—	4	1	no	—
319	C₂	—	4	2	no	—
320	C₂	#23771964	1	1	yes	characteristic
321	C₂	#23771995	1	1	yes	characteristic
322	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	C₇₉×#2922	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#2922	D₇₉
hypocenter	C₇₉	#17824
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
2. #23762189	C₂	semidirect
3. #23762185	C₂	semidirect
4. #23762180	C₂	semidirect
5. #23762171	C₂	semidirect
6. #23762167	C₂	semidirect
7. #23762163	C₂	semidirect
8. #23762156	C₂	semidirect
171. C₇₉	#17824	semidirect

Maximal subgroups

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

subgroup	index
#23772517	2
#23762189	2
#23762185	2
#23762180	2
#23762171	2
#23762167	2
#23762163	2
#23762156	2
C₇₉×#2922	2
#17824	79