Subgroups — #34188384

Order 20224.

538 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771797	C₂	1	1	yes	maximal, characteristic
2	#23763551	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	#23761284	C₂	1	1	yes	maximal, characteristic
5	#23761284	C₂	1	1	yes	maximal, characteristic
6	#23761283	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1296	C₂	1	1	yes	maximal, characteristic
8	C₂×#14579160	—	2	1	no	—
9	C₂×#14579160	—	2	1	no	—
10	C₂×#14579160	C₄	1	2	yes	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₇₉⋊C₈	—	2	1	no	—
14	C₂×Dic₇₉⋊C₈	—	2	1	no	—
15	C₂×C₈×Dic₇₉	—	2	1	no	—
16	#19098433	C₂²	1	1	yes	characteristic
17	#19098432	—	2	1	no	—
18	#19098432	—	2	1	no	—
19	#19098432	C₄	1	2	yes	—
20	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
21	#19098317	—	2	1	no	—
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#384×C₇₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₅₈	—	2	2	no	—
26	C₂²×C₇₉⋊C₈	—	2	1	no	—
27	C₂²×C₇₉⋊C₈	—	2	1	no	—
28	#14579160	—	4	2	no	—
29	#14579160	—	4	2	no	—
30	#14579160	—	4	4	no	—
31	#14579160	—	4	2	no	—
32	#14579160	—	4	2	no	—
33	C₂×C₄⋊Dic₇₉	—	2	1	no	—
34	C₂×C₄⋊Dic₇₉	—	2	1	no	—
35	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
36	C₂×Dic₇₉⋊C₄	—	2	2	no	—
37	C₂×Dic₇₉⋊C₄	—	2	2	no	—
38	C₂×Dic₇₉⋊C₄	—	2	2	no	—
39	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
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)	Q₈	1	2	yes	complemented
49	Dic₇₉⋊C₈	—	4	2	no	—
50	Dic₇₉⋊C₈	—	4	2	no	—
51	C₈×Dic₇₉	—	4	2	no	—
52	C₂²×C₈×C₇₉	—	2	1	no	—
53	C₂²×C₈×C₇₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₇₉	—	2	1	no	—
59	C₄⋊C₈×C₇₉	—	2	1	no	—
60	C₄⋊C₈×C₇₉	—	2	1	no	—
61	C₄⋊C₈×C₇₉	Q₈	1	2	yes	complemented
62	C₂×Dic₁₅₈	—	4	4	no	—
63	C₂×Dic₁₅₈	—	4	4	no	—
64	C₂×Dic₁₅₈	—	4	4	no	—
65	C₂²×Dic₇₉	—	2	2	no	—
66	C₂²×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₇₉⋊C₈	—	2	2	no	—
71	C₂×C₇₉⋊C₈	—	4	1	no	—
72	C₂×C₇₉⋊C₈	—	4	1	no	—
73	C₂×C₇₉⋊C₈	—	4	1	no	—
74	C₂×C₇₉⋊C₈	—	4	1	no	—
75	C₂×C₇₉⋊C₈	—	4	1	no	—
76	C₄⋊Dic₇₉	—	4	2	no	—
77	C₄⋊Dic₇₉	—	4	1	no	—
78	C₄⋊Dic₇₉	—	4	1	no	—
79	C₄⋊Dic₇₉	—	4	1	no	—
80	C₄⋊Dic₇₉	—	4	1	no	—
81	Dic₇₉⋊C₄	—	4	4	no	—
82	Dic₇₉⋊C₄	—	4	4	no	—
83	Dic₇₉⋊C₄	—	4	4	no	—
84	C₄×Dic₇₉	—	2	2	no	—
85	C₄×Dic₇₉	—	2	2	no	—
86	C₄×Dic₇₉	—	2	1	no	—
87	C₄×Dic₇₉	—	2	1	no	—
88	C₄×Dic₇₉	D₈	1	2	yes	—
89	C₄×Dic₇₉	SD₁₆	1	2	yes	—
90	C₄×Dic₇₉	SD₁₆	1	2	yes	—
91	C₄×Dic₇₉	SD₁₆	1	2	yes	—
92	C₄×Dic₇₉	Q₁₆	1	2	yes	—
93	C₄×Dic₇₉	Q₁₆	1	2	yes	—
94	C₂²×C₄×C₇₉	—	2	2	no	—
95	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₉	—	2	2	no	—
99	C₂×C₈×C₇₉	—	4	1	no	—
100	C₂×C₈×C₇₉	—	4	1	no	—
101	C₂×C₈×C₇₉	—	4	1	no	—
102	C₂×C₈×C₇₉	—	4	1	no	—
103	C₂×C₈×C₇₉	—	4	1	no	—
104	C₄⋊C₄×C₇₉	—	4	2	no	—
105	C₄⋊C₄×C₇₉	—	4	1	no	—
106	C₄⋊C₄×C₇₉	—	4	1	no	—
107	C₄⋊C₄×C₇₉	—	4	1	no	—
108	C₄⋊C₄×C₇₉	—	4	1	no	—
109	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
110	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₇₉	#41	1	1	yes	characteristic
112	C₄²×C₇₉	#41	1	1	yes	characteristic
113	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	4	2	no	—
117	C₂×C₄×C₇₉	—	4	2	no	—
118	C₂×C₄×C₇₉	—	4	2	no	—
119	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×Dic₇₉	—	2	1	no	—
134	C₂×Dic₇₉	—	2	1	no	—
135	C₂×Dic₇₉	—	2	1	no	—
136	C₂×Dic₇₉	—	2	1	no	—
137	C₂×Dic₇₉	—	4	2	no	—
138	C₂×Dic₇₉	—	4	2	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	#100	1	2	yes	—
141	C₂×Dic₇₉	#101	1	2	yes	—
142	C₂×Dic₇₉	#102	1	2	yes	—
143	C₂×Dic₇₉	#102	1	2	yes	—
144	C₂×Dic₇₉	#103	1	2	yes	—
145	C₂×Dic₇₉	#103	1	2	yes	—
146	C₂×Dic₇₉	#103	1	2	yes	—
147	C₂×Dic₇₉	#103	1	2	yes	—
148	Dic₁₅₈	—	8	8	no	—
149	C₈×C₇₉	—	4	2	no	—
150	C₈×C₇₉	—	8	1	no	—
151	C₇₉⋊C₈	—	4	2	no	—
152	C₇₉⋊C₈	—	8	1	no	—
153	C₂²×C₇₉	#391	1	1	yes	characteristic
154	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
155	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
156	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
157	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
158	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
159	C₂²×C₇₉	#418	1	1	yes	characteristic
160	C₄×C₇₉	—	2	1	no	—
161	C₄×C₇₉	—	2	1	no	—
162	C₄×C₇₉	—	8	2	no	—
163	C₄×C₇₉	#476	1	1	yes	characteristic
164	C₄×C₇₉	#476	1	1	yes	characteristic
165	C₄×C₇₉	#477	1	1	yes	characteristic
166	C₄×C₇₉	#477	1	1	yes	characteristic
167	C₄×C₇₉	#486	1	1	yes	characteristic
168	C₄×C₇₉	#487	1	1	yes	characteristic
169	C₄×C₇₉	#498	1	1	yes	characteristic
170	C₄×C₇₉	#500	1	1	yes	characteristic
171	Dic₇₉	—	2	2	no	—
172	Dic₇₉	—	2	2	no	—
173	Dic₇₉	—	8	2	no	—
174	Dic₇₉	#332	1	4	yes	—
175	Dic₇₉	#333	1	4	yes	—
176	#9896	—	79	1	no	maximal, Sylow
177	C₂×C₇₉	C₂×#332	1	1	yes	characteristic
178	C₂×C₇₉	C₂×#333	1	1	yes	characteristic
179	C₂×C₇₉	#1545	1	1	yes	characteristic
180	C₂×C₇₉	#1549	1	1	yes	characteristic
181	C₂×C₇₉	#1987	1	1	yes	characteristic
182	C₂×C₇₉	#1990	1	1	yes	characteristic
183	C₂×C₇₉	#1997	1	1	yes	characteristic
184	#2868	—	79	1	no	—
185	#1842	—	79	1	no	—
186	C₄×C₄⋊C₈	—	79	1	no	—
187	#1296	—	79	1	no	—
188	#1296	—	79	1	no	—
189	#1296	—	79	1	no	—
190	#1296	D₇₉	1	1	yes	characteristic
191	C₇₉	#9896	1	1	yes	characteristic, complemented, Sylow
192	C₂×C₄⋊Q₈	—	79	2	no	—
193	C₂×C₄⋊Q₈	—	158	1	no	—
194	C₂×C₄⋊Q₈	—	158	1	no	—
195	C₂×C₄⋊C₈	—	79	1	no	—
196	C₂×C₄⋊C₈	—	79	1	no	—
197	C₂×C₄⋊C₈	—	158	1	no	—
198	C₂×C₄⋊C₈	—	158	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	—	158	1	no	—
202	#384	—	79	2	no	—
203	#384	—	79	1	no	—
204	#384	—	158	1	no	—
205	#384	—	158	1	no	—
206	#384	C₂×D₇₉	1	1	yes	characteristic
207	C₄³	—	79	1	no	—
208	#336	—	158	1	no	—
209	C₂²×Q₈	—	158	2	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	158	1	no	—
213	C₂²×C₈	—	158	1	no	—
214	C₄⋊Q₈	—	316	2	no	—
215	C₄⋊Q₈	—	316	2	no	—
216	C₄⋊Q₈	—	316	2	no	—
217	C₄⋊Q₈	—	316	2	no	—
218	C₄⋊Q₈	—	316	4	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	79	2	no	—
222	C₂×C₄⋊C₄	—	158	2	no	—
223	C₂×C₄⋊C₄	—	158	2	no	—
224	C₂×C₄⋊C₄	—	158	2	no	—
225	C₂×C₄⋊C₄	—	158	1	no	—
226	C₂×C₄⋊C₄	—	158	1	no	—
227	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
228	C₂×C₄²	—	79	1	no	—
229	C₂×C₄²	—	79	1	no	—
230	C₂×C₄²	—	79	1	no	—
231	C₂×C₄²	—	79	1	no	—
232	C₂×C₄²	—	79	1	no	—
233	C₂×C₄²	—	79	1	no	—
234	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	316	2	no	—
239	C₄⋊C₈	—	316	2	no	—
240	C₄⋊C₈	—	79	2	no	—
241	C₄⋊C₈	—	158	1	no	—
242	C₄⋊C₈	—	158	1	no	—
243	C₄⋊C₈	—	158	1	no	—
244	C₄⋊C₈	Dic₁₅₈	1	2	yes	complemented
245	C₄×C₈	—	316	2	no	—
246	C₂×Q₈	—	316	4	no	—
247	C₂×Q₈	—	316	4	no	—
248	C₂×Q₈	—	316	4	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	79	1	no	—
251	C₂²×C₄	—	79	1	no	—
252	C₂²×C₄	—	79	1	no	—
253	C₂²×C₄	—	79	1	no	—
254	C₂²×C₄	—	158	2	no	—
255	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
256	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	316	1	no	—
265	C₂×C₈	—	316	1	no	—
266	C₂×C₈	—	316	1	no	—
267	C₂×C₈	—	316	1	no	—
268	C₂×C₈	—	316	1	no	—
269	C₂×C₈	—	158	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	316	2	no	—
276	C₄⋊C₄	—	316	4	no	—
277	C₄⋊C₄	—	316	4	no	—
278	C₄⋊C₄	—	316	4	no	—
279	C₄⋊C₄	—	316	1	no	—
280	C₄⋊C₄	—	316	1	no	—
281	C₄⋊C₄	—	316	1	no	—
282	C₄⋊C₄	—	316	1	no	—
283	C₄²	—	79	2	no	—
284	C₄²	—	79	2	no	—
285	C₄²	—	79	2	no	—
286	C₄²	—	79	2	no	—
287	C₄²	—	79	2	no	—
288	C₄²	—	79	2	no	—
289	C₄²	—	158	2	no	—
290	C₄²	—	158	2	no	—
291	C₄²	—	158	1	no	—
292	C₄²	—	158	1	no	—
293	C₄²	Q₈×D₇₉	1	1	yes	characteristic
294	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
295	C₄²	#11929623	1	1	yes	characteristic
296	C₄²	#11929624	1	1	yes	characteristic
297	C₂³	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
298	Q₈	—	632	8	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	—	316	2	no	—
305	C₂×C₄	—	316	2	no	—
306	C₂×C₄	—	316	2	no	—
307	C₂×C₄	—	79	2	no	—
308	C₂×C₄	—	79	2	no	—
309	C₂×C₄	—	79	2	no	—
310	C₂×C₄	—	79	2	no	—
311	C₂×C₄	—	79	2	no	—
312	C₂×C₄	—	79	2	no	—
313	C₂×C₄	—	79	2	no	—
314	C₂×C₄	—	79	2	no	—
315	C₂×C₄	—	158	1	no	—
316	C₂×C₄	—	158	1	no	—
317	C₂×C₄	—	158	1	no	—
318	C₂×C₄	—	158	1	no	—
319	C₂×C₄	#14579126	1	1	yes	characteristic
320	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
321	C₂×C₄	#14579138	1	1	yes	characteristic
322	C₂×C₄	#14579139	1	1	yes	characteristic
323	C₂×C₄	#14579147	1	1	yes	characteristic
324	C₂×C₄	#14579160	1	1	yes	characteristic
325	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
326	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
328	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
329	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
330	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
331	C₂×C₄	#14579185	1	1	yes	characteristic
332	C₂×C₄	#14579188	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	8	1	no	—
335	C₈	—	316	2	no	—
336	C₈	—	632	1	no	—
337	C₂²	#19098467	1	1	yes	characteristic
338	C₂²	#100×D₇₉	1	1	yes	characteristic
339	C₂²	#101×D₇₉	1	1	yes	characteristic
340	C₂²	#102×D₇₉	1	1	yes	characteristic
341	C₂²	#103×D₇₉	1	1	yes	characteristic
342	C₂²	#103×D₇₉	1	1	yes	characteristic
343	C₂²	#19098566	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	79	4	no	—
348	C₄	—	79	4	no	—
349	C₄	—	632	2	no	—
350	C₄	—	158	2	no	—
351	C₄	—	158	2	no	—
352	C₄	#19098851	1	1	yes	characteristic
353	C₄	#19098852	1	1	yes	characteristic
354	C₄	#19098856	1	1	yes	characteristic
355	C₄	#19098857	1	1	yes	characteristic
356	C₄	#19098908	1	1	yes	characteristic
357	C₄	#19098912	1	1	yes	characteristic
358	C₄	#19098960	1	1	yes	characteristic
359	C₄	#19098967	1	1	yes	characteristic
360	C₂	#332×D₇₉	1	1	yes	characteristic
361	C₂	#333×D₇₉	1	1	yes	characteristic
362	C₂	#23762066	1	1	yes	characteristic
363	C₂	#23762083	1	1	yes	characteristic
364	C₂	#23764346	1	1	yes	characteristic
365	C₂	#23764359	1	1	yes	characteristic
366	C₂	#23764387	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1296	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1296	D₇₉
hypocenter	C₇₉	#9896
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
48. C₄⋊(C₇₉⋊C₈)	Q₈	semidirect
61. C₄⋊C₈×C₇₉	Q₈	semidirect
191. C₇₉	#9896	semidirect
244. C₄⋊C₈	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23771797	2
#23763551	2
C₄⋊C₈×Dic₇₉	2
#23761284	2
#23761283	2
C₇₉×#1296	2
#9896	79