Subgroups — #34181078

Order 20224.

496 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579161	C₂	1	1	yes	maximal, characteristic
2	#23763552	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098311	C₂	1	2	yes	maximal
4	#23761284	C₂	1	2	yes	maximal
5	C₇₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579161	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₅₈	C₄	1	2	yes	—
8	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
11	#19098469	C₄	1	2	yes	—
12	#19098461	—	2	1	no	—
13	#19098461	—	2	1	no	—
14	#19098433	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₇₉	—	2	1	no	—
16	C₄×C₄⋊Dic₇₉	—	2	1	no	—
17	#19098311	—	2	4	no	—
18	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
19	C₄³×C₇₉	C₂²	1	1	yes	characteristic
20	#336×C₇₉	—	2	1	no	—
21	#336×C₇₉	—	2	1	no	—
22	C₂²×Dic₁₅₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	#14579161	—	2	1	no	—
25	#14579161	—	2	1	no	—
26	#14579161	D₄	1	2	yes	—
27	#14579161	Q₈	1	2	yes	—
28	C₄×Dic₁₅₈	—	4	4	no	—
29	C₂×C₄⋊Dic₇₉	—	2	1	no	—
30	C₂×C₄⋊Dic₇₉	—	2	1	no	—
31	C₂×C₄⋊Dic₇₉	—	2	1	no	—
32	C₂×C₄⋊Dic₇₉	—	2	1	no	—
33	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
34	C₂×Dic₇₉⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₇₉	—	2	2	no	—
36	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
39	C₄⋊(C₇₉⋊C₈)	—	4	2	no	—
40	C₇₉⋊#95	—	2	2	no	—
41	C₇₉⋊#95	—	2	2	no	—
42	C₂²×C₈×C₇₉	—	2	2	no	—
43	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₇₉	—	2	2	no	—
51	C₄⋊C₈×C₇₉	—	2	2	no	—
52	C₂×Dic₁₅₈	—	2	2	no	—
53	C₂×Dic₁₅₈	—	4	2	no	—
54	C₂×Dic₁₅₈	—	4	2	no	—
55	C₂×Dic₁₅₈	C₄⋊C₄	1	4	yes	—
56	C₂²×Dic₇₉	—	2	2	no	—
57	C₂²×Dic₇₉	—	2	2	no	—
58	C₂×C₇₉⋊C₈	—	4	2	no	—
59	C₂×C₇₉⋊C₈	—	4	2	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₄⋊Dic₇₉	—	2	2	no	—
62	C₄⋊Dic₇₉	—	2	2	no	—
63	C₄⋊Dic₇₉	—	2	2	no	—
64	C₄⋊Dic₇₉	—	2	2	no	—
65	C₄⋊Dic₇₉	—	4	2	no	—
66	C₄⋊Dic₇₉	—	4	2	no	—
67	C₄⋊Dic₇₉	—	4	1	no	—
68	C₄⋊Dic₇₉	—	4	1	no	—
69	Dic₇₉⋊C₄	—	4	4	no	—
70	C₄×Dic₇₉	—	4	4	no	—
71	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₇₉	—	4	2	no	—
79	C₂×C₈×C₇₉	—	4	2	no	—
80	C₂×C₈×C₇₉	—	4	2	no	—
81	C₄²×C₇₉	—	2	2	no	—
82	C₄²×C₇₉	—	2	2	no	—
83	C₄²×C₇₉	—	2	1	no	—
84	C₄²×C₇₉	—	2	1	no	—
85	C₄²×C₇₉	—	2	2	no	—
86	C₄²×C₇₉	—	4	1	no	—
87	C₄²×C₇₉	SD₁₆	1	2	yes	—
88	C₄²×C₇₉	Q₁₆	1	2	yes	—
89	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₇₉	#41	1	1	yes	characteristic
92	C₄²×C₇₉	#41	1	1	yes	characteristic
93	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₉	—	2	1	no	—
95	C₂×C₄×C₇₉	—	2	1	no	—
96	C₂×C₄×C₇₉	—	2	1	no	—
97	C₂×C₄×C₇₉	—	2	1	no	—
98	C₂×C₄×C₇₉	—	2	1	no	—
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	1	no	—
101	C₂×C₄×C₇₉	—	2	1	no	—
102	C₂×C₄×C₇₉	—	2	1	no	—
103	C₂×C₄×C₇₉	—	2	1	no	—
104	C₂×C₄×C₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	#100	1	2	yes	—
106	C₂×C₄×C₇₉	#103	1	2	yes	—
107	C₂×C₄×C₇₉	#103	1	2	yes	—
108	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#135	1	2	yes	—
117	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#137	1	2	yes	—
119	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
120	C₂×Dic₇₉	—	2	4	no	—
121	C₂×Dic₇₉	—	4	2	no	—
122	C₂×Dic₇₉	—	4	2	no	—
123	C₂×Dic₇₉	—	4	2	no	—
124	C₂×Dic₇₉	—	4	2	no	—
125	C₂×Dic₇₉	—	4	2	no	—
126	Dic₁₅₈	—	4	4	no	—
127	Dic₁₅₈	—	4	4	no	—
128	C₈×C₇₉	—	8	2	no	—
129	C₇₉⋊C₈	—	8	2	no	—
130	C₂²×C₇₉	#380	1	1	yes	characteristic
131	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
132	C₂²×C₇₉	#413	1	1	yes	characteristic
133	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
134	C₂²×C₇₉	#416	1	1	yes	characteristic
135	C₂²×C₇₉	#417	1	1	yes	characteristic
136	C₂²×C₇₉	#419	1	1	yes	characteristic
137	C₄×C₇₉	—	2	2	no	—
138	C₄×C₇₉	—	2	1	no	—
139	C₄×C₇₉	—	2	1	no	—
140	C₄×C₇₉	—	2	2	no	—
141	C₄×C₇₉	—	2	2	no	—
142	C₄×C₇₉	—	4	1	no	—
143	C₄×C₇₉	#460	1	1	yes	characteristic
144	C₄×C₇₉	#462	1	1	yes	characteristic
145	C₄×C₇₉	#464	1	2	yes	—
146	C₄×C₇₉	#475	1	1	yes	characteristic
147	C₄×C₇₉	#477	1	1	yes	characteristic
148	C₄×C₇₉	#479	1	2	yes	—
149	Dic₇₉	—	4	4	no	—
150	Dic₇₉	—	8	2	no	—
151	#8590	—	79	1	no	maximal, Sylow
152	C₂×C₇₉	#1542	1	1	yes	characteristic
153	C₂×C₇₉	#1548	1	1	yes	characteristic
154	C₂×C₇₉	#1551	1	1	yes	characteristic
155	C₂×C₇₉	#1559	1	1	yes	characteristic
156	C₂×C₇₉	#1800	1	1	yes	characteristic
157	C₂×C₇₉	#1803	1	1	yes	characteristic
158	C₂×C₇₉	#1805	1	1	yes	characteristic
159	C₄×C₄⋊Q₈	—	79	1	no	—
160	#1842	—	79	1	no	—
161	#1831	D₇₉	1	1	yes	characteristic
162	C₂×#333	—	79	2	no	—
163	#1296	—	79	2	no	—
164	C₇₉	#8590	1	1	yes	characteristic, complemented, Sylow
165	C₂×C₄⋊Q₈	—	79	1	no	—
166	C₂×C₄×Q₈	—	79	2	no	—
167	C₂×C₄⋊C₈	—	79	2	no	—
168	C₂×C₄⋊C₈	—	158	1	no	—
169	C₂×C₄⋊C₈	—	158	1	no	—
170	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
171	#391	—	79	2	no	—
172	#389	—	158	1	no	—
173	#389	—	158	1	no	—
174	#384	—	79	1	no	—
175	C₄×C₄⋊C₄	—	158	1	no	—
176	C₄×C₄⋊C₄	—	158	1	no	—
177	C₄³	C₂×D₇₉	1	1	yes	characteristic
178	#336	—	2	1	no	—
179	#336	—	2	1	no	—
180	#333	—	158	4	no	—
181	C₂²×Q₈	—	79	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	158	2	no	—
184	C₄⋊Q₈	—	79	2	no	—
185	C₄⋊Q₈	—	79	2	no	—
186	C₄⋊Q₈	—	158	1	no	—
187	C₄⋊Q₈	—	158	1	no	—
188	C₄×Q₈	—	316	4	no	—
189	C₂×C₄⋊C₄	—	79	2	no	—
190	C₂×C₄⋊C₄	—	158	2	no	—
191	C₂×C₄⋊C₄	—	158	1	no	—
192	C₂×C₄⋊C₄	—	158	1	no	—
193	C₂×C₄⋊C₄	—	158	1	no	—
194	C₂×C₄⋊C₄	—	158	1	no	—
195	C₂×C₄²	—	158	2	no	—
196	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
198	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
199	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	316	2	no	—
206	C₄⋊C₈	—	316	2	no	—
207	C₄⋊C₈	—	158	2	no	—
208	C₄⋊C₈	—	158	2	no	—
209	#95	—	158	2	no	—
210	#95	—	158	2	no	—
211	C₂×Q₈	—	316	2	no	—
212	C₂×Q₈	—	316	2	no	—
213	C₂×Q₈	—	79	4	no	—
214	C₂×Q₈	—	158	2	no	—
215	C₂²×C₄	—	158	2	no	—
216	C₂²×C₄	—	158	2	no	—
217	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	316	2	no	—
228	C₂×C₈	—	316	2	no	—
229	C₂×C₈	—	316	2	no	—
230	C₄⋊C₄	—	316	2	no	—
231	C₄⋊C₄	—	316	4	no	—
232	C₄⋊C₄	—	316	2	no	—
233	C₄⋊C₄	—	316	1	no	—
234	C₄⋊C₄	—	316	1	no	—
235	C₄⋊C₄	—	158	2	no	—
236	C₄⋊C₄	—	158	2	no	—
237	C₄⋊C₄	—	158	2	no	—
238	C₄⋊C₄	—	158	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	316	4	no	—
246	C₄²	#11929613	1	2	yes	—
247	C₄²	Dic₃₁₆	1	2	yes	—
248	C₄²	D₄×D₇₉	1	1	yes	characteristic
249	C₄²	Q₈×D₇₉	1	1	yes	characteristic
250	C₄²	#11929623	1	1	yes	characteristic
251	C₄²	#11929625	1	1	yes	characteristic
252	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
253	Q₈	—	316	4	no	—
254	Q₈	—	316	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	316	2	no	—
267	C₂×C₄	—	316	2	no	—
268	C₂×C₄	—	316	2	no	—
269	C₂×C₄	—	316	2	no	—
270	C₂×C₄	—	316	2	no	—
271	C₂×C₄	—	158	4	no	—
272	C₂×C₄	#14579077	1	2	yes	—
273	C₂×C₄	#14579086	1	2	yes	—
274	C₂×C₄	#14579086	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
276	C₂×C₄	#14579119	1	1	yes	characteristic
277	C₂×C₄	#14579132	1	1	yes	characteristic
278	C₂×C₄	#14579136	1	1	yes	characteristic
279	C₂×C₄	#14579137	1	1	yes	characteristic
280	C₂×C₄	#14579141	1	1	yes	characteristic
281	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
282	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
283	C₂×C₄	#14579181	1	2	yes	—
284	C₂×C₄	#14579189	1	1	yes	characteristic
285	C₂×C₄	#14579191	1	2	yes	—
286	C₂×C₄	#14579194	1	1	yes	characteristic
287	C₈	—	8	2	no	—
288	C₈	—	632	2	no	—
289	C₂²	#19098420	1	1	yes	characteristic
290	C₂²	C₂×#14579077	1	1	yes	characteristic
291	C₂²	#19098545	1	1	yes	characteristic
292	C₂²	C₂×#14579086	1	1	yes	characteristic
293	C₂²	#19098557	1	1	yes	characteristic
294	C₂²	#19098565	1	1	yes	characteristic
295	C₂²	#19098571	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	316	4	no	—
303	C₄	—	632	2	no	—
304	C₄	#19098764	1	1	yes	characteristic
305	C₄	#19098774	1	1	yes	characteristic
306	C₄	#19098782	1	2	yes	—
307	C₄	#19098849	1	1	yes	characteristic
308	C₄	#19098859	1	1	yes	characteristic
309	C₄	#19098867	1	2	yes	—
310	C₂	#23762055	1	1	yes	characteristic
311	C₂	#23762080	1	1	yes	characteristic
312	C₂	#23762096	1	1	yes	characteristic
313	C₂	#23762131	1	1	yes	characteristic
314	C₂	#23763389	1	1	yes	characteristic
315	C₂	#23763406	1	1	yes	characteristic
316	C₂	#23763412	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1831	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1831	D₇₉
hypocenter	C₇₉	#8590
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
164. C₇₉	#8590	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579161	2
#23763552	2
C₂×#19098311	2
#23761284	2
C₇₉×#1831	2
#8590	79