Subgroups — #34188250

Order 20224.

528 conjugacy classes of subgroups.

The subgroup lattice has 288 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	C₄×C₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#23761239	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₅₈	—	2	1	no	—
7	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
9	#19098469	—	2	1	no	—
10	#19098461	—	2	1	no	—
11	#19098461	C₄	1	2	yes	—
12	#19098433	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₇₉	—	2	1	no	—
14	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
15	#19098318	—	2	1	no	—
16	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	—	2	1	no	—
18	C₄³×C₇₉	C₂²	1	1	yes	characteristic
19	#336×C₇₉	—	2	1	no	—
20	C₂²×Dic₁₅₈	—	2	1	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	#14579161	—	2	2	no	—
23	#14579161	C₈	1	4	yes	complemented
24	C₄×Dic₁₅₈	—	4	4	no	—
25	C₂×C₄⋊Dic₇₉	—	2	1	no	—
26	C₂×C₄⋊Dic₇₉	—	2	1	no	—
27	C₂×C₄⋊Dic₇₉	—	2	1	no	—
28	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
30	C₂×Dic₇₉⋊C₄	—	2	2	no	—
31	C₂×C₄×Dic₇₉	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
33	C₄×C₇₉⋊C₈	—	2	4	no	—
34	C₇₉⋊#95	—	2	2	no	—
35	C₇₉⋊#95	—	2	2	no	—
36	C₂²×C₈×C₇₉	—	2	2	no	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₇₉	—	2	2	no	—
45	C₄⋊C₈×C₇₉	Q₈	1	4	yes	complemented
46	C₄×C₈×C₇₉	—	4	2	no	—
47	C₂×Dic₁₅₈	—	2	4	no	—
48	C₂×Dic₁₅₈	—	4	2	no	—
49	C₂×Dic₁₅₈	—	4	2	no	—
50	C₂²×Dic₇₉	—	2	2	no	—
51	C₂²×Dic₇₉	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₂×C₇₉⋊C₈	—	4	2	no	—
55	C₄⋊Dic₇₉	—	2	4	no	—
56	C₄⋊Dic₇₉	—	2	2	no	—
57	C₄⋊Dic₇₉	—	2	4	no	—
58	C₄⋊Dic₇₉	—	4	2	no	—
59	C₄⋊Dic₇₉	—	4	2	no	—
60	C₄⋊Dic₇₉	C₂×C₈	1	4	yes	—
61	Dic₇₉⋊C₄	—	4	4	no	—
62	C₄×Dic₇₉	—	4	4	no	—
63	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₇₉	—	2	4	no	—
71	C₂×C₈×C₇₉	—	4	2	no	—
72	C₂×C₈×C₇₉	—	4	2	no	—
73	C₄²×C₇₉	—	2	2	no	—
74	C₄²×C₇₉	—	2	2	no	—
75	C₄²×C₇₉	—	2	2	no	—
76	C₄²×C₇₉	—	4	1	no	—
77	C₄²×C₇₉	C₂×C₈	1	2	yes	—
78	C₄²×C₇₉	M₁₆	1	2	yes	—
79	C₄²×C₇₉	SD₁₆	1	2	yes	—
80	C₄²×C₇₉	Q₁₆	1	2	yes	—
81	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
82	C₄²×C₇₉	#41	1	2	yes	—
83	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	2	1	no	—
92	C₂×C₄×C₇₉	—	2	1	no	—
93	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₇₉	#103	1	2	yes	—
96	C₂×C₄×C₇₉	#103	1	2	yes	—
97	C₂×C₄×C₇₉	#104	1	2	yes	—
98	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₇₉	#131	1	2	yes	—
105	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×Dic₇₉	—	2	4	no	—
109	C₂×Dic₇₉	—	4	2	no	—
110	C₂×Dic₇₉	—	4	2	no	—
111	C₂×Dic₇₉	—	4	2	no	—
112	C₂×Dic₇₉	—	4	2	no	—
113	C₂×Dic₇₉	—	4	2	no	—
114	Dic₁₅₈	—	4	8	no	—
115	C₈×C₇₉	—	4	4	no	—
116	C₇₉⋊C₈	—	4	4	no	—
117	C₂²×C₇₉	#391	1	1	yes	characteristic
118	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₇₉	#408	1	1	yes	characteristic
120	C₂²×C₇₉	C₂×#103	1	1	yes	characteristic
121	C₂²×C₇₉	#418	1	1	yes	characteristic
122	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
123	C₂²×C₇₉	#421	1	1	yes	characteristic
124	C₄×C₇₉	—	2	1	no	—
125	C₄×C₇₉	—	2	1	no	—
126	C₄×C₇₉	—	2	2	no	—
127	C₄×C₇₉	—	2	2	no	—
128	C₄×C₇₉	—	4	1	no	—
129	C₄×C₇₉	Q₈⋊C₈	1	4	yes	—
130	C₄×C₇₉	C₈×Q₈	1	2	yes	—
131	C₄×C₇₉	#446	1	2	yes	—
132	C₄×C₇₉	#476	1	2	yes	—
133	C₄×C₇₉	#477	1	2	yes	—
134	Dic₇₉	—	4	4	no	—
135	Dic₇₉	—	8	2	no	—
136	#9864	—	79	1	no	maximal, Sylow
137	C₂×C₇₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
138	C₂×C₇₉	#1478	1	1	yes	characteristic
139	C₂×C₇₉	#1481	1	1	yes	characteristic
140	C₂×C₇₉	#1484	1	1	yes	characteristic
141	C₂×C₇₉	#1984	1	1	yes	characteristic
142	C₂×C₇₉	#1986	1	1	yes	characteristic
143	C₂×C₇₉	#1987	1	1	yes	characteristic
144	C₄×C₄⋊Q₈	—	79	1	no	—
145	C₄×C₄⋊C₈	—	79	1	no	—
146	C₄×C₄⋊C₈	D₇₉	1	1	yes	characteristic
147	#1279	—	79	4	no	—
148	C₇₉	#9864	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊Q₈	—	79	1	no	—
150	C₂×C₄×Q₈	—	158	1	no	—
151	C₂×C₄⋊C₈	—	158	1	no	—
152	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	C₂×C₄×C₈	—	79	2	no	—
155	#391	—	158	1	no	—
156	#389	—	79	2	no	—
157	#389	—	158	1	no	—
158	#384	—	79	1	no	—
159	C₄×C₄⋊C₄	—	79	2	no	—
160	C₄×C₄⋊C₄	—	158	1	no	—
161	C₄³	C₂×D₇₉	1	1	yes	characteristic
162	#336	—	2	1	no	—
163	#336	—	158	1	no	—
164	C₂²×Q₈	—	158	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	158	2	no	—
167	C₄⋊Q₈	—	79	4	no	—
168	C₄⋊Q₈	—	158	2	no	—
169	C₄×Q₈	—	316	4	no	—
170	C₂×C₄⋊C₄	—	79	2	no	—
171	C₂×C₄⋊C₄	—	79	2	no	—
172	C₂×C₄⋊C₄	—	158	1	no	—
173	C₂×C₄⋊C₄	—	158	2	no	—
174	C₂×C₄⋊C₄	—	158	1	no	—
175	C₂×C₄⋊C₄	—	158	1	no	—
176	C₂×C₄²	—	158	2	no	—
177	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
179	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
180	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	—	158	4	no	—
186	C₄⋊C₈	Dic₁₅₈	1	4	yes	complemented
187	C₄×C₈	—	4	2	no	—
188	C₄×C₈	—	158	4	no	—
189	#95	—	158	2	no	—
190	#95	—	158	2	no	—
191	C₂×Q₈	—	316	2	no	—
192	C₂×Q₈	—	316	2	no	—
193	C₂×Q₈	—	158	4	no	—
194	C₂²×C₄	—	158	2	no	—
195	C₂²×C₄	—	158	2	no	—
196	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	316	2	no	—
207	C₂×C₈	—	316	2	no	—
208	C₂×C₈	—	158	4	no	—
209	C₄⋊C₄	—	316	4	no	—
210	C₄⋊C₄	—	316	2	no	—
211	C₄⋊C₄	—	316	2	no	—
212	C₄⋊C₄	—	79	4	no	—
213	C₄⋊C₄	—	158	4	no	—
214	C₄⋊C₄	—	158	4	no	—
215	C₄⋊C₄	—	158	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	316	4	no	—
221	C₄²	C₈×D₇₉	1	2	yes	—
222	C₄²	#11929607	1	2	yes	—
223	C₄²	#11929612	1	2	yes	—
224	C₄²	#11929614	1	2	yes	—
225	C₄²	C₂×Dic₁₅₈	1	2	yes	—
226	C₄²	#11929624	1	2	yes	—
227	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
228	Q₈	—	316	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	316	2	no	—
239	C₂×C₄	—	316	2	no	—
240	C₂×C₄	—	316	2	no	—
241	C₂×C₄	—	316	2	no	—
242	C₂×C₄	—	316	2	no	—
243	C₂×C₄	—	158	4	no	—
244	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
245	C₂×C₄	(C₂×D₇₉)⋊C₈	1	2	yes	—
246	C₂×C₄	#14579084	1	2	yes	—
247	C₂×C₄	#14579084	1	2	yes	—
248	C₂×C₄	#14579087	1	2	yes	—
249	C₂×C₄	C₄×Dic₁₅₈	1	2	yes	—
250	C₂×C₄	#14579140	1	2	yes	—
251	C₂×C₄	#14579149	1	1	yes	characteristic
252	C₂×C₄	#14579161	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
254	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
255	C₂×C₄	#14579167	1	2	yes	—
256	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
257	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
258	C₂×C₄	#14579186	1	2	yes	—
259	C₈	—	4	4	no	—
260	C₈	—	316	4	no	—
261	C₂²	#19098469	1	1	yes	characteristic
262	C₂²	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
263	C₂²	#19098529	1	1	yes	characteristic
264	C₂²	C₂×#14579084	1	1	yes	characteristic
265	C₂²	#19098567	1	1	yes	characteristic
266	C₂²	C₂×#14579087	1	1	yes	characteristic
267	C₂²	#19098579	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	316	4	no	—
274	C₄	—	632	2	no	—
275	C₄	#19098293	1	4	yes	—
276	C₄	C₈×Dic₁₅₈	1	2	yes	—
277	C₄	#19098693	1	2	yes	—
278	C₄	#19098853	1	2	yes	—
279	C₄	#19098858	1	2	yes	—
280	C₂	C₂×#19098293	1	1	yes	characteristic
281	C₂	#23761790	1	1	yes	characteristic
282	C₂	#23761802	1	1	yes	characteristic
283	C₂	#23761816	1	1	yes	characteristic
284	C₂	#23764336	1	1	yes	characteristic
285	C₂	#23764344	1	1	yes	characteristic
286	C₂	#23764348	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#9864
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
23. #14579161	C₈	semidirect
45. C₄⋊C₈×C₇₉	Q₈	semidirect
148. C₇₉	#9864	semidirect
186. 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
C₄×#14579161	2
C₄×C₄⋊(C₇₉⋊C₈)	2
#23761239	2
C₄×C₄⋊C₈×C₇₉	2
#9864	79