Subgroups — #34187570

Order 20224.

642 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098940	C₂	1	1	yes	maximal, characteristic
2	#23764327	C₂	1	2	yes	maximal, complemented
3	C₂×#19098314	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098306	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₃₂	—	2	1	no	—
7	C₂²×D₆₃₂	—	2	1	no	—
8	C₂×#14579158	C₂²	1	2	yes	—
9	#19098940	—	2	2	no	—
10	#19098940	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
12	C₂×#14579081	—	2	4	no	—
13	#19098314	—	2	2	no	—
14	#19098306	—	2	4	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#97×C₇₉	—	2	1	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	#336×C₇₉	—	2	1	no	—
20	C₂²×D₃₁₆	—	2	2	no	—
21	C₂²×D₃₁₆	—	2	2	no	—
22	C₂²×D₃₁₆	—	2	2	no	—
23	C₂×D₆₃₂	—	2	4	no	—
24	C₂×D₆₃₂	—	2	4	no	—
25	C₂×D₆₃₂	—	4	2	no	—
26	C₂×D₆₃₂	—	4	2	no	—
27	C₂×D₆₃₂	—	4	2	no	—
28	C₂×D₆₃₂	—	4	2	no	—
29	C₂²×C₇₉⋊C₈	—	2	2	no	—
30	#14579158	—	2	2	no	—
31	#14579158	—	2	2	no	—
32	#14579158	—	2	2	no	—
33	#14579158	C₂×C₄	1	4	yes	—
34	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
36	#14579081	—	4	8	no	—
37	C₂²×C₈×C₇₉	—	2	1	no	—
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
41	C₂×C₄²×C₇₉	—	2	1	no	—
42	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₉	—	2	1	no	—
44	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
45	#97×C₇₉	—	4	2	no	—
46	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₇₉	D₄	1	2	yes	—
48	#95×C₇₉	—	2	1	no	—
49	C₂³×D₇₉	—	4	2	no	—
50	C₂×D₃₁₆	—	2	4	no	—
51	C₂×D₃₁₆	—	2	4	no	—
52	C₂×D₃₁₆	—	2	4	no	—
53	C₂×D₃₁₆	—	4	2	no	—
54	C₂×D₃₁₆	—	4	2	no	—
55	C₂×D₃₁₆	—	4	2	no	—
56	C₂×D₃₁₆	—	4	2	no	—
57	C₂×D₃₁₆	—	4	2	no	—
58	C₂×D₃₁₆	—	4	4	no	—
59	C₂×D₃₁₆	—	4	2	no	—
60	C₂×D₃₁₆	—	4	2	no	—
61	C₂×D₃₁₆	—	4	4	no	—
62	C₂×D₃₁₆	—	8	4	no	—
63	D₆₃₂	—	4	8	no	—
64	D₆₃₂	—	4	8	no	—
65	C₂×C₇₉⋊C₈	—	4	2	no	—
66	C₂×C₇₉⋊C₈	—	4	4	no	—
67	C₂²×C₄×C₇₉	—	2	2	no	—
68	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
71	C₂×C₈×C₇₉	—	2	2	no	—
72	C₂×C₈×C₇₉	—	2	1	no	—
73	C₂×C₈×C₇₉	—	2	1	no	—
74	C₂×C₈×C₇₉	—	4	1	no	—
75	C₂×C₈×C₇₉	—	4	1	no	—
76	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₇₉	D₈	1	2	yes	—
79	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₇₉	—	2	4	no	—
81	C₄²×C₇₉	—	4	1	no	—
82	C₄²×C₇₉	—	4	1	no	—
83	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇₉	#41	1	1	yes	characteristic
87	C₂³×C₇₉	#121	1	1	yes	characteristic
88	C₂²×D₇₉	—	4	4	no	—
89	C₂²×D₇₉	—	8	2	no	—
90	C₂²×D₇₉	—	8	2	no	—
91	C₂²×D₇₉	—	8	4	no	—
92	C₂²×D₇₉	—	8	4	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	—	4	2	no	—
95	C₂×C₄×C₇₉	—	4	2	no	—
96	C₂×C₄×C₇₉	—	4	2	no	—
97	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
108	D₃₁₆	—	4	4	no	—
109	D₃₁₆	—	4	4	no	—
110	D₃₁₆	—	4	8	no	—
111	D₃₁₆	—	8	4	no	—
112	D₃₁₆	—	8	4	no	—
113	D₃₁₆	—	8	2	no	—
114	D₃₁₆	—	8	2	no	—
115	D₃₁₆	—	8	2	no	—
116	D₃₁₆	—	8	2	no	—
117	C₈×C₇₉	—	2	2	no	—
118	C₈×C₇₉	—	4	2	no	—
119	C₈×C₇₉	#102	1	4	yes	complemented
120	C₇₉⋊C₈	—	8	2	no	—
121	C₂²×C₇₉	#390	1	1	yes	characteristic
122	C₂²×C₇₉	#459	1	1	yes	characteristic
123	C₂²×C₇₉	#460	1	1	yes	characteristic
124	C₂²×C₇₉	#469	1	2	yes	—
125	C₂²×C₇₉	#472	1	2	yes	—
126	C₂×D₇₉	—	8	4	no	—
127	C₂×D₇₉	—	8	4	no	—
128	C₂×D₇₉	—	8	4	no	—
129	C₂×D₇₉	—	16	4	no	—
130	C₂×D₇₉	—	16	4	no	—
131	C₄×C₇₉	—	2	1	no	—
132	C₄×C₇₉	—	2	1	no	—
133	C₄×C₇₉	—	8	2	no	—
134	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
135	C₄×C₇₉	#418	1	1	yes	characteristic
136	C₄×C₇₉	#442	1	2	yes	—
137	C₄×C₇₉	#459	1	1	yes	characteristic
138	C₄×C₇₉	#460	1	1	yes	characteristic
139	C₄×C₇₉	#496	1	2	yes	—
140	#9743	—	79	1	no	maximal, Sylow
141	C₂×C₇₉	#1687	1	2	yes	—
142	C₂×C₇₉	#1688	1	2	yes	—
143	C₂×C₇₉	#1972	1	1	yes	characteristic
144	C₂×C₇₉	#1978	1	1	yes	characteristic
145	C₂×C₇₉	#1982	1	1	yes	characteristic
146	D₇₉	—	16	4	no	—
147	C₂×#493	—	79	1	no	—
148	#1982	—	79	2	no	—
149	#1770	D₇₉	1	1	yes	characteristic, complemented
150	C₂×#334	—	79	1	no	—
151	C₂×#331	—	79	2	no	—
152	C₇₉	#9743	1	1	yes	characteristic, complemented, Sylow
153	C₂²×D₈	—	158	1	no	—
154	C₂²×D₈	—	158	1	no	—
155	C₂×#127	—	79	2	no	—
156	#493	—	79	4	no	—
157	#493	—	158	2	no	—
158	C₂×C₄⋊C₈	—	79	2	no	—
159	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
160	C₂×#102	—	158	4	no	—
161	C₂×#97	—	2	1	no	—
162	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
163	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
164	#336	—	2	1	no	—
165	#334	—	158	2	no	—
166	#331	—	158	4	no	—
167	C₂²×D₄	—	158	2	no	—
168	C₂²×D₄	—	158	2	no	—
169	C₂²×D₄	—	158	2	no	—
170	C₂×D₈	—	316	2	no	—
171	C₂×D₈	—	316	2	no	—
172	C₂×D₈	—	316	2	no	—
173	C₂×D₈	—	316	2	no	—
174	C₂×D₈	—	158	4	no	—
175	C₂×D₈	—	158	4	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	158	2	no	—
178	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
180	#127	—	79	4	no	—
181	#127	—	158	2	no	—
182	#127	—	158	2	no	—
183	#127	—	158	2	no	—
184	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	158	2	no	—
189	C₄⋊C₈	—	158	2	no	—
190	C₄⋊C₈	D₃₁₆	1	2	yes	—
191	#102	—	316	8	no	—
192	#97	—	4	2	no	—
193	C₄×C₈	C₄×D₇₉	1	2	yes	—
194	C₄×C₈	D₃₁₆	1	2	yes	—
195	#95	—	2	1	no	—
196	C₂⁴	—	316	2	no	—
197	C₂×D₄	—	316	2	no	—
198	C₂×D₄	—	316	2	no	—
199	C₂×D₄	—	316	2	no	—
200	C₂×D₄	—	316	2	no	—
201	C₂×D₄	—	316	2	no	—
202	C₂×D₄	—	316	2	no	—
203	C₂×D₄	—	316	2	no	—
204	C₂×D₄	—	316	4	no	—
205	C₂×D₄	—	316	4	no	—
206	C₂×D₄	—	632	4	no	—
207	C₂×D₄	—	158	4	no	—
208	C₂×D₄	—	158	4	no	—
209	C₂×D₄	—	158	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
213	C₂²×C₄	#11929624	1	1	yes	characteristic
214	D₈	—	316	8	no	—
215	D₈	—	316	8	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	316	2	no	—
222	C₂×C₈	—	316	4	no	—
223	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
224	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
225	C₂×C₈	#11929609	1	2	yes	—
226	C₂×C₈	#11929611	1	2	yes	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
231	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
232	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
233	C₄²	#11929624	1	1	yes	characteristic
234	C₂³	—	316	4	no	—
235	C₂³	—	632	2	no	—
236	C₂³	—	632	2	no	—
237	C₂³	—	632	4	no	—
238	C₂³	—	632	4	no	—
239	C₂³	#14579135	1	1	yes	characteristic
240	D₄	—	316	8	no	—
241	D₄	—	316	4	no	—
242	D₄	—	316	4	no	—
243	D₄	—	632	4	no	—
244	D₄	—	632	4	no	—
245	D₄	—	632	2	no	—
246	D₄	—	632	2	no	—
247	D₄	—	632	2	no	—
248	D₄	—	632	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
255	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
256	C₂×C₄	#14579135	1	1	yes	characteristic
257	C₂×C₄	#14579149	1	1	yes	characteristic
258	C₂×C₄	#14579158	1	1	yes	characteristic
259	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
260	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
261	C₂×C₄	#14579186	1	1	yes	characteristic
262	C₂×C₄	#14579186	1	1	yes	characteristic
263	C₂×C₄	#14579189	1	4	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	632	2	no	—
267	C₈	#14579081	1	4	yes	complemented
268	C₂²	—	632	4	no	—
269	C₂²	—	632	4	no	—
270	C₂²	—	632	4	no	—
271	C₂²	—	1264	4	no	—
272	C₂²	—	1264	4	no	—
273	C₂²	#19098465	1	1	yes	characteristic
274	C₂²	#19098758	1	1	yes	characteristic
275	C₂²	#19098763	1	1	yes	characteristic
276	C₂²	#19098816	1	2	yes	—
277	C₂²	#19098833	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	C₂×#14579081	1	1	yes	characteristic
282	C₄	#19098567	1	1	yes	characteristic
283	C₄	#19098676	1	2	yes	—
284	C₄	#19098758	1	1	yes	characteristic
285	C₄	#19098763	1	1	yes	characteristic
286	C₄	#19098952	1	2	yes	—
287	C₂	—	1264	4	no	—
288	C₂	#23762879	1	2	yes	—
289	C₂	#23762884	1	2	yes	—
290	C₂	#23764276	1	1	yes	characteristic
291	C₂	#23764310	1	1	yes	characteristic
292	C₂	#23764327	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579135
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1770	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂×#19098940	C₂
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1770	D₇₉
hypocenter	C₇₉	#9743
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #23764327	C₂	semidirect
3. C₂×#19098314	C₂	semidirect
4. C₂×#19098306	C₂	semidirect
5. C₇₉×#1770	C₂	semidirect
10. #19098940	C₄	semidirect
119. C₈×C₇₉	#102	semidirect
149. #1770	D₇₉	semidirect
152. C₇₉	#9743	semidirect
267. C₈	#14579081	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098940	2
#23764327	2
C₂×#19098314	2
C₂×#19098306	2
C₇₉×#1770	2
#9743	79