Subgroups — #34203505

Order 20224.

438 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098681	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098371	C₂	1	1	yes	maximal, characteristic
3	#23765108	C₂	1	1	yes	maximal, characteristic
4	#23763279	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761496	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579183	C₂²	1	1	yes	characteristic
9	C₂×#14579167	—	2	1	no	—
10	#19098681	—	2	2	no	—
11	#19098681	C₄	1	4	yes	complemented
12	C₂×#14579097	C₂²	1	1	yes	characteristic
13	C₂×#14579097	C₂²	1	1	yes	characteristic
14	#19098601	C₂²	1	1	yes	characteristic
15	C₂×#14579089	—	2	1	no	—
16	#19098518	—	2	1	no	—
17	#19098371	C₄	1	4	yes	complemented
18	#19098368	C₄	1	2	yes	complemented
19	#19098365	C₄	1	2	yes	complemented
20	#19098323	—	2	1	no	—
21	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₇₉	—	2	2	no	—
25	C₂×#11929624	—	2	1	no	—
26	#14579183	—	2	1	no	—
27	#14579183	—	2	1	no	—
28	#14579183	C₂×C₄	1	2	yes	—
29	#14579183	C₂×C₄	1	2	yes	—
30	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×#11929613	—	2	1	no	—
32	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
33	#14579167	—	2	4	no	—
34	#14579167	—	4	2	no	—
35	C₂×#11929608	—	2	1	no	—
36	C₂×#11929608	—	2	1	no	—
37	C₂×#11929607	—	4	1	no	—
38	C₂×C₈×D₇₉	—	4	1	no	—
39	#14579120	—	2	1	no	—
40	#14579097	—	2	1	no	—
41	#14579097	—	2	1	no	—
42	#14579097	C₂×C₄	1	2	yes	—
43	#14579097	C₂×C₄	1	2	yes	—
44	#14579095	C₂×C₄	1	2	yes	—
45	#14579093	—	2	1	no	—
46	#14579089	—	2	4	no	—
47	#14579071	—	4	1	no	—
48	C₈×Dic₇₉	—	4	1	no	—
49	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
52	C₂×C₁₆×C₇₉	—	2	1	no	—
53	C₂×C₁₆×C₇₉	—	2	1	no	—
54	C₂×C₁₆×C₇₉	—	2	1	no	—
55	C₂×C₁₆×C₇₉	—	2	1	no	—
56	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₇₉	D₄	1	2	yes	—
60	#11929624	—	2	2	no	—
61	#11929624	—	2	2	no	—
62	#11929624	—	4	1	no	—
63	#11929624	—	4	1	no	—
64	C₂×Dic₁₅₈	—	2	1	no	—
65	C₂×D₃₁₆	—	2	1	no	—
66	C₂×C₂²⋊D₇₉	—	4	1	no	—
67	C₂×C₄×D₇₉	—	4	1	no	—
68	Dic₃₁₆	—	2	1	no	—
69	Dic₃₁₆	C₄²	1	2	yes	—
70	#11929613	—	2	2	no	—
71	#11929613	—	2	2	no	—
72	D₆₃₂	—	2	1	no	—
73	D₆₃₂	C₄²	1	2	yes	—
74	#11929608	—	2	2	no	—
75	#11929608	—	2	2	no	—
76	#11929608	—	4	1	no	—
77	#11929608	—	4	1	no	—
78	#11929607	—	4	4	no	—
79	C₂×C₇₉⋊C₈	—	4	1	no	—
80	C₂×C₇₉⋊C₈	—	4	1	no	—
81	C₈×D₇₉	—	4	4	no	—
82	C₄⋊Dic₇₉	—	2	2	no	—
83	C₂²⋊Dic₇₉	—	4	1	no	—
84	C₄×Dic₇₉	—	4	1	no	—
85	C₂²×C₄×C₇₉	—	2	1	no	—
86	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
89	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
90	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
96	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
97	C₄²×C₇₉	—	2	1	no	—
98	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
99	C₁₆×C₇₉	—	2	2	no	—
100	C₁₆×C₇₉	—	2	2	no	—
101	C₂³×C₇₉	#135	1	1	yes	characteristic
102	C₂²×D₇₉	—	4	1	no	—
103	C₂×C₄×C₇₉	—	2	4	no	—
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
111	C₂²⋊D₇₉	—	4	2	no	—
112	C₂²⋊D₇₉	—	4	2	no	—
113	C₂×Dic₇₉	—	4	1	no	—
114	C₂×Dic₇₉	—	4	2	no	—
115	D₃₁₆	—	2	2	no	—
116	D₃₁₆	—	4	1	no	—
117	C₄×D₇₉	—	4	2	no	—
118	C₄×D₇₉	—	4	2	no	—
119	Dic₁₅₈	—	2	2	no	—
120	Dic₁₅₈	—	4	1	no	—
121	C₈×C₇₉	#102	1	2	yes	—
122	C₈×C₇₉	#102	1	2	yes	—
123	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
124	C₈×C₇₉	#117	1	1	yes	characteristic
125	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
126	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
127	C₇₉⋊C₈	—	4	2	no	—
128	C₇₉⋊C₈	—	4	2	no	—
129	C₂²×C₇₉	—	2	2	no	—
130	C₂²×C₇₉	#416	1	1	yes	characteristic
131	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
132	C₂²×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
133	C₂×D₇₉	—	4	2	no	—
134	C₂×D₇₉	—	8	2	no	—
135	C₄×C₇₉	—	2	4	no	—
136	C₄×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
138	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
139	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
140	Dic₇₉	—	4	2	no	—
141	Dic₇₉	—	8	2	no	—
142	#12641	—	79	1	no	maximal, Sylow
143	C₂×C₇₉	—	2	1	no	—
144	C₂×C₇₉	—	2	1	no	—
145	C₂×C₇₉	C₄×#102	1	1	yes	characteristic
146	C₂×C₇₉	#2178	1	2	yes	—
147	D₇₉	—	8	2	no	—
148	C₂×#443	—	79	1	no	—
149	C₂×#359	—	79	1	no	—
150	#2138	—	79	1	no	—
151	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic, complemented
152	#1772	—	79	1	no	—
153	#1382	—	79	1	no	—
154	#1381	—	79	1	no	—
155	C₇₉	#12641	1	1	yes	characteristic, complemented, Sylow
156	C₂×#135	—	79	1	no	—
157	C₂×#131	—	158	1	no	—
158	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
159	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
160	#443	—	79	4	no	—
161	#443	—	158	2	no	—
162	C₂×#108	—	79	1	no	—
163	C₂×#108	—	79	1	no	—
164	#427	—	79	1	no	—
165	C₂×#104	—	158	1	no	—
166	#405	—	158	1	no	—
167	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
168	#359	—	79	4	no	—
169	#358	—	79	2	no	—
170	#357	—	79	2	no	—
171	C₄×C₁₆	—	2	2	no	—
172	#339	—	158	1	no	—
173	C₂×#41	—	158	1	no	—
174	#135	—	79	2	no	—
175	#135	—	79	2	no	—
176	#135	—	158	1	no	—
177	#135	—	158	1	no	—
178	C₂×Q₁₆	—	79	1	no	—
179	C₂×SD₁₆	—	158	1	no	—
180	C₂×D₈	—	79	1	no	—
181	#131	—	316	2	no	—
182	#131	—	158	4	no	—
183	C₂×M₁₆	—	316	1	no	—
184	C₂×M₁₆	—	158	1	no	—
185	C₂×M₁₆	—	158	1	no	—
186	C₂²×C₈	—	316	1	no	—
187	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
189	#117	—	158	1	no	—
190	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
196	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
197	#108	—	79	2	no	—
198	#108	—	79	2	no	—
199	#108	—	158	1	no	—
200	#108	—	158	1	no	—
201	#107	—	79	2	no	—
202	#106	—	158	1	no	—
203	#104	—	158	4	no	—
204	#97	—	316	1	no	—
205	C₄×C₈	—	316	1	no	—
206	C₄×C₈	C₄×D₇₉	1	2	yes	—
207	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
208	#41	—	316	1	no	—
209	#41	—	316	1	no	—
210	#41	—	158	2	no	—
211	#41	—	158	2	no	—
212	C₂×Q₈	—	158	1	no	—
213	C₂×D₄	—	316	1	no	—
214	C₂×D₄	—	158	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	316	1	no	—
217	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
218	Q₁₆	—	79	2	no	—
219	Q₁₆	—	158	1	no	—
220	SD₁₆	—	158	2	no	—
221	SD₁₆	—	158	2	no	—
222	D₈	—	79	2	no	—
223	D₈	—	158	1	no	—
224	M₁₆	—	316	1	no	—
225	M₁₆	—	316	1	no	—
226	M₁₆	—	316	4	no	—
227	M₁₆	—	158	2	no	—
228	M₁₆	—	158	2	no	—
229	C₂×C₈	—	316	1	no	—
230	C₂×C₈	—	316	1	no	—
231	C₂×C₈	—	316	4	no	—
232	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
233	C₂×C₈	D₆₃₂	1	1	yes	characteristic
234	C₂×C₈	D₆₃₂	1	1	yes	characteristic
235	C₂×C₈	#11929613	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
239	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
240	C₂×C₈	#11929624	1	1	yes	characteristic
241	C₂×C₈	#11929624	1	1	yes	characteristic
242	C₄⋊C₄	—	158	2	no	—
243	C₂²⋊C₄	—	316	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	316	1	no	—
246	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	2	2	no	—
249	C₂³	—	316	1	no	—
250	C₂³	#14579183	1	1	yes	characteristic
251	Q₈	—	316	1	no	—
252	Q₈	—	158	2	no	—
253	D₄	—	316	2	no	—
254	D₄	—	316	2	no	—
255	D₄	—	316	1	no	—
256	D₄	—	158	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	316	1	no	—
260	C₂×C₄	—	316	2	no	—
261	C₂×C₄	—	316	2	no	—
262	C₂×C₄	—	316	2	no	—
263	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
265	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
266	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
267	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
268	C₂×C₄	#14579183	1	1	yes	characteristic
269	C₈	—	316	2	no	—
270	C₈	—	316	2	no	—
271	C₈	#14579083	1	2	yes	—
272	C₈	#14579083	1	2	yes	—
273	C₈	C₄²×D₇₉	1	1	yes	characteristic
274	C₈	#14579118	1	1	yes	characteristic
275	C₈	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
276	C₈	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	316	2	no	—
279	C₂²	—	632	2	no	—
280	C₂²	#19098560	1	1	yes	characteristic
281	C₂²	C₄×D₆₃₂	1	1	yes	characteristic
282	C₂²	C₄×#11929613	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	316	2	no	—
285	C₄	—	632	2	no	—
286	C₄	C₄×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
287	C₄	C₂×#14579083	1	1	yes	characteristic
288	C₄	C₄×D₆₃₂	1	1	yes	characteristic
289	C₄	C₄×#11929613	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	632	2	no	—
293	C₂	C₄×#14579083	1	1	yes	characteristic
294	C₂	#23765282	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₆₃₂
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₄×D₈
prime radical	C₂×#14579183	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#12641
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
4. #23763279	C₂	semidirect
5. #23761496	C₂	semidirect
6. #23761494	C₂	semidirect
7. C₂×C₄×C₁₆×C₇₉	C₂	semidirect
11. #19098681	C₄	semidirect
17. #19098371	C₄	semidirect
18. #19098368	C₄	semidirect
19. #19098365	C₄	semidirect
151. C₂×C₄×C₁₆	D₇₉	semidirect
155. C₇₉	#12641	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098681	2
C₂×#19098371	2
#23765108	2
#23763279	2
#23761496	2
#23761494	2
C₂×C₄×C₁₆×C₇₉	2
#12641	79