Subgroups — #34306753

Order 20224.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 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	#23772277	C₂	1	2	yes	maximal
3	#23771992	C₂	1	2	yes	maximal, complemented
4	#23764185	C₂	1	2	yes	maximal, complemented
5	#23763634	C₂	1	4	yes	maximal, complemented
6	#23763564	C₂	1	1	yes	maximal, characteristic, complemented
7	#23763456	C₂	1	2	yes	maximal, complemented
8	C₇₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₂²	1	1	yes	characteristic
10	C₂×#14579167	C₂²	1	2	yes	—
11	#19099043	C₂²	1	4	yes	—
12	#19098681	C₄	1	8	yes	complemented
13	#19098665	—	2	4	no	—
14	C₄×Dic₃₁₆	—	2	2	no	—
15	C₄×D₆₃₂	—	2	2	no	—
16	C₂×#14579097	C₂²	1	1	yes	characteristic
17	#19098606	C₂²	1	4	yes	—
18	#19098604	—	2	4	no	—
19	#19098598	—	2	2	no	—
20	C₂×#14579095	C₂²	1	2	yes	—
21	C₂×#14579093	C₂²	1	2	yes	—
22	#106×D₇₉	—	2	2	no	—
23	C₂×#14579089	C₂²	1	2	yes	—
24	#19098565	C₂²	1	4	yes	—
25	#19098429	C₂²	1	2	yes	—
26	#19098331	C₂²	1	4	yes	—
27	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
28	#427×C₇₉	C₂²	1	1	yes	characteristic
29	#405×C₇₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
31	#339×C₇₉	C₂²	1	2	yes	—
32	C₂×#11929624	C₂³	1	2	yes	—
33	#14579183	C₂×C₄	1	4	yes	—
34	#14579183	Q₈	1	4	yes	—
35	C₂×Dic₃₁₆	D₄	1	1	yes	characteristic
36	C₂×#11929613	D₄	1	2	yes	—
37	C₂×D₆₃₂	D₄	1	1	yes	characteristic
38	#14579167	—	2	4	no	—
39	#14579167	C₂×C₄	1	8	yes	complemented
40	C₂×#11929608	C₂³	1	2	yes	—
41	C₂×#11929607	—	2	2	no	—
42	C₂×C₈×D₇₉	—	2	2	no	—
43	C₄×Dic₁₅₈	—	2	4	no	—
44	C₄×D₃₁₆	—	2	4	no	—
45	(C₂²⋊D₇₉)⋊C₄	—	4	4	no	—
46	#14579119	—	2	4	no	—
47	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
48	C₄⋊C₄×D₇₉	—	2	4	no	—
49	#14579097	C₂×C₄	1	4	yes	complemented
50	#14579095	—	2	2	no	—
51	#14579095	—	2	2	no	—
52	#14579094	—	2	4	no	—
53	#14579093	—	2	2	no	—
54	#14579093	—	2	2	no	—
55	#14579092	—	2	4	no	—
56	#14579089	C₂×C₄	1	8	yes	complemented
57	#14579086	—	2	4	no	—
58	#14579083	—	2	4	no	—
59	C₇₉⋊#95	—	4	2	no	—
60	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
63	#117×C₇₉	C₂³	1	2	yes	—
64	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
65	#108×C₇₉	—	2	2	no	—
66	#107×C₇₉	D₄	1	2	yes	—
67	#106×C₇₉	D₄	1	1	yes	characteristic
68	#106×C₇₉	D₄	1	1	yes	characteristic
69	#97×C₇₉	—	2	2	no	—
70	C₄×C₈×C₇₉	—	2	2	no	—
71	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	complemented
72	#11929624	—	2	2	no	—
73	#11929624	—	2	2	no	—
74	#11929624	C₂²×C₄	1	4	yes	—
75	#11929624	C₂×Q₈	1	4	yes	—
76	C₂×Dic₁₅₈	C₂×D₄	1	2	yes	—
77	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
78	C₂×C₂²⋊D₇₉	—	2	2	no	—
79	C₂²×Dic₇₉	—	2	4	no	—
80	C₂×C₄×D₇₉	—	2	2	no	—
81	C₂×C₄×D₇₉	—	4	4	no	—
82	Dic₃₁₆	C₄⋊C₄	1	4	yes	—
83	#11929613	C₄⋊C₄	1	8	yes	—
84	D₆₃₂	C₄⋊C₄	1	4	yes	—
85	#11929608	—	2	2	no	—
86	#11929608	C₂²×C₄	1	4	yes	—
87	#11929607	—	2	8	no	—
88	C₂×C₇₉⋊C₈	—	2	2	no	—
89	C₈×D₇₉	—	2	8	no	—
90	C₄⋊Dic₇₉	—	2	4	no	—
91	C₄⋊Dic₇₉	—	2	4	no	—
92	Dic₇₉⋊C₄	—	4	4	no	—
93	(C₂×D₇₉)⋊C₄	—	4	4	no	—
94	C₄×Dic₇₉	—	4	4	no	—
95	C₂²×C₄×C₇₉	—	2	1	no	—
96	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₇₉	—	2	2	no	—
98	M₁₆×C₇₉	—	2	2	no	—
99	C₂×C₈×C₇₉	—	2	2	no	—
100	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₇₉	#41	1	2	yes	—
108	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
109	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄×C₇₉	—	2	2	no	—
111	C₄²×C₇₉	—	2	2	no	—
112	C₄²×C₇₉	—	2	1	no	—
113	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
114	C₂³×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×D₇₉	—	2	2	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	2	no	—
119	C₂×C₄×C₇₉	—	4	2	no	—
120	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂²⋊D₇₉	—	2	4	no	—
127	C₂²⋊D₇₉	—	2	4	no	—
128	C₂×Dic₇₉	—	2	2	no	—
129	C₂×Dic₇₉	—	4	4	no	—
130	C₂×Dic₇₉	—	8	4	no	—
131	D₃₁₆	—	2	2	no	—
132	D₃₁₆	C₂×C₄⋊C₄	1	4	yes	—
133	C₄×D₇₉	—	2	4	no	—
134	C₄×D₇₉	—	2	4	no	—
135	C₄×D₇₉	—	16	4	no	—
136	Dic₁₅₈	—	2	2	no	—
137	Dic₁₅₈	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₇₉	—	4	2	no	—
139	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
141	C₈×C₇₉	C₄×D₄	1	2	yes	—
142	C₈×C₇₉	C₄×D₄	1	2	yes	—
143	C₇₉⋊C₈	—	2	4	no	—
144	C₂²×C₇₉	—	2	2	no	—
145	C₂²×C₇₉	#520	1	1	yes	characteristic
146	C₂²×C₇₉	D₄²	1	1	yes	characteristic
147	C₂²×C₇₉	#554	1	1	yes	characteristic
148	C₂×D₇₉	—	2	4	no	—
149	C₂×D₇₉	—	4	4	no	—
150	C₄×C₇₉	—	4	2	no	—
151	C₄×C₇₉	—	4	2	no	—
152	C₄×C₇₉	—	4	2	no	—
153	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₇₉	#547	1	1	yes	characteristic
156	C₄×C₇₉	D₄×Q₈	1	1	yes	characteristic
157	Dic₇₉	—	2	4	no	—
158	Dic₇₉	—	8	4	no	—
159	#22690	—	79	1	no	maximal, Sylow
160	C₂×C₇₉	—	2	1	no	—
161	C₂×C₇₉	—	2	1	no	—
162	C₂×C₇₉	D₄×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₇₉	#3290	1	2	yes	—
164	D₇₉	—	4	4	no	—
165	C₂×#443	—	79	1	no	—
166	#2951	—	79	2	no	—
167	#2912	—	79	2	no	—
168	#1952	—	79	2	no	—
169	#1860	—	79	4	no	—
170	#1846	—	79	1	no	—
171	#1815	—	79	2	no	—
172	#1772	D₇₉	1	1	yes	characteristic, complemented
173	C₇₉	#22690	1	1	yes	characteristic, complemented, Sylow
174	C₂×#135	—	79	1	no	—
175	C₂×#131	—	79	2	no	—
176	#520	—	79	4	no	—
177	#443	—	79	8	no	—
178	#440	—	158	4	no	—
179	C₄×Q₁₆	—	158	2	no	—
180	C₄×D₈	—	158	2	no	—
181	C₂×#108	—	79	1	no	—
182	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
183	#428	—	79	4	no	—
184	#428	—	158	4	no	—
185	#427	—	158	2	no	—
186	#427	C₂×D₇₉	1	1	yes	characteristic
187	C₂×#107	—	79	2	no	—
188	C₂×#106	—	79	2	no	—
189	C₂×#106	—	158	2	no	—
190	C₂×#104	—	79	2	no	—
191	#417	—	79	4	no	—
192	#405	C₂×D₇₉	1	2	yes	—
193	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
194	#382	—	79	2	no	—
195	#343	—	79	4	no	—
196	#339	C₂×D₇₉	1	2	yes	—
197	C₂×#41	—	79	2	no	—
198	#135	—	79	4	no	—
199	#135	—	79	4	no	—
200	C₂×Q₁₆	—	79	1	no	—
201	C₂×SD₁₆	—	79	2	no	—
202	C₂×D₈	—	79	1	no	—
203	#131	—	79	8	no	—
204	#131	—	158	4	no	—
205	C₂×M₁₆	—	79	2	no	—
206	C₂×M₁₆	—	158	2	no	—
207	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
208	C₂²×C₈	—	158	2	no	—
209	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
211	C₄×Q₈	—	158	4	no	—
212	C₄×D₄	—	316	4	no	—
213	C₄×D₄	—	158	4	no	—
214	#117	—	158	4	no	—
215	#117	C₂²×D₇₉	1	2	yes	—
216	C₂×C₄⋊C₄	—	79	4	no	—
217	C₂×C₄⋊C₄	—	158	4	no	—
218	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
219	#108	—	2	2	no	—
220	#108	—	79	4	no	—
221	#107	—	158	2	no	—
222	#107	—	158	2	no	—
223	#107	—	158	4	no	—
224	#107	D₃₁₆	1	2	yes	—
225	#106	—	158	2	no	—
226	#106	—	158	2	no	—
227	#106	—	158	4	no	—
228	#106	D₃₁₆	1	1	yes	characteristic
229	#106	D₃₁₆	1	1	yes	characteristic
230	#104	—	79	8	no	—
231	#103	—	158	4	no	—
232	#102	—	158	4	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	C₄×D₇₉	1	4	yes	complemented
236	#95	—	316	2	no	—
237	#41	—	79	4	no	—
238	#41	—	79	4	no	—
239	#41	—	158	2	no	—
240	#41	—	158	2	no	—
241	C₂×Q₈	—	79	2	no	—
242	C₂×D₄	—	79	2	no	—
243	C₂×D₄	—	158	2	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	316	4	no	—
246	C₂²×C₄	—	158	4	no	—
247	C₂²×C₄	—	158	2	no	—
248	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
249	Q₁₆	—	79	4	no	—
250	SD₁₆	—	79	8	no	—
251	D₈	—	79	4	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	79	4	no	—
255	M₁₆	—	158	2	no	—
256	M₁₆	—	158	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	158	2	no	—
259	C₂×C₈	—	158	8	no	—
260	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
262	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
263	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
264	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
265	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
266	C₂×C₈	Q₈×D₇₉	1	2	yes	—
267	C₂×C₈	#11929624	1	2	yes	—
268	C₄⋊C₄	—	316	4	no	—
269	C₄⋊C₄	—	158	4	no	—
270	C₄⋊C₄	—	158	4	no	—
271	C₄⋊C₄	C₂×D₃₁₆	1	2	yes	—
272	C₄⋊C₄	C₂×D₃₁₆	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	316	4	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	316	4	no	—
278	C₄²	C₂×C₄×D₇₉	1	2	yes	—
279	C₂³	—	158	2	no	—
280	C₂³	#14579207	1	1	yes	characteristic
281	Q₈	—	79	4	no	—
282	Q₈	—	158	2	no	—
283	D₄	—	79	4	no	—
284	D₄	—	158	4	no	—
285	D₄	—	158	4	no	—
286	D₄	—	158	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	316	4	no	—
292	C₂×C₄	—	632	4	no	—
293	C₂×C₄	—	158	2	no	—
294	C₂×C₄	—	158	4	no	—
295	C₂×C₄	—	158	4	no	—
296	C₂×C₄	—	1264	4	no	—
297	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
301	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
302	C₂×C₄	#14579208	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	—	158	4	no	—
305	C₈	C₄⋊C₄×D₇₉	1	2	yes	—
306	C₈	C₄⋊C₄×D₇₉	1	2	yes	—
307	C₈	C₄×D₃₁₆	1	2	yes	—
308	C₈	C₄×D₃₁₆	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	316	4	no	—
311	C₂²	—	158	4	no	—
312	C₂²	#19099043	1	1	yes	characteristic
313	C₂²	D₄×D₃₁₆	1	1	yes	characteristic
314	C₂²	#19099273	1	1	yes	characteristic
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	632	4	no	—
319	C₄	—	158	4	no	—
320	C₄	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
321	C₄	C₂×C₄×D₃₁₆	1	1	yes	characteristic
322	C₄	#19099218	1	1	yes	characteristic
323	C₄	Q₈×D₃₁₆	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	316	4	no	—
327	C₂	C₄⋊C₄×D₃₁₆	1	1	yes	characteristic
328	C₂	#23775187	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1772	C₂
socle	C₂²×C₇₉	D₄²
prime radical	C₂×#14579183	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1772	D₇₉
hypocenter	C₇₉	#22690
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #23771992	C₂	semidirect
4. #23764185	C₂	semidirect
5. #23763634	C₂	semidirect
6. #23763564	C₂	semidirect
7. #23763456	C₂	semidirect
8. C₇₉×#1772	C₂	semidirect
12. #19098681	C₄	semidirect
39. #14579167	C₂×C₄	semidirect
49. #14579097	C₂×C₄	semidirect
56. #14579089	C₂×C₄	semidirect
71. C₄×C₈×C₇₉	C₂×C₄	semidirect
172. #1772	D₇₉	semidirect
173. C₇₉	#22690	semidirect
235. C₄×C₈	C₄×D₇₉	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
#23772277	2
#23771992	2
#23764185	2
#23763634	2
#23763564	2
#23763456	2
C₇₉×#1772	2
#22690	79