Subgroups — #34260262

Order 20224.

572 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773986	C₂	1	1	yes	maximal, characteristic
2	#23772501	C₂	1	2	yes	maximal
3	#23762298	C₂	1	2	yes	maximal, complemented
4	#23762273	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₃₁₆	C₂	1	2	yes	maximal, complemented
6	#23762148	C₂	1	1	yes	maximal, characteristic
7	#23761822	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#19099120	C₂²	1	2	yes	—
10	#19098968	C₄	1	1	yes	characteristic
11	#19098943	C₄	1	1	yes	characteristic
12	#19098914	C₄	1	2	yes	—
13	#19098804	C₄	1	2	yes	—
14	#19098804	C₄	1	2	yes	—
15	#19098693	C₂²	1	4	yes	—
16	#19098690	—	2	2	no	—
17	C₈×Dic₁₅₈	C₂²	1	4	yes	—
18	#19098687	—	2	2	no	—
19	C₄×Dic₃₁₆	C₂²	1	2	yes	—
20	C₄×Dic₃₁₆	C₂²	1	2	yes	—
21	#19098621	—	2	4	no	—
22	#19098616	C₂²	1	2	yes	—
23	#19098316	C₂²	1	1	yes	characteristic
24	#19098316	C₂²	1	1	yes	characteristic
25	#19098314	C₂²	1	2	yes	—
26	#19098295	C₂²	1	4	yes	—
27	#19098295	C₂²	1	4	yes	—
28	#431×C₇₉	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
31	#322×C₇₉	C₂²	1	2	yes	—
32	C₈²×C₇₉	C₂²	1	2	yes	—
33	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
35	#14579161	C₂×C₄	1	2	yes	—
36	#14579161	C₂×C₄	1	2	yes	—
37	#14579153	—	2	2	no	—
38	#14579140	—	2	2	no	—
39	#14579140	C₂×C₄	1	4	yes	—
40	C₄×Dic₁₅₈	—	2	2	no	—
41	C₄×Dic₁₅₈	C₂³	1	4	yes	—
42	#14579120	—	2	2	no	—
43	#14579095	C₂×C₄	1	2	yes	—
44	#14579095	C₂×C₄	1	2	yes	—
45	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
46	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
47	Dic₇₉⋊C₈	—	2	4	no	—
48	Dic₇₉⋊C₈	—	2	4	no	—
49	#14579086	C₂×C₄	1	4	yes	—
50	#14579086	C₂×C₄	1	4	yes	—
51	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
52	#14579071	—	2	4	no	—
53	C₄×C₇₉⋊C₈	—	2	2	no	—
54	C₈×Dic₇₉	—	2	4	no	—
55	C₂×M₁₆×C₇₉	—	2	1	no	—
56	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
60	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
61	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
62	#97×C₇₉	—	2	1	no	—
63	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₇₉	C₂³	1	2	yes	—
66	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₉	C₂³	1	2	yes	—
69	C₂×Dic₁₅₈	—	2	2	no	—
70	C₂×Dic₁₅₈	C₂²×C₄	1	4	yes	—
71	Dic₃₁₆	—	2	4	no	—
72	Dic₃₁₆	—	4	2	no	—
73	C₂×C₇₉⋊C₈	—	2	2	no	—
74	C₂×C₇₉⋊C₈	—	2	2	no	—
75	C₄⋊Dic₇₉	—	2	2	no	—
76	C₄⋊Dic₇₉	—	2	2	no	—
77	C₄⋊Dic₇₉	—	2	2	no	—
78	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
79	Dic₇₉⋊C₄	—	2	4	no	—
80	Dic₇₉⋊C₄	—	2	4	no	—
81	C₂²⋊Dic₇₉	—	2	4	no	—
82	C₄×Dic₇₉	—	2	4	no	—
83	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
86	M₁₆×C₇₉	—	2	4	no	—
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	4	1	no	—
90	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
95	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
96	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₉	#41	1	1	yes	characteristic
99	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
100	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
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	—	2	2	no	—
106	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₇₉	—	2	4	no	—
113	C₂×Dic₇₉	—	2	4	no	—
114	Dic₁₅₈	—	2	4	no	—
115	Dic₁₅₈	—	2	4	no	—
116	Dic₁₅₈	—	4	4	no	—
117	C₈×C₇₉	—	2	4	no	—
118	C₈×C₇₉	—	2	1	no	—
119	C₈×C₇₉	—	2	1	no	—
120	C₈×C₇₉	—	2	1	no	—
121	C₈×C₇₉	—	2	2	no	—
122	C₈×C₇₉	—	2	2	no	—
123	C₈×C₇₉	#131	1	2	yes	—
124	C₇₉⋊C₈	—	4	2	no	—
125	C₇₉⋊C₈	—	4	2	no	—
126	C₂²×C₇₉	—	2	1	no	—
127	C₂²×C₇₉	—	2	1	no	—
128	C₂²×C₇₉	—	2	1	no	—
129	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₇₉	—	2	1	no	—
131	C₄×C₇₉	—	2	1	no	—
132	C₄×C₇₉	—	2	1	no	—
133	C₄×C₇₉	#443	1	2	yes	—
134	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
135	C₄×C₇₉	#568	1	1	yes	characteristic
136	C₄×C₇₉	#578	1	1	yes	characteristic
137	C₄×C₇₉	#578	1	1	yes	characteristic
138	Dic₇₉	—	4	4	no	—
139	Dic₇₉	—	4	4	no	—
140	#18086	—	79	1	no	maximal, Sylow
141	C₂×C₇₉	—	4	1	no	—
142	C₂×C₇₉	#2935	1	1	yes	characteristic
143	C₂×C₇₉	#2950	1	1	yes	characteristic
144	C₂×C₇₉	C₂×#443	1	1	yes	characteristic
145	#3170	—	79	1	no	—
146	#2986	—	79	2	no	—
147	#1591	—	79	2	no	—
148	#1587	—	79	4	no	—
149	C₈×Q₁₆	—	79	2	no	—
150	#1564	—	79	1	no	—
151	#1485	—	79	2	no	—
152	C₈×M₁₆	D₇₉	1	1	yes	characteristic
153	C₇₉	#18086	1	1	yes	characteristic, complemented, Sylow
154	#533	—	79	2	no	—
155	#500	—	79	1	no	—
156	#494	—	79	1	no	—
157	#487	—	79	2	no	—
158	#467	—	79	2	no	—
159	#467	—	79	2	no	—
160	#446	—	79	4	no	—
161	#446	—	158	2	no	—
162	C₈×Q₈	—	79	4	no	—
163	C₈×Q₈	—	158	2	no	—
164	C₄×Q₁₆	—	79	2	no	—
165	C₄×Q₁₆	—	79	2	no	—
166	#433	—	158	4	no	—
167	#431	—	79	2	no	—
168	#431	C₂×D₇₉	1	1	yes	characteristic
169	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
171	#335	—	79	1	no	—
172	#335	—	79	1	no	—
173	#334	—	79	2	no	—
174	Q₈⋊C₈	—	79	4	no	—
175	Q₈⋊C₈	—	79	4	no	—
176	#322	C₂×D₇₉	1	2	yes	—
177	C₈²	C₂×D₇₉	1	2	yes	—
178	C₂×Q₁₆	—	79	2	no	—
179	C₂×Q₁₆	—	79	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
183	C₄⋊Q₈	—	79	2	no	—
184	C₄⋊Q₈	—	79	2	no	—
185	#125	—	158	2	no	—
186	C₂²⋊Q₈	—	79	4	no	—
187	C₂²⋊Q₈	—	158	2	no	—
188	C₄×Q₈	—	79	4	no	—
189	C₄×Q₈	—	158	2	no	—
190	#117	—	158	2	no	—
191	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
192	#107	—	79	2	no	—
193	#107	—	79	2	no	—
194	C₄⋊C₈	—	79	2	no	—
195	C₄⋊C₈	—	79	2	no	—
196	C₄⋊C₈	—	158	4	no	—
197	C₄⋊C₈	—	158	4	no	—
198	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
199	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
200	#103	—	79	4	no	—
201	#103	—	79	4	no	—
202	C₂²⋊C₈	—	158	2	no	—
203	C₂²⋊C₈	D₃₁₆	1	2	yes	—
204	#97	—	2	1	no	—
205	#97	—	158	4	no	—
206	C₄×C₈	—	158	4	no	—
207	C₄×C₈	—	158	2	no	—
208	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
209	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₇₉	1	2	yes	—
211	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₇₉	1	2	yes	—
214	C₂×Q₈	—	79	4	no	—
215	C₂×Q₈	—	158	2	no	—
216	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
218	C₂²×C₄	#11929624	1	1	yes	characteristic
219	Q₁₆	—	316	2	no	—
220	Q₁₆	—	158	4	no	—
221	M₁₆	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	158	2	no	—
226	C₂×C₈	—	158	2	no	—
227	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
231	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
232	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
233	C₄⋊C₄	—	79	4	no	—
234	C₄⋊C₄	—	158	4	no	—
235	C₄⋊C₄	—	158	4	no	—
236	C₄⋊C₄	—	158	2	no	—
237	C₄⋊C₄	—	158	2	no	—
238	C₄⋊C₄	—	158	2	no	—
239	C₂²⋊C₄	—	158	4	no	—
240	C₄²	—	158	4	no	—
241	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
242	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
243	C₄²	#11929624	1	1	yes	characteristic
244	C₄²	C₂³×D₇₉	1	1	yes	characteristic
245	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
246	Q₈	—	316	4	no	—
247	Q₈	—	158	4	no	—
248	Q₈	—	158	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	158	4	no	—
255	C₂×C₄	—	158	4	no	—
256	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
257	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
258	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
261	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	316	2	no	—
269	C₈	—	316	2	no	—
270	C₈	#14579169	1	2	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	316	4	no	—
279	C₄	—	316	4	no	—
280	C₄	#19098681	1	2	yes	—
281	C₄	C₂×#14579169	1	1	yes	characteristic
282	C₄	#19099329	1	1	yes	characteristic
283	C₄	#19099380	1	1	yes	characteristic
284	C₄	#19099380	1	1	yes	characteristic
285	C₂	—	4	1	no	—
286	C₂	#23772146	1	1	yes	characteristic
287	C₂	#23772269	1	1	yes	characteristic
288	C₂	C₂×#19098681	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₈×M₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂³×C₇₉	C₄×D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₈×M₁₆	D₇₉
hypocenter	C₇₉	#18086
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #23762298	C₂	semidirect
4. #23762273	C₂	semidirect
5. C₈×Dic₃₁₆	C₂	semidirect
153. C₇₉	#18086	semidirect

Maximal subgroups

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

subgroup	index
#23773986	2
#23772501	2
#23762298	2
#23762273	2
C₈×Dic₃₁₆	2
#23762148	2
#23761822	2
C₈×M₁₆×C₇₉	2
#18086	79