Subgroups — #34260301

Order 20224.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 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	#23762310	C₂	1	2	yes	maximal, complemented
4	#23762252	C₂	1	4	yes	maximal, complemented
5	#23762220	C₂	1	2	yes	maximal, complemented
6	#23762148	C₂	1	1	yes	maximal, characteristic
7	#23761822	C₂	1	2	yes	maximal
8	C₇₉×#1451	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	2	yes	—
24	#19098314	C₂²	1	1	yes	characteristic
25	#19098314	C₂²	1	1	yes	characteristic
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	#322×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	2	yes	—
60	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
61	#97×C₇₉	—	2	1	no	—
62	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₇₉	C₂³	1	2	yes	—
65	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₉	C₂³	1	2	yes	—
68	C₂×Dic₁₅₈	—	2	2	no	—
69	C₂×Dic₁₅₈	C₂²×C₄	1	4	yes	—
70	Dic₃₁₆	—	2	4	no	—
71	Dic₃₁₆	—	4	2	no	—
72	C₂×C₇₉⋊C₈	—	2	2	no	—
73	C₂×C₇₉⋊C₈	—	2	2	no	—
74	C₄⋊Dic₇₉	—	2	2	no	—
75	C₄⋊Dic₇₉	—	2	2	no	—
76	C₄⋊Dic₇₉	—	2	2	no	—
77	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
78	Dic₇₉⋊C₄	—	2	4	no	—
79	Dic₇₉⋊C₄	—	2	4	no	—
80	C₂²⋊Dic₇₉	—	2	4	no	—
81	C₄×Dic₇₉	—	2	4	no	—
82	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
85	M₁₆×C₇₉	—	4	2	no	—
86	C₂×C₈×C₇₉	—	2	2	no	—
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	4	1	no	—
89	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	4	yes	—
94	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₇₉	#41	1	1	yes	characteristic
97	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
98	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	1	no	—
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₇₉	—	2	4	no	—
110	C₂×Dic₇₉	—	2	4	no	—
111	Dic₁₅₈	—	2	4	no	—
112	Dic₁₅₈	—	2	4	no	—
113	Dic₁₅₈	—	4	4	no	—
114	C₈×C₇₉	—	2	1	no	—
115	C₈×C₇₉	—	2	1	no	—
116	C₈×C₇₉	—	2	1	no	—
117	C₈×C₇₉	—	2	4	no	—
118	C₈×C₇₉	—	4	2	no	—
119	C₈×C₇₉	#131	1	2	yes	—
120	C₇₉⋊C₈	—	4	2	no	—
121	C₇₉⋊C₈	—	4	2	no	—
122	C₂²×C₇₉	—	2	1	no	—
123	C₂²×C₇₉	—	2	1	no	—
124	C₂²×C₇₉	—	2	1	no	—
125	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₇₉	—	2	1	no	—
127	C₄×C₇₉	—	2	1	no	—
128	C₄×C₇₉	—	2	1	no	—
129	C₄×C₇₉	#444	1	2	yes	—
130	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
131	C₄×C₇₉	#568	1	1	yes	characteristic
132	C₄×C₇₉	#578	1	2	yes	—
133	Dic₇₉	—	4	4	no	—
134	Dic₇₉	—	4	4	no	—
135	#18090	—	79	1	no	maximal, Sylow
136	C₂×C₇₉	—	4	1	no	—
137	C₂×C₇₉	#2935	1	1	yes	characteristic
138	C₂×C₇₉	#2950	1	1	yes	characteristic
139	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
140	#3170	—	79	1	no	—
141	#2986	—	79	2	no	—
142	#1593	—	79	2	no	—
143	#1584	—	79	4	no	—
144	#1579	—	79	2	no	—
145	#1564	—	79	1	no	—
146	#1485	—	79	2	no	—
147	#1451	D₇₉	1	1	yes	characteristic
148	C₇₉	#18090	1	1	yes	characteristic, complemented, Sylow
149	#533	—	79	2	no	—
150	#500	—	79	1	no	—
151	#494	—	79	1	no	—
152	#487	—	79	2	no	—
153	#467	—	79	2	no	—
154	#467	—	79	2	no	—
155	#446	—	79	4	no	—
156	#446	—	158	2	no	—
157	C₈×Q₈	—	79	4	no	—
158	C₈×Q₈	—	158	2	no	—
159	C₄×Q₁₆	—	79	2	no	—
160	C₄×Q₁₆	—	79	2	no	—
161	#433	—	158	4	no	—
162	#431	—	79	2	no	—
163	#431	C₂×D₇₉	1	1	yes	characteristic
164	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
166	#335	—	79	2	no	—
167	#334	—	79	1	no	—
168	#334	—	79	1	no	—
169	Q₈⋊C₈	—	79	4	no	—
170	Q₈⋊C₈	—	79	4	no	—
171	#322	C₂×D₇₉	1	2	yes	—
172	#322	C₂×D₇₉	1	2	yes	—
173	C₂×Q₁₆	—	79	2	no	—
174	C₂×Q₁₆	—	79	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
178	C₄⋊Q₈	—	79	2	no	—
179	C₄⋊Q₈	—	79	2	no	—
180	#125	—	158	2	no	—
181	C₂²⋊Q₈	—	79	4	no	—
182	C₂²⋊Q₈	—	158	2	no	—
183	C₄×Q₈	—	79	4	no	—
184	C₄×Q₈	—	158	2	no	—
185	#117	—	158	2	no	—
186	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
187	#107	—	79	2	no	—
188	#107	—	79	2	no	—
189	C₄⋊C₈	—	79	2	no	—
190	C₄⋊C₈	—	79	2	no	—
191	C₄⋊C₈	—	158	4	no	—
192	C₄⋊C₈	—	158	4	no	—
193	C₄⋊C₈	D₃₁₆	1	2	yes	—
194	#103	—	79	4	no	—
195	#103	—	79	4	no	—
196	C₂²⋊C₈	—	158	2	no	—
197	C₂²⋊C₈	D₃₁₆	1	2	yes	—
198	#97	—	2	1	no	—
199	#97	—	158	4	no	—
200	C₄×C₈	—	158	4	no	—
201	C₄×C₈	—	158	2	no	—
202	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
203	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₇₉	1	2	yes	—
205	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₇₉	1	2	yes	—
208	C₂×Q₈	—	79	4	no	—
209	C₂×Q₈	—	158	2	no	—
210	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
212	C₂²×C₄	#11929624	1	1	yes	characteristic
213	Q₁₆	—	316	2	no	—
214	Q₁₆	—	158	4	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	158	2	no	—
220	C₂×C₈	—	158	2	no	—
221	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
225	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
226	C₄⋊C₄	—	79	4	no	—
227	C₄⋊C₄	—	158	4	no	—
228	C₄⋊C₄	—	158	4	no	—
229	C₄⋊C₄	—	158	2	no	—
230	C₄⋊C₄	—	158	2	no	—
231	C₄⋊C₄	—	158	2	no	—
232	C₂²⋊C₄	—	158	4	no	—
233	C₄²	—	158	4	no	—
234	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
235	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
236	C₄²	#11929624	1	1	yes	characteristic
237	C₄²	C₂³×D₇₉	1	1	yes	characteristic
238	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
239	Q₈	—	316	4	no	—
240	Q₈	—	158	4	no	—
241	Q₈	—	158	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	158	4	no	—
247	C₂×C₄	—	158	4	no	—
248	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
249	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
250	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
253	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₈	—	316	2	no	—
260	C₈	—	316	2	no	—
261	C₈	#14579169	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	316	4	no	—
270	C₄	—	316	4	no	—
271	C₄	#19098686	1	2	yes	—
272	C₄	C₂×#14579169	1	1	yes	characteristic
273	C₄	#19099329	1	1	yes	characteristic
274	C₄	#19099380	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	#23772146	1	1	yes	characteristic
277	C₂	#23772269	1	1	yes	characteristic
278	C₂	C₂×#19098686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1451	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂³×C₇₉	C₄×D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1451	D₇₉
hypocenter	C₇₉	#18090
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #23762310	C₂	semidirect
4. #23762252	C₂	semidirect
5. #23762220	C₂	semidirect
148. C₇₉	#18090	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
#23762310	2
#23762252	2
#23762220	2
#23762148	2
#23761822	2
C₇₉×#1451	2
#18090	79