Subgroups — #34275966

Order 20224.

922 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773975	C₂	1	4	yes	maximal, complemented
2	#23773828	C₂	1	1	yes	maximal, characteristic, complemented
3	#23763016	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763014	C₂	1	1	yes	maximal, characteristic, complemented
5	#23763007	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763005	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762990	C₂	1	4	yes	maximal, complemented
8	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579183	—	2	2	no	—
11	#19099113	C₂²	1	2	yes	—
12	#19098966	C₂²	1	4	yes	—
13	#19098947	—	2	2	no	—
14	#19098943	C₂²	1	1	yes	characteristic
15	#19098940	C₂²	1	1	yes	characteristic
16	#19098939	C₂²	1	4	yes	—
17	#19098936	C₂²	1	2	yes	—
18	#19098911	C₂²	1	4	yes	—
19	#19098909	C₂²	1	4	yes	—
20	#19098797	—	2	8	no	—
21	C₄×#11929609	—	2	8	no	—
22	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
23	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
24	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
25	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×#135×C₇₉	—	2	2	no	—
27	#532×C₇₉	C₂²	1	2	yes	—
28	#495×C₇₉	—	2	2	no	—
29	#494×C₇₉	C₂²	1	1	yes	characteristic
30	#493×C₇₉	C₂²	1	1	yes	characteristic
31	#492×C₇₉	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
33	C₂×#11929624	—	2	2	no	—
34	#14579183	—	4	8	no	—
35	C₂×Dic₃₁₆	—	2	2	no	—
36	C₂×#11929613	—	2	4	no	—
37	C₂×D₆₃₂	—	2	2	no	—
38	C₂×#11929609	—	2	8	no	—
39	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
40	#14579161	C₂³	1	2	yes	—
41	#14579158	C₂³	1	2	yes	—
42	#14579149	—	2	2	no	—
43	#14579135	—	2	4	no	—
44	C₄×D₃₁₆	—	2	4	no	—
45	#14579094	—	2	8	no	—
46	#14579082	—	2	8	no	—
47	#14579081	—	2	8	no	—
48	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
49	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
50	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
51	C₂×#41×C₇₉	—	2	2	no	—
52	#135×C₇₉	—	4	8	no	—
53	C₂×Q₁₆×C₇₉	—	2	2	no	—
54	C₂×SD₁₆×C₇₉	—	2	4	no	—
55	C₂×D₈×C₇₉	—	2	2	no	—
56	C₂²×C₈×C₇₉	D₄	1	2	yes	—
57	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
58	#127×C₇₉	C₂³	1	2	yes	—
59	#124×C₇₉	—	2	2	no	—
60	#121×C₇₉	—	2	4	no	—
61	C₄×D₄×C₇₉	—	2	4	no	—
62	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₇₉	D₄	1	2	yes	—
64	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	#11929624	—	4	8	no	—
67	C₂×Dic₁₅₈	—	2	2	no	—
68	C₂×D₃₁₆	—	2	2	no	—
69	C₂×D₃₁₆	—	2	4	no	—
70	C₂×C₂²⋊D₇₉	—	4	2	no	—
71	C₂×C₄×D₇₉	—	4	2	no	—
72	Dic₃₁₆	—	4	4	no	—
73	#11929613	—	4	8	no	—
74	D₆₃₂	—	4	4	no	—
75	#11929609	—	4	16	no	—
76	C₂×C₇₉⋊C₈	—	2	8	no	—
77	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
78	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
79	C₄⋊Dic₇₉	—	2	4	no	—
80	(C₂×D₇₉)⋊C₄	—	4	4	no	—
81	#41×C₇₉	—	4	8	no	—
82	C₂×Q₈×C₇₉	—	2	2	no	—
83	C₂×D₄×C₇₉	—	2	4	no	—
84	C₂×D₄×C₇₉	—	2	2	no	—
85	C₂×D₄×C₇₉	—	4	2	no	—
86	C₂²×C₄×C₇₉	—	4	2	no	—
87	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₇₉	—	4	4	no	—
90	SD₁₆×C₇₉	—	4	8	no	—
91	D₈×C₇₉	—	4	4	no	—
92	C₂×C₈×C₇₉	D₈	1	8	yes	complemented
93	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₇₉	—	2	4	no	—
96	C₂²⋊C₄×C₇₉	—	4	4	no	—
97	C₄²×C₇₉	C₂×D₄	1	2	yes	—
98	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
100	C₂³×C₇₉	—	4	2	no	—
101	C₂³×C₇₉	#127	1	1	yes	characteristic
102	C₂²×D₇₉	—	4	2	no	—
103	Q₈×C₇₉	—	4	4	no	—
104	D₄×C₇₉	—	4	4	no	—
105	D₄×C₇₉	—	4	8	no	—
106	D₄×C₇₉	—	8	4	no	—
107	C₂×C₄×C₇₉	—	2	4	no	—
108	C₂×C₄×C₇₉	—	4	2	no	—
109	C₂×C₄×C₇₉	—	8	4	no	—
110	C₂×C₄×C₇₉	#127	1	2	yes	—
111	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
113	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₇₉	—	8	4	no	—
116	C₂×Dic₇₉	—	4	2	no	—
117	D₃₁₆	—	4	4	no	—
118	D₃₁₆	—	4	8	no	—
119	C₄×D₇₉	—	8	4	no	—
120	Dic₁₅₈	—	4	4	no	—
121	C₈×C₇₉	C₂×D₈	1	4	yes	—
122	C₈×C₇₉	C₂×D₈	1	4	yes	—
123	C₇₉⋊C₈	—	2	4	no	—
124	C₇₉⋊C₈	#135	1	8	yes	complemented
125	C₂²×C₇₉	—	2	2	no	—
126	C₂²×C₇₉	—	8	2	no	—
127	C₂²×C₇₉	—	8	4	no	—
128	C₂²×C₇₉	#493	1	2	yes	—
129	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
130	C₂×D₇₉	—	8	2	no	—
131	C₂×D₇₉	—	8	4	no	—
132	C₄×C₇₉	—	2	2	no	—
133	C₄×C₇₉	—	8	2	no	—
134	C₄×C₇₉	#493	1	2	yes	—
135	C₄×C₇₉	C₂²×D₈	1	2	yes	—
136	C₄×C₇₉	C₂×#135	1	4	yes	—
137	Dic₇₉	—	8	2	no	—
138	#19603	—	79	1	no	maximal, Sylow
139	C₂×C₇₉	—	2	2	no	—
140	C₂×C₇₉	—	16	2	no	—
141	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
142	C₂×C₇₉	#3147	1	2	yes	—
143	D₇₉	—	16	2	no	—
144	#3169	—	79	4	no	—
145	#3147	—	79	1	no	—
146	#3147	D₇₉	1	1	yes	characteristic, complemented
147	#1713	—	79	1	no	—
148	#1712	—	79	1	no	—
149	#1709	—	79	1	no	—
150	#1709	—	79	1	no	—
151	#1706	—	79	4	no	—
152	C₂×C₈²	—	79	1	no	—
153	C₇₉	#19603	1	1	yes	characteristic, complemented, Sylow
154	C₂×#135	—	2	2	no	—
155	C₂×#135	—	158	2	no	—
156	#532	—	79	2	no	—
157	#532	C₂×D₇₉	1	2	yes	—
158	#500	—	79	4	no	—
159	#495	—	2	2	no	—
160	#495	—	158	2	no	—
161	#494	—	79	1	no	—
162	#494	C₂×D₇₉	1	1	yes	characteristic
163	#493	—	79	4	no	—
164	#493	—	79	1	no	—
165	#493	C₂×D₇₉	1	1	yes	characteristic
166	#492	—	79	2	no	—
167	#492	C₂×D₇₉	1	2	yes	—
168	#486	—	79	4	no	—
169	#486	—	79	4	no	—
170	#466	—	158	8	no	—
171	C₄×D₈	—	158	8	no	—
172	C₂×C₄×C₈	—	79	2	no	—
173	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
174	C₈²	—	79	2	no	—
175	C₈²	—	79	1	no	—
176	C₈²	—	79	1	no	—
177	C₂×#41	—	2	2	no	—
178	C₂×#41	—	158	2	no	—
179	#135	—	4	8	no	—
180	#135	—	316	8	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	158	2	no	—
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×SD₁₆	—	158	4	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	158	8	no	—
187	C₂×D₈	—	158	2	no	—
188	C₂²×C₈	—	79	4	no	—
189	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
190	C₄⋊Q₈	—	79	2	no	—
191	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
192	#127	—	79	2	no	—
193	#127	C₂²×D₇₉	1	2	yes	—
194	#124	—	2	2	no	—
195	#124	—	158	2	no	—
196	#121	—	2	4	no	—
197	#121	—	158	4	no	—
198	C₄×D₄	—	2	4	no	—
199	C₄×D₄	—	158	4	no	—
200	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
201	#107	—	158	8	no	—
202	#102	—	158	8	no	—
203	#102	—	158	8	no	—
204	C₄×C₈	—	79	4	no	—
205	C₄×C₈	—	79	2	no	—
206	C₄×C₈	—	79	2	no	—
207	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
208	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
210	#41	—	4	8	no	—
211	#41	—	316	8	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	158	2	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	316	2	no	—
218	C₂×D₄	—	158	4	no	—
219	C₂×D₄	—	158	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	316	2	no	—
222	C₂²×C₄	D₄×D₇₉	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
224	Q₁₆	—	4	4	no	—
225	Q₁₆	—	316	4	no	—
226	SD₁₆	—	4	8	no	—
227	SD₁₆	—	316	8	no	—
228	D₈	—	4	4	no	—
229	D₈	—	316	16	no	—
230	D₈	—	316	4	no	—
231	C₂×C₈	—	79	4	no	—
232	C₂×C₈	—	79	4	no	—
233	C₂×C₈	—	158	8	no	—
234	C₂×C₈	#11929609	1	8	yes	complemented
235	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	158	4	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₂²⋊C₄	—	316	4	no	—
241	C₄²	D₄×D₇₉	1	2	yes	—
242	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
243	C₄²	C₂³×D₇₉	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	—	316	2	no	—
246	C₂³	#14579157	1	1	yes	characteristic
247	Q₈	—	4	4	no	—
248	Q₈	—	316	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	8	no	—
251	D₄	—	8	4	no	—
252	D₄	—	316	4	no	—
253	D₄	—	316	8	no	—
254	D₄	—	632	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	316	2	no	—
259	C₂×C₄	—	632	4	no	—
260	C₂×C₄	#14579157	1	2	yes	—
261	C₂×C₄	#14579157	1	1	yes	characteristic
262	C₂×C₄	C₂×#11929609	1	4	yes	—
263	C₂×C₄	C₂×D₄×D₇₉	1	2	yes	—
264	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
265	C₈	—	79	8	no	—
266	C₈	—	158	4	no	—
267	C₈	C₂×#11929609	1	4	yes	—
268	C₈	C₂×#11929609	1	4	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	632	2	no	—
273	C₂²	—	632	4	no	—
274	C₂²	#19098939	1	2	yes	—
275	C₂²	C₂×#14579157	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	632	2	no	—
279	C₄	#19098939	1	2	yes	—
280	C₄	C₂²×#11929609	1	2	yes	—
281	C₄	#135×D₇₉	1	4	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	16	2	no	—
284	C₂	—	1264	2	no	—
285	C₂	C₂×#19098939	1	1	yes	characteristic
286	C₂	#23773824	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579157
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#3147	C₂
socle	C₂²×C₇₉	C₂×#127
hypercenter	#3147	D₇₉
hypocenter	C₇₉	#19603
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #23773975	C₂	semidirect
2. #23773828	C₂	semidirect
3. #23763016	C₂	semidirect
4. #23763014	C₂	semidirect
5. #23763007	C₂	semidirect
6. #23763005	C₂	semidirect
7. #23762990	C₂	semidirect
8. C₂×C₈×C₇₉⋊C₈	C₂	semidirect
9. C₇₉×#3147	C₂	semidirect
92. C₂×C₈×C₇₉	D₈	semidirect
124. C₇₉⋊C₈	#135	semidirect
146. #3147	D₇₉	semidirect
153. C₇₉	#19603	semidirect
234. C₂×C₈	#11929609	semidirect

Maximal subgroups

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

subgroup	index
#23773975	2
#23773828	2
#23763016	2
#23763014	2
#23763007	2
#23763005	2
#23762990	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#3147	2
#19603	79