Subgroups — #34181314

Order 20224.

542 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₁₆	C₂	1	1	yes	maximal, characteristic
2	#23763557	C₂	1	1	yes	maximal, characteristic, complemented
3	#23763401	C₂	1	2	yes	maximal
4	#23761284	C₂	1	2	yes	maximal, complemented
5	C₂×#334×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₇₉	—	2	1	no	—
7	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
8	C₈×D₃₁₆	—	2	2	no	—
9	C₈×D₃₁₆	C₄	1	4	yes	complemented
10	C₂×#14579093	—	2	1	no	—
11	C₂×#14579093	—	2	1	no	—
12	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#14579083	—	2	2	no	—
14	C₂×(C₂×D₇₉)⋊C₈	—	2	1	no	—
15	#19098433	C₂²	1	2	yes	—
16	#19098326	—	2	2	no	—
17	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#334×C₇₉	—	2	2	no	—
20	C₂²×D₃₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₇₉	—	2	1	no	—
22	C₂²×C₇₉⋊C₈	—	2	1	no	—
23	C₂×C₈×D₇₉	—	2	4	no	—
24	C₂×C₈×D₇₉	—	4	2	no	—
25	C₂×C₈×D₇₉	—	4	2	no	—
26	C₄×D₃₁₆	—	2	1	no	—
27	C₄×D₃₁₆	—	2	1	no	—
28	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
29	C₄×D₃₁₆	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₇₉	—	2	2	no	—
31	C₂×C₄⋊Dic₇₉	—	2	2	no	—
32	C₂×C₄⋊Dic₇₉	—	2	2	no	—
33	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
34	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
35	#14579093	—	4	2	no	—
36	#14579093	—	4	2	no	—
37	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
38	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
39	#14579083	—	4	4	no	—
40	(C₂×D₇₉)⋊C₈	—	2	4	no	—
41	C₂²×C₈×C₇₉	—	2	2	no	—
42	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₇₉	—	2	2	no	—
46	C₄⋊C₈×C₇₉	—	2	2	no	—
47	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₇₉	D₄	1	2	yes	—
49	C₂³×D₇₉	—	2	1	no	—
50	C₂×D₃₁₆	—	2	1	no	—
51	C₂×D₃₁₆	—	2	2	no	—
52	C₂×D₃₁₆	C₄⋊C₄	1	2	yes	—
53	C₂×D₃₁₆	SD₁₆	1	4	yes	—
54	C₂²×Dic₇₉	—	2	1	no	—
55	C₂²×Dic₇₉	—	4	2	no	—
56	C₂×C₄×D₇₉	—	2	2	no	—
57	C₂×C₄×D₇₉	—	2	2	no	—
58	C₂×C₄×D₇₉	—	4	1	no	—
59	C₂×C₄×D₇₉	—	4	1	no	—
60	C₂×C₄×D₇₉	—	4	1	no	—
61	C₂×C₄×D₇₉	—	4	1	no	—
62	C₂×C₇₉⋊C₈	—	2	2	no	—
63	C₂×C₇₉⋊C₈	—	4	1	no	—
64	C₂×C₇₉⋊C₈	—	4	1	no	—
65	C₈×D₇₉	—	4	8	no	—
66	C₄⋊Dic₇₉	—	2	1	no	—
67	C₄⋊Dic₇₉	—	4	2	no	—
68	C₄⋊Dic₇₉	—	4	2	no	—
69	C₄⋊Dic₇₉	—	4	2	no	—
70	C₄⋊Dic₇₉	—	4	2	no	—
71	C₄⋊Dic₇₉	—	8	2	no	—
72	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
73	(C₂×D₇₉)⋊C₄	—	2	2	no	—
74	(C₂×D₇₉)⋊C₄	—	2	2	no	—
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
78	C₂×C₈×C₇₉	—	2	1	no	—
79	C₂×C₈×C₇₉	—	2	1	no	—
80	C₂×C₈×C₇₉	—	4	2	no	—
81	C₂×C₈×C₇₉	—	4	2	no	—
82	C₂×C₈×C₇₉	—	4	2	no	—
83	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₇₉	D₈	1	2	yes	—
86	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
87	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₇₉	#41	1	1	yes	characteristic
91	C₂³×C₇₉	#121	1	1	yes	characteristic
92	C₂²×D₇₉	—	2	2	no	—
93	C₂²×D₇₉	—	2	4	no	—
94	C₂²×D₇₉	—	4	1	no	—
95	C₂²×D₇₉	—	4	1	no	—
96	C₂²×D₇₉	—	4	2	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₉	#135	1	2	yes	—
108	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
110	C₂×Dic₇₉	—	2	2	no	—
111	C₂×Dic₇₉	—	4	1	no	—
112	C₂×Dic₇₉	—	4	1	no	—
113	C₂×Dic₇₉	—	8	2	no	—
114	C₂×Dic₇₉	—	8	2	no	—
115	C₂×Dic₇₉	—	8	2	no	—
116	D₃₁₆	—	2	2	no	—
117	D₃₁₆	—	2	4	no	—
118	D₃₁₆	#106	1	4	yes	complemented
119	C₄×D₇₉	—	4	2	no	—
120	C₄×D₇₉	—	4	2	no	—
121	C₄×D₇₉	—	4	2	no	—
122	C₄×D₇₉	—	4	2	no	—
123	C₈×C₇₉	—	2	2	no	—
124	C₈×C₇₉	—	8	2	no	—
125	C₈×C₇₉	#102	1	4	yes	complemented
126	C₇₉⋊C₈	—	4	2	no	—
127	C₂²×C₇₉	#380	1	1	yes	characteristic
128	C₂²×C₇₉	#459	1	1	yes	characteristic
129	C₂²×C₇₉	#461	1	1	yes	characteristic
130	C₂²×C₇₉	#465	1	2	yes	—
131	C₂²×C₇₉	#471	1	2	yes	—
132	C₂×D₇₉	—	2	4	no	—
133	C₂×D₇₉	—	4	2	no	—
134	C₂×D₇₉	—	4	2	no	—
135	C₂×D₇₉	—	4	4	no	—
136	C₂×D₇₉	—	4	4	no	—
137	C₄×C₇₉	—	2	1	no	—
138	C₄×C₇₉	—	2	1	no	—
139	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
140	C₄×C₇₉	#417	1	1	yes	characteristic
141	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
142	C₄×C₇₉	#427	1	1	yes	characteristic
143	C₄×C₇₉	#465	1	2	yes	—
144	C₄×C₇₉	#474	1	1	yes	characteristic
145	C₄×C₇₉	#476	1	1	yes	characteristic
146	Dic₇₉	—	4	2	no	—
147	Dic₇₉	—	16	2	no	—
148	#8642	—	79	1	no	maximal, Sylow
149	C₂×C₇₉	#1665	1	2	yes	—
150	C₂×C₇₉	#1671	1	2	yes	—
151	C₂×C₇₉	#1802	1	1	yes	characteristic
152	C₂×C₇₉	#1807	1	1	yes	characteristic
153	C₂×C₇₉	#1815	1	1	yes	characteristic
154	D₇₉	—	4	4	no	—
155	C₂×C₈×D₄	—	79	1	no	—
156	#1844	—	79	1	no	—
157	#1802	—	79	2	no	—
158	C₂×#334	D₇₉	1	1	yes	characteristic, complemented
159	#1296	—	79	2	no	—
160	C₇₉	#8642	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₈	—	158	1	no	—
162	C₂×C₄×D₄	—	79	1	no	—
163	C₈×D₄	—	79	4	no	—
164	C₈×D₄	—	158	2	no	—
165	C₂×#106	—	158	1	no	—
166	C₂×#106	—	158	1	no	—
167	C₂×C₄⋊C₈	—	79	1	no	—
168	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
169	C₂×#102	—	158	2	no	—
170	C₂×C₂²⋊C₈	—	158	1	no	—
171	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
172	#384	—	79	2	no	—
173	#340	—	158	2	no	—
174	#334	—	2	2	no	—
175	C₂²×D₄	—	79	1	no	—
176	C₂³×C₄	—	158	1	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	316	2	no	—
179	C₂²×C₈	—	316	2	no	—
180	C₂²×C₈	—	158	1	no	—
181	C₂²×C₈	—	158	4	no	—
182	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
183	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
184	C₄×D₄	—	79	2	no	—
185	C₄×D₄	—	79	2	no	—
186	C₄×D₄	—	158	1	no	—
187	C₄×D₄	—	158	1	no	—
188	C₂×C₄⋊C₄	—	79	1	no	—
189	C₂×C₄⋊C₄	—	158	2	no	—
190	C₂×C₄⋊C₄	—	158	2	no	—
191	C₂×C₄⋊C₄	—	158	2	no	—
192	C₂×C₂²⋊C₄	—	158	1	no	—
193	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
194	#106	—	316	2	no	—
195	#106	—	316	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	79	2	no	—
199	C₄⋊C₈	—	158	1	no	—
200	#102	—	316	4	no	—
201	C₂²⋊C₈	—	158	4	no	—
202	C₄×C₈	C₄×D₇₉	1	2	yes	—
203	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
204	C₂⁴	—	158	1	no	—
205	C₂×D₄	—	79	2	no	—
206	C₂×D₄	—	79	4	no	—
207	C₂×D₄	—	158	1	no	—
208	C₂×D₄	—	158	2	no	—
209	C₂²×C₄	—	316	1	no	—
210	C₂²×C₄	—	316	1	no	—
211	C₂²×C₄	—	316	1	no	—
212	C₂²×C₄	—	316	1	no	—
213	C₂²×C₄	—	316	2	no	—
214	C₂²×C₄	—	158	1	no	—
215	C₂²×C₄	—	158	2	no	—
216	C₂²×C₄	—	158	2	no	—
217	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
219	C₂²×C₄	#11929625	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	316	1	no	—
226	C₂×C₈	—	316	1	no	—
227	C₂×C₈	—	316	8	no	—
228	C₂×C₈	—	158	2	no	—
229	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
230	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
231	C₂×C₈	D₆₃₂	1	2	yes	—
232	C₂×C₈	#11929613	1	2	yes	—
233	C₄⋊C₄	—	316	2	no	—
234	C₄⋊C₄	—	316	2	no	—
235	C₄⋊C₄	—	316	2	no	—
236	C₄⋊C₄	—	316	2	no	—
237	C₄⋊C₄	—	79	2	no	—
238	C₄⋊C₄	—	632	2	no	—
239	C₄⋊C₄	—	158	1	no	—
240	C₂²⋊C₄	—	158	2	no	—
241	C₂²⋊C₄	—	158	2	no	—
242	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
243	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
244	C₄²	Q₈×D₇₉	1	1	yes	characteristic
245	C₄²	#11929623	1	1	yes	characteristic
246	C₂³	—	316	1	no	—
247	C₂³	—	316	1	no	—
248	C₂³	—	316	2	no	—
249	C₂³	—	158	2	no	—
250	C₂³	—	158	4	no	—
251	C₂³	#14579136	1	1	yes	characteristic
252	D₄	—	79	4	no	—
253	D₄	—	158	4	no	—
254	D₄	—	158	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	316	1	no	—
257	C₂×C₄	—	316	1	no	—
258	C₂×C₄	—	316	2	no	—
259	C₂×C₄	—	316	2	no	—
260	C₂×C₄	—	316	2	no	—
261	C₂×C₄	—	316	2	no	—
262	C₂×C₄	—	632	2	no	—
263	C₂×C₄	—	632	2	no	—
264	C₂×C₄	—	632	2	no	—
265	C₂×C₄	—	158	2	no	—
266	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
268	C₂×C₄	#14579119	1	1	yes	characteristic
269	C₂×C₄	#14579132	1	1	yes	characteristic
270	C₂×C₄	#14579137	1	1	yes	characteristic
271	C₂×C₄	#14579141	1	1	yes	characteristic
272	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
273	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
274	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
275	C₂×C₄	#14579182	1	2	yes	—
276	C₂×C₄	#14579189	1	1	yes	characteristic
277	C₂×C₄	#14579194	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	8	2	no	—
280	C₈	—	316	2	no	—
281	C₈	#14579083	1	4	yes	complemented
282	C₂²	—	316	2	no	—
283	C₂²	—	316	4	no	—
284	C₂²	—	316	4	no	—
285	C₂²	—	316	2	no	—
286	C₂²	—	158	4	no	—
287	C₂²	#19098420	1	1	yes	characteristic
288	C₂²	#19098759	1	1	yes	characteristic
289	C₂²	#19098769	1	1	yes	characteristic
290	C₂²	#19098793	1	2	yes	—
291	C₂²	#19098827	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	316	2	no	—
295	C₄	—	1264	2	no	—
296	C₄	C₂×#14579083	1	1	yes	characteristic
297	C₄	#19098565	1	1	yes	characteristic
298	C₄	#106×D₇₉	1	1	yes	characteristic
299	C₄	#19098598	1	1	yes	characteristic
300	C₄	#19098789	1	2	yes	—
301	C₄	#19098844	1	1	yes	characteristic
302	C₄	#19098854	1	1	yes	characteristic
303	C₂	—	316	4	no	—
304	C₂	#23762742	1	2	yes	—
305	C₂	#23762776	1	2	yes	—
306	C₂	#23763401	1	1	yes	characteristic
307	C₂	#23763422	1	1	yes	characteristic
308	C₂	#23763456	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579136
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#334×C₇₉	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂²×D₃₁₆	D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#334	D₇₉
hypocenter	C₇₉	#8642
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
2. #23763557	C₂	semidirect
4. #23761284	C₂	semidirect
5. C₂×#334×C₇₉	C₂	semidirect
9. C₈×D₃₁₆	C₄	semidirect
118. D₃₁₆	#106	semidirect
125. C₈×C₇₉	#102	semidirect
158. C₂×#334	D₇₉	semidirect
160. C₇₉	#8642	semidirect
281. C₈	#14579083	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₁₆	2
#23763557	2
#23763401	2
#23761284	2
C₂×#334×C₇₉	2
#8642	79