Subgroups — #34443801

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773974	C₂	1	2	yes	maximal, complemented
2	#23765674	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765648	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765622	C₂	1	2	yes	maximal, complemented
5	#23765486	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765484	C₂	1	2	yes	maximal, complemented
7	D₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765246	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765244	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2215	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099108	—	2	2	no	—
13	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
14	C₂×#14579102	C₂²	1	2	yes	—
15	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#19098967	C₂²	1	2	yes	—
17	#19098938	C₂²	1	2	yes	—
18	#19098908	—	2	2	no	—
19	#19098799	C₂²	1	2	yes	—
20	#19098797	—	2	2	no	—
21	D₈×Dic₇₉	C₂²	1	2	yes	—
22	#19098645	C₂²	1	2	yes	—
23	#19098643	—	2	2	no	—
24	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
25	#19098389	C₂²	1	1	yes	characteristic
26	#19098388	C₂²	1	2	yes	—
27	#19098365	C₂²	1	2	yes	—
28	#19098364	C₂²	1	2	yes	—
29	#19098363	C₂²	1	2	yes	—
30	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
36	#466×C₇₉	C₂²	1	2	yes	—
37	#366×C₇₉	C₂²	1	1	yes	characteristic
38	#357×C₇₉	C₂²	1	2	yes	—
39	C₂×#11929623	—	4	2	no	—
40	C₂×D₆₃₂	C₂³	1	2	yes	—
41	C₂×#11929609	—	2	2	no	—
42	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
43	#14579160	—	2	2	no	—
44	#14579157	—	2	2	no	—
45	#14579147	—	4	2	no	—
46	#14579135	—	2	2	no	—
47	#14579133	—	4	2	no	—
48	D₄×Dic₇₉	—	2	2	no	—
49	#14579123	—	2	2	no	—
50	(C₂²⋊D₇₉)⋊C₄	—	4	2	no	—
51	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
52	D₁₂₆₄	—	2	2	no	—
53	#14579102	—	2	4	no	—
54	C₂×C₇₉⋊C₁₆	—	2	2	no	—
55	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14579095	C₂³	1	2	yes	—
58	#14579094	—	2	2	no	—
59	#14579083	—	2	2	no	—
60	#14579082	—	2	2	no	—
61	#14579081	—	2	2	no	—
62	C₈×Dic₇₉	D₄	1	1	yes	characteristic
63	C₈×Dic₇₉	D₄	1	1	yes	characteristic
64	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
65	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
66	C₂×D₈×C₇₉	C₂³	1	2	yes	—
67	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
68	#121×C₇₉	—	2	2	no	—
69	D₁₆×C₇₉	—	2	2	no	—
70	C₂×C₁₆×C₇₉	—	2	2	no	—
71	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
73	#107×C₇₉	C₂³	1	2	yes	—
74	#102×C₇₉	—	2	2	no	—
75	#11929623	—	8	8	no	—
76	C₂×Dic₁₅₈	—	4	2	no	—
77	C₂×D₃₁₆	—	2	2	no	—
78	C₂×C₂²⋊D₇₉	—	4	2	no	—
79	C₂×C₂²⋊D₇₉	—	4	2	no	—
80	C₂×C₂²⋊D₇₉	—	4	2	no	—
81	C₂²×Dic₇₉	—	4	2	no	—
82	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
83	C₂²×Dic₇₉	D₈	1	1	yes	characteristic
84	C₂×C₄×D₇₉	—	4	2	no	—
85	D₆₃₂	—	2	2	no	—
86	D₆₃₂	—	2	2	no	—
87	#11929609	—	4	4	no	—
88	C₂×C₇₉⋊C₈	—	2	2	no	—
89	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
91	C₄⋊Dic₇₉	—	2	2	no	—
92	Dic₇₉⋊C₄	—	4	2	no	—
93	C₂²⋊Dic₇₉	—	4	2	no	—
94	(C₂×D₇₉)⋊C₄	—	4	2	no	—
95	(C₂×D₇₉)⋊C₄	—	4	2	no	—
96	C₄×Dic₇₉	D₈	1	1	yes	characteristic
97	C₄×Dic₇₉	D₈	1	1	yes	characteristic
98	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
99	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
100	C₇₉⋊C₁₆	—	2	1	no	—
101	C₇₉⋊C₁₆	#41	1	2	yes	complemented
102	C₂×D₄×C₇₉	—	2	2	no	—
103	C₂×D₄×C₇₉	—	4	2	no	—
104	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
105	D₈×C₇₉	—	2	2	no	—
106	D₈×C₇₉	—	2	2	no	—
107	C₂×C₈×C₇₉	—	2	1	no	—
108	C₂×C₈×C₇₉	—	2	1	no	—
109	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₇₉	—	2	2	no	—
112	C₂²⋊C₄×C₇₉	—	4	2	no	—
113	C₁₆×C₇₉	—	2	1	no	—
114	C₁₆×C₇₉	#41	1	2	yes	complemented
115	C₂³×C₇₉	—	4	2	no	—
116	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₇₉	—	4	2	no	—
118	D₄×C₇₉	—	4	2	no	—
119	D₄×C₇₉	—	4	2	no	—
120	D₄×C₇₉	—	8	4	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	4	2	no	—
124	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₇₉	—	8	4	no	—
127	C₂²⋊D₇₉	—	8	4	no	—
128	C₂²⋊D₇₉	—	8	4	no	—
129	C₂×Dic₇₉	—	2	2	no	—
130	C₂×Dic₇₉	—	4	2	no	—
131	C₂×Dic₇₉	—	8	4	no	—
132	C₂×Dic₇₉	D₁₆	1	4	yes	complemented
133	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
136	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
137	D₃₁₆	—	4	2	no	—
138	D₃₁₆	—	4	2	no	—
139	C₄×D₇₉	—	8	4	no	—
140	Dic₁₅₈	—	8	4	no	—
141	C₈×C₇₉	—	2	1	no	—
142	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
144	C₇₉⋊C₈	—	2	1	no	—
145	C₇₉⋊C₈	#135	1	2	yes	—
146	C₂²×C₇₉	—	2	1	no	—
147	C₂²×C₇₉	—	2	1	no	—
148	C₂²×C₇₉	—	8	2	no	—
149	C₂²×C₇₉	—	8	2	no	—
150	C₂²×C₇₉	—	8	2	no	—
151	C₂²×C₇₉	C₂×D₁₆	1	2	yes	—
152	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₇₉	—	8	2	no	—
154	C₂×D₇₉	—	8	2	no	—
155	C₂×D₇₉	—	8	2	no	—
156	C₄×C₇₉	—	2	1	no	—
157	C₄×C₇₉	—	8	2	no	—
158	C₄×C₇₉	#532	1	1	yes	characteristic
159	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
160	Dic₇₉	—	2	1	no	—
161	Dic₇₉	—	8	2	no	—
162	Dic₇₉	C₂×D₁₆	1	2	yes	—
163	Dic₇₉	C₂×D₁₆	1	2	yes	—
164	Dic₇₉	#508	1	2	yes	—
165	#33733	—	79	1	no	maximal, Sylow
166	C₂×C₇₉	—	2	2	no	—
167	C₂×C₇₉	—	16	2	no	—
168	C₂×C₇₉	#3169	1	1	yes	characteristic
169	C₂×C₇₉	C₂²×D₁₆	1	1	yes	characteristic
170	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
171	D₇₉	—	16	2	no	—
172	#3169	—	79	2	no	—
173	#2252	—	79	1	no	—
174	#2246	—	79	1	no	—
175	#2240	—	79	2	no	—
176	#2215	—	79	1	no	—
177	#2215	—	79	2	no	—
178	#2215	D₇₉	1	1	yes	characteristic, complemented
179	C₄×D₁₆	—	79	1	no	—
180	C₄×D₁₆	—	79	1	no	—
181	C₄×D₁₆	—	79	2	no	—
182	C₂×C₄×C₁₆	—	79	1	no	—
183	C₇₉	#33733	1	1	yes	characteristic, complemented, Sylow
184	#532	—	158	2	no	—
185	C₂×D₁₆	—	79	2	no	—
186	C₂×D₁₆	—	79	1	no	—
187	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
188	C₂²×C₁₆	—	79	1	no	—
189	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
190	#500	—	79	2	no	—
191	#493	—	79	2	no	—
192	#486	—	158	2	no	—
193	#466	—	79	2	no	—
194	#466	—	158	2	no	—
195	#466	C₂×D₇₉	1	2	yes	—
196	C₄×D₈	—	79	2	no	—
197	C₄×D₈	—	79	2	no	—
198	C₄×D₈	—	158	2	no	—
199	C₂×C₄×C₈	—	79	1	no	—
200	#366	—	79	1	no	—
201	#366	—	79	2	no	—
202	#366	C₂×D₇₉	1	1	yes	characteristic
203	#357	—	79	2	no	—
204	#357	—	79	2	no	—
205	#357	—	79	2	no	—
206	#357	C₂×D₇₉	1	2	yes	—
207	C₄×C₁₆	—	79	1	no	—
208	C₄×C₁₆	—	79	1	no	—
209	C₄×C₁₆	—	79	1	no	—
210	C₄×C₁₆	—	79	1	no	—
211	C₂×#41	—	316	2	no	—
212	C₂×D₈	—	79	2	no	—
213	C₂×D₈	—	158	2	no	—
214	C₂×D₈	C₂²×D₇₉	1	2	yes	—
215	C₂²×C₈	—	79	1	no	—
216	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
217	C₄⋊Q₈	—	158	2	no	—
218	#127	—	158	2	no	—
219	#124	—	316	2	no	—
220	#121	—	2	2	no	—
221	#121	—	316	2	no	—
222	#121	—	158	2	no	—
223	C₄×D₄	—	316	2	no	—
224	C₄×D₄	—	158	2	no	—
225	C₄×D₄	—	158	2	no	—
226	C₂×C₄²	—	79	1	no	—
227	D₁₆	—	2	2	no	—
228	D₁₆	—	158	4	no	—
229	D₁₆	—	158	2	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	—	79	1	no	—
232	C₂×C₁₆	—	79	1	no	—
233	C₂×C₁₆	—	158	2	no	—
234	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
236	#107	—	79	2	no	—
237	#107	—	158	2	no	—
238	#107	C₂²×D₇₉	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	158	2	no	—
241	#102	—	158	2	no	—
242	#102	—	158	2	no	—
243	C₄×C₈	—	79	1	no	—
244	C₄×C₈	—	79	1	no	—
245	C₄×C₈	—	79	1	no	—
246	C₄×C₈	—	79	1	no	—
247	#41	—	632	8	no	—
248	C₂×Q₈	—	316	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	316	2	no	—
252	C₂×D₄	—	316	2	no	—
253	C₂×D₄	—	316	2	no	—
254	C₂×D₄	—	158	2	no	—
255	C₂²×C₄	—	316	2	no	—
256	C₂²×C₄	—	316	2	no	—
257	C₂²×C₄	—	79	1	no	—
258	C₂²×C₄	—	79	1	no	—
259	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	316	4	no	—
263	D₈	—	158	2	no	—
264	D₈	—	158	2	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	79	1	no	—
268	C₂×C₈	—	79	1	no	—
269	C₂×C₈	—	158	2	no	—
270	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	316	2	no	—
274	C₄⋊C₄	—	158	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	316	2	no	—
277	C₂²⋊C₄	—	316	2	no	—
278	C₂²⋊C₄	—	316	2	no	—
279	C₄²	—	79	1	no	—
280	C₄²	—	79	1	no	—
281	C₄²	—	79	1	no	—
282	C₄²	—	79	1	no	—
283	C₁₆	—	2	1	no	—
284	C₁₆	—	79	2	no	—
285	C₁₆	—	158	1	no	—
286	C₁₆	#11929623	1	2	yes	complemented
287	C₂³	—	4	2	no	—
288	C₂³	—	316	2	no	—
289	C₂³	D₈×D₇₉	1	1	yes	characteristic
290	Q₈	—	632	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	316	2	no	—
295	D₄	—	316	2	no	—
296	D₄	—	632	4	no	—
297	D₄	—	632	4	no	—
298	D₄	—	632	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	316	2	no	—
303	C₂×C₄	—	79	4	no	—
304	C₂×C₄	—	79	1	no	—
305	C₂×C₄	—	79	1	no	—
306	C₂×C₄	—	79	1	no	—
307	C₂×C₄	—	79	1	no	—
308	C₂×C₄	—	632	4	no	—
309	C₂×C₄	—	632	4	no	—
310	C₂×C₄	—	158	2	no	—
311	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	79	2	no	—
315	C₈	—	158	1	no	—
316	C₈	#41×D₇₉	1	1	yes	characteristic
317	C₈	C₂×#11929623	1	1	yes	characteristic
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	632	2	no	—
324	C₂²	—	632	2	no	—
325	C₂²	—	632	2	no	—
326	C₂²	D₁₆×D₇₉	1	2	yes	—
327	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	79	2	no	—
331	C₄	—	79	2	no	—
332	C₄	—	79	2	no	—
333	C₄	—	632	2	no	—
334	C₄	—	158	1	no	—
335	C₄	#19099108	1	1	yes	characteristic
336	C₄	#135×D₇₉	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	16	2	no	—
339	C₂	—	1264	2	no	—
340	C₂	#23773974	1	1	yes	characteristic
341	C₂	C₂×D₁₆×D₇₉	1	1	yes	characteristic
342	C₂	#508×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2215	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	#2215	D₇₉
hypocenter	C₇₉	#33733
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #23773974	C₂	semidirect
2. #23765674	C₂	semidirect
3. #23765648	C₂	semidirect
4. #23765622	C₂	semidirect
5. #23765486	C₂	semidirect
6. #23765484	C₂	semidirect
7. D₁₆×Dic₇₉	C₂	semidirect
8. #23765246	C₂	semidirect
9. #23765244	C₂	semidirect
10. C₂×C₁₆×Dic₇₉	C₂	semidirect
11. C₇₉×#2215	C₂	semidirect
101. C₇₉⋊C₁₆	#41	semidirect
114. C₁₆×C₇₉	#41	semidirect
132. C₂×Dic₇₉	D₁₆	semidirect
178. #2215	D₇₉	semidirect
183. C₇₉	#33733	semidirect
286. C₁₆	#11929623	semidirect

Maximal subgroups

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

subgroup	index
#23773974	2
#23765674	2
#23765648	2
#23765622	2
#23765486	2
#23765484	2
D₁₆×Dic₇₉	2
#23765246	2
#23765244	2
C₂×C₁₆×Dic₇₉	2
C₇₉×#2215	2
#33733	79