Subgroups — #34445876

Order 20224.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775824	C₂	1	1	yes	maximal, characteristic, complemented
2	#23775749	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765682	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765590	C₂	1	2	yes	maximal, complemented
5	#23765584	C₂	1	2	yes	maximal, complemented
6	#23765547	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765537	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765500	C₂	1	2	yes	maximal, complemented
9	#23765233	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765185	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099278	—	2	1	no	—
13	#19099218	—	2	1	no	—
14	C₂×#14579100	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098885	—	2	2	no	—
18	#19098875	—	2	2	no	—
19	#19098853	—	2	1	no	—
20	#19098843	—	2	1	no	—
21	#19098796	C₂²	1	2	yes	—
22	#19098796	C₂²	1	2	yes	—
23	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
24	#19098600	C₂²	1	2	yes	—
25	#107×D₇₉	C₂²	1	2	yes	—
26	#19098391	C₂²	1	2	yes	—
27	#19098391	C₂²	1	2	yes	—
28	#19098390	C₂²	1	2	yes	—
29	#19098388	C₂²	1	2	yes	—
30	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
31	#19098364	C₂²	1	2	yes	—
32	#19098364	C₂²	1	2	yes	—
33	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
34	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
37	#366×C₇₉	C₂²	1	1	yes	characteristic
38	#357×C₇₉	C₂²	1	1	yes	characteristic
39	#357×C₇₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×#11929624	—	4	1	no	—
42	C₂×C₈×D₇₉	C₂³	1	2	yes	—
43	#14579161	—	2	1	no	—
44	#14579161	—	2	1	no	—
45	#14579147	—	4	2	no	—
46	#14579143	—	4	2	no	—
47	#14579140	—	4	1	no	—
48	#14579135	—	4	1	no	—
49	#14579132	—	2	2	no	—
50	#14579132	—	2	2	no	—
51	C₄×D₃₁₆	D₄	1	1	yes	characteristic
52	#14579120	—	4	1	no	—
53	#14579119	—	2	2	no	—
54	C₄⋊C₄×D₇₉	—	2	2	no	—
55	#14579100	—	2	4	no	—
56	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
57	#14579095	—	2	1	no	—
58	#14579095	—	2	1	no	—
59	#14579095	C₂³	1	2	yes	—
60	#14579095	C₂³	1	2	yes	—
61	#14579094	—	2	2	no	—
62	#14579092	—	2	2	no	—
63	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
64	#14579082	—	2	2	no	—
65	#14579082	—	2	2	no	—
66	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
67	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₇₉	—	2	1	no	—
70	C₄×D₄×C₇₉	—	2	1	no	—
71	D₁₆×C₇₉	—	2	2	no	—
72	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	#107×C₇₉	C₂³	1	1	yes	characteristic
75	#107×C₇₉	C₂³	1	1	yes	characteristic
76	#102×C₇₉	—	2	1	no	—
77	#102×C₇₉	—	2	1	no	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	#11929624	—	8	4	no	—
80	C₂×Dic₁₅₈	—	4	1	no	—
81	C₂×Dic₁₅₈	—	4	1	no	—
82	C₂×D₃₁₆	D₈	1	1	yes	characteristic
83	C₂×C₂²⋊D₇₉	—	4	2	no	—
84	C₂×C₂²⋊D₇₉	—	4	2	no	—
85	C₂×C₄×D₇₉	—	4	2	no	—
86	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
87	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
88	C₈×D₇₉	—	2	2	no	—
89	C₈×D₇₉	—	2	2	no	—
90	C₄⋊Dic₇₉	—	2	2	no	—
91	C₄⋊Dic₇₉	—	2	2	no	—
92	C₄⋊Dic₇₉	—	4	1	no	—
93	C₄⋊Dic₇₉	—	4	1	no	—
94	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
95	Dic₇₉⋊C₄	—	4	2	no	—
96	C₂²⋊Dic₇₉	—	4	2	no	—
97	C₂²⋊Dic₇₉	—	4	2	no	—
98	(C₂×D₇₉)⋊C₄	—	4	2	no	—
99	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
100	C₄×Dic₇₉	—	4	2	no	—
101	C₇₉⋊C₁₆	—	2	2	no	—
102	C₂×D₄×C₇₉	—	2	1	no	—
103	C₂×D₄×C₇₉	—	2	1	no	—
104	C₂²×C₄×C₇₉	—	4	1	no	—
105	C₂²×C₄×C₇₉	—	4	1	no	—
106	D₈×C₇₉	—	2	1	no	—
107	D₈×C₇₉	—	2	1	no	—
108	D₈×C₇₉	—	2	1	no	—
109	D₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₇₉	—	2	1	no	—
113	C₄⋊C₄×C₇₉	—	2	1	no	—
114	C₂²⋊C₄×C₇₉	—	4	1	no	—
115	C₂²⋊C₄×C₇₉	—	4	1	no	—
116	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₉	—	2	1	no	—
118	C₁₆×C₇₉	#41	1	2	yes	complemented
119	C₂³×C₇₉	—	4	1	no	—
120	C₂³×C₇₉	—	4	1	no	—
121	C₂²×D₇₉	C₂×D₈	1	2	yes	—
122	D₄×C₇₉	—	4	1	no	—
123	D₄×C₇₉	—	4	1	no	—
124	D₄×C₇₉	—	4	1	no	—
125	D₄×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	1	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	8	2	no	—
129	C₂×C₄×C₇₉	—	8	2	no	—
130	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₇₉	—	8	4	no	—
134	C₂²⋊D₇₉	—	16	2	no	—
135	C₂×Dic₇₉	—	4	2	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
138	D₃₁₆	—	2	2	no	—
139	C₄×D₇₉	—	2	2	no	—
140	C₄×D₇₉	—	2	2	no	—
141	C₄×D₇₉	—	8	4	no	—
142	Dic₁₅₈	—	8	2	no	—
143	Dic₁₅₈	—	16	1	no	—
144	C₈×C₇₉	—	2	1	no	—
145	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
146	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
147	C₇₉⋊C₈	—	2	2	no	—
148	C₂²×C₇₉	—	8	1	no	—
149	C₂²×C₇₉	—	8	1	no	—
150	C₂²×C₇₉	—	8	1	no	—
151	C₂²×C₇₉	—	8	1	no	—
152	C₂²×C₇₉	—	8	1	no	—
153	C₂²×C₇₉	—	8	1	no	—
154	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₇₉	—	2	2	no	—
156	C₂×D₇₉	—	2	2	no	—
157	C₂×D₇₉	—	2	2	no	—
158	C₄×C₇₉	—	2	1	no	—
159	C₄×C₇₉	—	8	1	no	—
160	C₄×C₇₉	—	8	1	no	—
161	C₄×C₇₉	#509	1	2	yes	—
162	C₄×C₇₉	#547	1	1	yes	characteristic
163	C₄×C₇₉	#578	1	1	yes	characteristic
164	Dic₇₉	—	2	2	no	—
165	Dic₇₉	—	8	2	no	—
166	Dic₇₉	—	8	2	no	—
167	#33921	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	—	16	1	no	—
169	C₂×C₇₉	—	16	1	no	—
170	C₂×C₇₉	#3334	1	1	yes	characteristic
171	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
172	C₂×C₇₉	#3416	1	1	yes	characteristic
173	D₇₉	—	4	2	no	—
174	#3341	—	79	1	no	—
175	#3334	—	79	1	no	—
176	#2254	—	79	1	no	—
177	#2234	—	79	2	no	—
178	#2233	—	79	2	no	—
179	#2226	—	79	1	no	—
180	#2224	—	79	1	no	—
181	#2218	—	79	2	no	—
182	C₄×D₁₆	D₇₉	1	1	yes	characteristic, complemented
183	#2169	—	79	1	no	—
184	#2159	—	79	2	no	—
185	C₇₉	#33921	1	1	yes	characteristic, complemented, Sylow
186	#555	—	158	1	no	—
187	#547	—	158	1	no	—
188	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
189	C₂×M₃₂	—	79	2	no	—
190	#500	—	79	1	no	—
191	#500	—	79	1	no	—
192	#482	—	158	2	no	—
193	#480	—	158	2	no	—
194	#476	—	158	1	no	—
195	#474	—	158	1	no	—
196	#466	—	79	2	no	—
197	#466	—	79	2	no	—
198	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
199	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
200	C₈×D₄	—	79	1	no	—
201	#427	—	79	2	no	—
202	C₂×#107	—	79	2	no	—
203	#367	—	79	2	no	—
204	#367	—	79	2	no	—
205	#367	—	79	2	no	—
206	#366	—	79	2	no	—
207	#366	C₂×D₇₉	1	1	yes	characteristic
208	C₄⋊C₁₆	—	79	1	no	—
209	#357	—	79	2	no	—
210	#357	—	79	2	no	—
211	#357	C₂×D₇₉	1	1	yes	characteristic
212	#357	C₂×D₇₉	1	1	yes	characteristic
213	C₂²⋊C₁₆	—	79	2	no	—
214	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
215	C₂×#41	—	316	1	no	—
216	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
217	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
218	C₂²×C₈	—	79	2	no	—
219	C₄⋊Q₈	—	158	1	no	—
220	C₄⋊Q₈	—	158	1	no	—
221	#124	—	316	2	no	—
222	#123	—	316	2	no	—
223	C₂²⋊Q₈	—	316	1	no	—
224	#121	—	316	1	no	—
225	#121	—	158	2	no	—
226	#121	—	158	2	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	—	79	1	no	—
230	#117	—	316	1	no	—
231	#117	—	158	2	no	—
232	C₂×C₄⋊C₄	—	158	2	no	—
233	D₁₆	—	2	2	no	—
234	M₃₂	—	158	4	no	—
235	C₂×C₁₆	—	79	2	no	—
236	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
238	#107	—	79	2	no	—
239	#107	—	79	2	no	—
240	#107	—	158	2	no	—
241	#107	—	158	1	no	—
242	#107	—	158	1	no	—
243	#107	C₂²×D₇₉	1	1	yes	characteristic
244	#107	C₂²×D₇₉	1	1	yes	characteristic
245	#106	—	158	2	no	—
246	C₄⋊C₈	—	79	1	no	—
247	#102	—	2	1	no	—
248	#102	—	2	1	no	—
249	#102	—	158	2	no	—
250	#102	—	158	2	no	—
251	C₂²⋊C₈	—	79	2	no	—
252	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
253	#41	—	632	4	no	—
254	C₂×Q₈	—	316	1	no	—
255	C₂×Q₈	—	316	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	316	2	no	—
259	C₂×D₄	—	316	2	no	—
260	C₂×D₄	—	79	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	316	2	no	—
264	C₂²×C₄	—	79	2	no	—
265	D₈	—	2	1	no	—
266	D₈	—	2	1	no	—
267	D₈	—	2	1	no	—
268	D₈	—	2	1	no	—
269	C₂×C₈	—	79	2	no	—
270	C₂×C₈	—	158	2	no	—
271	C₂×C₈	—	158	2	no	—
272	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	316	2	no	—
277	C₄⋊C₄	—	316	1	no	—
278	C₄⋊C₄	—	316	1	no	—
279	C₄⋊C₄	—	79	1	no	—
280	C₄⋊C₄	—	158	2	no	—
281	C₄⋊C₄	—	158	2	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	316	2	no	—
285	C₂²⋊C₄	—	316	2	no	—
286	C₂²⋊C₄	—	316	2	no	—
287	C₂²⋊C₄	—	79	2	no	—
288	C₄²	—	316	2	no	—
289	C₄²	D₄×D₇₉	1	1	yes	characteristic
290	C₁₆	—	2	1	no	—
291	C₁₆	—	158	2	no	—
292	C₁₆	#11929624	1	2	yes	complemented
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	79	2	no	—
296	Q₈	—	632	2	no	—
297	Q₈	—	1264	1	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	632	4	no	—
303	D₄	—	158	2	no	—
304	D₄	—	1264	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	316	2	no	—
310	C₂×C₄	—	316	2	no	—
311	C₂×C₄	—	79	2	no	—
312	C₂×C₄	—	632	4	no	—
313	C₂×C₄	—	158	2	no	—
314	C₂×C₄	—	158	2	no	—
315	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
316	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	158	2	no	—
320	C₈	C₂×#11929624	1	1	yes	characteristic
321	C₈	#14579208	1	1	yes	characteristic
322	C₂²	—	8	1	no	—
323	C₂²	—	8	1	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	158	2	no	—
329	C₂²	—	158	2	no	—
330	C₂²	—	158	2	no	—
331	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	632	2	no	—
336	C₄	—	632	2	no	—
337	C₄	—	158	2	no	—
338	C₄	#19098997	1	2	yes	—
339	C₄	#19099218	1	1	yes	characteristic
340	C₄	#19099377	1	1	yes	characteristic
341	C₂	—	16	1	no	—
342	C₂	—	16	1	no	—
343	C₂	—	316	2	no	—
344	C₂	#23775749	1	1	yes	characteristic
345	C₂	C₂×#19098997	1	1	yes	characteristic
346	C₂	#23776543	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₄×D₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	C₄×D₁₆	D₇₉
hypocenter	C₇₉	#33921
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #23775824	C₂	semidirect
2. #23775749	C₂	semidirect
3. #23765682	C₂	semidirect
4. #23765590	C₂	semidirect
5. #23765584	C₂	semidirect
6. #23765547	C₂	semidirect
7. #23765537	C₂	semidirect
8. #23765500	C₂	semidirect
9. #23765233	C₂	semidirect
10. #23765185	C₂	semidirect
11. C₄×D₁₆×C₇₉	C₂	semidirect
118. C₁₆×C₇₉	#41	semidirect
182. C₄×D₁₆	D₇₉	semidirect
185. C₇₉	#33921	semidirect
292. C₁₆	#11929624	semidirect

Maximal subgroups

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

subgroup	index
#23775824	2
#23775749	2
#23765682	2
#23765590	2
#23765584	2
#23765547	2
#23765537	2
#23765500	2
#23765233	2
#23765185	2
C₄×D₁₆×C₇₉	2
#33921	79