Subgroups — #34442961

Order 20224.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773940	C₂	1	1	yes	maximal, characteristic, complemented
2	#23772251	C₂	1	1	yes	maximal, characteristic
3	#23765560	C₂	1	2	yes	maximal, complemented
4	#23765556	C₂	1	2	yes	maximal, complemented
5	#23765546	C₂	1	2	yes	maximal, complemented
6	#23765536	C₂	1	2	yes	maximal, complemented
7	#23765189	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765158	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765115	C₂	1	1	yes	maximal, characteristic
10	C₂×#19098365	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
13	#19099191	—	2	1	no	—
14	#19099037	—	2	1	no	—
15	C₂×#14579101	C₂²	1	1	yes	characteristic
16	#19098970	—	2	2	no	—
17	#19098967	C₂²	1	2	yes	complemented
18	#19098964	C₂²	1	2	yes	complemented
19	#19098674	—	2	2	no	—
20	#19098645	C₂²	1	2	yes	—
21	#19098645	C₂²	1	2	yes	—
22	#19098606	—	2	1	no	—
23	C₂×#14579095	C₂²	1	1	yes	characteristic
24	#19098567	—	2	1	no	—
25	#19098518	C₂²	1	1	yes	characteristic
26	#19098390	C₂²	1	2	yes	complemented
27	#19098388	C₂²	1	2	yes	complemented
28	Dic₇₉⋊C₁₆	C₂²	1	2	yes	complemented
29	Dic₇₉⋊C₁₆	C₂²	1	2	yes	complemented
30	#19098365	C₂²	1	2	yes	—
31	#19098365	C₂²	1	2	yes	—
32	#19098363	C₂²	1	2	yes	—
33	#19098363	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
35	#427×C₇₉	C₂²	1	1	yes	characteristic
36	#427×C₇₉	C₂²	1	1	yes	characteristic
37	#367×C₇₉	C₂²	1	1	yes	characteristic, complemented
38	#367×C₇₉	C₂²	1	1	yes	characteristic, complemented
39	#366×C₇₉	C₂²	1	1	yes	characteristic, complemented
40	#366×C₇₉	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₃₁₆	—	2	1	no	—
42	C₂×D₆₃₂	—	2	1	no	—
43	C₂×D₆₃₂	—	2	2	no	—
44	C₂×D₆₃₂	Q₈	1	4	yes	complemented
45	C₂×D₆₃₂	C₂³	1	2	yes	—
46	C₂×#11929608	D₄	1	1	yes	characteristic
47	#14579161	—	4	1	no	—
48	#14579160	—	2	2	no	—
49	#14579153	—	4	1	no	—
50	#14579152	—	2	2	no	—
51	#14579138	—	4	2	no	—
52	C₄×D₃₁₆	—	4	1	no	—
53	C₄×D₃₁₆	—	4	1	no	—
54	#14579123	—	2	2	no	—
55	#14579123	—	2	2	no	—
56	#14579120	D₄	1	1	yes	characteristic
57	C₂×C₄⋊Dic₇₉	—	2	1	no	—
58	C₂²⋊C₄×D₇₉	—	4	2	no	—
59	#14579101	—	2	2	no	—
60	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
61	#14579095	—	2	1	no	—
62	#14579095	C₂³	1	2	yes	—
63	#14579094	—	2	2	no	—
64	#14579092	—	2	2	no	—
65	#14579081	—	2	2	no	—
66	#14579081	—	2	2	no	—
67	#14579071	D₄	1	2	yes	—
68	C₈×Dic₇₉	C₂³	1	2	yes	—
69	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
70	#117×C₇₉	—	2	1	no	—
71	#117×C₇₉	—	2	1	no	—
72	C₂×C₁₆×C₇₉	—	2	2	no	—
73	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
75	#107×C₇₉	C₂³	1	1	yes	characteristic
76	#107×C₇₉	C₂³	1	1	yes	characteristic
77	#107×C₇₉	C₂³	1	1	yes	characteristic
78	#107×C₇₉	C₂³	1	1	yes	characteristic
79	#106×C₇₉	—	2	1	no	—
80	#106×C₇₉	—	2	1	no	—
81	C₂³×D₇₉	—	4	1	no	—
82	C₂×Dic₁₅₈	—	4	2	no	—
83	C₂×D₃₁₆	—	2	4	no	—
84	C₂×D₃₁₆	—	2	2	no	—
85	C₂×D₃₁₆	—	4	1	no	—
86	C₂×D₃₁₆	—	4	2	no	—
87	C₂²×Dic₇₉	—	4	1	no	—
88	C₂×C₄×D₇₉	—	4	2	no	—
89	C₂×C₄×D₇₉	—	4	2	no	—
90	D₆₃₂	—	2	4	no	—
91	D₆₃₂	—	2	2	no	—
92	D₆₃₂	C₂×Q₈	1	4	yes	—
93	#11929608	—	2	2	no	—
94	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	C₄⋊Dic₇₉	—	4	2	no	—
97	C₄⋊Dic₇₉	—	4	1	no	—
98	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
99	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
100	Dic₇₉⋊C₄	—	4	2	no	—
101	Dic₇₉⋊C₄	—	4	2	no	—
102	C₂²⋊Dic₇₉	D₈	1	2	yes	—
103	(C₂×D₇₉)⋊C₄	—	4	2	no	—
104	(C₂×D₇₉)⋊C₄	—	4	2	no	—
105	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
106	C₇₉⋊C₁₆	—	2	2	no	—
107	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
110	C₂×C₈×C₇₉	#41	1	2	yes	—
111	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₇₉	—	2	1	no	—
113	C₄⋊C₄×C₇₉	—	2	1	no	—
114	C₄⋊C₄×C₇₉	—	2	1	no	—
115	C₄⋊C₄×C₇₉	—	2	1	no	—
116	C₂²⋊C₄×C₇₉	—	4	1	no	—
117	C₂²⋊C₄×C₇₉	—	4	1	no	—
118	C₄²×C₇₉	—	4	1	no	—
119	C₄²×C₇₉	—	4	1	no	—
120	C₁₆×C₇₉	—	2	1	no	—
121	C₁₆×C₇₉	—	2	1	no	—
122	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₇₉	—	4	2	no	—
124	C₂²×D₇₉	—	4	4	no	—
125	C₂²×D₇₉	—	16	1	no	—
126	C₂²×D₇₉	—	16	1	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	4	1	no	—
131	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#137	1	2	yes	—
134	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₇₉	—	4	2	no	—
136	C₂×Dic₇₉	—	16	1	no	—
137	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
138	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
139	D₃₁₆	—	2	4	no	—
140	D₃₁₆	—	4	2	no	—
141	D₃₁₆	—	4	4	no	—
142	C₄×D₇₉	—	16	2	no	—
143	C₄×D₇₉	—	16	2	no	—
144	Dic₁₅₈	—	16	2	no	—
145	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
146	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
148	C₇₉⋊C₈	—	2	2	no	—
149	C₂²×C₇₉	—	2	1	no	—
150	C₂²×C₇₉	—	2	1	no	—
151	C₂²×C₇₉	#474	1	2	yes	—
152	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₇₉	—	4	4	no	—
154	C₂×D₇₉	—	8	2	no	—
155	C₂×D₇₉	—	8	2	no	—
156	C₂×D₇₉	—	16	4	no	—
157	C₄×C₇₉	—	8	1	no	—
158	C₄×C₇₉	—	8	1	no	—
159	C₄×C₇₉	—	8	1	no	—
160	C₄×C₇₉	—	8	1	no	—
161	C₄×C₇₉	#474	1	2	yes	—
162	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
164	Dic₇₉	—	2	2	no	—
165	Dic₇₉	—	2	2	no	—
166	Dic₇₉	—	8	2	no	—
167	#33631	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	—	2	2	no	—
169	C₂×C₇₉	C₂×#474	1	1	yes	characteristic
170	C₂×C₇₉	#3415	1	1	yes	characteristic
171	C₂×C₇₉	#3416	1	1	yes	characteristic
172	D₇₉	—	8	4	no	—
173	#3164	—	79	1	no	—
174	#2948	—	79	1	no	—
175	#2228	—	79	2	no	—
176	#2228	—	79	2	no	—
177	#2226	—	79	2	no	—
178	#2224	—	79	2	no	—
179	#2159	—	79	1	no	—
180	#2158	D₇₉	1	1	yes	characteristic, complemented
181	#2151	—	79	1	no	—
182	#2140	—	79	1	no	—
183	C₂×#357	—	79	1	no	—
184	C₇₉	#33631	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	—	79	1	no	—
186	#544	—	158	1	no	—
187	#518	—	158	1	no	—
188	C₂×M₃₂	—	79	1	no	—
189	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
190	#501	—	158	2	no	—
191	#500	—	79	2	no	—
192	#499	—	79	2	no	—
193	#442	—	158	2	no	—
194	C₄×D₈	—	79	2	no	—
195	C₄×D₈	—	79	2	no	—
196	#428	—	158	1	no	—
197	#427	C₂×D₇₉	1	1	yes	characteristic
198	#427	C₂×D₇₉	1	1	yes	characteristic
199	C₂×#107	—	79	1	no	—
200	#418	—	158	1	no	—
201	#405	—	79	1	no	—
202	#367	—	79	2	no	—
203	#367	C₂×D₇₉	1	1	yes	characteristic, complemented
204	#367	C₂×D₇₉	1	1	yes	characteristic, complemented
205	#366	—	79	2	no	—
206	#366	C₂×D₇₉	1	1	yes	characteristic, complemented
207	#366	C₂×D₇₉	1	1	yes	characteristic, complemented
208	C₄⋊C₁₆	—	79	2	no	—
209	C₄⋊C₁₆	—	79	2	no	—
210	#357	—	79	2	no	—
211	#357	—	79	2	no	—
212	#357	—	79	2	no	—
213	#357	—	79	2	no	—
214	C₂²×D₄	—	158	1	no	—
215	C₂×D₈	—	79	4	no	—
216	C₂×D₈	—	79	2	no	—
217	C₂×D₈	—	158	1	no	—
218	C₂×D₈	—	158	2	no	—
219	C₂×M₁₆	—	79	1	no	—
220	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
221	C₄⋊Q₈	—	316	1	no	—
222	C₄⋊Q₈	—	158	2	no	—
223	#125	—	316	1	no	—
224	#125	—	158	2	no	—
225	C₂²⋊Q₈	—	316	2	no	—
226	C₄×D₄	—	316	1	no	—
227	C₄×D₄	—	316	1	no	—
228	C₄×D₄	—	158	2	no	—
229	C₄×D₄	—	158	2	no	—
230	#117	—	2	1	no	—
231	#117	—	2	1	no	—
232	#117	—	79	1	no	—
233	C₂×C₄⋊C₄	—	158	1	no	—
234	C₂×C₂²⋊C₄	—	316	2	no	—
235	M₃₂	—	158	2	no	—
236	C₂×C₁₆	—	2	2	no	—
237	C₂×C₁₆	—	79	2	no	—
238	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
240	#107	—	79	2	no	—
241	#107	—	158	2	no	—
242	#107	—	158	1	no	—
243	#107	C₂²×D₇₉	1	1	yes	characteristic
244	#107	C₂²×D₇₉	1	1	yes	characteristic
245	#107	C₂²×D₇₉	1	1	yes	characteristic
246	#107	C₂²×D₇₉	1	1	yes	characteristic
247	#106	—	2	1	no	—
248	#106	—	2	1	no	—
249	#106	—	158	2	no	—
250	#102	—	158	2	no	—
251	#102	—	158	2	no	—
252	#97	—	79	2	no	—
253	C₄×C₈	—	79	2	no	—
254	C₂⁴	—	316	1	no	—
255	C₂×Q₈	—	316	2	no	—
256	C₂×D₄	—	316	1	no	—
257	C₂×D₄	—	316	2	no	—
258	C₂×D₄	—	158	4	no	—
259	C₂×D₄	—	158	2	no	—
260	C₂²×C₄	—	316	2	no	—
261	C₂²×C₄	—	316	2	no	—
262	C₂²×C₄	—	316	1	no	—
263	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
264	D₈	—	79	4	no	—
265	D₈	—	158	2	no	—
266	D₈	—	158	4	no	—
267	M₁₆	—	158	2	no	—
268	C₂×C₈	—	79	2	no	—
269	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₇₉	1	2	yes	—
271	C₂×C₈	#11929624	1	2	yes	—
272	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	316	2	no	—
278	C₄⋊C₄	—	316	2	no	—
279	C₄⋊C₄	—	316	2	no	—
280	C₄⋊C₄	—	316	1	no	—
281	C₄⋊C₄	—	79	1	no	—
282	C₄⋊C₄	—	79	1	no	—
283	C₄⋊C₄	—	158	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	316	2	no	—
287	C₂²⋊C₄	—	316	2	no	—
288	C₂²⋊C₄	—	79	2	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	79	2	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	158	2	no	—
295	C₂³	—	316	2	no	—
296	C₂³	—	316	4	no	—
297	C₂³	—	1264	1	no	—
298	C₂³	—	1264	1	no	—
299	C₂³	D₈×D₇₉	1	1	yes	characteristic
300	Q₈	—	1264	2	no	—
301	D₄	—	316	4	no	—
302	D₄	—	316	2	no	—
303	D₄	—	158	4	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	316	2	no	—
309	C₂×C₄	—	79	2	no	—
310	C₂×C₄	—	79	2	no	—
311	C₂×C₄	—	1264	2	no	—
312	C₂×C₄	—	1264	2	no	—
313	C₂×C₄	—	1264	1	no	—
314	C₂×C₄	#14579139	1	2	yes	—
315	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
316	C₂×C₄	#14579193	1	2	yes	—
317	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
318	C₈	—	158	2	no	—
319	C₈	#14579139	1	2	yes	—
320	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
321	C₈	C₂×#11929624	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	316	4	no	—
325	C₂²	—	632	2	no	—
326	C₂²	—	632	2	no	—
327	C₂²	—	1264	4	no	—
328	C₂²	#19098842	1	2	yes	—
329	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
330	C₄	—	8	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	632	2	no	—
335	C₄	—	158	2	no	—
336	C₄	—	158	2	no	—
337	C₄	#19098842	1	2	yes	—
338	C₄	C₂×#14579139	1	1	yes	characteristic
339	C₄	C₂×#14579193	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	632	4	no	—
342	C₂	C₂×#19098842	1	1	yes	characteristic
343	C₂	#23776538	1	1	yes	characteristic
344	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	C₇₉×#2158	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂²×D₆₃₂	C₂²
hypercenter	#2158	D₇₉
hypocenter	C₇₉	#33631
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
1. #23773940	C₂	semidirect
3. #23765560	C₂	semidirect
4. #23765556	C₂	semidirect
5. #23765546	C₂	semidirect
6. #23765536	C₂	semidirect
7. #23765189	C₂	semidirect
8. #23765158	C₂	semidirect
11. C₇₉×#2158	C₂	semidirect
17. #19098967	C₂²	semidirect
18. #19098964	C₂²	semidirect
26. #19098390	C₂²	semidirect
27. #19098388	C₂²	semidirect
28. Dic₇₉⋊C₁₆	C₂²	semidirect
29. Dic₇₉⋊C₁₆	C₂²	semidirect
37. #367×C₇₉	C₂²	semidirect
38. #367×C₇₉	C₂²	semidirect
39. #366×C₇₉	C₂²	semidirect
40. #366×C₇₉	C₂²	semidirect
44. C₂×D₆₃₂	Q₈	semidirect
180. #2158	D₇₉	semidirect
184. C₇₉	#33631	semidirect
203. #367	C₂×D₇₉	semidirect
204. #367	C₂×D₇₉	semidirect
206. #366	C₂×D₇₉	semidirect
207. #366	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23773940	2
#23772251	2
#23765560	2
#23765556	2
#23765546	2
#23765536	2
#23765189	2
#23765158	2
#23765115	2
C₂×#19098365	2
C₇₉×#2158	2
#33631	79