Subgroups — #34442988

Order 20224.

534 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	#23773940	C₂	1	1	yes	maximal, characteristic, complemented
2	#23772274	C₂	1	1	yes	maximal, characteristic
3	#23765563	C₂	1	2	yes	maximal, complemented
4	#23765560	C₂	1	2	yes	maximal, complemented
5	#23765546	C₂	1	2	yes	maximal, complemented
6	#23765541	C₂	1	2	yes	maximal, complemented
7	C₂×#19098390	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765160	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765113	C₂	1	1	yes	maximal, characteristic
10	#23765105	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579183	C₂²	1	1	yes	characteristic
13	#19099191	—	2	1	no	—
14	#19099044	—	2	1	no	—
15	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	#19098970	—	2	2	no	—
17	#19098967	C₂²	1	2	yes	—
18	#19098964	C₂²	1	2	yes	—
19	#19098662	—	2	2	no	—
20	#19098657	C₂²	1	2	yes	—
21	#19098645	C₂²	1	2	yes	—
22	#19098606	—	2	1	no	—
23	C₂×#14579095	C₂²	1	1	yes	characteristic
24	#19098562	—	2	1	no	—
25	#19098518	C₂²	1	1	yes	characteristic
26	#19098390	C₂²	1	2	yes	—
27	#19098390	C₂²	1	2	yes	—
28	Dic₇₉⋊C₁₆	C₂²	1	2	yes	—
29	Dic₇₉⋊C₁₆	C₂²	1	2	yes	—
30	#19098368	C₂²	1	2	yes	—
31	#19098366	C₂²	1	1	yes	characteristic
32	#19098366	C₂²	1	1	yes	characteristic
33	#19098365	C₂²	1	2	yes	—
34	#19098363	C₂²	1	1	yes	characteristic
35	#19098363	C₂²	1	1	yes	characteristic
36	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
37	#427×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
39	#367×C₇₉	C₂²	1	2	yes	—
40	#366×C₇₉	C₂²	1	2	yes	—
41	C₂×#11929624	—	2	1	no	—
42	#14579183	—	2	2	no	—
43	#14579183	Q₈	1	4	yes	complemented
44	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
45	C₂×#11929613	—	2	1	no	—
46	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
47	C₂×#11929608	D₄	1	1	yes	characteristic
48	#14579161	—	4	1	no	—
49	#14579160	—	2	2	no	—
50	#14579153	—	4	1	no	—
51	#14579152	—	2	2	no	—
52	#14579138	—	4	2	no	—
53	#14579126	—	2	2	no	—
54	#14579123	—	2	2	no	—
55	C₄×C₂²⋊D₇₉	—	4	2	no	—
56	#14579120	D₄	1	1	yes	characteristic
57	#14579119	—	4	1	no	—
58	C₂×C₄⋊Dic₇₉	—	2	1	no	—
59	C₄⋊C₄×D₇₉	—	4	1	no	—
60	C₂×C₇₉⋊C₁₆	—	2	2	no	—
61	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
63	#14579095	—	2	1	no	—
64	#14579095	C₂³	1	2	yes	—
65	#14579094	—	2	2	no	—
66	#14579092	—	2	2	no	—
67	#14579084	—	2	2	no	—
68	#14579081	—	2	2	no	—
69	#14579071	D₄	1	2	yes	—
70	C₈×Dic₇₉	C₂³	1	2	yes	—
71	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
72	#117×C₇₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
74	M₃₂×C₇₉	—	2	2	no	—
75	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
76	#107×C₇₉	—	2	1	no	—
77	#107×C₇₉	C₂³	1	2	yes	—
78	#107×C₇₉	C₂³	1	1	yes	characteristic
79	#107×C₇₉	C₂³	1	1	yes	characteristic
80	#106×C₇₉	—	2	1	no	—
81	#11929624	—	2	4	no	—
82	#11929624	—	4	2	no	—
83	C₂×Dic₁₅₈	—	2	1	no	—
84	C₂×Dic₁₅₈	—	4	2	no	—
85	C₂×D₃₁₆	—	2	1	no	—
86	C₂×C₂²⋊D₇₉	—	4	1	no	—
87	C₂²×Dic₇₉	—	4	1	no	—
88	C₂×C₄×D₇₉	—	4	2	no	—
89	C₂×C₄×D₇₉	—	4	1	no	—
90	Dic₃₁₆	—	2	1	no	—
91	Dic₃₁₆	C₂×Q₈	1	2	yes	—
92	#11929613	—	2	4	no	—
93	D₆₃₂	—	2	1	no	—
94	D₆₃₂	C₂×Q₈	1	2	yes	—
95	#11929608	—	2	2	no	—
96	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
97	C₄⋊Dic₇₉	—	2	2	no	—
98	C₄⋊Dic₇₉	—	4	2	no	—
99	C₄⋊Dic₇₉	—	4	1	no	—
100	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
101	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
102	Dic₇₉⋊C₄	—	4	2	no	—
103	Dic₇₉⋊C₄	—	4	2	no	—
104	Dic₇₉⋊C₄	—	4	2	no	—
105	C₂²⋊Dic₇₉	D₈	1	2	yes	—
106	(C₂×D₇₉)⋊C₄	—	4	2	no	—
107	C₄×Dic₇₉	—	4	2	no	—
108	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
109	C₇₉⋊C₁₆	—	2	1	no	—
110	C₇₉⋊C₁₆	—	2	1	no	—
111	C₂²×C₄×C₇₉	—	4	1	no	—
112	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₇₉	#41	1	2	yes	—
116	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₇₉	—	2	1	no	—
118	C₄⋊C₄×C₇₉	—	2	1	no	—
119	C₄⋊C₄×C₇₉	—	2	2	no	—
120	C₄⋊C₄×C₇₉	—	4	1	no	—
121	C₂²⋊C₄×C₇₉	—	4	1	no	—
122	C₄²×C₇₉	—	4	1	no	—
123	C₁₆×C₇₉	—	2	2	no	—
124	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	16	1	no	—
130	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#137	1	2	yes	—
133	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₇₉	—	4	4	no	—
135	C₂×Dic₇₉	—	4	1	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	—	16	1	no	—
138	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
139	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
140	D₃₁₆	—	2	2	no	—
141	D₃₁₆	—	4	1	no	—
142	C₄×D₇₉	—	4	4	no	—
143	C₄×D₇₉	—	16	2	no	—
144	Dic₁₅₈	—	2	2	no	—
145	Dic₁₅₈	—	4	1	no	—
146	Dic₁₅₈	—	16	2	no	—
147	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
148	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
150	C₇₉⋊C₈	—	2	2	no	—
151	C₂²×C₇₉	—	2	1	no	—
152	C₂²×C₇₉	—	2	1	no	—
153	C₂²×C₇₉	#474	1	2	yes	—
154	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₇₉	—	4	2	no	—
156	C₂×D₇₉	—	8	1	no	—
157	C₂×D₇₉	—	8	1	no	—
158	C₄×C₇₉	—	8	1	no	—
159	C₄×C₇₉	—	8	2	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	—	4	2	no	—
167	Dic₇₉	—	8	2	no	—
168	#33634	—	79	1	no	maximal, Sylow
169	C₂×C₇₉	—	2	2	no	—
170	C₂×C₇₉	C₂×#474	1	1	yes	characteristic
171	C₂×C₇₉	#3416	1	1	yes	characteristic
172	C₂×C₇₉	#3416	1	1	yes	characteristic
173	D₇₉	—	8	2	no	—
174	#3164	—	79	1	no	—
175	#2951	—	79	1	no	—
176	#2229	—	79	2	no	—
177	#2228	—	79	2	no	—
178	#2226	—	79	2	no	—
179	#2225	—	79	2	no	—
180	#2159	D₇₉	1	1	yes	characteristic, complemented
181	C₂×#367	—	79	1	no	—
182	#2151	—	79	1	no	—
183	#2139	—	79	1	no	—
184	#2138	—	79	1	no	—
185	C₇₉	#33634	1	1	yes	characteristic, complemented, Sylow
186	C₂×#135	—	79	1	no	—
187	#544	—	158	1	no	—
188	#520	—	158	1	no	—
189	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
190	C₂²×C₁₆	—	79	1	no	—
191	#501	—	158	2	no	—
192	#500	—	79	2	no	—
193	#499	—	79	2	no	—
194	#440	—	158	2	no	—
195	C₄×Q₁₆	—	79	2	no	—
196	C₄×D₈	—	79	2	no	—
197	#428	—	158	1	no	—
198	#427	C₂×D₇₉	1	1	yes	characteristic
199	C₂×#107	—	79	1	no	—
200	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
201	#417	—	158	1	no	—
202	#405	—	79	1	no	—
203	#367	—	79	2	no	—
204	#367	—	79	2	no	—
205	#367	C₂×D₇₉	1	2	yes	—
206	#366	C₂×D₇₉	1	2	yes	—
207	C₄⋊C₁₆	—	79	2	no	—
208	C₄⋊C₁₆	—	79	2	no	—
209	#358	—	79	2	no	—
210	#358	—	79	1	no	—
211	#358	—	79	1	no	—
212	#357	—	79	2	no	—
213	#357	—	79	1	no	—
214	#357	—	79	1	no	—
215	C₂×#41	—	158	1	no	—
216	#135	—	79	4	no	—
217	#135	—	158	2	no	—
218	C₂×Q₁₆	—	79	1	no	—
219	C₂×SD₁₆	—	158	1	no	—
220	C₂×D₈	—	79	1	no	—
221	C₂×M₁₆	—	79	1	no	—
222	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
223	C₄⋊Q₈	—	316	1	no	—
224	C₄⋊Q₈	—	158	2	no	—
225	#125	—	316	1	no	—
226	#125	—	158	2	no	—
227	C₂²⋊Q₈	—	316	2	no	—
228	C₄×Q₈	—	158	2	no	—
229	C₄×D₄	—	316	2	no	—
230	C₄×D₄	—	158	2	no	—
231	#117	—	2	1	no	—
232	#117	—	316	1	no	—
233	#117	—	79	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	316	1	no	—
236	C₂×C₄⋊C₄	—	158	1	no	—
237	M₃₂	—	2	2	no	—
238	C₂×C₁₆	—	79	1	no	—
239	C₂×C₁₆	—	79	1	no	—
240	C₂×C₁₆	—	158	2	no	—
241	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
242	#107	—	2	1	no	—
243	#107	—	79	2	no	—
244	#107	—	158	2	no	—
245	#107	—	158	1	no	—
246	#107	C₂²×D₇₉	1	1	yes	characteristic
247	#107	C₂²×D₇₉	1	1	yes	characteristic
248	#107	C₂²×D₇₉	1	2	yes	—
249	#106	—	2	1	no	—
250	#106	—	158	2	no	—
251	#103	—	158	2	no	—
252	#102	—	158	2	no	—
253	#97	—	79	2	no	—
254	C₄×C₈	—	79	2	no	—
255	#41	—	316	2	no	—
256	#41	—	158	4	no	—
257	C₂×Q₈	—	316	2	no	—
258	C₂×Q₈	—	158	1	no	—
259	C₂×D₄	—	316	1	no	—
260	C₂×D₄	—	158	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	316	2	no	—
263	C₂²×C₄	—	316	1	no	—
264	C₂²×C₄	—	316	1	no	—
265	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
266	Q₁₆	—	79	2	no	—
267	Q₁₆	—	158	1	no	—
268	SD₁₆	—	158	4	no	—
269	D₈	—	79	2	no	—
270	D₈	—	158	1	no	—
271	M₁₆	—	158	2	no	—
272	C₂×C₈	—	79	2	no	—
273	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₇₉	1	2	yes	—
275	C₂×C₈	#11929624	1	2	yes	—
276	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	316	2	no	—
282	C₄⋊C₄	—	316	2	no	—
283	C₄⋊C₄	—	316	2	no	—
284	C₄⋊C₄	—	316	2	no	—
285	C₄⋊C₄	—	316	1	no	—
286	C₄⋊C₄	—	79	1	no	—
287	C₄⋊C₄	—	79	1	no	—
288	C₄⋊C₄	—	158	2	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	316	2	no	—
291	C₂²⋊C₄	—	79	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	316	2	no	—
294	C₄²	—	79	2	no	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	158	1	no	—
297	C₁₆	—	158	1	no	—
298	C₂³	—	316	1	no	—
299	C₂³	D₈×D₇₉	1	1	yes	characteristic
300	Q₈	—	316	1	no	—
301	Q₈	—	158	2	no	—
302	Q₈	—	1264	2	no	—
303	D₄	—	316	4	no	—
304	D₄	—	316	1	no	—
305	D₄	—	158	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	—	316	1	no	—
311	C₂×C₄	—	316	4	no	—
312	C₂×C₄	—	316	2	no	—
313	C₂×C₄	—	79	2	no	—
314	C₂×C₄	—	79	2	no	—
315	C₂×C₄	—	1264	2	no	—
316	C₂×C₄	—	1264	1	no	—
317	C₂×C₄	#14579139	1	2	yes	—
318	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
319	C₂×C₄	#14579193	1	2	yes	—
320	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
321	C₈	—	158	2	no	—
322	C₈	#14579139	1	2	yes	—
323	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
324	C₈	C₂×#11929624	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	316	2	no	—
328	C₂²	—	632	1	no	—
329	C₂²	—	632	1	no	—
330	C₂²	#19098842	1	2	yes	—
331	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	316	2	no	—
336	C₄	—	632	2	no	—
337	C₄	—	158	2	no	—
338	C₄	—	158	2	no	—
339	C₄	#19098842	1	2	yes	—
340	C₄	C₂×#14579139	1	1	yes	characteristic
341	C₄	C₂×#14579193	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	632	2	no	—
344	C₂	C₂×#19098842	1	1	yes	characteristic
345	C₂	#23776545	1	1	yes	characteristic
346	C₂	#23776546	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×#2159	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#14579183	C₂²
hypercenter	#2159	D₇₉
hypocenter	C₇₉	#33634
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
1. #23773940	C₂	semidirect
3. #23765563	C₂	semidirect
4. #23765560	C₂	semidirect
5. #23765546	C₂	semidirect
6. #23765541	C₂	semidirect
7. C₂×#19098390	C₂	semidirect
8. #23765160	C₂	semidirect
11. C₇₉×#2159	C₂	semidirect
43. #14579183	Q₈	semidirect
180. #2159	D₇₉	semidirect
185. C₇₉	#33634	semidirect

Maximal subgroups

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

subgroup	index
#23773940	2
#23772274	2
#23765563	2
#23765560	2
#23765546	2
#23765541	2
C₂×#19098390	2
#23765160	2
#23765113	2
#23765105	2
C₇₉×#2159	2
#33634	79