Subgroups — #34443022

Order 20224.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773932	C₂	1	1	yes	maximal, characteristic, complemented
2	#23772316	C₂	1	1	yes	maximal, characteristic
3	#23765575	C₂	1	4	yes	maximal, complemented
4	#23765570	C₂	1	4	yes	maximal, complemented
5	C₂×#19098387	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765165	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765135	C₂	1	1	yes	maximal, characteristic
8	C₂×#19098374	C₂	1	1	yes	maximal, characteristic
9	C₇₉×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579183	C₂²	1	1	yes	characteristic
11	#19099329	—	2	1	no	—
12	#19099191	—	2	1	no	—
13	C₂×#14579101	C₂²	1	1	yes	characteristic
14	#19098971	—	2	2	no	—
15	#19098963	C₂²	1	2	yes	complemented
16	#19098960	C₂²	1	2	yes	complemented
17	#19098684	—	2	2	no	—
18	#19098679	C₂²	1	4	yes	—
19	#19098606	—	2	1	no	—
20	C₂×#14579093	C₂²	1	1	yes	characteristic
21	#19098579	—	2	1	no	—
22	#19098518	C₂²	1	1	yes	characteristic
23	#19098387	C₂²	1	4	yes	complemented
24	#19098383	C₂²	1	4	yes	complemented
25	#19098374	C₂²	1	4	yes	—
26	#19098372	C₂²	1	2	yes	—
27	#19098372	C₂²	1	2	yes	—
28	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
30	#427×C₇₉	C₂²	1	1	yes	characteristic
31	#365×C₇₉	C₂²	1	2	yes	complemented
32	#365×C₇₉	C₂²	1	2	yes	complemented
33	C₂×#11929624	—	2	1	no	—
34	#14579183	—	2	1	no	—
35	#14579183	—	2	1	no	—
36	#14579183	Q₈	1	2	yes	complemented
37	#14579183	C₂³	1	2	yes	—
38	C₂×Dic₃₁₆	Q₈	1	1	yes	characteristic, complemented
39	C₂×#11929613	—	2	1	no	—
40	C₂×D₆₃₂	Q₈	1	1	yes	characteristic, complemented
41	#14579169	—	2	4	no	—
42	#14579167	—	4	2	no	—
43	C₂×#11929608	D₄	1	1	yes	characteristic
44	M₁₆×D₇₉	—	4	2	no	—
45	#14579161	—	4	1	no	—
46	#14579160	—	2	2	no	—
47	#14579153	—	4	1	no	—
48	#14579152	—	2	2	no	—
49	#14579138	—	4	2	no	—
50	#14579120	D₄	1	1	yes	characteristic
51	C₂×C₄⋊Dic₇₉	—	2	1	no	—
52	#14579101	—	2	1	no	—
53	#14579101	C₂³	1	2	yes	—
54	C₂×C₇₉⋊C₁₆	—	2	1	no	—
55	#14579094	—	2	2	no	—
56	#14579093	—	2	1	no	—
57	#14579093	C₂³	1	2	yes	—
58	#14579092	—	2	2	no	—
59	#14579087	—	2	4	no	—
60	#14579071	D₄	1	2	yes	—
61	C₈×Dic₇₉	C₂³	1	2	yes	—
62	C₂×M₁₆×C₇₉	—	2	1	no	—
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	#117×C₇₉	—	2	1	no	—
65	M₃₂×C₇₉	—	2	1	no	—
66	M₃₂×C₇₉	C₂³	1	2	yes	—
67	C₂×C₁₆×C₇₉	—	2	1	no	—
68	#108×C₇₉	—	2	1	no	—
69	#108×C₇₉	C₂³	1	2	yes	—
70	#107×C₇₉	—	2	1	no	—
71	#106×C₇₉	C₂³	1	1	yes	characteristic
72	#106×C₇₉	C₂³	1	1	yes	characteristic
73	#11929624	—	2	2	no	—
74	#11929624	—	2	2	no	—
75	#11929624	—	4	1	no	—
76	#11929624	—	4	1	no	—
77	C₂×Dic₁₅₈	—	2	1	no	—
78	C₂×Dic₁₅₈	—	4	2	no	—
79	C₂×D₃₁₆	—	2	1	no	—
80	C₂×C₂²⋊D₇₉	—	4	1	no	—
81	C₂²×Dic₇₉	—	4	1	no	—
82	C₂×C₄×D₇₉	—	4	1	no	—
83	Dic₃₁₆	—	2	1	no	—
84	Dic₃₁₆	C₂×Q₈	1	2	yes	—
85	#11929613	—	2	2	no	—
86	#11929613	—	2	2	no	—
87	D₆₃₂	—	2	1	no	—
88	D₆₃₂	C₂×Q₈	1	2	yes	—
89	#11929608	SD₁₆	1	2	yes	—
90	#11929608	SD₁₆	1	2	yes	—
91	#11929607	—	4	4	no	—
92	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
93	C₈×D₇₉	—	4	4	no	—
94	C₄⋊Dic₇₉	—	2	1	no	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	C₄⋊Dic₇₉	—	4	2	no	—
97	C₄⋊Dic₇₉	—	4	1	no	—
98	Dic₇₉⋊C₄	—	4	2	no	—
99	Dic₇₉⋊C₄	—	4	2	no	—
100	C₂²⋊Dic₇₉	—	2	1	no	—
101	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
102	C₇₉⋊C₁₆	—	2	2	no	—
103	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
104	M₁₆×C₇₉	—	2	2	no	—
105	M₁₆×C₇₉	—	4	1	no	—
106	C₂×C₈×C₇₉	—	4	1	no	—
107	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₉	—	2	1	no	—
114	C₄⋊C₄×C₇₉	—	2	1	no	—
115	C₂²⋊C₄×C₇₉	—	4	1	no	—
116	C₄²×C₇₉	—	4	1	no	—
117	C₁₆×C₇₉	—	2	2	no	—
118	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₇₉	—	4	1	no	—
120	C₂×C₄×C₇₉	—	4	1	no	—
121	C₂×C₄×C₇₉	—	4	1	no	—
122	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₇₉	—	4	2	no	—
129	C₂²⋊D₇₉	—	4	2	no	—
130	C₂×Dic₇₉	—	2	2	no	—
131	C₂×Dic₇₉	—	4	1	no	—
132	C₂×Dic₇₉	—	4	2	no	—
133	C₂×Dic₇₉	—	16	1	no	—
134	D₃₁₆	—	2	2	no	—
135	D₃₁₆	—	4	1	no	—
136	C₄×D₇₉	—	4	2	no	—
137	C₄×D₇₉	—	4	2	no	—
138	Dic₁₅₈	—	2	2	no	—
139	Dic₁₅₈	—	4	1	no	—
140	Dic₁₅₈	—	16	2	no	—
141	C₈×C₇₉	—	4	2	no	—
142	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
145	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
146	C₇₉⋊C₈	C₂×SD₁₆	1	4	yes	—
147	C₂²×C₇₉	—	4	1	no	—
148	C₂²×C₇₉	#476	1	1	yes	characteristic
149	C₂²×C₇₉	#476	1	1	yes	characteristic
150	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×D₇₉	—	4	2	no	—
152	C₂×D₇₉	—	16	1	no	—
153	C₄×C₇₉	—	8	1	no	—
154	C₄×C₇₉	—	8	1	no	—
155	C₄×C₇₉	#476	1	1	yes	characteristic
156	C₄×C₇₉	#476	1	1	yes	characteristic
157	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₇₉	—	4	2	no	—
160	Dic₇₉	—	4	2	no	—
161	Dic₇₉	—	8	2	no	—
162	#33638	—	79	1	no	maximal, Sylow
163	C₂×C₇₉	—	2	1	no	—
164	C₂×C₇₉	—	2	1	no	—
165	C₂×C₇₉	—	2	1	no	—
166	C₂×C₇₉	C₂×#476	1	1	yes	characteristic
167	D₇₉	—	8	2	no	—
168	#3163	—	79	1	no	—
169	#2957	—	79	1	no	—
170	#2231	—	79	4	no	—
171	#2230	—	79	4	no	—
172	#2155	D₇₉	1	1	yes	characteristic, complemented
173	C₂×#365	—	79	1	no	—
174	#2153	—	79	1	no	—
175	#2145	—	79	1	no	—
176	C₂×#360	—	79	1	no	—
177	C₇₉	#33638	1	1	yes	characteristic, complemented, Sylow
178	C₂×#135	—	79	1	no	—
179	#568	—	158	1	no	—
180	#544	—	158	1	no	—
181	C₂×M₃₂	—	79	1	no	—
182	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
183	#501	—	158	2	no	—
184	#499	—	79	2	no	—
185	#498	—	79	2	no	—
186	#444	—	158	2	no	—
187	#443	—	79	4	no	—
188	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
189	#428	—	158	1	no	—
190	#427	C₂×D₇₉	1	1	yes	characteristic
191	C₂×#106	—	79	1	no	—
192	#421	—	158	1	no	—
193	#405	—	79	1	no	—
194	#365	—	79	4	no	—
195	#365	C₂×D₇₉	1	2	yes	complemented
196	#365	C₂×D₇₉	1	2	yes	complemented
197	#364	—	79	4	no	—
198	#360	—	79	4	no	—
199	#360	—	79	2	no	—
200	#360	—	79	2	no	—
201	C₂×#41	—	158	1	no	—
202	#135	—	79	2	no	—
203	#135	—	79	2	no	—
204	#135	—	158	1	no	—
205	#135	—	158	1	no	—
206	C₂×Q₁₆	—	79	1	no	—
207	C₂×SD₁₆	—	158	1	no	—
208	C₂×D₈	—	79	1	no	—
209	#131	—	316	2	no	—
210	#131	—	158	4	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	316	2	no	—
213	C₂×M₁₆	—	79	1	no	—
214	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	316	1	no	—
216	C₄⋊Q₈	—	158	2	no	—
217	#125	—	316	1	no	—
218	#125	—	158	2	no	—
219	C₂²⋊Q₈	—	316	2	no	—
220	#117	—	2	1	no	—
221	#117	—	79	1	no	—
222	C₂×C₄⋊C₄	—	158	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	79	2	no	—
225	M₃₂	—	158	1	no	—
226	M₃₂	C₂²×D₇₉	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	158	1	no	—
229	#108	—	2	1	no	—
230	#108	C₂²×D₇₉	1	2	yes	—
231	#107	—	2	1	no	—
232	#107	—	158	2	no	—
233	#106	—	79	2	no	—
234	#106	—	158	2	no	—
235	#106	—	158	1	no	—
236	#106	C₂²×D₇₉	1	1	yes	characteristic
237	#106	C₂²×D₇₉	1	1	yes	characteristic
238	#104	—	158	4	no	—
239	#97	—	79	2	no	—
240	C₄×C₈	—	79	2	no	—
241	#41	—	316	1	no	—
242	#41	—	316	1	no	—
243	#41	—	158	2	no	—
244	#41	—	158	2	no	—
245	C₂×Q₈	—	316	2	no	—
246	C₂×Q₈	—	158	1	no	—
247	C₂×D₄	—	316	1	no	—
248	C₂×D₄	—	158	1	no	—
249	C₂²×C₄	—	316	1	no	—
250	C₂²×C₄	—	316	1	no	—
251	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
252	Q₁₆	—	79	2	no	—
253	Q₁₆	—	158	1	no	—
254	SD₁₆	—	158	2	no	—
255	SD₁₆	—	158	2	no	—
256	D₈	—	79	2	no	—
257	D₈	—	158	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	316	4	no	—
261	M₁₆	—	79	2	no	—
262	M₁₆	—	79	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	316	4	no	—
265	C₂×C₈	—	79	2	no	—
266	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
267	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
268	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
269	C₂×C₈	#11929624	1	1	yes	characteristic
270	C₂×C₈	#11929624	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	316	2	no	—
275	C₄⋊C₄	—	316	2	no	—
276	C₄⋊C₄	—	316	2	no	—
277	C₄⋊C₄	—	316	1	no	—
278	C₄⋊C₄	—	158	2	no	—
279	C₄⋊C₄	—	158	1	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	158	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	79	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	158	2	no	—
286	C₂³	—	316	1	no	—
287	C₂³	#14579188	1	1	yes	characteristic
288	Q₈	—	316	1	no	—
289	Q₈	—	158	2	no	—
290	Q₈	—	1264	2	no	—
291	D₄	—	316	2	no	—
292	D₄	—	316	2	no	—
293	D₄	—	316	1	no	—
294	D₄	—	158	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	316	1	no	—
298	C₂×C₄	—	316	2	no	—
299	C₂×C₄	—	316	2	no	—
300	C₂×C₄	—	316	2	no	—
301	C₂×C₄	—	158	2	no	—
302	C₂×C₄	—	1264	1	no	—
303	C₂×C₄	#14579139	1	1	yes	characteristic
304	C₂×C₄	#14579139	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
307	C₂×C₄	#14579188	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	79	4	no	—
311	C₈	#14579139	1	1	yes	characteristic
312	C₈	#14579139	1	1	yes	characteristic
313	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
314	C₈	C₂×#11929624	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	316	2	no	—
317	C₂²	—	1264	1	no	—
318	C₂²	#19098852	1	1	yes	characteristic
319	C₂²	#19098852	1	1	yes	characteristic
320	C₂²	C₂×#14579188	1	1	yes	characteristic
321	C₄	—	8	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	316	2	no	—
324	C₄	—	316	2	no	—
325	C₄	—	632	2	no	—
326	C₄	#19098852	1	1	yes	characteristic
327	C₄	#19098852	1	1	yes	characteristic
328	C₄	C₂×#14579139	1	1	yes	characteristic
329	C₄	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	632	2	no	—
334	C₂	C₂×#19098852	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19098852
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2155	C₂
socle	C₂×C₇₉	C₂×#476
prime radical	C₂×#14579183	C₂²
hypercenter	#2155	D₇₉
hypocenter	C₇₉	#33638
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
1. #23773932	C₂	semidirect
3. #23765575	C₂	semidirect
4. #23765570	C₂	semidirect
5. C₂×#19098387	C₂	semidirect
6. #23765165	C₂	semidirect
9. C₇₉×#2155	C₂	semidirect
15. #19098963	C₂²	semidirect
16. #19098960	C₂²	semidirect
23. #19098387	C₂²	semidirect
24. #19098383	C₂²	semidirect
31. #365×C₇₉	C₂²	semidirect
32. #365×C₇₉	C₂²	semidirect
36. #14579183	Q₈	semidirect
38. C₂×Dic₃₁₆	Q₈	semidirect
40. C₂×D₆₃₂	Q₈	semidirect
172. #2155	D₇₉	semidirect
177. C₇₉	#33638	semidirect
195. #365	C₂×D₇₉	semidirect
196. #365	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23773932	2
#23772316	2
#23765575	2
#23765570	2
C₂×#19098387	2
#23765165	2
#23765135	2
C₂×#19098374	2
C₇₉×#2155	2
#33638	79