Subgroups — #34445860

Order 20224.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775830	C₂	1	1	yes	maximal, characteristic, complemented
2	#23775749	C₂	1	1	yes	maximal, characteristic
3	#23765673	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765604	C₂	1	2	yes	maximal, complemented
5	#23765590	C₂	1	2	yes	maximal, complemented
6	#23765543	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765512	C₂	1	2	yes	maximal, complemented
8	C₄×#14579105	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765231	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765185	C₂	1	2	yes	maximal, complemented
11	C₇₉×#2224	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099287	—	2	1	no	—
13	#19099218	—	2	1	no	—
14	C₂×#14579105	C₂²	1	1	yes	characteristic
15	C₂×#14579100	C₂²	1	2	yes	—
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098906	—	2	2	no	—
18	#19098875	—	2	2	no	—
19	#19098845	—	2	1	no	—
20	#19098843	—	2	1	no	—
21	#19098805	C₂²	1	2	yes	—
22	#19098796	C₂²	1	2	yes	—
23	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
24	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
25	#19098600	C₂²	1	2	yes	—
26	#107×D₇₉	C₂²	1	2	yes	—
27	#19098391	C₂²	1	2	yes	—
28	#19098390	C₂²	1	1	yes	characteristic
29	#19098388	C₂²	1	2	yes	—
30	#19098388	C₂²	1	2	yes	—
31	#19098368	C₂²	1	2	yes	—
32	#19098366	C₂²	1	1	yes	characteristic
33	#19098364	C₂²	1	1	yes	characteristic
34	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
36	#500×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
38	#366×C₇₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#357×C₇₉	C₂²	1	2	yes	—
41	C₂×#11929624	—	4	1	no	—
42	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₇₉	C₂³	1	2	yes	—
44	#14579161	—	2	1	no	—
45	#14579156	—	4	2	no	—
46	#14579150	—	4	1	no	—
47	#14579143	—	4	2	no	—
48	#14579141	—	2	2	no	—
49	#14579140	—	4	1	no	—
50	#14579132	—	2	2	no	—
51	Q₈×Dic₇₉	—	4	1	no	—
52	C₄×Dic₁₅₈	—	2	1	no	—
53	C₄×D₃₁₆	D₄	1	1	yes	characteristic
54	#14579119	—	2	2	no	—
55	C₄⋊C₄×D₇₉	—	2	2	no	—
56	#14579105	—	2	2	no	—
57	#14579100	—	2	4	no	—
58	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
60	#14579095	—	2	1	no	—
61	#14579095	C₂³	1	2	yes	—
62	#14579095	C₂³	1	1	yes	characteristic
63	#14579094	—	2	2	no	—
64	#14579092	—	2	2	no	—
65	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
66	#14579086	—	2	1	no	—
67	#14579084	—	2	2	no	—
68	#14579082	—	2	2	no	—
69	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
70	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₇₉	—	2	1	no	—
72	C₄×D₄×C₇₉	—	2	1	no	—
73	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
74	#107×C₇₉	—	2	1	no	—
75	#107×C₇₉	C₂³	1	1	yes	characteristic
76	#107×C₇₉	C₂³	1	2	yes	—
77	#102×C₇₉	—	2	1	no	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	#11929624	—	8	4	no	—
80	C₂×Dic₁₅₈	—	2	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×C₄×D₇₉	—	4	2	no	—
85	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
86	Dic₃₁₆	—	2	1	no	—
87	Dic₃₁₆	—	2	1	no	—
88	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
89	C₈×D₇₉	—	2	2	no	—
90	C₈×D₇₉	—	2	2	no	—
91	C₄⋊Dic₇₉	—	2	2	no	—
92	C₄⋊Dic₇₉	—	2	1	no	—
93	C₄⋊Dic₇₉	—	4	2	no	—
94	C₄⋊Dic₇₉	—	4	1	no	—
95	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
96	Dic₇₉⋊C₄	—	4	2	no	—
97	Dic₇₉⋊C₄	—	4	1	no	—
98	C₂²⋊Dic₇₉	—	4	2	no	—
99	(C₂×D₇₉)⋊C₄	—	4	2	no	—
100	(C₂×D₇₉)⋊C₄	—	4	2	no	—
101	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
102	C₄×Dic₇₉	—	4	2	no	—
103	C₄×Dic₇₉	—	4	1	no	—
104	C₇₉⋊C₁₆	—	2	1	no	—
105	C₇₉⋊C₁₆	#41	1	2	yes	complemented
106	C₂×Q₈×C₇₉	—	4	1	no	—
107	C₂×D₄×C₇₉	—	2	1	no	—
108	C₂²×C₄×C₇₉	—	4	1	no	—
109	D₈×C₇₉	—	2	1	no	—
110	D₈×C₇₉	—	2	1	no	—
111	C₂×C₈×C₇₉	C₂×D₄	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	2	no	—
115	C₄⋊C₄×C₇₉	—	4	1	no	—
116	C₂²⋊C₄×C₇₉	—	4	1	no	—
117	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₉	—	2	2	no	—
119	C₂³×C₇₉	—	4	1	no	—
120	C₂²×D₇₉	C₂×D₈	1	2	yes	—
121	Q₈×C₇₉	—	8	2	no	—
122	D₄×C₇₉	—	4	1	no	—
123	D₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	8	2	no	—
127	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₇₉	—	8	4	no	—
131	C₂×Dic₇₉	—	4	1	no	—
132	C₂×Dic₇₉	—	4	1	no	—
133	C₂×Dic₇₉	—	4	2	no	—
134	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
135	D₃₁₆	—	2	2	no	—
136	C₄×D₇₉	—	2	2	no	—
137	C₄×D₇₉	—	2	2	no	—
138	C₄×D₇₉	—	8	4	no	—
139	Dic₁₅₈	—	4	1	no	—
140	Dic₁₅₈	—	4	1	no	—
141	Dic₁₅₈	—	8	2	no	—
142	C₈×C₇₉	—	2	1	no	—
143	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₇₉⋊C₈	—	2	2	no	—
146	C₂²×C₇₉	—	8	1	no	—
147	C₂²×C₇₉	—	8	1	no	—
148	C₂²×C₇₉	—	8	1	no	—
149	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₇₉	—	2	2	no	—
151	C₂×D₇₉	—	2	2	no	—
152	C₂×D₇₉	—	2	2	no	—
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	8	1	no	—
155	C₄×C₇₉	—	8	2	no	—
156	C₄×C₇₉	#509	1	2	yes	—
157	C₄×C₇₉	#547	1	1	yes	characteristic
158	C₄×C₇₉	#578	1	1	yes	characteristic
159	Dic₇₉	—	2	2	no	—
160	Dic₇₉	—	8	1	no	—
161	Dic₇₉	—	8	2	no	—
162	Dic₇₉	—	8	1	no	—
163	#33920	—	79	1	no	maximal, Sylow
164	C₂×C₇₉	—	16	1	no	—
165	C₂×C₇₉	#3334	1	1	yes	characteristic
166	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
167	C₂×C₇₉	#3417	1	1	yes	characteristic
168	D₇₉	—	4	2	no	—
169	#3342	—	79	1	no	—
170	#3334	—	79	1	no	—
171	#2252	—	79	1	no	—
172	#2236	—	79	2	no	—
173	#2234	—	79	2	no	—
174	#2225	—	79	1	no	—
175	#2224	D₇₉	1	1	yes	characteristic, complemented
176	#2220	—	79	2	no	—
177	C₄×SD₃₂	—	79	1	no	—
178	#2169	—	79	1	no	—
179	#2159	—	79	2	no	—
180	C₇₉	#33920	1	1	yes	characteristic, complemented, Sylow
181	#556	—	158	1	no	—
182	#547	—	158	1	no	—
183	C₂×SD₃₂	—	79	1	no	—
184	C₂×M₃₂	—	79	2	no	—
185	#500	—	79	1	no	—
186	#500	C₂×D₇₉	1	1	yes	characteristic
187	#485	—	158	2	no	—
188	#480	—	158	2	no	—
189	#475	—	158	1	no	—
190	#474	—	158	1	no	—
191	#467	—	79	2	no	—
192	#466	—	79	2	no	—
193	C₄×Q₁₆	—	79	1	no	—
194	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
195	C₈×D₄	—	79	1	no	—
196	#427	—	79	2	no	—
197	C₂×#107	—	79	2	no	—
198	#367	—	79	2	no	—
199	#367	—	79	1	no	—
200	#366	—	79	2	no	—
201	#366	—	79	2	no	—
202	#366	C₂×D₇₉	1	2	yes	—
203	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
204	#358	—	79	2	no	—
205	#358	—	79	1	no	—
206	#357	—	79	1	no	—
207	#357	C₂×D₇₉	1	2	yes	—
208	C₂²⋊C₁₆	—	79	2	no	—
209	C₄×C₁₆	—	79	1	no	—
210	C₂×#41	—	316	1	no	—
211	C₂×Q₁₆	—	79	1	no	—
212	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
213	C₂²×C₈	—	79	2	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	158	1	no	—
216	#126	—	316	2	no	—
217	#124	—	316	1	no	—
218	#123	—	316	2	no	—
219	C₂²⋊Q₈	—	316	1	no	—
220	C₂²⋊Q₈	—	158	2	no	—
221	#121	—	158	2	no	—
222	C₄×Q₈	—	316	1	no	—
223	C₄×Q₈	—	158	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	79	1	no	—
226	#117	—	158	2	no	—
227	C₂×C₄⋊C₄	—	158	2	no	—
228	SD₃₂	—	158	2	no	—
229	M₃₂	—	158	4	no	—
230	C₂×C₁₆	—	79	1	no	—
231	C₂×C₁₆	—	79	1	no	—
232	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
233	#107	—	2	1	no	—
234	#107	—	79	1	no	—
235	#107	—	79	2	no	—
236	#107	—	158	2	no	—
237	#107	—	158	1	no	—
238	#107	C₂²×D₇₉	1	2	yes	—
239	#107	C₂²×D₇₉	1	1	yes	characteristic
240	#106	—	158	2	no	—
241	C₄⋊C₈	—	79	1	no	—
242	#103	—	158	2	no	—
243	#103	—	158	1	no	—
244	#102	—	2	1	no	—
245	#102	—	158	2	no	—
246	C₂²⋊C₈	—	79	2	no	—
247	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
248	#41	—	632	4	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	316	1	no	—
251	C₂×Q₈	—	158	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	316	2	no	—
254	C₂×D₄	—	79	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	316	2	no	—
257	C₂²×C₄	—	79	2	no	—
258	Q₁₆	—	158	1	no	—
259	Q₁₆	—	158	1	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	C₂×C₈	—	79	2	no	—
263	C₂×C₈	—	158	2	no	—
264	C₂×C₈	—	158	2	no	—
265	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
266	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	316	2	no	—
271	C₄⋊C₄	—	316	2	no	—
272	C₄⋊C₄	—	316	1	no	—
273	C₄⋊C₄	—	316	1	no	—
274	C₄⋊C₄	—	79	1	no	—
275	C₄⋊C₄	—	158	2	no	—
276	C₄⋊C₄	—	158	1	no	—
277	C₂²⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	316	2	no	—
279	C₂²⋊C₄	—	316	2	no	—
280	C₂²⋊C₄	—	316	2	no	—
281	C₂²⋊C₄	—	79	2	no	—
282	C₄²	—	316	2	no	—
283	C₄²	—	316	1	no	—
284	C₄²	D₄×D₇₉	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	79	2	no	—
287	C₁₆	—	158	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	79	2	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	316	1	no	—
292	Q₈	—	316	1	no	—
293	Q₈	—	632	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	632	4	no	—
297	D₄	—	158	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	316	1	no	—
302	C₂×C₄	—	316	1	no	—
303	C₂×C₄	—	316	2	no	—
304	C₂×C₄	—	79	2	no	—
305	C₂×C₄	—	632	4	no	—
306	C₂×C₄	—	158	2	no	—
307	C₂×C₄	—	158	2	no	—
308	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
309	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	158	2	no	—
313	C₈	#41×D₇₉	1	1	yes	characteristic
314	C₈	#14579209	1	1	yes	characteristic
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	158	2	no	—
319	C₂²	—	158	2	no	—
320	C₂²	—	158	2	no	—
321	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	632	1	no	—
326	C₄	—	632	2	no	—
327	C₄	—	632	1	no	—
328	C₄	—	158	2	no	—
329	C₄	#19098997	1	2	yes	—
330	C₄	#19099215	1	1	yes	characteristic
331	C₄	#19099379	1	1	yes	characteristic
332	C₂	—	16	1	no	—
333	C₂	—	316	2	no	—
334	C₂	#23775746	1	1	yes	characteristic
335	C₂	C₂×#19098997	1	1	yes	characteristic
336	C₂	#23776552	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×#2224	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23775749	C₂
hypercenter	#2224	D₇₉
hypocenter	C₇₉	#33920
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2224

Complements

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

N	H ≅ G/N	type
1. #23775830	C₂	semidirect
3. #23765673	C₂	semidirect
4. #23765604	C₂	semidirect
5. #23765590	C₂	semidirect
6. #23765543	C₂	semidirect
7. #23765512	C₂	semidirect
8. C₄×#14579105	C₂	semidirect
9. #23765231	C₂	semidirect
10. #23765185	C₂	semidirect
11. C₇₉×#2224	C₂	semidirect
105. C₇₉⋊C₁₆	#41	semidirect
175. #2224	D₇₉	semidirect
180. C₇₉	#33920	semidirect

Maximal subgroups

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

subgroup	index
#23775830	2
#23775749	2
#23765673	2
#23765604	2
#23765590	2
#23765543	2
#23765512	2
C₄×#14579105	2
#23765231	2
#23765185	2
C₇₉×#2224	2
#33920	79