Subgroups — #34445938

Order 20224.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776410	C₂	1	1	yes	maximal, characteristic, complemented
2	#23776269	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765633	C₂	1	2	yes	maximal, complemented
4	#23765625	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765560	C₂	1	2	yes	maximal, complemented
6	#23765458	C₂	1	2	yes	maximal, complemented
7	#23765436	C₂	1	1	yes	maximal, characteristic
8	#23765302	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14579104	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765246	C₂	1	2	yes	maximal, complemented
11	C₇₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099298	—	2	1	no	—
13	#19099274	—	2	1	no	—
14	C₂×#14579104	C₂²	1	1	yes	characteristic
15	C₂×D₁₂₆₄	C₂²	1	2	yes	—
16	#19098964	C₂²	1	2	yes	—
17	#19098948	C₂²	1	2	yes	—
18	#19098940	C₂²	1	1	yes	characteristic
19	#19098866	—	2	2	no	—
20	#19098850	—	2	1	no	—
21	#19098781	—	2	2	no	—
22	#19098763	—	2	1	no	—
23	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
24	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
25	#19098645	C₂²	1	2	yes	—
26	#19098645	C₂²	1	2	yes	—
27	#19098390	C₂²	1	1	yes	characteristic
28	Dic₇₉⋊C₁₆	C₂²	1	2	yes	—
29	#19098367	C₂²	1	1	yes	characteristic
30	#19098365	C₂²	1	2	yes	—
31	#19098363	C₂²	1	1	yes	characteristic
32	#19098363	C₂²	1	2	yes	—
33	#19098363	C₂²	1	2	yes	—
34	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₉	C₂²	1	2	yes	—
36	#500×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	#366×C₇₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#358×C₇₉	C₂²	1	2	yes	—
41	C₂×#11929611	—	2	2	no	—
42	C₂×D₆₃₂	—	2	1	no	—
43	C₂×D₆₃₂	C₂³	1	2	yes	—
44	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
45	#14579158	—	2	1	no	—
46	#14579152	—	2	2	no	—
47	#14579150	—	2	2	no	—
48	#14579149	—	4	1	no	—
49	#14579146	—	4	2	no	—
50	#14579141	—	4	1	no	—
51	#14579137	—	4	2	no	—
52	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
53	C₄×D₃₁₆	—	2	1	no	—
54	C₄×D₃₁₆	—	4	1	no	—
55	#14579123	—	2	2	no	—
56	#14579123	—	2	2	no	—
57	C₄⋊C₄×D₇₉	—	4	1	no	—
58	#14579104	—	2	2	no	—
59	D₁₂₆₄	—	2	4	no	—
60	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
62	#14579095	C₂³	1	1	yes	characteristic
63	#14579092	—	2	2	no	—
64	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
65	Dic₇₉⋊C₈	D₄	1	2	yes	—
66	#14579083	—	2	1	no	—
67	#14579081	—	2	2	no	—
68	#14579081	—	2	2	no	—
69	C₈×Dic₇₉	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₇₉	—	2	1	no	—
72	C₄×Q₈×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	#103×C₇₉	—	2	1	no	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
80	C₂×D₃₁₆	—	2	1	no	—
81	C₂×D₃₁₆	—	2	2	no	—
82	C₂×D₃₁₆	—	4	1	no	—
83	C₂×D₃₁₆	—	4	1	no	—
84	C₂×C₄×D₇₉	—	4	2	no	—
85	C₂×C₄×D₇₉	—	4	2	no	—
86	C₂×C₄×D₇₉	—	4	1	no	—
87	#11929611	—	8	2	no	—
88	D₆₃₂	—	2	2	no	—
89	D₆₃₂	—	2	1	no	—
90	D₆₃₂	—	2	2	no	—
91	D₆₃₂	—	8	1	no	—
92	D₆₃₂	#41	1	2	yes	—
93	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
94	C₄⋊Dic₇₉	—	2	1	no	—
95	C₄⋊Dic₇₉	—	4	2	no	—
96	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
97	Dic₇₉⋊C₄	—	4	2	no	—
98	Dic₇₉⋊C₄	D₈	1	2	yes	—
99	(C₂×D₇₉)⋊C₄	—	4	2	no	—
100	(C₂×D₇₉)⋊C₄	—	4	2	no	—
101	(C₂×D₇₉)⋊C₄	—	4	2	no	—
102	(C₂×D₇₉)⋊C₄	—	4	2	no	—
103	(C₂×D₇₉)⋊C₄	—	4	1	no	—
104	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
105	C₇₉⋊C₁₆	—	2	1	no	—
106	C₇₉⋊C₁₆	—	2	1	no	—
107	C₂×Q₈×C₇₉	—	2	1	no	—
108	C₂×Q₈×C₇₉	—	4	1	no	—
109	Q₁₆×C₇₉	—	2	1	no	—
110	Q₁₆×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	C₄⋊C₄×C₇₉	—	4	1	no	—
117	C₄²×C₇₉	—	4	1	no	—
118	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₇₉	—	2	2	no	—
120	C₂²×D₇₉	—	4	1	no	—
121	C₂²×D₇₉	—	4	2	no	—
122	Q₈×C₇₉	—	4	1	no	—
123	Q₈×C₇₉	—	4	1	no	—
124	Q₈×C₇₉	—	16	1	no	—
125	C₂×C₄×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	1	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₇₉	—	4	1	no	—
132	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
133	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
134	D₃₁₆	—	2	2	no	—
135	D₃₁₆	—	4	2	no	—
136	D₃₁₆	—	4	1	no	—
137	D₃₁₆	—	4	2	no	—
138	D₃₁₆	—	8	2	no	—
139	D₃₁₆	—	16	1	no	—
140	C₄×D₇₉	—	8	2	no	—
141	C₄×D₇₉	—	16	2	no	—
142	C₄×D₇₉	—	16	2	no	—
143	Dic₁₅₈	—	2	2	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	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₇₉	—	4	2	no	—
150	C₂×D₇₉	—	8	1	no	—
151	C₂×D₇₉	—	8	1	no	—
152	C₂×D₇₉	—	8	2	no	—
153	C₂×D₇₉	—	8	2	no	—
154	C₂×D₇₉	—	8	2	no	—
155	C₄×C₇₉	—	2	1	no	—
156	C₄×C₇₉	—	8	1	no	—
157	C₄×C₇₉	—	8	2	no	—
158	C₄×C₇₉	—	8	1	no	—
159	C₄×C₇₉	#508	1	2	yes	—
160	C₄×C₇₉	#548	1	1	yes	characteristic
161	C₄×C₇₉	#577	1	1	yes	characteristic
162	Dic₇₉	—	2	2	no	—
163	Dic₇₉	—	2	2	no	—
164	Dic₇₉	—	8	1	no	—
165	#33928	—	79	1	no	maximal, Sylow
166	C₂×C₇₉	#3358	1	1	yes	characteristic
167	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
168	C₂×C₇₉	#3415	1	1	yes	characteristic
169	D₇₉	—	8	2	no	—
170	D₇₉	—	16	2	no	—
171	#3396	—	79	1	no	—
172	#3380	—	79	1	no	—
173	#2242	—	79	2	no	—
174	#2240	—	79	1	no	—
175	#2228	—	79	2	no	—
176	#2227	D₇₉	1	1	yes	characteristic, complemented
177	#2210	—	79	2	no	—
178	#2206	—	79	1	no	—
179	Q₈×C₁₆	—	79	1	no	—
180	C₄×SD₃₂	—	79	1	no	—
181	C₄×D₁₆	—	79	2	no	—
182	C₇₉	#33928	1	1	yes	characteristic, complemented, Sylow
183	#559	—	158	1	no	—
184	#554	—	158	1	no	—
185	C₂×SD₃₂	—	79	1	no	—
186	C₂×D₁₆	—	79	2	no	—
187	#500	C₂×D₇₉	1	1	yes	characteristic
188	#499	—	79	2	no	—
189	#495	—	79	2	no	—
190	#493	—	79	1	no	—
191	#478	—	158	2	no	—
192	#476	—	158	1	no	—
193	#463	—	158	2	no	—
194	#460	—	158	1	no	—
195	C₈×Q₈	—	79	1	no	—
196	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
197	C₄×D₈	—	79	2	no	—
198	C₄×D₈	—	79	2	no	—
199	C₄×D₈	—	79	1	no	—
200	#367	—	79	1	no	—
201	#366	C₂×D₇₉	1	2	yes	—
202	C₄⋊C₁₆	—	79	2	no	—
203	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
204	#358	—	79	1	no	—
205	#358	C₂×D₇₉	1	2	yes	—
206	#357	—	79	2	no	—
207	#357	—	79	1	no	—
208	#357	—	79	2	no	—
209	#357	—	79	2	no	—
210	C₄×C₁₆	—	79	2	no	—
211	C₄×C₁₆	—	79	1	no	—
212	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
213	C₂×SD₁₆	—	158	2	no	—
214	C₂×D₈	—	79	1	no	—
215	C₂×D₈	—	79	2	no	—
216	C₂×D₈	—	158	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	#127	—	158	1	no	—
219	#125	—	158	2	no	—
220	#124	—	316	1	no	—
221	#124	—	158	2	no	—
222	#123	—	316	2	no	—
223	C₂²⋊Q₈	—	316	1	no	—
224	C₂²⋊Q₈	—	316	2	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	79	1	no	—
227	C₄×D₄	—	316	1	no	—
228	C₄×D₄	—	158	2	no	—
229	C₄×D₄	—	158	2	no	—
230	C₄×D₄	—	158	1	no	—
231	C₂×C₄⋊C₄	—	316	1	no	—
232	SD₃₂	—	158	2	no	—
233	D₁₆	—	158	4	no	—
234	C₂×C₁₆	—	79	1	no	—
235	C₂×C₁₆	—	79	1	no	—
236	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
237	#107	—	2	1	no	—
238	#107	—	79	1	no	—
239	#107	C₂²×D₇₉	1	2	yes	—
240	#107	C₂²×D₇₉	1	1	yes	characteristic
241	#106	—	158	2	no	—
242	C₄⋊C₈	—	79	2	no	—
243	C₄⋊C₈	—	79	1	no	—
244	#103	—	2	1	no	—
245	#102	—	158	2	no	—
246	#102	—	158	2	no	—
247	#102	—	158	1	no	—
248	C₄×C₈	—	79	2	no	—
249	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	79	1	no	—
253	C₂×D₄	—	316	1	no	—
254	C₂×D₄	—	316	1	no	—
255	C₂×D₄	—	158	2	no	—
256	C₂×D₄	—	158	1	no	—
257	C₂²×C₄	—	316	2	no	—
258	C₂²×C₄	—	316	2	no	—
259	C₂²×C₄	—	316	1	no	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	1	no	—
262	SD₁₆	—	632	2	no	—
263	D₈	—	79	2	no	—
264	D₈	—	632	1	no	—
265	D₈	—	158	2	no	—
266	D₈	—	158	2	no	—
267	D₈	—	158	1	no	—
268	C₂×C₈	—	79	2	no	—
269	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	316	2	no	—
276	C₄⋊C₄	—	316	2	no	—
277	C₄⋊C₄	—	79	2	no	—
278	C₄⋊C₄	—	79	1	no	—
279	C₄⋊C₄	—	158	1	no	—
280	C₂²⋊C₄	—	316	2	no	—
281	C₂²⋊C₄	—	316	2	no	—
282	C₂²⋊C₄	—	316	2	no	—
283	C₂²⋊C₄	—	316	2	no	—
284	C₂²⋊C₄	—	316	1	no	—
285	C₄²	—	4	1	no	—
286	C₄²	—	79	2	no	—
287	C₄²	D₄×D₇₉	1	1	yes	characteristic
288	C₁₆	—	2	2	no	—
289	C₁₆	—	158	1	no	—
290	C₁₆	—	158	1	no	—
291	C₂³	—	316	1	no	—
292	C₂³	—	316	2	no	—
293	Q₈	—	4	1	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	16	1	no	—
296	Q₈	—	158	2	no	—
297	D₄	—	316	2	no	—
298	D₄	—	316	1	no	—
299	D₄	—	316	2	no	—
300	D₄	—	632	2	no	—
301	D₄	—	158	2	no	—
302	D₄	—	1264	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	316	1	no	—
307	C₂×C₄	—	79	2	no	—
308	C₂×C₄	—	79	2	no	—
309	C₂×C₄	—	632	2	no	—
310	C₂×C₄	—	1264	2	no	—
311	C₂×C₄	—	1264	2	no	—
312	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
313	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	158	2	no	—
317	C₈	#41×D₇₉	1	1	yes	characteristic
318	C₈	#14579209	1	1	yes	characteristic
319	C₂²	—	316	2	no	—
320	C₂²	—	632	1	no	—
321	C₂²	—	632	1	no	—
322	C₂²	—	632	2	no	—
323	C₂²	—	632	2	no	—
324	C₂²	—	632	2	no	—
325	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	1	no	—
330	C₄	—	632	1	no	—
331	C₄	—	158	2	no	—
332	C₄	—	158	2	no	—
333	C₄	#19098996	1	2	yes	—
334	C₄	#19099223	1	1	yes	characteristic
335	C₄	#19099372	1	1	yes	characteristic
336	C₂	—	632	2	no	—
337	C₂	—	1264	2	no	—
338	C₂	#23776034	1	1	yes	characteristic
339	C₂	C₂×#19098996	1	1	yes	characteristic
340	C₂	#23776540	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×#2227	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23765436	C₂
hypercenter	#2227	D₇₉
hypocenter	C₇₉	#33928
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
1. #23776410	C₂	semidirect
2. #23776269	C₂	semidirect
3. #23765633	C₂	semidirect
4. #23765625	C₂	semidirect
5. #23765560	C₂	semidirect
6. #23765458	C₂	semidirect
8. #23765302	C₂	semidirect
9. C₄×#14579104	C₂	semidirect
10. #23765246	C₂	semidirect
11. C₇₉×#2227	C₂	semidirect
176. #2227	D₇₉	semidirect
182. C₇₉	#33928	semidirect

Maximal subgroups

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

subgroup	index
#23776410	2
#23776269	2
#23765633	2
#23765625	2
#23765560	2
#23765458	2
#23765436	2
#23765302	2
C₄×#14579104	2
#23765246	2
C₇₉×#2227	2
#33928	79