Subgroups — #34446010

Order 20224.

518 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	#23776375	C₂	1	1	yes	maximal, characteristic
2	#23776292	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765642	C₂	1	2	yes	maximal, complemented
4	#23765624	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765558	C₂	1	2	yes	maximal, complemented
6	#23765538	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765441	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765441	C₂	1	2	yes	maximal, complemented
9	#23765266	C₂	1	2	yes	maximal, complemented
10	C₄×#14579106	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099291	—	2	1	no	—
13	#19099235	—	2	1	no	—
14	C₂×#14579106	C₂²	1	1	yes	characteristic
15	C₂×#14579106	C₂²	1	2	yes	—
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098965	C₂²	1	2	yes	—
18	#19098940	C₂²	1	1	yes	characteristic
19	#19098940	C₂²	1	2	yes	—
20	#19098871	—	2	2	no	—
21	#19098849	—	2	1	no	—
22	#19098759	—	2	1	no	—
23	#19098759	—	2	2	no	—
24	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₁₆	C₂²	1	2	yes	—
26	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
27	C₄×D₆₃₂	C₂²	1	2	yes	—
28	#19098391	C₂²	1	1	yes	characteristic
29	#19098389	C₂²	1	2	yes	—
30	#19098368	C₂²	1	1	yes	characteristic
31	#19098365	C₂²	1	1	yes	characteristic
32	#19098365	C₂²	1	2	yes	—
33	#19098365	C₂²	1	2	yes	—
34	#19098365	C₂²	1	2	yes	—
35	#19098365	C₂²	1	2	yes	—
36	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₇₉	C₂²	1	2	yes	—
39	#346×C₇₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×#11929625	—	4	1	no	—
42	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
43	C₂×D₆₃₂	—	2	2	no	—
44	C₂×D₆₃₂	—	2	1	no	—
45	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
46	C₂×D₆₃₂	C₂³	1	2	yes	—
47	#14579161	—	2	1	no	—
48	#14579159	—	4	1	no	—
49	#14579158	—	2	2	no	—
50	#14579158	—	2	1	no	—
51	#14579153	—	2	2	no	—
52	#14579151	—	4	2	no	—
53	#14579136	—	4	1	no	—
54	#14579136	—	4	2	no	—
55	C₄×Dic₁₅₈	—	2	1	no	—
56	C₄×Dic₁₅₈	—	2	2	no	—
57	#14579126	—	4	1	no	—
58	C₄×D₃₁₆	—	2	2	no	—
59	C₄×D₃₁₆	—	2	1	no	—
60	#14579106	—	2	2	no	—
61	#14579106	—	2	4	no	—
62	#14579095	—	2	1	no	—
63	#14579095	C₂³	1	1	yes	characteristic
64	#14579095	C₂³	1	2	yes	—
65	#14579095	C₂³	1	1	yes	characteristic
66	#14579093	—	2	2	no	—
67	#14579086	—	2	2	no	—
68	#14579086	—	2	1	no	—
69	#14579083	—	2	2	no	—
70	#14579083	—	2	1	no	—
71	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
72	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
74	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
77	C₄×C₈×C₇₉	C₂³	1	2	yes	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	#11929625	—	8	4	no	—
80	C₂×Dic₁₅₈	—	2	1	no	—
81	C₂×Dic₁₅₈	—	4	1	no	—
82	C₂×D₃₁₆	—	2	2	no	—
83	C₂×D₃₁₆	—	2	1	no	—
84	C₂×D₃₁₆	—	4	2	no	—
85	C₂×D₃₁₆	—	4	1	no	—
86	C₂×D₃₁₆	—	4	2	no	—
87	C₂×D₃₁₆	—	4	1	no	—
88	C₂×C₄×D₇₉	—	4	2	no	—
89	C₂×C₄×D₇₉	—	4	1	no	—
90	Dic₃₁₆	—	2	1	no	—
91	Dic₃₁₆	#41	1	2	yes	complemented
92	D₆₃₂	—	2	1	no	—
93	D₆₃₂	—	2	2	no	—
94	D₆₃₂	—	2	1	no	—
95	D₆₃₂	—	2	2	no	—
96	D₆₃₂	—	8	2	no	—
97	D₆₃₂	—	8	1	no	—
98	C₄⋊Dic₇₉	—	2	1	no	—
99	C₄⋊Dic₇₉	—	2	1	no	—
100	C₄⋊Dic₇₉	—	2	2	no	—
101	C₄⋊Dic₇₉	—	4	1	no	—
102	Dic₇₉⋊C₄	—	4	2	no	—
103	Dic₇₉⋊C₄	—	4	1	no	—
104	Dic₇₉⋊C₄	—	4	2	no	—
105	Dic₇₉⋊C₄	—	4	2	no	—
106	(C₂×D₇₉)⋊C₄	—	4	2	no	—
107	(C₂×D₇₉)⋊C₄	—	4	1	no	—
108	C₄×Dic₇₉	—	4	2	no	—
109	C₄×Dic₇₉	—	4	1	no	—
110	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
115	C₄⋊C₄×C₇₉	D₈	1	2	yes	—
116	C₄²×C₇₉	C₂×D₄	1	2	yes	—
117	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₉	—	2	2	no	—
119	C₁₆×C₇₉	—	2	1	no	—
120	C₁₆×C₇₉	—	2	1	no	—
121	C₂²×D₇₉	—	4	2	no	—
122	C₂²×D₇₉	—	4	1	no	—
123	Q₈×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₇₉	—	4	1	no	—
130	C₂×Dic₇₉	—	4	1	no	—
131	C₂×Dic₇₉	—	4	2	no	—
132	C₂×Dic₇₉	—	4	1	no	—
133	D₃₁₆	—	4	1	no	—
134	D₃₁₆	—	4	2	no	—
135	D₃₁₆	—	4	1	no	—
136	D₃₁₆	—	4	2	no	—
137	D₃₁₆	—	8	4	no	—
138	D₃₁₆	—	8	2	no	—
139	D₃₁₆	—	16	2	no	—
140	D₃₁₆	—	16	1	no	—
141	C₄×D₇₉	—	8	4	no	—
142	C₄×D₇₉	—	8	2	no	—
143	Dic₁₅₈	—	2	2	no	—
144	Dic₁₅₈	—	4	1	no	—
145	Dic₁₅₈	—	8	2	no	—
146	C₈×C₇₉	—	2	2	no	—
147	C₈×C₇₉	—	2	1	no	—
148	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
149	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₇₉	—	8	2	no	—
152	C₂×D₇₉	—	8	1	no	—
153	C₂×D₇₉	—	8	2	no	—
154	C₂×D₇₉	—	8	1	no	—
155	C₂×D₇₉	—	8	2	no	—
156	C₂×D₇₉	—	8	1	no	—
157	C₄×C₇₉	—	2	2	no	—
158	C₄×C₇₉	—	2	2	no	—
159	C₄×C₇₉	—	2	1	no	—
160	C₄×C₇₉	#509	1	2	yes	—
161	C₄×C₇₉	#548	1	1	yes	characteristic
162	C₄×C₇₉	#577	1	1	yes	characteristic
163	Dic₇₉	—	4	2	no	—
164	Dic₇₉	—	8	2	no	—
165	Dic₇₉	—	8	1	no	—
166	Dic₇₉	—	8	1	no	—
167	#33935	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	#3358	1	1	yes	characteristic
169	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
170	C₂×C₇₉	#3416	1	1	yes	characteristic
171	D₇₉	—	16	2	no	—
172	D₇₉	—	16	1	no	—
173	#3391	—	79	1	no	—
174	#3383	—	79	1	no	—
175	#2244	—	79	2	no	—
176	#2240	—	79	1	no	—
177	#2228	—	79	2	no	—
178	#2224	—	79	1	no	—
179	#2207	—	79	2	no	—
180	#2207	—	79	1	no	—
181	#2183	D₇₉	1	1	yes	characteristic, complemented
182	#2175	—	79	2	no	—
183	C₄×SD₃₂	—	79	1	no	—
184	C₇₉	#33935	1	1	yes	characteristic, complemented, Sylow
185	#557	—	158	1	no	—
186	#550	—	158	1	no	—
187	C₂×SD₃₂	—	79	1	no	—
188	C₂×SD₃₂	—	79	2	no	—
189	#500	—	79	1	no	—
190	#499	—	79	2	no	—
191	#493	—	79	1	no	—
192	#493	—	79	2	no	—
193	#479	—	158	2	no	—
194	#475	—	158	1	no	—
195	#459	—	158	1	no	—
196	#459	—	158	2	no	—
197	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
198	C₄×Q₁₆	—	79	1	no	—
199	C₄×Q₁₆	—	79	2	no	—
200	C₄×D₈	—	79	1	no	—
201	C₄×D₈	—	79	2	no	—
202	#367	—	79	1	no	—
203	#366	—	79	2	no	—
204	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
205	C₄⋊C₁₆	C₂×D₇₉	1	2	yes	—
206	#358	—	79	1	no	—
207	#357	—	79	1	no	—
208	#357	—	79	2	no	—
209	#357	—	79	2	no	—
210	#357	—	79	2	no	—
211	#357	—	79	2	no	—
212	#346	C₂×D₇₉	1	2	yes	—
213	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
214	C₂×#41	—	316	1	no	—
215	C₂×Q₁₆	—	79	1	no	—
216	C₂×D₈	—	79	1	no	—
217	C₂×D₈	—	79	2	no	—
218	C₂×D₈	—	158	2	no	—
219	C₂×D₈	—	158	1	no	—
220	C₄⋊Q₈	—	316	1	no	—
221	C₄⋊Q₈	—	158	1	no	—
222	#127	—	158	2	no	—
223	#127	—	158	1	no	—
224	#125	—	316	2	no	—
225	#125	—	158	2	no	—
226	#121	—	316	1	no	—
227	#121	—	316	2	no	—
228	C₄×Q₈	—	316	1	no	—
229	C₄×Q₈	—	158	1	no	—
230	C₄×Q₈	—	158	2	no	—
231	C₄×Q₈	D₃₁₆	1	1	yes	characteristic
232	C₄×D₄	—	158	2	no	—
233	C₄×D₄	—	158	1	no	—
234	SD₃₂	—	158	2	no	—
235	SD₃₂	—	158	4	no	—
236	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
239	#107	—	79	2	no	—
240	#107	—	79	1	no	—
241	#107	—	79	1	no	—
242	#107	—	158	1	no	—
243	#106	—	158	2	no	—
244	C₄⋊C₈	D₃₁₆	1	1	yes	characteristic
245	C₄⋊C₈	D₃₁₆	1	2	yes	—
246	#103	—	158	2	no	—
247	#103	—	158	1	no	—
248	#102	—	158	2	no	—
249	#102	—	158	1	no	—
250	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₇₉	1	2	yes	—
252	#41	—	632	4	no	—
253	C₂×Q₈	—	316	1	no	—
254	C₂×Q₈	—	158	1	no	—
255	C₂×Q₈	D₆₃₂	1	1	yes	characteristic
256	C₂×D₄	—	316	2	no	—
257	C₂×D₄	—	316	1	no	—
258	C₂×D₄	—	316	2	no	—
259	C₂×D₄	—	316	1	no	—
260	C₂×D₄	—	158	2	no	—
261	C₂×D₄	—	158	1	no	—
262	C₂²×C₄	—	316	2	no	—
263	C₂²×C₄	—	316	1	no	—
264	Q₁₆	—	79	2	no	—
265	Q₁₆	—	158	1	no	—
266	D₈	—	632	2	no	—
267	D₈	—	632	1	no	—
268	D₈	—	158	1	no	—
269	D₈	—	158	2	no	—
270	D₈	—	158	1	no	—
271	D₈	—	158	2	no	—
272	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
273	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
275	C₄⋊C₄	—	316	2	no	—
276	C₄⋊C₄	—	316	1	no	—
277	C₄⋊C₄	—	316	2	no	—
278	C₄⋊C₄	—	316	2	no	—
279	C₄⋊C₄	—	316	1	no	—
280	C₄⋊C₄	—	158	1	no	—
281	C₄⋊C₄	—	158	1	no	—
282	C₄⋊C₄	—	158	2	no	—
283	C₄⋊C₄	D₆₃₂	1	1	yes	characteristic
284	C₄⋊C₄	D₆₃₂	1	2	yes	—
285	C₂²⋊C₄	—	316	2	no	—
286	C₂²⋊C₄	—	316	1	no	—
287	C₄²	—	316	2	no	—
288	C₄²	—	316	1	no	—
289	C₄²	C₂×D₃₁₆	1	2	yes	—
290	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	—	2	1	no	—
294	C₂³	—	316	2	no	—
295	C₂³	—	316	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	316	1	no	—
298	Q₈	—	632	2	no	—
299	Q₈	—	158	2	no	—
300	D₄	—	316	1	no	—
301	D₄	—	316	2	no	—
302	D₄	—	316	1	no	—
303	D₄	—	316	2	no	—
304	D₄	—	632	4	no	—
305	D₄	—	632	2	no	—
306	D₄	—	1264	2	no	—
307	D₄	—	1264	1	no	—
308	C₂×C₄	—	316	1	no	—
309	C₂×C₄	—	316	1	no	—
310	C₂×C₄	—	316	2	no	—
311	C₂×C₄	—	316	1	no	—
312	C₂×C₄	—	632	4	no	—
313	C₂×C₄	—	632	2	no	—
314	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
315	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
316	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
317	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	#41×D₇₉	1	1	yes	characteristic
322	C₈	#14579209	1	1	yes	characteristic
323	C₂²	—	632	2	no	—
324	C₂²	—	632	1	no	—
325	C₂²	—	632	2	no	—
326	C₂²	—	632	1	no	—
327	C₂²	—	632	2	no	—
328	C₂²	—	632	1	no	—
329	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	316	2	no	—
334	C₄	—	632	1	no	—
335	C₄	—	632	2	no	—
336	C₄	—	632	1	no	—
337	C₄	#19099001	1	2	yes	—
338	C₄	#19099225	1	1	yes	characteristic
339	C₄	#19099375	1	1	yes	characteristic
340	C₂	—	1264	2	no	—
341	C₂	—	1264	1	no	—
342	C₂	#23776036	1	1	yes	characteristic
343	C₂	C₂×#19099001	1	1	yes	characteristic
344	C₂	#23776548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2183	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23776375	C₂
hypercenter	#2183	D₇₉
hypocenter	C₇₉	#33935
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #23776292	C₂	semidirect
3. #23765642	C₂	semidirect
4. #23765624	C₂	semidirect
5. #23765558	C₂	semidirect
6. #23765538	C₂	semidirect
7. #23765441	C₂	semidirect
8. #23765441	C₂	semidirect
9. #23765266	C₂	semidirect
10. C₄×#14579106	C₂	semidirect
11. C₇₉×#2183	C₂	semidirect
91. Dic₃₁₆	#41	semidirect
181. #2183	D₇₉	semidirect
184. C₇₉	#33935	semidirect

Maximal subgroups

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

subgroup	index
#23776375	2
#23776292	2
#23765642	2
#23765624	2
#23765558	2
#23765538	2
#23765441	2
#23765266	2
C₄×#14579106	2
C₇₉×#2183	2
#33935	79