Subgroups — #34204878

Order 20224.

490 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098391	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098365	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098365	C₂	1	1	yes	maximal, characteristic
5	#23764133	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₁₆	—	2	1	no	—
10	C₄×D₆₃₂	—	2	2	no	—
11	C₄×D₆₃₂	—	2	2	no	—
12	C₂×#14579095	C₂²	1	1	yes	characteristic
13	C₂×#14579095	C₂²	1	1	yes	characteristic
14	C₂×#14579095	C₂²	1	1	yes	characteristic
15	C₂×#14579083	—	2	1	no	—
16	#19098461	—	2	1	no	—
17	#19098391	—	2	2	no	—
18	#19098365	—	2	2	no	—
19	#19098365	C₄	1	4	yes	complemented
20	#19098326	—	2	1	no	—
21	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂²×D₃₁₆	—	2	1	no	—
25	C₂²×C₄×D₇₉	—	4	1	no	—
26	C₂×D₆₃₂	—	2	1	no	—
27	C₂×D₆₃₂	—	2	1	no	—
28	C₂×D₆₃₂	—	2	1	no	—
29	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
30	C₂×D₆₃₂	D₄	1	2	yes	—
31	C₂×D₆₃₂	Q₈	1	2	yes	—
32	C₄×D₃₁₆	—	2	4	no	—
33	C₄×D₃₁₆	—	4	2	no	—
34	C₂×C₄⋊Dic₇₉	—	2	1	no	—
35	C₂×C₄⋊Dic₇₉	—	2	1	no	—
36	C₂×C₄⋊Dic₇₉	—	2	1	no	—
37	C₂×Dic₇₉⋊C₄	—	4	1	no	—
38	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
39	#14579095	—	2	1	no	—
40	#14579095	—	2	1	no	—
41	#14579095	—	2	1	no	—
42	#14579095	—	2	1	no	—
43	#14579095	C₂×C₄	1	2	yes	complemented
44	#14579095	D₄	1	2	yes	—
45	#14579083	—	2	4	no	—
46	C₇₉⋊#95	—	4	1	no	—
47	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₇₉	—	2	1	no	—
51	C₂×C₁₆×C₇₉	—	2	1	no	—
52	C₂×C₁₆×C₇₉	—	2	1	no	—
53	C₂×C₁₆×C₇₉	—	2	1	no	—
54	C₂×C₁₆×C₇₉	—	2	1	no	—
55	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	complemented
56	C₄×C₈×C₇₉	—	2	1	no	—
57	C₄×C₈×C₇₉	—	2	1	no	—
58	C₂³×D₇₉	—	4	1	no	—
59	C₂×D₃₁₆	—	2	2	no	—
60	C₂×D₃₁₆	—	2	2	no	—
61	C₂×D₃₁₆	—	2	2	no	—
62	C₂×D₃₁₆	—	4	1	no	—
63	C₂×D₃₁₆	—	4	1	no	—
64	C₂×D₃₁₆	—	4	1	no	—
65	C₂²×Dic₇₉	—	4	1	no	—
66	C₂²×Dic₇₉	—	4	1	no	—
67	C₂²×Dic₇₉	—	4	1	no	—
68	C₂×C₄×D₇₉	—	4	4	no	—
69	C₂×C₄×D₇₉	—	8	2	no	—
70	C₂×C₄×D₇₉	—	8	2	no	—
71	D₆₃₂	—	2	2	no	—
72	D₆₃₂	—	2	2	no	—
73	D₆₃₂	—	2	2	no	—
74	D₆₃₂	C₄⋊C₄	1	4	yes	complemented
75	C₄⋊Dic₇₉	—	2	2	no	—
76	C₄⋊Dic₇₉	—	2	2	no	—
77	C₄⋊Dic₇₉	—	4	1	no	—
78	C₄⋊Dic₇₉	—	4	1	no	—
79	C₄⋊Dic₇₉	—	4	1	no	—
80	C₄⋊Dic₇₉	—	4	1	no	—
81	Dic₇₉⋊C₄	—	8	2	no	—
82	(C₂×D₇₉)⋊C₄	—	4	4	no	—
83	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₉	—	2	1	no	—
87	C₂×C₈×C₇₉	—	2	1	no	—
88	C₂×C₈×C₇₉	—	2	1	no	—
89	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
94	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
95	C₄²×C₇₉	—	2	1	no	—
96	C₄²×C₇₉	—	2	1	no	—
97	C₁₆×C₇₉	—	2	2	no	—
98	C₁₆×C₇₉	—	4	1	no	—
99	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₇₉	—	4	2	no	—
101	C₂²×D₇₉	—	4	2	no	—
102	C₂²×D₇₉	—	4	2	no	—
103	C₂²×D₇₉	—	8	1	no	—
104	C₂²×D₇₉	—	8	1	no	—
105	C₂²×D₇₉	—	8	1	no	—
106	C₂²×D₇₉	—	8	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
113	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
117	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
119	C₂×Dic₇₉	—	4	2	no	—
120	C₂×Dic₇₉	—	4	2	no	—
121	C₂×Dic₇₉	—	8	1	no	—
122	C₂×Dic₇₉	—	8	1	no	—
123	C₂×Dic₇₉	—	8	1	no	—
124	C₂×Dic₇₉	—	8	1	no	—
125	C₂×Dic₇₉	—	8	1	no	—
126	C₂×Dic₇₉	—	8	1	no	—
127	C₂×Dic₇₉	—	8	1	no	—
128	D₃₁₆	—	2	4	no	—
129	D₃₁₆	—	4	2	no	—
130	D₃₁₆	—	4	2	no	—
131	D₃₁₆	—	4	2	no	—
132	C₄×D₇₉	—	8	8	no	—
133	C₈×C₇₉	—	4	1	no	—
134	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
136	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₇₉	#123	1	1	yes	characteristic
138	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
139	C₂²×C₇₉	#447	1	1	yes	characteristic
140	C₂²×C₇₉	#474	1	1	yes	characteristic
141	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
142	C₂²×C₇₉	#508	1	1	yes	characteristic
143	C₂²×C₇₉	#509	1	1	yes	characteristic
144	C₂²×C₇₉	#509	1	1	yes	characteristic
145	C₂×D₇₉	—	4	4	no	—
146	C₂×D₇₉	—	8	2	no	—
147	C₂×D₇₉	—	8	2	no	—
148	C₂×D₇₉	—	8	2	no	—
149	C₂×D₇₉	—	8	2	no	—
150	C₂×D₇₉	—	8	2	no	—
151	C₂×D₇₉	—	8	2	no	—
152	C₄×C₇₉	—	2	2	no	—
153	C₄×C₇₉	—	4	1	no	—
154	C₄×C₇₉	#385	1	1	yes	characteristic
155	C₄×C₇₉	#440	1	1	yes	characteristic
156	C₄×C₇₉	#452	1	1	yes	characteristic
157	C₄×C₇₉	#478	1	1	yes	characteristic
158	Dic₇₉	—	8	2	no	—
159	Dic₇₉	—	8	2	no	—
160	Dic₇₉	—	16	1	no	—
161	#12937	—	79	1	no	maximal, Sylow
162	C₂×C₇₉	#1862	1	1	yes	characteristic
163	C₂×C₇₉	C₄×SD₃₂	1	1	yes	characteristic
164	C₂×C₇₉	#2177	1	1	yes	characteristic
165	C₂×C₇₉	#2186	1	1	yes	characteristic
166	C₂×C₇₉	#2187	1	1	yes	characteristic
167	C₂×C₇₉	#2226	1	1	yes	characteristic
168	C₂×C₇₉	#2228	1	1	yes	characteristic
169	D₇₉	—	8	4	no	—
170	C₂×C₄×D₈	—	79	1	no	—
171	C₂×#367	—	79	1	no	—
172	C₂×#357	—	79	1	no	—
173	C₂×#357	—	79	1	no	—
174	#1942	—	79	1	no	—
175	#1380	—	79	1	no	—
176	#1374	D₇₉	1	1	yes	characteristic, complemented
177	C₇₉	#12937	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₈	—	79	1	no	—
179	C₂×C₄×D₄	—	158	1	no	—
180	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
181	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
182	C₄×D₈	—	158	2	no	—
183	C₄×D₈	—	158	2	no	—
184	C₂×#107	—	79	1	no	—
185	C₂×#107	—	79	1	no	—
186	C₂×#107	—	79	1	no	—
187	C₂×#102	—	158	1	no	—
188	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
189	#389	—	158	1	no	—
190	#367	—	158	2	no	—
191	#357	—	79	4	no	—
192	#357	—	158	2	no	—
193	#340	—	158	1	no	—
194	C₂²×D₄	—	158	1	no	—
195	C₂³×C₄	—	316	1	no	—
196	C₂×D₈	—	79	2	no	—
197	C₂×D₈	—	79	2	no	—
198	C₂×D₈	—	79	2	no	—
199	C₂×D₈	—	158	1	no	—
200	C₂×D₈	—	158	1	no	—
201	C₂×D₈	—	158	1	no	—
202	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
204	C₄×D₄	—	316	2	no	—
205	C₄×D₄	—	158	4	no	—
206	C₂×C₄⋊C₄	—	316	1	no	—
207	C₂×C₄⋊C₄	—	158	1	no	—
208	C₂×C₄⋊C₄	—	158	1	no	—
209	C₂×C₄⋊C₄	—	158	1	no	—
210	C₂×C₂²⋊C₄	—	316	1	no	—
211	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	C₄×D₇₉	1	2	yes	complemented
218	#107	—	79	2	no	—
219	#107	—	79	2	no	—
220	#107	—	158	1	no	—
221	#107	—	158	1	no	—
222	#107	—	158	1	no	—
223	#107	—	158	1	no	—
224	#102	—	158	4	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	316	1	no	—
228	C₂⁴	—	316	1	no	—
229	C₂×D₄	—	316	1	no	—
230	C₂×D₄	—	316	1	no	—
231	C₂×D₄	—	316	1	no	—
232	C₂×D₄	—	158	2	no	—
233	C₂×D₄	—	158	2	no	—
234	C₂×D₄	—	158	2	no	—
235	C₂²×C₄	—	316	4	no	—
236	C₂²×C₄	—	316	1	no	—
237	C₂²×C₄	—	316	1	no	—
238	C₂²×C₄	—	316	1	no	—
239	C₂²×C₄	—	632	2	no	—
240	C₂²×C₄	—	632	2	no	—
241	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
242	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
244	D₈	—	79	4	no	—
245	D₈	—	158	2	no	—
246	D₈	—	158	2	no	—
247	D₈	—	158	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
252	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
253	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
254	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
255	C₂×C₈	#11929623	1	1	yes	characteristic
256	C₂×C₈	#11929624	1	1	yes	characteristic
257	C₄⋊C₄	—	316	1	no	—
258	C₄⋊C₄	—	316	1	no	—
259	C₄⋊C₄	—	316	1	no	—
260	C₄⋊C₄	—	316	1	no	—
261	C₄⋊C₄	—	632	2	no	—
262	C₄⋊C₄	—	158	2	no	—
263	C₄⋊C₄	—	158	2	no	—
264	C₂²⋊C₄	—	316	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	4	1	no	—
269	C₂³	—	316	2	no	—
270	C₂³	—	316	2	no	—
271	C₂³	—	316	2	no	—
272	C₂³	—	632	1	no	—
273	C₂³	—	632	1	no	—
274	C₂³	—	632	1	no	—
275	C₂³	—	632	1	no	—
276	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
277	D₄	—	316	2	no	—
278	D₄	—	316	2	no	—
279	D₄	—	316	2	no	—
280	D₄	—	158	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	316	2	no	—
287	C₂×C₄	—	316	2	no	—
288	C₂×C₄	—	632	8	no	—
289	C₂×C₄	—	632	1	no	—
290	C₂×C₄	—	632	1	no	—
291	C₂×C₄	—	632	1	no	—
292	C₂×C₄	—	632	1	no	—
293	C₂×C₄	—	632	1	no	—
294	C₂×C₄	—	632	1	no	—
295	C₂×C₄	—	632	1	no	—
296	C₂×C₄	#14579106	1	2	yes	—
297	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
298	C₂×C₄	#14579131	1	1	yes	characteristic
299	C₂×C₄	#14579141	1	1	yes	characteristic
300	C₂×C₄	#14579183	1	1	yes	characteristic
301	C₂×C₄	#14579189	1	1	yes	characteristic
302	C₂×C₄	#14579194	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₄⋊C₄×D₇₉	1	1	yes	characteristic
305	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
306	C₈	#14579139	1	1	yes	characteristic
307	C₈	#14579143	1	1	yes	characteristic
308	C₂²	—	316	4	no	—
309	C₂²	—	632	2	no	—
310	C₂²	—	632	2	no	—
311	C₂²	—	632	2	no	—
312	C₂²	—	632	2	no	—
313	C₂²	—	632	2	no	—
314	C₂²	—	632	2	no	—
315	C₂²	C₄×D₆₃₂	1	1	yes	characteristic
316	C₂²	#19098699	1	1	yes	characteristic
317	C₂²	#19098844	1	1	yes	characteristic
318	C₂²	C₂×#14579106	1	1	yes	characteristic
319	C₂²	#19098995	1	1	yes	characteristic
320	C₂²	#19099001	1	1	yes	characteristic
321	C₂²	#19099001	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	632	2	no	—
325	C₄	—	632	2	no	—
326	C₄	—	1264	1	no	—
327	C₄	#19098437	1	1	yes	characteristic
328	C₄	#19098665	1	1	yes	characteristic
329	C₄	#19098727	1	1	yes	characteristic
330	C₄	#19098864	1	1	yes	characteristic
331	C₂	—	632	4	no	—
332	C₂	#23763650	1	1	yes	characteristic
333	C₂	C₄×#14579106	1	1	yes	characteristic
334	C₂	#23765277	1	1	yes	characteristic
335	C₂	#23765325	1	1	yes	characteristic
336	C₂	#23765331	1	1	yes	characteristic
337	C₂	#23765548	1	1	yes	characteristic
338	C₂	#23765558	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1374	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₆₃₂	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1374	D₇₉
hypocenter	C₇₉	#12937
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#19098391	C₂	semidirect
5. #23764133	C₂	semidirect
6. #23761492	C₂	semidirect
7. C₇₉×#1374	C₂	semidirect
19. #19098365	C₄	semidirect
43. #14579095	C₂×C₄	semidirect
55. C₂×C₁₆×C₇₉	C₂×C₄	semidirect
74. D₆₃₂	C₄⋊C₄	semidirect
176. #1374	D₇₉	semidirect
177. C₇₉	#12937	semidirect
217. C₂×C₁₆	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₆₃₂	2
C₂×#19098391	2
C₂×#19098365	2
#23764133	2
#23761492	2
C₇₉×#1374	2
#12937	79