Subgroups — #34205713

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098980	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098376	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098364	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763464	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761510	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#472×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19098980	—	2	2	no	—
9	#19098980	—	2	2	no	—
10	C₂×#14579101	—	2	1	no	—
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
13	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×#14579097	C₂²	1	1	yes	characteristic
15	C₂×#14579095	C₂²	1	1	yes	characteristic
16	#19098564	—	2	1	no	—
17	#19098376	—	2	2	no	—
18	#19098364	—	2	1	no	—
19	#19098364	—	2	1	no	—
20	#19098331	—	2	1	no	—
21	C₂×AGL(1,ℤ/8)×C₇₉	—	2	1	no	—
22	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
24	#472×C₇₉	—	2	2	no	—
25	#472×C₇₉	C₄	1	4	yes	complemented
26	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#100×C₇₉	—	2	1	no	—
28	C₂×#11929608	—	2	1	no	—
29	C₂×C₄⋊Dic₇₉	—	2	1	no	—
30	#14579101	—	2	4	no	—
31	#14579101	—	2	1	no	—
32	#14579101	D₄	1	2	yes	—
33	C₂×C₇₉⋊C₁₆	—	2	4	no	—
34	C₂×C₇₉⋊C₁₆	—	2	1	no	—
35	C₂×C₇₉⋊C₁₆	—	2	1	no	—
36	C₂×C₇₉⋊C₁₆	—	2	1	no	—
37	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
38	C₂×C₇₉⋊C₁₆	D₄	1	1	yes	characteristic
39	#14579097	—	2	1	no	—
40	#14579097	—	2	1	no	—
41	#14579095	—	2	1	no	—
42	#14579095	—	2	1	no	—
43	#14579085	—	4	1	no	—
44	#14579082	—	4	1	no	—
45	C₂×#41×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₇₉	—	2	1	no	—
47	AGL(1,ℤ/8)×C₇₉	—	2	4	no	—
48	AGL(1,ℤ/8)×C₇₉	—	4	2	no	—
49	C₂×SD₁₆×C₇₉	—	4	1	no	—
50	C₂×D₈×C₇₉	—	2	1	no	—
51	C₂×D₈×C₇₉	—	2	1	no	—
52	C₂×D₈×C₇₉	—	2	1	no	—
53	C₂×D₈×C₇₉	—	2	1	no	—
54	C₂×D₈×C₇₉	—	2	1	no	—
55	C₂×D₈×C₇₉	—	4	1	no	—
56	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
57	#131×C₇₉	—	2	1	no	—
58	#131×C₇₉	—	2	1	no	—
59	#131×C₇₉	C₂×C₄	1	2	yes	—
60	#131×C₇₉	D₄	1	2	yes	—
61	C₂×M₁₆×C₇₉	—	2	1	no	—
62	C₂×M₁₆×C₇₉	—	2	1	no	—
63	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₉	—	2	1	no	—
65	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	#108×C₇₉	—	2	1	no	—
67	#108×C₇₉	C₂×C₄	1	2	yes	—
68	#100×C₇₉	—	2	4	no	—
69	C₂²×Dic₇₉	—	4	1	no	—
70	#11929608	—	4	1	no	—
71	#11929608	—	4	1	no	—
72	C₂×C₇₉⋊C₈	—	4	1	no	—
73	C₄⋊Dic₇₉	—	4	1	no	—
74	C₄⋊Dic₇₉	—	4	1	no	—
75	C₇₉⋊C₁₆	—	2	1	no	—
76	C₇₉⋊C₁₆	—	2	2	no	—
77	C₇₉⋊C₁₆	—	2	1	no	—
78	C₂⁴×C₇₉	—	4	1	no	—
79	#41×C₇₉	—	2	1	no	—
80	#41×C₇₉	—	2	1	no	—
81	#41×C₇₉	C₂²⋊C₄	1	2	yes	—
82	#41×C₇₉	SD₁₆	1	2	yes	—
83	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
84	C₂×D₄×C₇₉	—	2	1	no	—
85	C₂×D₄×C₇₉	—	2	2	no	—
86	C₂×D₄×C₇₉	—	4	1	no	—
87	C₂×D₄×C₇₉	—	4	1	no	—
88	C₂×D₄×C₇₉	—	4	1	no	—
89	C₂×D₄×C₇₉	—	4	1	no	—
90	C₂×D₄×C₇₉	—	4	1	no	—
91	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₇₉	—	2	1	no	—
93	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
94	SD₁₆×C₇₉	—	4	4	no	—
95	D₈×C₇₉	—	2	2	no	—
96	D₈×C₇₉	—	2	2	no	—
97	D₈×C₇₉	—	4	1	no	—
98	D₈×C₇₉	—	4	1	no	—
99	D₈×C₇₉	—	4	4	no	—
100	M₁₆×C₇₉	—	2	2	no	—
101	M₁₆×C₇₉	—	2	2	no	—
102	M₁₆×C₇₉	—	2	2	no	—
103	M₁₆×C₇₉	—	2	1	no	—
104	M₁₆×C₇₉	—	4	1	no	—
105	M₁₆×C₇₉	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₇₉	—	2	2	no	—
107	C₂×C₈×C₇₉	—	2	2	no	—
108	C₂×C₈×C₇₉	—	2	1	no	—
109	C₂×C₈×C₇₉	—	4	1	no	—
110	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
115	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
116	C₂³×C₇₉	—	2	1	no	—
117	C₂³×C₇₉	—	4	2	no	—
118	C₂³×C₇₉	—	8	1	no	—
119	C₂³×C₇₉	—	8	1	no	—
120	C₂³×C₇₉	—	16	1	no	—
121	C₂³×C₇₉	—	16	1	no	—
122	C₂³×C₇₉	#135	1	1	yes	characteristic
123	Q₈×C₇₉	—	2	1	no	—
124	Q₈×C₇₉	#102	1	2	yes	—
125	D₄×C₇₉	—	2	2	no	—
126	D₄×C₇₉	—	2	2	no	—
127	D₄×C₇₉	—	2	1	no	—
128	D₄×C₇₉	—	4	2	no	—
129	D₄×C₇₉	—	4	2	no	—
130	D₄×C₇₉	—	8	1	no	—
131	D₄×C₇₉	—	8	1	no	—
132	D₄×C₇₉	#102	1	2	yes	—
133	C₂×C₄×C₇₉	—	2	1	no	—
134	C₂×C₄×C₇₉	—	2	2	no	—
135	C₂×C₄×C₇₉	—	2	2	no	—
136	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
142	C₂×Dic₇₉	—	8	1	no	—
143	C₂×Dic₇₉	—	16	1	no	—
144	C₈×C₇₉	—	2	2	no	—
145	C₈×C₇₉	—	4	2	no	—
146	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
147	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
148	C₈×C₇₉	#120	1	1	yes	characteristic
149	C₈×C₇₉	#123	1	1	yes	characteristic
150	C₇₉⋊C₈	—	8	1	no	—
151	C₂²×C₇₉	—	2	1	no	—
152	C₂²×C₇₉	—	2	1	no	—
153	C₂²×C₇₉	—	2	2	no	—
154	C₂²×C₇₉	—	4	1	no	—
155	C₂²×C₇₉	—	4	1	no	—
156	C₂²×C₇₉	—	8	2	no	—
157	C₂²×C₇₉	—	16	2	no	—
158	C₂²×C₇₉	—	16	2	no	—
159	C₂²×C₇₉	—	16	1	no	—
160	C₂²×C₇₉	—	16	1	no	—
161	C₂²×C₇₉	#416	1	1	yes	characteristic
162	C₂²×C₇₉	#465	1	1	yes	characteristic
163	C₂²×C₇₉	#466	1	1	yes	characteristic
164	C₄×C₇₉	—	2	2	no	—
165	C₄×C₇₉	#386	1	1	yes	characteristic
166	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
167	C₄×C₇₉	#465	1	1	yes	characteristic
168	C₄×C₇₉	#466	1	1	yes	characteristic
169	Dic₇₉	—	16	1	no	—
170	#13073	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	—	2	1	no	—
172	C₂×C₇₉	—	2	1	no	—
173	C₂×C₇₉	—	4	2	no	—
174	C₂×C₇₉	—	16	2	no	—
175	C₂×C₇₉	#1893	1	1	yes	characteristic
176	C₂×C₇₉	#2221	1	1	yes	characteristic
177	C₂×C₇₉	#2223	1	1	yes	characteristic
178	C₂×#504	—	79	1	no	—
179	C₂×#472	D₇₉	1	1	yes	characteristic
180	C₂×#361	—	79	1	no	—
181	C₂×#357	—	79	1	no	—
182	#1816	—	79	1	no	—
183	#1387	—	79	1	no	—
184	#1381	—	79	1	no	—
185	C₇₉	#13073	1	1	yes	characteristic, complemented, Sylow
186	C₂×AGL(1,ℤ/8)	—	2	1	no	—
187	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
188	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
189	#504	—	158	2	no	—
190	#504	—	158	2	no	—
191	C₂×M₃₂	—	79	1	no	—
192	C₂×M₃₂	—	158	1	no	—
193	C₂²×C₁₆	—	79	1	no	—
194	C₂²×C₁₆	—	158	1	no	—
195	#472	—	2	2	no	—
196	#472	Dic₇₉	1	4	yes	complemented
197	C₂×#108	—	79	1	no	—
198	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
199	C₂×#107	—	79	1	no	—
200	#417	—	158	1	no	—
201	C₂×#100	—	2	1	no	—
202	#361	—	158	2	no	—
203	#357	—	158	1	no	—
204	#357	—	158	1	no	—
205	#343	—	158	1	no	—
206	C₂×#41	C₂²⋊D₇₉	1	1	yes	characteristic
207	C₂²×D₄	—	2	1	no	—
208	AGL(1,ℤ/8)	—	2	4	no	—
209	AGL(1,ℤ/8)	—	4	2	no	—
210	C₂×SD₁₆	—	4	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	4	1	no	—
217	C₂×D₈	C₂×Dic₇₉	1	2	yes	—
218	#131	—	2	1	no	—
219	#131	—	2	1	no	—
220	#131	C₂×Dic₇₉	1	2	yes	—
221	#131	C₂²⋊D₇₉	1	2	yes	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	—	2	1	no	—
224	C₂×M₁₆	—	158	1	no	—
225	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
228	C₂×C₄⋊C₄	—	158	1	no	—
229	M₃₂	—	79	2	no	—
230	M₃₂	—	158	4	no	—
231	M₃₂	—	158	1	no	—
232	C₂×C₁₆	—	79	1	no	—
233	C₂×C₁₆	—	79	1	no	—
234	C₂×C₁₆	—	158	4	no	—
235	C₂×C₁₆	—	158	1	no	—
236	C₂×C₁₆	—	158	1	no	—
237	C₂×C₁₆	—	158	1	no	—
238	#108	—	2	1	no	—
239	#108	—	158	1	no	—
240	#108	—	158	1	no	—
241	#108	C₂×Dic₇₉	1	2	yes	—
242	#107	—	158	1	no	—
243	#107	—	158	1	no	—
244	#103	—	316	1	no	—
245	#102	—	316	1	no	—
246	#100	—	2	4	no	—
247	C₂⁴	—	4	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	C₂²⋊Dic₇₉	1	2	yes	—
251	#41	#11929612	1	2	yes	—
252	C₂×Q₈	#11929609	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	#11929609	1	1	yes	characteristic
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	316	1	no	—
263	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
264	SD₁₆	—	4	4	no	—
265	D₈	—	2	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	4	4	no	—
268	D₈	—	4	1	no	—
269	D₈	—	4	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	316	1	no	—
276	M₁₆	—	316	1	no	—
277	M₁₆	C₂²⋊Dic₇₉	1	2	yes	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	316	1	no	—
283	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
284	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
285	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
287	C₂×C₈	#11929623	1	1	yes	characteristic
288	C₂×C₈	#11929623	1	1	yes	characteristic
289	C₄⋊C₄	—	316	1	no	—
290	C₄⋊C₄	—	316	1	no	—
291	C₁₆	—	158	1	no	—
292	C₁₆	—	158	1	no	—
293	C₁₆	—	158	2	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	1	no	—
297	C₂³	—	8	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	—	16	1	no	—
300	C₂³	#14579181	1	1	yes	characteristic
301	Q₈	—	2	1	no	—
302	Q₈	#14579082	1	2	yes	—
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	1	no	—
309	D₄	—	8	1	no	—
310	D₄	#14579082	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	632	1	no	—
315	C₂×C₄	—	1264	1	no	—
316	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
317	C₂×C₄	#14579133	1	1	yes	characteristic
318	C₂×C₄	#14579133	1	1	yes	characteristic
319	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
320	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
321	C₂×C₄	#14579181	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	4	2	no	—
324	C₈	—	632	1	no	—
325	C₈	D₄×Dic₇₉	1	1	yes	characteristic
326	C₈	D₄×Dic₇₉	1	1	yes	characteristic
327	C₈	#14579129	1	1	yes	characteristic
328	C₈	#14579142	1	1	yes	characteristic
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	16	2	no	—
336	C₂²	—	16	2	no	—
337	C₂²	—	16	1	no	—
338	C₂²	—	16	1	no	—
339	C₂²	#19098559	1	1	yes	characteristic
340	C₂²	#19098790	1	1	yes	characteristic
341	C₂²	#19098797	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	1264	1	no	—
344	C₄	#19098443	1	1	yes	characteristic
345	C₄	C₂×#14579082	1	1	yes	characteristic
346	C₄	#19098790	1	1	yes	characteristic
347	C₄	#19098797	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	4	2	no	—
351	C₂	—	16	2	no	—
352	C₂	#23763829	1	1	yes	characteristic
353	C₂	#23765518	1	1	yes	characteristic
354	C₂	#23765528	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098797
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₂×#472×C₇₉	C₂
socle	C₂²×C₇₉	#466
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#472	D₇₉
hypocenter	C₇₉	#13073
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#472

Complements

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

N	H ≅ G/N	type
1. C₂×#19098980	C₂	semidirect
2. C₂×#19098376	C₂	semidirect
3. C₂×#19098364	C₂	semidirect
4. #23763464	C₂	semidirect
5. #23761510	C₂	semidirect
6. #23761493	C₂	semidirect
25. #472×C₇₉	C₄	semidirect
185. C₇₉	#13073	semidirect
196. #472	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098980	2
C₂×#19098376	2
C₂×#19098364	2
#23763464	2
#23761510	2
#23761493	2
C₂×#472×C₇₉	2
#13073	79