Subgroups — #34206531

Order 20224.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098948	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098379	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098377	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098375	C₂	1	1	yes	maximal, characteristic
5	M₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763570	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#362×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11929611	—	2	1	no	—
9	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
10	C₂×#14579150	—	2	1	no	—
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	#19098948	—	2	1	no	—
13	#19098948	—	2	1	no	—
14	#19098948	C₄	1	2	yes	—
15	#19098948	C₄	1	2	yes	—
16	C₂×#14579097	C₂²	1	1	yes	characteristic
17	C₂×#14579096	—	2	1	no	—
18	#19098586	—	2	1	no	—
19	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
20	#19098379	—	2	2	no	—
21	#19098377	—	2	2	no	—
22	#19098375	—	2	2	no	—
23	#19098338	—	2	1	no	—
24	C₁₆×Dic₇₉	—	2	1	no	—
25	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
28	#362×C₇₉	—	2	2	no	—
29	C₂²×D₃₁₆	—	2	1	no	—
30	C₂×#11929611	—	2	2	no	—
31	C₂×#11929611	—	2	2	no	—
32	C₂×#11929611	—	4	2	no	—
33	C₂×#11929611	—	4	2	no	—
34	C₂×D₆₃₂	—	2	1	no	—
35	C₂×D₆₃₂	—	2	1	no	—
36	C₂×D₆₃₂	—	2	1	no	—
37	C₂×D₆₃₂	—	2	1	no	—
38	C₂×D₆₃₂	—	2	1	no	—
39	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
40	C₂×#11929608	—	2	1	no	—
41	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
42	#14579150	—	2	2	no	—
43	#14579150	—	2	2	no	—
44	#14579150	—	4	2	no	—
45	C₂×(C₂×D₇₉)⋊C₄	—	4	2	no	—
46	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
47	#14579101	—	2	1	no	—
48	#14579101	D₄	1	2	yes	—
49	C₂×C₇₉⋊C₁₆	—	2	1	no	—
50	#14579097	—	2	1	no	—
51	#14579097	—	2	1	no	—
52	#14579096	—	4	2	no	—
53	Dic₇₉⋊C₈	—	4	2	no	—
54	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₇₉	D₄	1	1	yes	characteristic
57	C₈×Dic₇₉	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₇₉	—	2	1	no	—
59	C₂×Q₁₆×C₇₉	—	2	1	no	—
60	C₂×Q₁₆×C₇₉	—	2	1	no	—
61	C₂×Q₁₆×C₇₉	—	2	1	no	—
62	C₂×Q₁₆×C₇₉	—	2	1	no	—
63	C₂×Q₁₆×C₇₉	—	2	1	no	—
64	C₂×Q₁₆×C₇₉	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₇₉	—	2	1	no	—
66	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	M₃₂×C₇₉	—	2	1	no	—
68	M₃₂×C₇₉	D₄	1	2	yes	—
69	C₂×C₁₆×C₇₉	—	2	1	no	—
70	#108×C₇₉	—	2	1	no	—
71	#108×C₇₉	—	2	1	no	—
72	C₂³×D₇₉	—	4	1	no	—
73	C₂×D₃₁₆	—	2	2	no	—
74	C₂×D₃₁₆	—	4	1	no	—
75	C₂×D₃₁₆	—	4	1	no	—
76	C₂×D₃₁₆	—	4	1	no	—
77	C₂×D₃₁₆	—	4	1	no	—
78	C₂×D₃₁₆	—	4	1	no	—
79	C₂²×Dic₇₉	SD₁₆	1	2	yes	—
80	#11929611	—	4	4	no	—
81	#11929611	—	4	4	no	—
82	D₆₃₂	—	2	2	no	—
83	D₆₃₂	—	2	2	no	—
84	D₆₃₂	—	4	1	no	—
85	D₆₃₂	—	4	1	no	—
86	#11929608	—	4	1	no	—
87	#11929608	—	4	1	no	—
88	C₂×C₇₉⋊C₈	—	4	1	no	—
89	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
92	C₂×C₇₉⋊C₈	SD₁₆	1	2	yes	—
93	(C₂×D₇₉)⋊C₄	—	4	4	no	—
94	(C₂×D₇₉)⋊C₄	—	4	4	no	—
95	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×Dic₇₉	D₈	1	2	yes	—
98	C₇₉⋊C₁₆	—	2	2	no	—
99	C₂×Q₈×C₇₉	—	2	2	no	—
100	C₂×Q₈×C₇₉	—	4	1	no	—
101	C₂×Q₈×C₇₉	—	4	1	no	—
102	C₂×Q₈×C₇₉	—	4	1	no	—
103	C₂×Q₈×C₇₉	—	4	1	no	—
104	C₂×Q₈×C₇₉	—	4	1	no	—
105	C₂²×C₄×C₇₉	—	4	1	no	—
106	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
107	Q₁₆×C₇₉	—	2	2	no	—
108	Q₁₆×C₇₉	—	2	2	no	—
109	Q₁₆×C₇₉	—	4	1	no	—
110	Q₁₆×C₇₉	—	4	1	no	—
111	M₁₆×C₇₉	—	4	1	no	—
112	M₁₆×C₇₉	—	4	1	no	—
113	C₂×C₈×C₇₉	—	4	1	no	—
114	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
119	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
120	C₁₆×C₇₉	—	2	2	no	—
121	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×D₇₉	—	4	2	no	—
123	C₂²×D₇₉	—	8	4	no	—
124	C₂²×D₇₉	—	8	1	no	—
125	C₂²×D₇₉	—	8	1	no	—
126	Q₈×C₇₉	—	4	2	no	—
127	Q₈×C₇₉	—	4	2	no	—
128	Q₈×C₇₉	—	8	1	no	—
129	Q₈×C₇₉	—	8	1	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	—	8	1	no	—
132	C₂×C₄×C₇₉	—	8	1	no	—
133	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
135	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	#102	1	2	yes	—
141	C₂×Dic₇₉	#102	1	2	yes	—
142	D₃₁₆	—	4	2	no	—
143	D₃₁₆	—	4	2	no	—
144	D₃₁₆	—	8	1	no	—
145	D₃₁₆	—	8	1	no	—
146	C₈×C₇₉	—	8	1	no	—
147	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
148	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
149	C₈×C₇₉	#121	1	1	yes	characteristic
150	C₈×C₇₉	#121	1	1	yes	characteristic
151	C₇₉⋊C₈	—	8	1	no	—
152	C₇₉⋊C₈	#102	1	2	yes	—
153	C₇₉⋊C₈	#102	1	2	yes	—
154	C₂²×C₇₉	—	2	2	no	—
155	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
156	C₂²×C₇₉	#486	1	1	yes	characteristic
157	C₂²×C₇₉	#492	1	1	yes	characteristic
158	C₂×D₇₉	—	8	4	no	—
159	C₂×D₇₉	—	8	2	no	—
160	C₂×D₇₉	—	16	2	no	—
161	C₂×D₇₉	—	16	2	no	—
162	C₄×C₇₉	—	8	2	no	—
163	C₄×C₇₉	#390	1	1	yes	characteristic
164	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
165	C₄×C₇₉	#486	1	1	yes	characteristic
166	C₄×C₇₉	#493	1	1	yes	characteristic
167	Dic₇₉	—	2	2	no	—
168	Dic₇₉	—	2	2	no	—
169	#13214	—	79	1	no	maximal, Sylow
170	C₂×C₇₉	—	2	1	no	—
171	C₂×C₇₉	—	2	1	no	—
172	C₂×C₇₉	#1957	1	1	yes	characteristic
173	C₂×C₇₉	#2148	1	2	yes	—
174	D₇₉	—	16	2	no	—
175	C₂×#495	—	79	1	no	—
176	C₂×#362	—	79	1	no	—
177	C₂×#362	D₇₉	1	1	yes	characteristic, complemented
178	C₂×#361	—	79	1	no	—
179	C₂×#361	—	79	1	no	—
180	C₄×M₃₂	—	79	1	no	—
181	#1848	—	79	1	no	—
182	C₇₉	#13214	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
184	C₂²×SD₁₆	—	158	1	no	—
185	C₂²×D₈	—	79	1	no	—
186	C₂×#124	—	158	1	no	—
187	C₂×M₃₂	—	79	1	no	—
188	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
189	#495	—	79	2	no	—
190	#495	—	79	2	no	—
191	#495	—	158	1	no	—
192	#495	—	158	1	no	—
193	C₂×#108	—	79	1	no	—
194	C₂×#108	—	158	1	no	—
195	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
196	#423	—	158	1	no	—
197	C₂×C₄×C₈	—	79	1	no	—
198	#362	—	2	2	no	—
199	#362	—	158	2	no	—
200	#361	—	158	2	no	—
201	#361	—	158	2	no	—
202	#346	—	158	1	no	—
203	C₄×C₁₆	—	158	1	no	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×D₄	—	158	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₄×D₇₉	1	2	yes	—
212	C₂×SD₁₆	—	316	2	no	—
213	C₂×SD₁₆	—	316	2	no	—
214	C₂×SD₁₆	—	158	2	no	—
215	C₂×SD₁₆	—	158	2	no	—
216	C₂×D₈	—	79	2	no	—
217	C₂×D₈	—	158	1	no	—
218	C₂×D₈	—	158	1	no	—
219	C₂×D₈	—	158	1	no	—
220	C₂×D₈	—	158	1	no	—
221	C₂×D₈	—	158	1	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	—	158	1	no	—
224	C₂²×C₈	—	79	1	no	—
225	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
226	#124	—	316	2	no	—
227	#124	—	158	2	no	—
228	#124	—	158	2	no	—
229	C₂×C₂²⋊C₄	—	316	2	no	—
230	C₂×C₄²	—	79	1	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	79	2	no	—
233	M₃₂	—	158	1	no	—
234	M₃₂	C₂²⋊D₇₉	1	2	yes	—
235	C₂×C₁₆	—	2	1	no	—
236	C₂×C₁₆	—	158	1	no	—
237	#108	—	2	1	no	—
238	#108	—	2	1	no	—
239	#108	—	316	2	no	—
240	#108	—	158	1	no	—
241	#108	—	158	1	no	—
242	C₄⋊C₈	—	316	2	no	—
243	C₄×C₈	—	79	1	no	—
244	C₄×C₈	—	79	1	no	—
245	C₄×C₈	—	79	1	no	—
246	C₄×C₈	—	79	1	no	—
247	C₂⁴	—	316	1	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×D₄	—	316	1	no	—
255	C₂×D₄	—	316	1	no	—
256	C₂×D₄	—	316	1	no	—
257	C₂×D₄	—	316	1	no	—
258	C₂×D₄	—	316	1	no	—
259	C₂×D₄	—	158	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	79	2	no	—
262	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	4	1	no	—
266	Q₁₆	—	4	1	no	—
267	SD₁₆	—	316	4	no	—
268	SD₁₆	—	316	4	no	—
269	D₈	—	316	1	no	—
270	D₈	—	316	1	no	—
271	D₈	—	158	2	no	—
272	D₈	—	158	2	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	316	1	no	—
276	M₁₆	—	316	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	316	1	no	—
279	C₂×C₈	—	79	1	no	—
280	C₂×C₈	—	79	1	no	—
281	C₂×C₈	—	79	2	no	—
282	C₂×C₈	—	79	2	no	—
283	C₂×C₈	C₂×C₄×D₇₉	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	#11929624	1	1	yes	characteristic
289	C₂²⋊C₄	—	316	4	no	—
290	C₂²⋊C₄	—	316	4	no	—
291	C₄²	—	79	1	no	—
292	C₄²	—	79	1	no	—
293	C₄²	—	79	2	no	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	158	2	no	—
296	C₂³	—	316	2	no	—
297	C₂³	—	632	4	no	—
298	C₂³	—	632	1	no	—
299	C₂³	—	632	1	no	—
300	C₂³	SD₁₆×D₇₉	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	1	no	—
304	Q₈	—	8	1	no	—
305	D₄	—	316	2	no	—
306	D₄	—	316	2	no	—
307	D₄	—	632	1	no	—
308	D₄	—	632	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	316	2	no	—
313	C₂×C₄	—	79	2	no	—
314	C₂×C₄	—	79	2	no	—
315	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
316	C₂×C₄	#14579147	1	1	yes	characteristic
317	C₂×C₄	#14579157	1	1	yes	characteristic
318	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
319	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
320	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
321	C₈	—	8	1	no	—
322	C₈	—	79	2	no	—
323	C₈	—	79	2	no	—
324	C₈	—	632	1	no	—
325	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
326	C₈	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
327	C₈	#14579133	1	1	yes	characteristic
328	C₈	#14579134	1	1	yes	characteristic
329	C₂²	—	2	2	no	—
330	C₂²	—	632	4	no	—
331	C₂²	—	632	2	no	—
332	C₂²	—	1264	2	no	—
333	C₂²	—	1264	2	no	—
334	C₂²	#102×D₇₉	1	1	yes	characteristic
335	C₂²	#19098908	1	1	yes	characteristic
336	C₂²	#19098934	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	158	2	no	—
339	C₄	—	158	2	no	—
340	C₄	#19098463	1	1	yes	characteristic
341	C₄	#102×D₇₉	1	1	yes	characteristic
342	C₄	#19098908	1	1	yes	characteristic
343	C₄	#19098938	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	1264	2	no	—
347	C₂	#23764204	1	1	yes	characteristic
348	C₂	#23765148	1	2	yes	—

	subgroup	G/H
center	C₂²	#102×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#362×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂²×D₆₃₂	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#362	D₇₉
hypocenter	C₇₉	#13214
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
2. C₂×#19098379	C₂	semidirect
5. M₃₂×Dic₇₉	C₂	semidirect
6. #23763570	C₂	semidirect
7. C₂×#362×C₇₉	C₂	semidirect
177. C₂×#362	D₇₉	semidirect
182. C₇₉	#13214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098948	2
C₂×#19098379	2
C₂×#19098377	2
C₂×#19098375	2
M₃₂×Dic₇₉	2
#23763570	2
C₂×#362×C₇₉	2
#13214	79