Subgroups — #34206527

Order 20224.

574 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098938	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098377	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098376	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098375	C₂	1	1	yes	maximal, characteristic, complemented
5	M₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763570	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#361×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
9	C₂²×#11929609	—	2	1	no	—
10	C₂×#14579157	—	2	1	no	—
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	#19098938	—	2	1	no	—
13	#19098938	—	2	1	no	—
14	#19098938	C₄	1	2	yes	—
15	#19098938	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	#19098377	—	2	2	no	—
21	#19098376	—	2	2	no	—
22	#19098375	—	2	2	no	—
23	#19098338	—	2	1	no	—
24	C₁₆×Dic₇₉	—	2	1	no	—
25	C₂²×D₈×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	#361×C₇₉	—	2	2	no	—
29	C₂²×D₃₁₆	—	2	1	no	—
30	C₂²×C₂²⋊D₇₉	—	4	2	no	—
31	C₂×D₆₃₂	—	2	1	no	—
32	C₂×D₆₃₂	—	2	1	no	—
33	C₂×D₆₃₂	—	2	1	no	—
34	C₂×D₆₃₂	—	2	1	no	—
35	C₂×D₆₃₂	—	2	1	no	—
36	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
37	C₂×#11929609	—	2	2	no	—
38	C₂×#11929609	—	2	2	no	—
39	C₂×#11929609	—	4	2	no	—
40	C₂×#11929609	—	4	2	no	—
41	C₂×#11929608	—	2	1	no	—
42	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
43	#14579157	—	2	2	no	—
44	#14579157	—	2	2	no	—
45	#14579157	—	4	2	no	—
46	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
47	#14579101	—	2	1	no	—
48	#14579101	D₄	1	2	yes	complemented
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×D₄×C₇₉	—	2	1	no	—
59	C₂×D₈×C₇₉	—	2	1	no	—
60	C₂×D₈×C₇₉	—	2	1	no	—
61	C₂×D₈×C₇₉	—	2	1	no	—
62	C₂×D₈×C₇₉	—	2	1	no	—
63	C₂×D₈×C₇₉	—	2	1	no	—
64	C₂×D₈×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	complemented
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	C₂×C₂²⋊D₇₉	—	4	4	no	—
80	C₂×C₂²⋊D₇₉	—	4	4	no	—
81	C₂×C₂²⋊D₇₉	—	8	8	no	—
82	C₂²×Dic₇₉	SD₁₆	1	2	yes	—
83	D₆₃₂	—	2	2	no	—
84	D₆₃₂	—	2	2	no	—
85	D₆₃₂	—	4	1	no	—
86	D₆₃₂	—	4	1	no	—
87	#11929609	—	4	4	no	—
88	#11929609	—	4	4	no	—
89	#11929608	—	4	1	no	—
90	#11929608	—	4	1	no	—
91	C₂×C₇₉⋊C₈	—	4	1	no	—
92	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
95	C₂×C₇₉⋊C₈	SD₁₆	1	2	yes	—
96	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×Dic₇₉	D₈	1	2	yes	—
99	C₇₉⋊C₁₆	—	2	2	no	—
100	C₂⁴×C₇₉	—	4	1	no	—
101	C₂×D₄×C₇₉	—	2	2	no	—
102	C₂×D₄×C₇₉	—	4	1	no	—
103	C₂×D₄×C₇₉	—	4	1	no	—
104	C₂×D₄×C₇₉	—	4	1	no	—
105	C₂×D₄×C₇₉	—	4	1	no	—
106	C₂×D₄×C₇₉	—	4	1	no	—
107	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
108	D₈×C₇₉	—	2	2	no	—
109	D₈×C₇₉	—	2	2	no	—
110	D₈×C₇₉	—	4	1	no	—
111	D₈×C₇₉	—	4	1	no	—
112	M₁₆×C₇₉	—	4	1	no	—
113	M₁₆×C₇₉	—	4	1	no	—
114	C₂×C₈×C₇₉	—	4	1	no	—
115	C₂×C₈×C₇₉	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
120	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
121	C₁₆×C₇₉	—	2	2	no	—
122	C₂³×C₇₉	—	4	2	no	—
123	C₂³×C₇₉	—	8	4	no	—
124	C₂³×C₇₉	—	8	1	no	—
125	C₂³×C₇₉	—	8	1	no	—
126	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂²×D₇₉	—	4	2	no	—
128	C₂²×D₇₉	—	8	4	no	—
129	C₂²×D₇₉	—	8	1	no	—
130	C₂²×D₇₉	—	8	1	no	—
131	D₄×C₇₉	—	4	2	no	—
132	D₄×C₇₉	—	4	2	no	—
133	D₄×C₇₉	—	8	1	no	—
134	D₄×C₇₉	—	8	1	no	—
135	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
137	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×D₈	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	C₂²⋊D₇₉	—	8	8	no	—
142	C₂²⋊D₇₉	—	8	8	no	—
143	C₂×Dic₇₉	—	2	4	no	—
144	C₂×Dic₇₉	#102	1	2	yes	—
145	C₂×Dic₇₉	#102	1	2	yes	—
146	D₃₁₆	—	4	2	no	—
147	D₃₁₆	—	4	2	no	—
148	D₃₁₆	—	8	1	no	—
149	D₃₁₆	—	8	1	no	—
150	C₈×C₇₉	—	8	1	no	—
151	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
152	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
153	C₈×C₇₉	#121	1	1	yes	characteristic
154	C₈×C₇₉	#121	1	1	yes	characteristic
155	C₇₉⋊C₈	—	8	1	no	—
156	C₇₉⋊C₈	#102	1	2	yes	—
157	C₇₉⋊C₈	#102	1	2	yes	—
158	C₂²×C₇₉	—	2	2	no	—
159	C₂²×C₇₉	—	8	4	no	—
160	C₂²×C₇₉	—	8	2	no	—
161	C₂²×C₇₉	—	16	2	no	—
162	C₂²×C₇₉	—	16	2	no	—
163	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
164	C₂²×C₇₉	#486	1	1	yes	characteristic
165	C₂²×C₇₉	#492	1	1	yes	characteristic
166	C₂×D₇₉	—	8	4	no	—
167	C₂×D₇₉	—	8	2	no	—
168	C₂×D₇₉	—	16	2	no	—
169	C₂×D₇₉	—	16	2	no	—
170	C₄×C₇₉	#390	1	1	yes	characteristic
171	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
172	C₄×C₇₉	#486	1	1	yes	characteristic
173	C₄×C₇₉	#493	1	1	yes	characteristic
174	Dic₇₉	—	2	2	no	—
175	Dic₇₉	#361	1	4	yes	complemented
176	#13213	—	79	1	no	maximal, Sylow
177	C₂×C₇₉	—	2	1	no	—
178	C₂×C₇₉	—	2	1	no	—
179	C₂×C₇₉	—	16	2	no	—
180	C₂×C₇₉	#1957	1	1	yes	characteristic
181	C₂×C₇₉	C₂×#361	1	2	yes	—
182	D₇₉	—	16	2	no	—
183	C₂×#493	—	79	1	no	—
184	C₂×#361	—	79	1	no	—
185	C₂×#361	—	79	1	no	—
186	C₂×#361	—	79	1	no	—
187	C₂×#361	D₇₉	1	1	yes	characteristic, complemented
188	C₄×M₃₂	—	79	1	no	—
189	#1848	—	79	1	no	—
190	C₇₉	#13213	1	1	yes	characteristic, complemented, Sylow
191	C₂²×D₈	—	79	1	no	—
192	C₂²×D₈	—	158	1	no	—
193	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
194	C₂×#127	—	158	1	no	—
195	C₂×M₃₂	—	79	1	no	—
196	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
197	#493	—	79	2	no	—
198	#493	—	79	2	no	—
199	#493	—	158	1	no	—
200	#493	—	158	1	no	—
201	C₂×#108	—	79	1	no	—
202	C₂×#108	—	158	1	no	—
203	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
204	#423	—	158	1	no	—
205	C₂×C₄×C₈	—	79	1	no	—
206	#361	—	2	2	no	—
207	#361	—	158	2	no	—
208	#361	—	158	2	no	—
209	#361	—	158	2	no	—
210	#346	—	158	1	no	—
211	C₄×C₁₆	—	158	1	no	—
212	C₂²×D₄	—	2	1	no	—
213	C₂²×D₄	—	316	2	no	—
214	C₂²×D₄	—	158	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	—	2	1	no	—
220	C₂×D₈	—	316	2	no	—
221	C₂×D₈	—	316	2	no	—
222	C₂×D₈	—	79	2	no	—
223	C₂×D₈	—	158	2	no	—
224	C₂×D₈	—	158	2	no	—
225	C₂×D₈	—	158	1	no	—
226	C₂×D₈	—	158	1	no	—
227	C₂×D₈	—	158	1	no	—
228	C₂×D₈	—	158	1	no	—
229	C₂×D₈	—	158	1	no	—
230	C₂×D₈	C₄×D₇₉	1	2	yes	—
231	C₂×M₁₆	—	2	1	no	—
232	C₂×M₁₆	—	158	1	no	—
233	C₂²×C₈	—	79	1	no	—
234	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
235	#127	—	316	2	no	—
236	#127	—	158	2	no	—
237	#127	—	158	2	no	—
238	C₂×C₄²	—	79	1	no	—
239	M₃₂	—	2	1	no	—
240	M₃₂	—	79	2	no	—
241	M₃₂	—	158	1	no	—
242	M₃₂	C₂²⋊D₇₉	1	2	yes	complemented
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	158	1	no	—
245	#108	—	2	1	no	—
246	#108	—	2	1	no	—
247	#108	—	316	2	no	—
248	#108	—	158	1	no	—
249	#108	—	158	1	no	—
250	C₄⋊C₈	—	316	2	no	—
251	C₄×C₈	—	79	1	no	—
252	C₄×C₈	—	79	1	no	—
253	C₄×C₈	—	79	1	no	—
254	C₄×C₈	—	79	1	no	—
255	C₂⁴	—	4	1	no	—
256	C₂⁴	—	316	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	316	4	no	—
264	C₂×D₄	—	316	4	no	—
265	C₂×D₄	—	316	1	no	—
266	C₂×D₄	—	316	1	no	—
267	C₂×D₄	—	316	1	no	—
268	C₂×D₄	—	316	1	no	—
269	C₂×D₄	—	316	1	no	—
270	C₂×D₄	—	632	8	no	—
271	C₂×D₄	—	158	2	no	—
272	C₂²×C₄	—	79	2	no	—
273	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
274	D₈	—	2	2	no	—
275	D₈	—	2	2	no	—
276	D₈	—	4	1	no	—
277	D₈	—	4	1	no	—
278	D₈	—	316	4	no	—
279	D₈	—	316	4	no	—
280	D₈	—	316	1	no	—
281	D₈	—	316	1	no	—
282	D₈	—	158	2	no	—
283	D₈	—	158	2	no	—
284	M₁₆	—	4	1	no	—
285	M₁₆	—	4	1	no	—
286	M₁₆	—	316	1	no	—
287	M₁₆	—	316	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	316	1	no	—
290	C₂×C₈	—	79	1	no	—
291	C₂×C₈	—	79	1	no	—
292	C₂×C₈	—	79	2	no	—
293	C₂×C₈	—	79	2	no	—
294	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
295	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
296	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
298	C₂×C₈	#11929623	1	1	yes	characteristic
299	C₂×C₈	#11929624	1	1	yes	characteristic
300	C₄²	—	79	1	no	—
301	C₄²	—	79	1	no	—
302	C₄²	—	79	2	no	—
303	C₁₆	—	2	2	no	—
304	C₁₆	—	158	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	8	4	no	—
307	C₂³	—	8	1	no	—
308	C₂³	—	8	1	no	—
309	C₂³	—	316	2	no	—
310	C₂³	—	632	4	no	—
311	C₂³	—	632	1	no	—
312	C₂³	—	632	1	no	—
313	C₂³	SD₁₆×D₇₉	1	1	yes	characteristic
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	8	1	no	—
317	D₄	—	8	1	no	—
318	D₄	—	316	2	no	—
319	D₄	—	316	2	no	—
320	D₄	—	632	8	no	—
321	D₄	—	632	8	no	—
322	D₄	—	632	1	no	—
323	D₄	—	632	1	no	—
324	C₂×C₄	—	79	2	no	—
325	C₂×C₄	—	79	2	no	—
326	C₂×C₄	—	158	4	no	—
327	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
328	C₂×C₄	#14579147	1	1	yes	characteristic
329	C₂×C₄	#14579157	1	1	yes	characteristic
330	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
331	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
332	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
333	C₈	—	8	1	no	—
334	C₈	—	79	2	no	—
335	C₈	—	79	2	no	—
336	C₈	—	632	1	no	—
337	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
338	C₈	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
339	C₈	#14579133	1	1	yes	characteristic
340	C₈	#14579134	1	1	yes	characteristic
341	C₂²	—	2	2	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	16	2	no	—
346	C₂²	—	632	4	no	—
347	C₂²	—	632	2	no	—
348	C₂²	—	1264	2	no	—
349	C₂²	—	1264	2	no	—
350	C₂²	#102×D₇₉	1	1	yes	characteristic
351	C₂²	#19098908	1	1	yes	characteristic
352	C₂²	#19098934	1	1	yes	characteristic
353	C₄	—	79	4	no	—
354	C₄	—	158	2	no	—
355	C₄	#19098463	1	1	yes	characteristic
356	C₄	#102×D₇₉	1	1	yes	characteristic
357	C₄	#19098908	1	1	yes	characteristic
358	C₄	#19098938	1	1	yes	characteristic
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	16	2	no	—
362	C₂	—	1264	2	no	—
363	C₂	#23764204	1	1	yes	characteristic
364	C₂	#361×D₇₉	1	2	yes	—

	subgroup	G/H
center	C₂²	#102×D₇₉
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#361×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂×#19098938	C₂
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#361	D₇₉
hypocenter	C₇₉	#13213
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. C₂×#19098377	C₂	semidirect
3. C₂×#19098376	C₂	semidirect
4. C₂×#19098375	C₂	semidirect
5. M₃₂×Dic₇₉	C₂	semidirect
6. #23763570	C₂	semidirect
7. C₂×#361×C₇₉	C₂	semidirect
48. #14579101	D₄	semidirect
68. M₃₂×C₇₉	D₄	semidirect
175. Dic₇₉	#361	semidirect
187. C₂×#361	D₇₉	semidirect
190. C₇₉	#13213	semidirect
242. M₃₂	C₂²⋊D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098938	2
C₂×#19098377	2
C₂×#19098376	2
C₂×#19098375	2
M₃₂×Dic₇₉	2
#23763570	2
C₂×#361×C₇₉	2
#13213	79