Subgroups — C₂×#23763019 #34275979

Order 20224.

914 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098968	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098959	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098935	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098915	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098909	C₂	1	1	yes	maximal, characteristic
6	#23763019	C₂	1	8	yes	maximal, direct product factor
7	C₂×#19098284	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#492×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11929612	—	2	2	no	—
10	C₂×#14579161	C₂²	1	1	yes	characteristic
11	C₂×#14579161	C₂²	1	1	yes	characteristic
12	#19098968	—	2	2	no	—
13	#19098968	C₂²	1	4	yes	complemented
14	#19098959	—	2	2	no	—
15	#19098959	C₂²	1	4	yes	complemented
16	#19098935	—	2	2	no	—
17	#19098935	C₂²	1	4	yes	—
18	#19098915	—	2	2	no	—
19	#19098915	C₂²	1	4	yes	complemented
20	#19098909	—	2	2	no	—
21	#19098909	C₂²	1	4	yes	—
22	C₂×#14579095	—	2	2	no	—
23	C₂×#14579092	—	2	2	no	—
24	C₂×#14579084	—	2	2	no	—
25	C₂×#14579082	—	2	2	no	—
26	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
28	#19098284	C₂²	1	4	yes	complemented
29	C₂²×SD₁₆×C₇₉	—	2	2	no	—
30	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×#127×C₇₉	C₂²	1	1	yes	characteristic
32	#492×C₇₉	—	2	2	no	—
33	#492×C₇₉	C₂²	1	4	yes	—
34	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	C₂²×Dic₁₅₈	—	2	1	no	—
36	C₂²×Dic₁₅₈	—	2	1	no	—
37	C₂×#11929612	—	2	8	no	—
38	C₂×#11929612	—	4	8	no	—
39	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
40	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
41	#14579161	—	2	1	no	—
42	#14579161	—	2	1	no	—
43	#14579161	—	2	1	no	—
44	#14579161	—	2	1	no	—
45	#14579161	—	2	1	no	—
46	#14579161	—	2	1	no	—
47	#14579161	C₂³	1	2	yes	—
48	#14579161	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₇₉	—	2	1	no	—
50	C₂×C₄⋊Dic₇₉	—	2	1	no	—
51	C₂×C₄⋊Dic₇₉	—	2	1	no	—
52	C₂×C₄⋊Dic₇₉	—	2	1	no	—
53	#14579095	—	2	8	no	—
54	#14579092	—	2	8	no	—
55	#14579084	—	2	8	no	—
56	#14579082	—	2	8	no	—
57	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
58	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
59	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
60	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
61	C₂²×Q₈×C₇₉	—	2	1	no	—
62	C₂²×D₄×C₇₉	—	2	1	no	—
63	C₂²×D₄×C₇₉	—	2	1	no	—
64	C₂²×D₄×C₇₉	—	2	1	no	—
65	C₂×SD₁₆×C₇₉	—	2	8	no	—
66	C₂×SD₁₆×C₇₉	—	4	8	no	—
67	C₂²×C₈×C₇₉	D₄	1	2	yes	—
68	C₄⋊Q₈×C₇₉	—	2	1	no	—
69	C₄⋊Q₈×C₇₉	—	2	1	no	—
70	C₄⋊Q₈×C₇₉	—	2	1	no	—
71	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
72	#127×C₇₉	—	2	1	no	—
73	#127×C₇₉	—	2	1	no	—
74	#127×C₇₉	—	2	1	no	—
75	#127×C₇₉	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
77	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
78	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₉	D₄	1	2	yes	—
80	C₄×C₈×C₇₉	C₂³	1	2	yes	—
81	C₂×Dic₁₅₈	—	2	2	no	—
82	C₂×Dic₁₅₈	—	2	2	no	—
83	C₂×Dic₁₅₈	—	4	1	no	—
84	C₂×Dic₁₅₈	—	4	1	no	—
85	C₂×Dic₁₅₈	—	4	4	no	—
86	C₂×Dic₁₅₈	—	4	4	no	—
87	C₂²×Dic₇₉	—	4	1	no	—
88	C₂²×Dic₇₉	—	4	1	no	—
89	#11929612	—	4	16	no	—
90	C₂×C₇₉⋊C₈	—	2	4	no	—
91	C₂×C₇₉⋊C₈	—	2	4	no	—
92	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
93	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
94	C₄⋊Dic₇₉	—	2	2	no	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	C₄⋊Dic₇₉	—	2	2	no	—
97	C₄⋊Dic₇₉	—	2	2	no	—
98	C₄⋊Dic₇₉	—	4	1	no	—
99	C₄⋊Dic₇₉	—	4	1	no	—
100	C₄⋊Dic₇₉	—	4	1	no	—
101	C₄⋊Dic₇₉	—	4	1	no	—
102	C₂⁴×C₇₉	—	4	1	no	—
103	C₂×Q₈×C₇₉	—	2	2	no	—
104	C₂×Q₈×C₇₉	—	4	1	no	—
105	C₂×Q₈×C₇₉	—	4	4	no	—
106	C₂×D₄×C₇₉	—	2	2	no	—
107	C₂×D₄×C₇₉	—	2	2	no	—
108	C₂×D₄×C₇₉	—	2	2	no	—
109	C₂×D₄×C₇₉	—	4	1	no	—
110	C₂×D₄×C₇₉	—	4	1	no	—
111	C₂×D₄×C₇₉	—	4	1	no	—
112	C₂×D₄×C₇₉	—	4	4	no	—
113	C₂×D₄×C₇₉	—	4	4	no	—
114	C₂×D₄×C₇₉	—	4	4	no	—
115	C₂²×C₄×C₇₉	—	4	1	no	—
116	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
119	SD₁₆×C₇₉	—	4	16	no	—
120	C₂×C₈×C₇₉	SD₁₆	1	8	yes	complemented
121	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
122	C₄⋊C₄×C₇₉	—	2	2	no	—
123	C₄⋊C₄×C₇₉	—	2	2	no	—
124	C₄⋊C₄×C₇₉	—	4	1	no	—
125	C₄⋊C₄×C₇₉	—	4	1	no	—
126	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
130	C₂³×C₇₉	—	4	2	no	—
131	C₂³×C₇₉	—	8	2	no	—
132	C₂³×C₇₉	—	8	2	no	—
133	C₂³×C₇₉	—	8	2	no	—
134	C₂³×C₇₉	#127	1	1	yes	characteristic
135	Q₈×C₇₉	—	4	4	no	—
136	Q₈×C₇₉	—	8	2	no	—
137	D₄×C₇₉	—	4	4	no	—
138	D₄×C₇₉	—	4	4	no	—
139	D₄×C₇₉	—	4	4	no	—
140	D₄×C₇₉	—	8	2	no	—
141	D₄×C₇₉	—	8	2	no	—
142	D₄×C₇₉	—	8	2	no	—
143	C₂×C₄×C₇₉	—	4	2	no	—
144	C₂×C₄×C₇₉	—	8	2	no	—
145	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
146	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
147	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
149	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
150	C₂×C₄×C₇₉	#137	1	4	yes	—
151	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×Dic₇₉	—	4	2	no	—
155	C₂×Dic₇₉	—	4	2	no	—
156	C₂×Dic₇₉	—	8	2	no	—
157	C₂×Dic₇₉	—	8	2	no	—
158	Dic₁₅₈	—	4	4	no	—
159	Dic₁₅₈	—	4	4	no	—
160	Dic₁₅₈	—	8	2	no	—
161	Dic₁₅₈	—	8	2	no	—
162	C₈×C₇₉	C₂×SD₁₆	1	8	yes	complemented
163	C₇₉⋊C₈	—	2	4	no	—
164	C₇₉⋊C₈	—	2	4	no	—
165	C₂²×C₇₉	—	8	2	no	—
166	C₂²×C₇₉	—	8	2	no	—
167	C₂²×C₇₉	—	8	2	no	—
168	C₂²×C₇₉	—	16	4	no	—
169	C₂²×C₇₉	#492	1	2	yes	—
170	C₂²×C₇₉	#496	1	2	yes	—
171	C₂²×C₇₉	#497	1	2	yes	—
172	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
173	C₄×C₇₉	—	8	2	no	—
174	C₄×C₇₉	#492	1	2	yes	—
175	C₄×C₇₉	#496	1	2	yes	—
176	C₄×C₇₉	#497	1	2	yes	—
177	C₄×C₇₉	C₂²×SD₁₆	1	2	yes	—
178	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₄×C₇₉	C₂×#137	1	2	yes	—
180	Dic₇₉	—	8	2	no	—
181	Dic₇₉	—	8	2	no	—
182	C₂×#1714	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	16	2	no	—
184	C₂×C₇₉	#1714	1	4	yes	complemented
185	C₂×C₇₉	C₂×#492	1	1	yes	characteristic
186	C₂×C₇₉	C₂×#496	1	1	yes	characteristic
187	C₂×C₇₉	C₂×#497	1	1	yes	characteristic
188	C₂×#500	—	79	1	no	—
189	C₂×#498	—	79	1	no	—
190	C₂×#492	—	79	1	no	—
191	C₂×#492	D₇₉	1	1	yes	characteristic
192	C₂×#487	—	79	1	no	—
193	C₂×#486	—	79	1	no	—
194	#1714	—	79	8	no	—
195	C₂×#322	—	79	1	no	—
196	C₇₉	C₂×#1714	1	1	yes	characteristic, complemented, Sylow
197	C₂²×SD₁₆	—	2	2	no	—
198	C₂²×SD₁₆	—	158	2	no	—
199	C₂×C₄⋊Q₈	—	79	1	no	—
200	C₂×C₄⋊Q₈	—	79	1	no	—
201	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
202	C₂×#127	C₂×D₇₉	1	1	yes	characteristic
203	#500	—	79	4	no	—
204	#500	—	158	2	no	—
205	#498	—	79	4	no	—
206	#498	—	158	2	no	—
207	#492	—	2	2	no	—
208	#492	—	79	4	no	—
209	#492	—	158	2	no	—
210	#492	C₂×D₇₉	1	4	yes	—
211	#487	—	79	4	no	—
212	#487	—	158	2	no	—
213	#486	—	79	4	no	—
214	#486	—	158	2	no	—
215	C₂×#107	—	158	2	no	—
216	C₂×#106	—	158	2	no	—
217	C₂×#103	—	158	2	no	—
218	C₂×#102	—	158	2	no	—
219	C₂×C₄×C₈	—	79	1	no	—
220	C₂×C₄×C₈	—	79	1	no	—
221	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
222	#322	—	79	4	no	—
223	C₂²×Q₈	—	2	1	no	—
224	C₂²×Q₈	—	158	1	no	—
225	C₂²×Q₈	—	158	1	no	—
226	C₂²×D₄	—	2	1	no	—
227	C₂²×D₄	—	2	1	no	—
228	C₂²×D₄	—	2	1	no	—
229	C₂×SD₁₆	—	2	8	no	—
230	C₂×SD₁₆	—	4	8	no	—
231	C₂×SD₁₆	—	316	8	no	—
232	C₂×SD₁₆	—	158	8	no	—
233	C₂²×C₈	—	79	2	no	—
234	C₂²×C₈	—	79	2	no	—
235	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	79	2	no	—
240	C₄⋊Q₈	—	79	2	no	—
241	C₄⋊Q₈	—	158	1	no	—
242	C₄⋊Q₈	—	158	1	no	—
243	C₄⋊Q₈	—	158	1	no	—
244	C₄⋊Q₈	—	158	1	no	—
245	C₄⋊Q₈	—	158	1	no	—
246	C₄⋊Q₈	—	158	1	no	—
247	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
248	#127	—	2	1	no	—
249	#127	—	2	1	no	—
250	#127	—	2	1	no	—
251	#127	C₂²×D₇₉	1	2	yes	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	158	1	no	—
255	C₂×C₄⋊C₄	—	158	1	no	—
256	C₂×C₄⋊C₄	—	158	1	no	—
257	C₂×C₄⋊C₄	—	158	1	no	—
258	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
259	#107	—	158	8	no	—
260	#106	—	158	8	no	—
261	#103	—	158	8	no	—
262	#102	—	158	8	no	—
263	C₄×C₈	—	79	2	no	—
264	C₄×C₈	—	79	2	no	—
265	C₄×C₈	—	79	2	no	—
266	C₄×C₈	—	79	2	no	—
267	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
268	C₄×C₈	C₂²×D₇₉	1	2	yes	—
269	C₂⁴	—	4	1	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	4	1	no	—
272	C₂×Q₈	—	4	4	no	—
273	C₂×Q₈	—	316	1	no	—
274	C₂×Q₈	—	316	1	no	—
275	C₂×Q₈	—	316	4	no	—
276	C₂×Q₈	—	316	4	no	—
277	C₂×Q₈	—	158	2	no	—
278	C₂×Q₈	—	158	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂×D₄	—	4	1	no	—
284	C₂×D₄	—	4	1	no	—
285	C₂×D₄	—	4	4	no	—
286	C₂×D₄	—	4	4	no	—
287	C₂×D₄	—	4	4	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	316	1	no	—
290	C₂²×C₄	—	316	1	no	—
291	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
294	SD₁₆	—	4	16	no	—
295	SD₁₆	—	316	16	no	—
296	C₂×C₈	—	79	4	no	—
297	C₂×C₈	—	79	4	no	—
298	C₂×C₈	—	158	4	no	—
299	C₂×C₈	—	158	4	no	—
300	C₂×C₈	#11929612	1	8	yes	complemented
301	C₂×C₈	C₂×C₂²⋊D₇₉	1	4	yes	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	316	1	no	—
307	C₄⋊C₄	—	316	1	no	—
308	C₄⋊C₄	—	316	1	no	—
309	C₄⋊C₄	—	316	1	no	—
310	C₄⋊C₄	—	158	2	no	—
311	C₄⋊C₄	—	158	2	no	—
312	C₄⋊C₄	—	158	2	no	—
313	C₄⋊C₄	—	158	2	no	—
314	C₄²	D₄×D₇₉	1	1	yes	characteristic
315	C₄²	D₄×D₇₉	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
317	C₄²	C₂³×D₇₉	1	1	yes	characteristic
318	C₂³	—	4	2	no	—
319	C₂³	—	8	2	no	—
320	C₂³	—	8	2	no	—
321	C₂³	—	8	2	no	—
322	C₂³	#14579157	1	1	yes	characteristic
323	Q₈	—	4	4	no	—
324	Q₈	—	8	2	no	—
325	Q₈	—	316	4	no	—
326	Q₈	—	316	4	no	—
327	Q₈	—	632	2	no	—
328	Q₈	—	632	2	no	—
329	D₄	—	4	4	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	8	2	no	—
333	D₄	—	8	2	no	—
334	D₄	—	8	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	316	2	no	—
338	C₂×C₄	—	316	2	no	—
339	C₂×C₄	—	632	2	no	—
340	C₂×C₄	—	632	2	no	—
341	C₂×C₄	#14579157	1	1	yes	characteristic
342	C₂×C₄	#14579157	1	1	yes	characteristic
343	C₂×C₄	#14579157	1	1	yes	characteristic
344	C₂×C₄	C₂×#11929612	1	4	yes	—
345	C₂×C₄	#14579188	1	4	yes	—
346	C₂×C₄	#14579190	1	4	yes	—
347	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
349	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
350	C₈	—	158	4	no	—
351	C₈	—	158	4	no	—
352	C₈	C₂×#11929612	1	8	yes	complemented
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	16	4	no	—
357	C₂²	#19098935	1	2	yes	—
358	C₂²	#19098949	1	2	yes	—
359	C₂²	#19098954	1	2	yes	—
360	C₂²	C₂×#14579157	1	1	yes	characteristic
361	C₄	—	8	2	no	—
362	C₄	—	632	2	no	—
363	C₄	—	632	2	no	—
364	C₄	#19098935	1	2	yes	—
365	C₄	#19098949	1	2	yes	—
366	C₄	#19098954	1	2	yes	—
367	C₄	C₂²×#11929612	1	2	yes	—
368	C₄	C₂×#14579188	1	2	yes	—
369	C₄	C₂×#14579190	1	2	yes	—
370	C₂	—	16	2	no	—
371	C₂	#23763019	1	4	yes	direct product factor
372	C₂	C₂×#19098935	1	1	yes	characteristic
373	C₂	C₂×#19098949	1	1	yes	characteristic
374	C₂	C₂×#19098954	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579157
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#492×C₇₉	C₂
socle	C₂³×C₇₉	#127
prime radical	C₂×#127×C₇₉	C₂²
hypercenter	C₂×#492	D₇₉
hypocenter	C₇₉	C₂×#1714
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#492

Complements

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

N	H ≅ G/N	type
1. C₂×#19098968	C₂	semidirect
2. C₂×#19098959	C₂	semidirect
4. C₂×#19098915	C₂	semidirect
6. #23763019	C₂	direct (N × H)
7. C₂×#19098284	C₂	semidirect
13. #19098968	C₂²	semidirect
15. #19098959	C₂²	semidirect
19. #19098915	C₂²	semidirect
28. #19098284	C₂²	semidirect
120. C₂×C₈×C₇₉	SD₁₆	semidirect
162. C₈×C₇₉	C₂×SD₁₆	semidirect
184. C₂×C₇₉	#1714	semidirect
196. C₇₉	C₂×#1714	semidirect
300. C₂×C₈	#11929612	semidirect
352. C₈	C₂×#11929612	semidirect
371. C₂	#23763019	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098968	2
C₂×#19098959	2
C₂×#19098935	2
C₂×#19098915	2
C₂×#19098909	2
#23763019	2
C₂×#19098284	2
C₂×#492×C₇₉	2
C₂×#1714	79

Subgroups — C2×#23763019 #34275979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763019 #34275979