Subgroups — C₂×#23773913 #35157873

Order 20224.

1942 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099119	C₂	1	2	yes	maximal
2	#23773913	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19098967	C₂	1	2	yes	maximal, complemented
4	C₂×#19098965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098962	C₂	1	2	yes	maximal, complemented
6	C₂×#19098960	C₂	1	2	yes	maximal, complemented
7	C₂×#19098600	C₂	1	2	yes	maximal
8	C₂×#19098598	C₂	1	2	yes	maximal
9	C₂×C₄×C₈×D₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂×#499×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₇₉	C₂²	1	1	yes	characteristic
13	#19099119	—	2	8	no	—
14	#19099119	C₂²	1	16	yes	—
15	C₂×#14579160	C₂²	1	4	yes	—
16	C₂×#14579153	C₂²	1	2	yes	—
17	C₂×#14579139	—	2	4	no	—
18	C₂×#14579126	—	2	4	no	—
19	C₂×#14579119	C₂²	1	4	yes	—
20	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
21	#19098967	C₂²	1	16	yes	complemented
22	#19098965	C₂²	1	8	yes	complemented
23	#19098962	C₂²	1	16	yes	complemented
24	#19098960	C₂²	1	16	yes	complemented
25	#19098600	C₂²	1	16	yes	—
26	#19098598	C₂²	1	16	yes	—
27	C₂×#14579095	C₂²	1	2	yes	—
28	C₂×#14579094	C₂²	1	4	yes	—
29	C₂×#14579093	C₂²	1	2	yes	—
30	C₂×#14579092	C₂²	1	4	yes	—
31	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₇₉	C₂²	1	8	yes	—
35	C₂×#125×C₇₉	C₂²	1	2	yes	—
36	#499×C₇₉	C₂²	1	8	yes	complemented
37	C₂×#107×C₇₉	C₂²	1	2	yes	—
38	C₂×#106×C₇₉	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
40	C₂²×Dic₁₅₈	—	2	4	no	—
41	C₂²×C₄×D₇₉	D₄	1	2	yes	—
42	C₂²×C₄×D₇₉	C₂³	1	1	yes	characteristic
43	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₇₉	Q₈	1	8	yes	complemented
46	C₂×C₈×D₇₉	Q₈	1	8	yes	complemented
47	C₂×C₈×D₇₉	C₂³	1	4	yes	—
48	C₂×C₈×D₇₉	C₂³	1	4	yes	—
49	#14579160	—	2	8	no	—
50	#14579160	C₂³	1	16	yes	—
51	#14579153	—	2	4	no	—
52	#14579153	C₂³	1	8	yes	—
53	#14579139	—	2	32	no	—
54	#14579126	—	2	32	no	—
55	#14579119	—	2	8	no	—
56	#14579119	C₂³	1	16	yes	—
57	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
58	C₂×Dic₇₉⋊C₄	—	2	4	no	—
59	C₂×Dic₇₉⋊C₄	—	2	4	no	—
60	C₂×(C₂×D₇₉)⋊C₄	—	2	4	no	—
61	C₂×C₄×Dic₇₉	—	2	4	no	—
62	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
63	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
64	C₄²×D₇₉	D₄	1	4	yes	—
65	C₄²×D₇₉	C₂³	1	4	yes	—
66	#14579095	C₂³	1	8	yes	—
67	#14579094	C₂³	1	16	yes	—
68	#14579093	C₂³	1	8	yes	—
69	#14579092	C₂³	1	16	yes	—
70	C₄×C₇₉⋊C₈	C₂³	1	4	yes	—
71	C₈×Dic₇₉	C₂³	1	4	yes	—
72	C₈×Dic₇₉	C₂³	1	4	yes	—
73	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	#125×C₇₉	—	2	4	no	—
76	#125×C₇₉	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₇₉	—	2	4	no	—
78	C₂×C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
79	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
80	#107×C₇₉	C₂³	1	8	yes	—
81	#106×C₇₉	C₂³	1	8	yes	—
82	C₄×C₈×C₇₉	C₂³	1	4	yes	—
83	C₂³×D₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂×Dic₁₅₈	—	2	16	no	—
85	C₂×Dic₁₅₈	—	4	16	no	—
86	C₂²×Dic₇₉	—	2	4	no	—
87	C₂²×Dic₇₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×D₇₉	—	2	8	no	—
90	C₂×C₄×D₇₉	C₂×D₄	1	8	yes	—
91	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×D₇₉	C₂×Q₈	1	4	yes	—
93	C₂×C₄×D₇₉	C₂×Q₈	1	4	yes	—
94	C₂×C₄×D₇₉	C₂⁴	1	2	yes	—
95	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
96	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
97	C₂×C₇₉⋊C₈	C₂⁴	1	2	yes	—
98	C₂×C₇₉⋊C₈	C₂⁴	1	2	yes	—
99	C₈×D₇₉	C₂×Q₈	1	16	yes	complemented
100	C₈×D₇₉	C₂×Q₈	1	16	yes	complemented
101	C₄⋊Dic₇₉	—	2	4	no	—
102	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
103	Dic₇₉⋊C₄	—	2	16	no	—
104	Dic₇₉⋊C₄	—	2	16	no	—
105	(C₂×D₇₉)⋊C₄	—	2	16	no	—
106	C₄×Dic₇₉	—	2	16	no	—
107	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
108	C₄×Dic₇₉	C₂×D₄	1	8	yes	—
109	C₄×Dic₇₉	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₇₉	—	2	4	no	—
111	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
114	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
115	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
116	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
117	C₄⋊C₄×C₇₉	—	2	16	no	—
118	C₄⋊C₄×C₇₉	—	2	4	no	—
119	C₄⋊C₄×C₇₉	C₂⁴	1	8	yes	—
120	C₄²×C₇₉	C₂×D₄	1	2	yes	—
121	C₄²×C₇₉	C₂⁴	1	2	yes	—
122	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	C₂²×D₇₉	—	2	4	no	—
124	C₂²×D₇₉	C₄⋊Q₈	1	4	yes	—
125	C₂²×D₇₉	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₇₉	—	2	4	no	—
127	C₂×C₄×C₇₉	—	2	8	no	—
128	C₂×C₄×C₇₉	—	4	8	no	—
129	C₂×C₄×C₇₉	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
134	C₂×Dic₇₉	—	2	8	no	—
135	C₂×Dic₇₉	—	4	8	no	—
136	C₂×Dic₇₉	C₄⋊Q₈	1	4	yes	—
137	C₂×Dic₇₉	#135	1	8	yes	—
138	C₂×Dic₇₉	C₂²×D₄	1	2	yes	—
139	C₂×Dic₇₉	C₂²×D₄	1	4	yes	—
140	C₄×D₇₉	—	2	16	no	—
141	C₄×D₇₉	C₄⋊Q₈	1	8	yes	—
142	C₄×D₇₉	C₂²×Q₈	1	4	yes	—
143	C₄×D₇₉	C₂²×Q₈	1	4	yes	—
144	Dic₁₅₈	—	4	32	no	—
145	C₈×C₇₉	C₂²×Q₈	1	4	yes	—
146	C₈×C₇₉	C₂²×Q₈	1	4	yes	—
147	C₇₉⋊C₈	C₂²×Q₈	1	4	yes	—
148	C₇₉⋊C₈	C₂²×Q₈	1	4	yes	—
149	C₂²×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
150	C₂²×C₇₉	C₂×#135	1	4	yes	—
151	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
152	C₂×D₇₉	—	2	8	no	—
153	C₂×D₇₉	—	2	4	no	—
154	C₂×D₇₉	C₂×C₄⋊Q₈	1	4	yes	—
155	C₄×C₇₉	—	2	4	no	—
156	C₄×C₇₉	—	4	8	no	—
157	C₄×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
158	C₄×C₇₉	C₂³×Q₈	1	1	yes	characteristic
159	C₄×C₇₉	C₂³×Q₈	1	1	yes	characteristic
160	Dic₇₉	—	4	8	no	—
161	Dic₇₉	C₂×C₄⋊Q₈	1	4	yes	—
162	Dic₇₉	C₂×#135	1	8	yes	—
163	C₂×#3160	—	79	1	no	maximal, Sylow
164	C₂×C₇₉	#3160	1	4	yes	complemented
165	C₂×C₇₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₇₉	C₂²×#135	1	2	yes	—
167	D₇₉	—	2	4	no	—
168	C₂×#533	—	79	2	no	—
169	#3160	—	79	16	no	—
170	C₂×#500	—	79	2	no	—
171	C₂×#499	—	79	1	no	—
172	C₂×#499	—	79	2	no	—
173	C₂×#499	D₇₉	1	1	yes	characteristic, complemented
174	C₂×#498	—	79	2	no	—
175	C₂×#427	—	79	2	no	—
176	C₂×#427	—	79	2	no	—
177	C₂²×C₄×C₈	—	79	1	no	—
178	C₇₉	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₈	—	79	1	no	—
180	C₂³×C₈	—	79	1	no	—
181	#533	—	79	16	no	—
182	#533	—	158	8	no	—
183	C₂×C₄⋊Q₈	—	79	4	no	—
184	C₂×#125	—	79	2	no	—
185	C₂×#125	C₂×D₇₉	1	2	yes	—
186	C₂×C₂²⋊Q₈	—	158	4	no	—
187	C₂×C₄×Q₈	—	158	4	no	—
188	C₂×#117	—	79	4	no	—
189	C₂²×C₄²	—	79	1	no	—
190	#500	—	79	16	no	—
191	#499	—	79	16	no	—
192	#499	—	79	8	no	—
193	#499	C₂×D₇₉	1	8	yes	complemented
194	#498	—	79	16	no	—
195	#427	—	79	16	no	—
196	#427	—	79	16	no	—
197	C₂×#107	—	79	2	no	—
198	C₂×#107	—	79	4	no	—
199	C₂×#107	C₂×D₇₉	1	2	yes	—
200	C₂×#106	—	79	2	no	—
201	C₂×#106	—	79	4	no	—
202	C₂×#106	C₂×D₇₉	1	2	yes	—
203	C₂×C₄×C₈	—	79	1	no	—
204	C₂×C₄×C₈	—	79	1	no	—
205	C₂×C₄×C₈	—	79	1	no	—
206	C₂×C₄×C₈	—	79	8	no	—
207	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
208	C₂²×Q₈	—	158	4	no	—
209	C₂³×C₄	—	79	1	no	—
210	C₂³×C₄	—	79	2	no	—
211	C₂²×C₈	—	79	1	no	—
212	C₂²×C₈	—	79	1	no	—
213	C₂²×C₈	—	79	8	no	—
214	C₂²×C₈	—	79	8	no	—
215	C₂²×C₈	—	79	4	no	—
216	C₂²×C₈	—	79	4	no	—
217	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
219	C₄⋊Q₈	—	79	16	no	—
220	C₄⋊Q₈	—	158	8	no	—
221	#125	—	2	4	no	—
222	#125	—	79	8	no	—
223	#125	—	158	4	no	—
224	#125	C₂²×D₇₉	1	8	yes	—
225	C₂²⋊Q₈	—	158	32	no	—
226	C₄×Q₈	—	158	32	no	—
227	#117	—	79	16	no	—
228	#117	—	158	8	no	—
229	C₂×C₄⋊C₄	—	2	4	no	—
230	C₂×C₄⋊C₄	—	79	4	no	—
231	C₂×C₄⋊C₄	—	158	4	no	—
232	C₂×C₄⋊C₄	—	158	4	no	—
233	C₂×C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
234	C₂×C₂²⋊C₄	—	158	4	no	—
235	C₂×C₄²	—	79	2	no	—
236	C₂×C₄²	—	79	1	no	—
237	C₂×C₄²	—	79	4	no	—
238	C₂×C₄²	—	79	4	no	—
239	C₂×C₄²	—	158	4	no	—
240	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
241	#107	—	79	8	no	—
242	#107	—	79	16	no	—
243	#107	C₂²×D₇₉	1	8	yes	—
244	#106	—	79	8	no	—
245	#106	—	79	16	no	—
246	#106	C₂²×D₇₉	1	8	yes	—
247	C₄×C₈	—	79	4	no	—
248	C₄×C₈	—	79	4	no	—
249	C₄×C₈	—	79	4	no	—
250	C₄×C₈	C₂²×D₇₉	1	4	yes	—
251	C₂⁴	—	79	1	no	—
252	C₂×Q₈	—	316	16	no	—
253	C₂×Q₈	—	158	16	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	79	1	no	—
256	C₂²×C₄	—	79	2	no	—
257	C₂²×C₄	—	79	4	no	—
258	C₂²×C₄	—	79	4	no	—
259	C₂²×C₄	—	79	2	no	—
260	C₂²×C₄	—	79	2	no	—
261	C₂²×C₄	—	79	8	no	—
262	C₂²×C₄	—	158	8	no	—
263	C₂²×C₄	—	158	4	no	—
264	C₂²×C₄	D₄×D₇₉	1	2	yes	—
265	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
266	C₂×C₈	—	79	4	no	—
267	C₂×C₈	—	79	4	no	—
268	C₂×C₈	—	79	2	no	—
269	C₂×C₈	—	79	2	no	—
270	C₂×C₈	—	79	16	no	—
271	C₂×C₈	—	79	16	no	—
272	C₂×C₈	Q₈×D₇₉	1	4	yes	—
273	C₂×C₈	Q₈×D₇₉	1	4	yes	—
274	C₂×C₈	C₂³×D₇₉	1	2	yes	—
275	C₂×C₈	C₂³×D₇₉	1	2	yes	—
276	C₄⋊C₄	—	2	16	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	79	8	no	—
279	C₄⋊C₄	—	158	16	no	—
280	C₄⋊C₄	—	158	16	no	—
281	C₄⋊C₄	—	158	4	no	—
282	C₄⋊C₄	C₂³×D₇₉	1	8	yes	—
283	C₂²⋊C₄	—	158	16	no	—
284	C₄²	—	79	2	no	—
285	C₄²	—	79	8	no	—
286	C₄²	—	79	2	no	—
287	C₄²	—	158	16	no	—
288	C₄²	D₄×D₇₉	1	2	yes	—
289	C₄²	C₂³×D₇₉	1	2	yes	—
290	C₂³	—	79	4	no	—
291	C₂³	—	79	2	no	—
292	C₂³	—	158	4	no	—
293	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
294	Q₈	—	316	32	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	316	8	no	—
299	C₂×C₄	—	79	8	no	—
300	C₂×C₄	—	79	4	no	—
301	C₂×C₄	—	79	2	no	—
302	C₂×C₄	—	79	4	no	—
303	C₂×C₄	—	79	8	no	—
304	C₂×C₄	—	79	4	no	—
305	C₂×C₄	—	79	4	no	—
306	C₂×C₄	—	158	16	no	—
307	C₂×C₄	—	158	8	no	—
308	C₂×C₄	C₂×D₄×D₇₉	1	4	yes	—
309	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
310	C₂×C₄	C₂×Q₈×D₇₉	1	2	yes	—
311	C₂×C₄	C₂×Q₈×D₇₉	1	2	yes	—
312	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
313	C₈	—	79	4	no	—
314	C₈	—	79	4	no	—
315	C₈	C₂×Q₈×D₇₉	1	4	yes	—
316	C₈	C₂×Q₈×D₇₉	1	4	yes	—
317	C₂²	—	79	4	no	—
318	C₂²	—	158	4	no	—
319	C₂²	—	158	8	no	—
320	C₂²	C₄⋊Q₈×D₇₉	1	2	yes	—
321	C₂²	#135×D₇₉	1	4	yes	—
322	C₂²	C₂²×D₄×D₇₉	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	4	8	no	—
325	C₄	—	316	8	no	—
326	C₄	—	79	8	no	—
327	C₄	—	79	4	no	—
328	C₄	C₄⋊Q₈×D₇₉	1	2	yes	—
329	C₄	C₂²×Q₈×D₇₉	1	1	yes	characteristic
330	C₄	C₂²×Q₈×D₇₉	1	1	yes	characteristic
331	C₂	—	158	4	no	—
332	C₂	#23773913	1	4	yes	direct product factor
333	C₂	C₂×C₄⋊Q₈×D₇₉	1	1	yes	characteristic
334	C₂	C₂×#135×D₇₉	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂×#499×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂³×D₇₉	C₂×D₄
hypercenter	C₂×#499	D₇₉
hypocenter	C₇₉	C₂×#3160
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #23773913	C₂	direct (N × H)
3. C₂×#19098967	C₂	semidirect
4. C₂×#19098965	C₂	semidirect
5. C₂×#19098962	C₂	semidirect
6. C₂×#19098960	C₂	semidirect
10. C₂×#499×C₇₉	C₂	semidirect
21. #19098967	C₂²	semidirect
22. #19098965	C₂²	semidirect
23. #19098962	C₂²	semidirect
24. #19098960	C₂²	semidirect
36. #499×C₇₉	C₂²	semidirect
45. C₂×C₈×D₇₉	Q₈	semidirect
46. C₂×C₈×D₇₉	Q₈	semidirect
99. C₈×D₇₉	C₂×Q₈	semidirect
100. C₈×D₇₉	C₂×Q₈	semidirect
164. C₂×C₇₉	#3160	semidirect
173. C₂×#499	D₇₉	semidirect
178. C₇₉	C₂×#3160	semidirect
193. #499	C₂×D₇₉	semidirect
332. C₂	#23773913	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×#19099119	2
#23773913	2
C₂×#19098967	2
C₂×#19098965	2
C₂×#19098962	2
C₂×#19098960	2
C₂×#19098600	2
C₂×#19098598	2
C₂×C₄×C₈×D₇₉	2
C₂×#499×C₇₉	2
C₂×#3160	79

Subgroups — C2×#23773913 #35157873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23773913 #35157873