Subgroups — C₂×#23763011 #34275944

Order 20224.

802 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098915	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098914	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098913	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098909	C₂	1	1	yes	maximal, characteristic
5	C₂×#19098909	C₂	1	1	yes	maximal, characteristic
6	#23763011	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#486×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14579161	C₂²	1	1	yes	characteristic
10	C₂×#14579161	C₂²	1	1	yes	characteristic
11	#19098915	—	2	2	no	—
12	#19098915	C₂²	1	4	yes	complemented
13	#19098914	—	2	2	no	—
14	#19098914	C₂²	1	4	yes	complemented
15	#19098913	—	2	2	no	—
16	#19098913	C₂²	1	4	yes	complemented
17	#19098909	—	2	2	no	—
18	#19098909	—	2	2	no	—
19	#19098909	C₂²	1	4	yes	—
20	#19098909	C₂²	1	4	yes	—
21	C₂×#14579086	—	2	2	no	—
22	C₂×#14579085	—	2	2	no	—
23	C₂×#14579084	—	2	2	no	—
24	C₂×#14579082	—	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	C₈×C₇₉⋊C₈	C₂²	1	4	yes	complemented
29	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#127×C₇₉	C₂²	1	1	yes	characteristic
31	#486×C₇₉	—	2	2	no	—
32	#486×C₇₉	C₂²	1	4	yes	—
33	C₂×#102×C₇₉	—	2	2	no	—
34	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	C₂²×Dic₁₅₈	—	2	1	no	—
36	C₂²×Dic₁₅₈	—	2	1	no	—
37	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
38	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
39	#14579161	—	2	1	no	—
40	#14579161	—	2	1	no	—
41	#14579161	—	2	1	no	—
42	#14579161	—	2	1	no	—
43	#14579161	—	2	1	no	—
44	#14579161	—	2	1	no	—
45	#14579161	C₂³	1	2	yes	—
46	#14579161	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₉	—	2	1	no	—
48	C₂×C₄⋊Dic₇₉	—	2	1	no	—
49	C₂×C₄⋊Dic₇₉	—	2	1	no	—
50	C₂×C₄⋊Dic₇₉	—	2	1	no	—
51	#14579086	—	2	8	no	—
52	#14579085	—	2	8	no	—
53	#14579084	—	2	8	no	—
54	#14579082	—	2	8	no	—
55	#14579082	—	2	8	no	—
56	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
57	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
58	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
59	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
60	C₂²×Q₈×C₇₉	—	2	1	no	—
61	C₂²×D₄×C₇₉	—	2	1	no	—
62	C₂²×D₄×C₇₉	—	2	1	no	—
63	C₂²×D₄×C₇₉	—	2	1	no	—
64	C₂²×C₈×C₇₉	D₄	1	2	yes	—
65	C₄⋊Q₈×C₇₉	—	2	1	no	—
66	C₄⋊Q₈×C₇₉	—	2	1	no	—
67	C₄⋊Q₈×C₇₉	—	2	1	no	—
68	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
69	#127×C₇₉	—	2	1	no	—
70	#127×C₇₉	—	2	1	no	—
71	#127×C₇₉	—	2	1	no	—
72	#127×C₇₉	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
74	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
75	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
76	#102×C₇₉	—	2	8	no	—
77	C₄×C₈×C₇₉	D₄	1	2	yes	—
78	C₄×C₈×C₇₉	C₂³	1	2	yes	—
79	C₂×Dic₁₅₈	—	2	2	no	—
80	C₂×Dic₁₅₈	—	2	2	no	—
81	C₂×Dic₁₅₈	—	4	1	no	—
82	C₂×Dic₁₅₈	—	4	1	no	—
83	C₂×Dic₁₅₈	—	4	4	no	—
84	C₂×Dic₁₅₈	—	4	4	no	—
85	C₂²×Dic₇₉	—	4	1	no	—
86	C₂²×Dic₇₉	—	4	1	no	—
87	C₂×C₇₉⋊C₈	—	2	4	no	—
88	C₂×C₇₉⋊C₈	—	2	4	no	—
89	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
90	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
91	C₄⋊Dic₇₉	—	2	2	no	—
92	C₄⋊Dic₇₉	—	2	2	no	—
93	C₄⋊Dic₇₉	—	2	2	no	—
94	C₄⋊Dic₇₉	—	2	2	no	—
95	C₄⋊Dic₇₉	—	4	1	no	—
96	C₄⋊Dic₇₉	—	4	1	no	—
97	C₄⋊Dic₇₉	—	4	1	no	—
98	C₄⋊Dic₇₉	—	4	1	no	—
99	C₂⁴×C₇₉	—	4	1	no	—
100	C₂×Q₈×C₇₉	—	2	2	no	—
101	C₂×Q₈×C₇₉	—	4	1	no	—
102	C₂×Q₈×C₇₉	—	4	4	no	—
103	C₂×D₄×C₇₉	—	2	2	no	—
104	C₂×D₄×C₇₉	—	2	2	no	—
105	C₂×D₄×C₇₉	—	2	2	no	—
106	C₂×D₄×C₇₉	—	4	1	no	—
107	C₂×D₄×C₇₉	—	4	1	no	—
108	C₂×D₄×C₇₉	—	4	1	no	—
109	C₂×D₄×C₇₉	—	4	4	no	—
110	C₂×D₄×C₇₉	—	4	4	no	—
111	C₂×D₄×C₇₉	—	4	4	no	—
112	C₂²×C₄×C₇₉	—	4	1	no	—
113	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	—	2	4	no	—
117	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
118	C₄⋊C₄×C₇₉	—	2	2	no	—
119	C₄⋊C₄×C₇₉	—	2	2	no	—
120	C₄⋊C₄×C₇₉	—	4	1	no	—
121	C₄⋊C₄×C₇₉	—	4	1	no	—
122	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
126	C₂³×C₇₉	—	4	2	no	—
127	C₂³×C₇₉	—	8	2	no	—
128	C₂³×C₇₉	—	8	2	no	—
129	C₂³×C₇₉	—	8	2	no	—
130	C₂³×C₇₉	#127	1	1	yes	characteristic
131	Q₈×C₇₉	—	4	4	no	—
132	Q₈×C₇₉	—	8	2	no	—
133	D₄×C₇₉	—	4	4	no	—
134	D₄×C₇₉	—	4	4	no	—
135	D₄×C₇₉	—	4	4	no	—
136	D₄×C₇₉	—	8	2	no	—
137	D₄×C₇₉	—	8	2	no	—
138	D₄×C₇₉	—	8	2	no	—
139	C₂×C₄×C₇₉	—	4	2	no	—
140	C₂×C₄×C₇₉	—	8	2	no	—
141	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
143	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
144	C₂×C₄×C₇₉	#135	1	4	yes	—
145	C₂×C₄×C₇₉	#135	1	4	yes	—
146	C₂×C₄×C₇₉	#135	1	4	yes	—
147	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₇₉	—	4	2	no	—
151	C₂×Dic₇₉	—	4	2	no	—
152	C₂×Dic₇₉	—	8	2	no	—
153	C₂×Dic₇₉	—	8	2	no	—
154	Dic₁₅₈	—	4	4	no	—
155	Dic₁₅₈	—	4	4	no	—
156	Dic₁₅₈	—	8	2	no	—
157	Dic₁₅₈	—	8	2	no	—
158	C₈×C₇₉	—	2	4	no	—
159	C₇₉⋊C₈	—	2	4	no	—
160	C₇₉⋊C₈	—	2	4	no	—
161	C₂²×C₇₉	—	8	2	no	—
162	C₂²×C₇₉	—	8	2	no	—
163	C₂²×C₇₉	—	8	2	no	—
164	C₂²×C₇₉	—	16	4	no	—
165	C₂²×C₇₉	#495	1	2	yes	—
166	C₂²×C₇₉	#495	1	2	yes	—
167	C₂²×C₇₉	#495	1	2	yes	—
168	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
169	C₄×C₇₉	—	8	2	no	—
170	C₄×C₇₉	#495	1	2	yes	—
171	C₄×C₇₉	#495	1	2	yes	—
172	C₄×C₇₉	#495	1	2	yes	—
173	C₄×C₇₉	C₂×#135	1	2	yes	—
174	C₄×C₇₉	C₂×#135	1	2	yes	—
175	C₄×C₇₉	C₂×#135	1	2	yes	—
176	Dic₇₉	—	8	2	no	—
177	Dic₇₉	—	8	2	no	—
178	C₂×#1711	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	16	2	no	—
180	C₂×C₇₉	#1711	1	4	yes	complemented
181	C₂×C₇₉	C₂×#495	1	1	yes	characteristic
182	C₂×C₇₉	C₂×#495	1	1	yes	characteristic
183	C₂×C₇₉	C₂×#495	1	1	yes	characteristic
184	C₂×#487	—	79	1	no	—
185	C₂×#487	—	79	1	no	—
186	C₂×#487	—	79	1	no	—
187	C₂×#486	—	79	1	no	—
188	C₂×#486	—	79	1	no	—
189	C₂×#486	D₇₉	1	1	yes	characteristic
190	#1711	—	79	8	no	—
191	C₂×C₈²	—	79	1	no	—
192	C₇₉	C₂×#1711	1	1	yes	characteristic, complemented, Sylow
193	C₂×C₄⋊Q₈	—	79	1	no	—
194	C₂×C₄⋊Q₈	—	79	1	no	—
195	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
196	C₂×#127	C₂×D₇₉	1	1	yes	characteristic
197	#487	—	79	4	no	—
198	#487	—	79	4	no	—
199	#487	—	79	4	no	—
200	#487	—	158	2	no	—
201	#487	—	158	2	no	—
202	#487	—	158	2	no	—
203	#486	—	2	2	no	—
204	#486	—	79	4	no	—
205	#486	—	79	4	no	—
206	#486	—	158	2	no	—
207	#486	—	158	2	no	—
208	#486	C₂×D₇₉	1	4	yes	—
209	C₂×#103	—	158	2	no	—
210	C₂×#103	—	158	2	no	—
211	C₂×#103	—	158	2	no	—
212	C₂×#102	—	2	2	no	—
213	C₂×#102	—	158	2	no	—
214	C₂×#102	—	158	2	no	—
215	C₂×C₄×C₈	—	79	1	no	—
216	C₂×C₄×C₈	—	79	1	no	—
217	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
218	C₈²	—	79	4	no	—
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×Q₈	—	158	1	no	—
221	C₂²×Q₈	—	158	1	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂²×D₄	—	2	1	no	—
224	C₂²×D₄	—	2	1	no	—
225	C₂²×C₈	—	79	2	no	—
226	C₂²×C₈	—	79	2	no	—
227	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	79	2	no	—
232	C₄⋊Q₈	—	79	2	no	—
233	C₄⋊Q₈	—	158	1	no	—
234	C₄⋊Q₈	—	158	1	no	—
235	C₄⋊Q₈	—	158	1	no	—
236	C₄⋊Q₈	—	158	1	no	—
237	C₄⋊Q₈	—	158	1	no	—
238	C₄⋊Q₈	—	158	1	no	—
239	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
240	#127	—	2	1	no	—
241	#127	—	2	1	no	—
242	#127	—	2	1	no	—
243	#127	C₂²×D₇₉	1	2	yes	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	158	1	no	—
247	C₂×C₄⋊C₄	—	158	1	no	—
248	C₂×C₄⋊C₄	—	158	1	no	—
249	C₂×C₄⋊C₄	—	158	1	no	—
250	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
251	#103	—	158	8	no	—
252	#103	—	158	8	no	—
253	#103	—	158	8	no	—
254	#102	—	2	8	no	—
255	#102	—	158	8	no	—
256	#102	—	158	8	no	—
257	C₄×C₈	—	79	2	no	—
258	C₄×C₈	—	79	2	no	—
259	C₄×C₈	—	79	2	no	—
260	C₄×C₈	—	79	2	no	—
261	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
262	C₄×C₈	C₂²×D₇₉	1	2	yes	—
263	C₂⁴	—	4	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	4	no	—
267	C₂×Q₈	—	316	1	no	—
268	C₂×Q₈	—	316	1	no	—
269	C₂×Q₈	—	316	4	no	—
270	C₂×Q₈	—	316	4	no	—
271	C₂×Q₈	—	158	2	no	—
272	C₂×Q₈	—	158	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	4	no	—
280	C₂×D₄	—	4	4	no	—
281	C₂×D₄	—	4	4	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	316	1	no	—
284	C₂²×C₄	—	316	1	no	—
285	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	79	4	no	—
290	C₂×C₈	—	79	4	no	—
291	C₂×C₈	—	158	4	no	—
292	C₂×C₈	—	158	4	no	—
293	C₂×C₈	C₂×C₂²⋊D₇₉	1	4	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	316	1	no	—
299	C₄⋊C₄	—	316	1	no	—
300	C₄⋊C₄	—	316	1	no	—
301	C₄⋊C₄	—	316	1	no	—
302	C₄⋊C₄	—	158	2	no	—
303	C₄⋊C₄	—	158	2	no	—
304	C₄⋊C₄	—	158	2	no	—
305	C₄⋊C₄	—	158	2	no	—
306	C₄²	D₄×D₇₉	1	1	yes	characteristic
307	C₄²	D₄×D₇₉	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
309	C₄²	C₂³×D₇₉	1	1	yes	characteristic
310	C₂³	—	4	2	no	—
311	C₂³	—	8	2	no	—
312	C₂³	—	8	2	no	—
313	C₂³	—	8	2	no	—
314	C₂³	#14579157	1	1	yes	characteristic
315	Q₈	—	4	4	no	—
316	Q₈	—	8	2	no	—
317	Q₈	—	316	4	no	—
318	Q₈	—	316	4	no	—
319	Q₈	—	632	2	no	—
320	Q₈	—	632	2	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	4	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	316	2	no	—
330	C₂×C₄	—	316	2	no	—
331	C₂×C₄	—	632	2	no	—
332	C₂×C₄	—	632	2	no	—
333	C₂×C₄	#14579157	1	1	yes	characteristic
334	C₂×C₄	#14579157	1	1	yes	characteristic
335	C₂×C₄	#14579157	1	1	yes	characteristic
336	C₂×C₄	#14579180	1	4	yes	—
337	C₂×C₄	#14579181	1	4	yes	—
338	C₂×C₄	#14579182	1	4	yes	—
339	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
342	C₈	—	2	4	no	—
343	C₈	—	158	4	no	—
344	C₈	—	158	4	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	16	4	no	—
349	C₂²	#19098944	1	2	yes	—
350	C₂²	#19098945	1	2	yes	—
351	C₂²	#19098946	1	2	yes	—
352	C₂²	C₂×#14579157	1	1	yes	characteristic
353	C₄	—	8	2	no	—
354	C₄	—	632	2	no	—
355	C₄	—	632	2	no	—
356	C₄	#19098944	1	2	yes	—
357	C₄	#19098945	1	2	yes	—
358	C₄	#19098946	1	2	yes	—
359	C₄	C₂×#14579180	1	2	yes	—
360	C₄	C₂×#14579181	1	2	yes	—
361	C₄	C₂×#14579182	1	2	yes	—
362	C₂	—	16	2	no	—
363	C₂	#23763011	1	4	yes	direct product factor
364	C₂	C₂×#19098944	1	1	yes	characteristic
365	C₂	C₂×#19098945	1	1	yes	characteristic
366	C₂	C₂×#19098946	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579157
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#486×C₇₉	C₂
socle	C₂³×C₇₉	#127
prime radical	C₂×#127×C₇₉	C₂²
hypercenter	C₂×#486	D₇₉
hypocenter	C₇₉	C₂×#1711
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#486

Complements

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

N	H ≅ G/N	type
1. C₂×#19098915	C₂	semidirect
2. C₂×#19098914	C₂	semidirect
3. C₂×#19098913	C₂	semidirect
6. #23763011	C₂	direct (N × H)
7. C₂×C₈×C₇₉⋊C₈	C₂	semidirect
12. #19098915	C₂²	semidirect
14. #19098914	C₂²	semidirect
16. #19098913	C₂²	semidirect
28. C₈×C₇₉⋊C₈	C₂²	semidirect
180. C₂×C₇₉	#1711	semidirect
192. C₇₉	C₂×#1711	semidirect
363. C₂	#23763011	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×#19098915	2
C₂×#19098914	2
C₂×#19098913	2
C₂×#19098909	2
#23763011	2
C₂×C₈×C₇₉⋊C₈	2
C₂×#486×C₇₉	2
C₂×#1711	79

Subgroups — C2×#23763011 #34275944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763011 #34275944