Subgroups — C₂×#23765621 #34443161

Order 20224.

552 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098822	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098610	C₂	1	1	yes	maximal, characteristic
3	#23765621	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098385	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098380	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19098378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19098344	C₂	1	1	yes	maximal, characteristic
8	C₂×#365×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
10	C₂²×#11929607	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₇₉	—	2	1	no	—
12	C₂×#14579141	—	2	1	no	—
13	C₂×#14579101	C₂²	1	1	yes	characteristic
14	#19098822	—	2	2	no	—
15	#19098822	C₂²	1	4	yes	—
16	#19098610	—	2	2	no	—
17	#19098610	C₂²	1	4	yes	—
18	C₂×#14579097	C₂²	1	1	yes	characteristic
19	C₂×#14579096	—	2	1	no	—
20	C₂×#14579084	—	2	1	no	—
21	#19098385	C₂²	1	4	yes	complemented
22	#19098380	C₂²	1	4	yes	complemented
23	#19098378	C₂²	1	4	yes	complemented
24	#19098344	C₂²	1	4	yes	—
25	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
28	#365×C₇₉	C₂²	1	4	yes	complemented
29	C₂²×Dic₁₅₈	—	2	1	no	—
30	C₂²×C₄×D₇₉	D₄	1	1	yes	characteristic
31	C₂×Dic₃₁₆	—	2	1	no	—
32	C₂×Dic₃₁₆	—	2	1	no	—
33	C₂×Dic₃₁₆	—	2	1	no	—
34	C₂×Dic₃₁₆	—	2	1	no	—
35	C₂×Dic₃₁₆	—	2	1	no	—
36	C₂×Dic₃₁₆	C₂³	1	2	yes	—
37	C₂×#11929608	—	2	1	no	—
38	C₂×#11929607	—	2	1	no	—
39	C₂×#11929607	—	2	1	no	—
40	C₂×#11929607	—	2	1	no	—
41	C₂×#11929607	—	2	1	no	—
42	C₂×#11929607	—	4	1	no	—
43	C₂×#11929607	D₄	1	2	yes	—
44	C₂×#11929607	C₂³	1	2	yes	—
45	M₁₆×D₇₉	—	2	4	no	—
46	M₁₆×D₇₉	—	4	2	no	—
47	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
48	C₂×C₈×D₇₉	—	4	1	no	—
49	#14579141	—	2	4	no	—
50	#14579141	—	4	2	no	—
51	C₂×C₄⋊Dic₇₉	—	4	1	no	—
52	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
53	#14579101	—	2	1	no	—
54	#14579101	C₂³	1	2	yes	—
55	C₂×C₇₉⋊C₁₆	—	2	1	no	—
56	#14579097	—	2	1	no	—
57	#14579097	C₂³	1	2	yes	—
58	#14579096	—	2	4	no	—
59	#14579084	—	2	4	no	—
60	C₂×M₁₆×C₇₉	—	2	1	no	—
61	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
63	M₃₂×C₇₉	—	2	1	no	—
64	M₃₂×C₇₉	C₂³	1	2	yes	—
65	C₂×C₁₆×C₇₉	—	2	1	no	—
66	#108×C₇₉	—	2	1	no	—
67	#108×C₇₉	C₂³	1	2	yes	—
68	#106×C₇₉	—	2	1	no	—
69	#106×C₇₉	C₂³	1	2	yes	—
70	C₂³×D₇₉	SD₁₆	1	1	yes	characteristic
71	C₂×Dic₁₅₈	—	2	2	no	—
72	C₂×Dic₁₅₈	—	4	1	no	—
73	C₂×Dic₁₅₈	—	4	1	no	—
74	C₂×Dic₁₅₈	—	4	1	no	—
75	C₂×Dic₁₅₈	—	4	1	no	—
76	C₂×Dic₁₅₈	—	4	1	no	—
77	C₂²×Dic₇₉	—	4	1	no	—
78	C₂²×Dic₇₉	SD₁₆	1	1	yes	characteristic
79	C₂×C₄×D₇₉	—	2	1	no	—
80	C₂×C₄×D₇₉	—	2	1	no	—
81	C₂×C₄×D₇₉	—	2	1	no	—
82	C₂×C₄×D₇₉	—	2	1	no	—
83	C₂×C₄×D₇₉	SD₁₆	1	2	yes	—
84	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
85	Dic₃₁₆	—	2	2	no	—
86	Dic₃₁₆	—	2	2	no	—
87	Dic₃₁₆	—	4	1	no	—
88	Dic₃₁₆	—	4	1	no	—
89	#11929608	—	2	2	no	—
90	#11929608	—	4	1	no	—
91	#11929607	—	2	2	no	—
92	#11929607	—	2	2	no	—
93	#11929607	—	2	2	no	—
94	#11929607	—	2	2	no	—
95	#11929607	—	4	4	no	—
96	C₂×C₇₉⋊C₈	—	2	1	no	—
97	C₂×C₇₉⋊C₈	—	2	1	no	—
98	C₂×C₇₉⋊C₈	—	4	1	no	—
99	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
100	C₈×D₇₉	—	4	4	no	—
101	C₄⋊Dic₇₉	—	4	4	no	—
102	(C₂×D₇₉)⋊C₄	—	4	4	no	—
103	C₇₉⋊C₁₆	—	2	2	no	—
104	C₂²×C₄×C₇₉	—	4	1	no	—
105	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
106	M₁₆×C₇₉	—	2	2	no	—
107	M₁₆×C₇₉	—	4	1	no	—
108	C₂×C₈×C₇₉	—	4	1	no	—
109	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₇₉	—	2	2	no	—
116	C₄⋊C₄×C₇₉	—	4	1	no	—
117	C₁₆×C₇₉	—	2	2	no	—
118	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²×D₇₉	—	2	1	no	—
120	C₂²×D₇₉	—	2	1	no	—
121	C₂²×D₇₉	—	4	2	no	—
122	C₂²×D₇₉	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₇₉	—	4	2	no	—
124	C₂×C₄×C₇₉	—	8	2	no	—
125	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₇₉	—	2	1	no	—
132	C₂×Dic₇₉	—	2	1	no	—
133	C₂×Dic₇₉	—	4	2	no	—
134	C₂×Dic₇₉	—	8	1	no	—
135	C₂×Dic₇₉	—	8	1	no	—
136	C₂×Dic₇₉	C₂×SD₁₆	1	2	yes	—
137	C₄×D₇₉	—	2	2	no	—
138	C₄×D₇₉	—	2	2	no	—
139	C₄×D₇₉	—	2	2	no	—
140	C₄×D₇₉	—	2	2	no	—
141	Dic₁₅₈	—	4	2	no	—
142	Dic₁₅₈	—	4	2	no	—
143	Dic₁₅₈	—	8	1	no	—
144	Dic₁₅₈	—	8	1	no	—
145	C₈×C₇₉	—	4	2	no	—
146	C₈×C₇₉	#123	1	1	yes	characteristic
147	C₈×C₇₉	#123	1	1	yes	characteristic
148	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
149	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
150	C₇₉⋊C₈	—	2	2	no	—
151	C₇₉⋊C₈	—	4	2	no	—
152	C₂²×C₇₉	—	2	2	no	—
153	C₂²×C₇₉	#481	1	1	yes	characteristic
154	C₂²×C₇₉	#481	1	1	yes	characteristic
155	C₂²×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
156	C₂×D₇₉	—	2	2	no	—
157	C₂×D₇₉	—	4	2	no	—
158	C₂×D₇₉	—	4	2	no	—
159	C₂×D₇₉	—	8	1	no	—
160	C₄×C₇₉	—	8	2	no	—
161	C₄×C₇₉	#481	1	1	yes	characteristic
162	C₄×C₇₉	#481	1	1	yes	characteristic
163	C₄×C₇₉	C₂×#123	1	1	yes	characteristic
164	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	Dic₇₉	—	2	2	no	—
166	Dic₇₉	—	8	2	no	—
167	C₂×#2239	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	—	2	1	no	—
169	C₂×C₇₉	—	2	1	no	—
170	C₂×C₇₉	#2239	1	2	yes	complemented
171	C₂×C₇₉	C₂×#481	1	1	yes	characteristic
172	D₇₉	—	4	2	no	—
173	C₂×#470	—	79	1	no	—
174	C₂×#430	—	79	1	no	—
175	#2239	—	79	8	no	—
176	C₂×#365	—	79	1	no	—
177	C₂×#365	D₇₉	1	1	yes	characteristic, complemented
178	C₂×#362	—	79	1	no	—
179	C₂×#362	—	79	1	no	—
180	C₂×#349	—	79	1	no	—
181	C₇₉	C₂×#2239	1	1	yes	characteristic, complemented, Sylow
182	C₂²×Q₁₆	—	79	1	no	—
183	C₂²×M₁₆	—	79	1	no	—
184	C₂²×M₁₆	—	158	1	no	—
185	C₂×C₂²⋊Q₈	—	158	1	no	—
186	C₂×M₃₂	—	79	1	no	—
187	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
188	#470	—	79	4	no	—
189	#470	—	158	2	no	—
190	#430	—	79	4	no	—
191	#430	—	158	2	no	—
192	C₂×#108	—	79	1	no	—
193	C₂×#108	—	158	1	no	—
194	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
195	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
196	C₂×#103	—	158	1	no	—
197	#365	—	79	4	no	—
198	#365	C₂×D₇₉	1	4	yes	complemented
199	#362	—	79	4	no	—
200	#362	—	79	4	no	—
201	#349	—	79	4	no	—
202	C₂²×Q₈	—	158	1	no	—
203	C₂³×C₄	—	79	1	no	—
204	C₂×Q₁₆	—	79	2	no	—
205	C₂×Q₁₆	—	158	1	no	—
206	C₂×Q₁₆	—	158	1	no	—
207	C₂×Q₁₆	—	158	1	no	—
208	C₂×Q₁₆	—	158	1	no	—
209	C₂×Q₁₆	—	158	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	316	1	no	—
212	C₂×M₁₆	—	316	2	no	—
213	C₂×M₁₆	—	79	2	no	—
214	C₂×M₁₆	—	79	2	no	—
215	C₂×M₁₆	—	158	4	no	—
216	C₂×M₁₆	—	158	1	no	—
217	C₂×M₁₆	—	158	1	no	—
218	C₂×M₁₆	—	158	1	no	—
219	C₂×M₁₆	—	158	1	no	—
220	C₂×M₁₆	—	158	1	no	—
221	C₂²×C₈	—	316	1	no	—
222	C₂²×C₈	—	79	1	no	—
223	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
224	C₂²⋊Q₈	—	316	2	no	—
225	C₂²⋊Q₈	—	158	4	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	316	1	no	—
228	C₂×C₂²⋊C₄	—	316	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	79	2	no	—
231	M₃₂	—	158	1	no	—
232	M₃₂	C₂²×D₇₉	1	2	yes	—
233	C₂×C₁₆	—	2	1	no	—
234	C₂×C₁₆	—	158	1	no	—
235	#108	—	2	1	no	—
236	#108	—	79	2	no	—
237	#108	—	158	4	no	—
238	#108	—	158	1	no	—
239	#108	C₂²×D₇₉	1	2	yes	—
240	#106	—	2	1	no	—
241	#106	C₂²×D₇₉	1	2	yes	—
242	#103	—	158	4	no	—
243	C₂⁴	—	79	1	no	—
244	C₂×Q₈	—	316	1	no	—
245	C₂×Q₈	—	316	1	no	—
246	C₂×Q₈	—	316	1	no	—
247	C₂×Q₈	—	316	1	no	—
248	C₂×Q₈	—	316	1	no	—
249	C₂×Q₈	—	158	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	316	1	no	—
252	C₂²×C₄	—	79	1	no	—
253	C₂²×C₄	—	79	2	no	—
254	C₂²×C₄	—	79	2	no	—
255	C₂²×C₄	—	158	1	no	—
256	C₂²×C₄	—	158	1	no	—
257	C₂²×C₄	—	158	1	no	—
258	C₂²×C₄	—	158	1	no	—
259	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
260	Q₁₆	—	316	1	no	—
261	Q₁₆	—	316	1	no	—
262	Q₁₆	—	158	2	no	—
263	Q₁₆	—	158	2	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	316	4	no	—
267	M₁₆	—	316	1	no	—
268	M₁₆	—	158	2	no	—
269	M₁₆	—	158	2	no	—
270	M₁₆	—	158	2	no	—
271	M₁₆	—	158	2	no	—
272	M₁₆	—	158	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	316	4	no	—
275	C₂×C₈	—	316	1	no	—
276	C₂×C₈	—	79	2	no	—
277	C₂×C₈	—	158	1	no	—
278	C₂×C₈	—	158	1	no	—
279	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
280	C₂×C₈	#11929624	1	1	yes	characteristic
281	C₂×C₈	#11929624	1	1	yes	characteristic
282	C₂×C₈	#11929625	1	1	yes	characteristic
283	C₂×C₈	#11929625	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	316	4	no	—
288	C₂²⋊C₄	—	316	4	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	158	2	no	—
291	C₂³	—	316	2	no	—
292	C₂³	—	79	2	no	—
293	C₂³	—	158	1	no	—
294	C₂³	—	158	1	no	—
295	C₂³	SD₁₆×D₇₉	1	1	yes	characteristic
296	Q₈	—	316	2	no	—
297	Q₈	—	316	2	no	—
298	Q₈	—	632	1	no	—
299	Q₈	—	632	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	316	2	no	—
303	C₂×C₄	—	79	2	no	—
304	C₂×C₄	—	632	1	no	—
305	C₂×C₄	—	632	1	no	—
306	C₂×C₄	—	158	1	no	—
307	C₂×C₄	—	158	1	no	—
308	C₂×C₄	—	158	2	no	—
309	C₂×C₄	—	158	2	no	—
310	C₂×C₄	—	158	2	no	—
311	C₂×C₄	—	158	2	no	—
312	C₂×C₄	#14579146	1	1	yes	characteristic
313	C₂×C₄	#14579146	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
315	C₂×C₄	#14579188	1	1	yes	characteristic
316	C₂×C₄	#14579188	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	316	2	no	—
320	C₈	—	158	2	no	—
321	C₈	#14579146	1	1	yes	characteristic
322	C₈	#14579146	1	1	yes	characteristic
323	C₈	C₂×#11929624	1	1	yes	characteristic
324	C₈	C₂×#11929625	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	316	2	no	—
327	C₂²	—	316	2	no	—
328	C₂²	—	632	1	no	—
329	C₂²	—	158	2	no	—
330	C₂²	#19098883	1	1	yes	characteristic
331	C₂²	#19098883	1	1	yes	characteristic
332	C₂²	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
333	C₄	—	8	2	no	—
334	C₄	—	632	2	no	—
335	C₄	—	158	2	no	—
336	C₄	#19098883	1	1	yes	characteristic
337	C₄	#19098883	1	1	yes	characteristic
338	C₄	C₂×#14579146	1	1	yes	characteristic
339	C₄	C₂×#14579188	1	1	yes	characteristic
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	—	316	2	no	—
343	C₂	#23765621	1	2	yes	direct product factor
344	C₂	C₂×#19098883	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098883
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#365×C₇₉	C₂
socle	C₂²×C₇₉	#481
prime radical	C₂³×D₇₉	SD₁₆
hypercenter	C₂×#365	D₇₉
hypocenter	C₇₉	C₂×#2239
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
3. #23765621	C₂	direct (N × H)
4. C₂×#19098385	C₂	semidirect
5. C₂×#19098380	C₂	semidirect
6. C₂×#19098378	C₂	semidirect
8. C₂×#365×C₇₉	C₂	semidirect
21. #19098385	C₂²	semidirect
22. #19098380	C₂²	semidirect
23. #19098378	C₂²	semidirect
28. #365×C₇₉	C₂²	semidirect
170. C₂×C₇₉	#2239	semidirect
177. C₂×#365	D₇₉	semidirect
181. C₇₉	C₂×#2239	semidirect
198. #365	C₂×D₇₉	semidirect
343. C₂	#23765621	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×#19098822	2
C₂×#19098610	2
#23765621	2
C₂×#19098385	2
C₂×#19098380	2
C₂×#19098378	2
C₂×#19098344	2
C₂×#365×C₇₉	2
C₂×#2239	79

Subgroups — C2×#23765621 #34443161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765621 #34443161