Subgroups — C₂×#31771230 #46824279

Order 34432.

1296 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771230	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22575433	C₂	1	2	yes	maximal, complemented
3	C₂×#22575432	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#22575431	C₂	1	2	yes	maximal, complemented
5	C₂×#22575431	C₂	1	2	yes	maximal, complemented
6	C₂×#22575430	C₂	1	2	yes	maximal, complemented
7	C₂×#22575429	C₂	1	2	yes	maximal, complemented
8	C₂²×#16864379	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂²⋊C₄×D₂₆₉	C₂	1	2	yes	maximal
10	C₂×C₂²⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₂₆₉	C₂²	1	1	yes	characteristic
12	C₂²×Dic₅₃₈	C₂²	1	2	yes	complemented
13	C₂²×Dic₅₃₈	C₂²	1	1	yes	characteristic, complemented
14	C₂³×Dic₂₆₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₂₆₉	C₂²	1	2	yes	—
16	C₂²×C₄×D₂₆₉	C₂²	1	2	yes	—
17	#22575433	C₂²	1	16	yes	complemented
18	#22575432	C₂²	1	8	yes	complemented
19	#22575431	C₂²	1	16	yes	complemented
20	#22575431	C₂²	1	16	yes	complemented
21	#22575430	C₂²	1	16	yes	complemented
22	#22575429	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	complemented
24	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
25	C₂×#16864381	C₂²	1	2	yes	complemented
26	C₂×#16864381	C₂²	1	2	yes	complemented
27	C₂×#16864381	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
29	C₂×#16864379	C₂²	1	8	yes	—
30	C₂×#16864379	C₂²	1	2	yes	—
31	C₂×#16864379	C₂²	1	2	yes	—
32	C₂×#16864379	C₂²	1	2	yes	—
33	C₂×#16864379	C₂²	1	2	yes	—
34	C₂²⋊C₄×D₂₆₉	C₂²	1	16	yes	—
35	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
36	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₂₆₉	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	complemented
40	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
41	C₂³×D₂₆₉	—	4	2	no	—
42	C₂³×D₂₆₉	—	4	2	no	—
43	C₂³×D₂₆₉	Q₈	1	8	yes	complemented
44	C₂³×D₂₆₉	C₂³	1	2	yes	—
45	C₂³×D₂₆₉	C₂³	1	4	yes	—
46	C₂×Dic₅₃₈	—	4	2	no	—
47	C₂×Dic₅₃₈	—	4	4	no	—
48	C₂×Dic₅₃₈	C₂³	1	8	yes	complemented
49	C₂×Dic₅₃₈	C₂³	1	4	yes	complemented
50	C₂²×Dic₂₆₉	—	4	2	no	—
51	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
52	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
53	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
54	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
55	C₂×C₄×D₂₆₉	—	4	4	no	—
56	C₂×C₄×D₂₆₉	—	4	4	no	—
57	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
58	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
59	C₄⋊Dic₂₆₉	C₂³	1	8	yes	complemented
60	C₄⋊Dic₂₆₉	C₂³	1	4	yes	complemented
61	#16864381	C₂³	1	8	yes	complemented
62	#16864381	C₂³	1	8	yes	complemented
63	#16864381	C₂³	1	8	yes	complemented
64	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
65	#16864379	C₂³	1	8	yes	—
66	#16864379	C₂³	1	8	yes	—
67	#16864379	C₂³	1	8	yes	—
68	#16864379	C₂³	1	8	yes	—
69	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₂₆₉	—	4	2	no	—
71	C₂×Q₈×C₂₆₉	C₂³	1	4	yes	complemented
72	C₂²×C₄×C₂₆₉	—	4	2	no	—
73	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
76	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
77	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
78	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	complemented
79	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	complemented
80	C₂²⋊C₄×C₂₆₉	C₂³	1	8	yes	—
81	C₂²×D₂₆₉	—	2	4	no	—
82	C₂²×D₂₆₉	—	2	4	no	—
83	C₂²×D₂₆₉	—	4	16	no	—
84	C₂²×D₂₆₉	—	4	4	no	—
85	C₂²×D₂₆₉	—	4	4	no	—
86	C₂²×D₂₆₉	C₂×Q₈	1	8	yes	—
87	C₂²×D₂₆₉	C₂×Q₈	1	16	yes	complemented
88	C₂²×D₂₆₉	C₂⁴	1	4	yes	—
89	Dic₅₃₈	—	4	4	no	—
90	Dic₅₃₈	—	4	8	no	—
91	C₂×Dic₂₆₉	—	2	4	no	—
92	C₂×Dic₂₆₉	—	2	4	no	—
93	C₂×Dic₂₆₉	—	2	4	no	—
94	C₂×Dic₂₆₉	—	4	4	no	—
95	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
96	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
97	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
98	C₄×D₂₆₉	—	4	8	no	—
99	C₄×D₂₆₉	—	4	8	no	—
100	C₂³×C₂₆₉	—	2	2	no	—
101	C₂³×C₂₆₉	—	2	2	no	—
102	C₂³×C₂₆₉	C₂×Q₈	1	4	yes	—
103	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
104	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
105	Q₈×C₂₆₉	—	4	4	no	—
106	C₂×C₄×C₂₆₉	—	2	2	no	—
107	C₂×C₄×C₂₆₉	—	2	2	no	—
108	C₂×C₄×C₂₆₉	—	2	4	no	—
109	C₂×C₄×C₂₆₉	—	2	4	no	—
110	C₂×C₄×C₂₆₉	—	4	4	no	—
111	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
112	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
113	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
114	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
115	C₂²×C₂₆₉	—	2	8	no	—
116	C₂²×C₂₆₉	—	2	2	no	—
117	C₂²×C₂₆₉	—	2	2	no	—
118	C₂²×C₂₆₉	C₂²×Q₈	1	4	yes	—
119	C₂²×C₂₆₉	C₂²×Q₈	1	2	yes	—
120	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
123	C₂×D₂₆₉	—	2	16	no	—
124	C₂×D₂₆₉	—	2	4	no	—
125	C₂×D₂₆₉	—	2	4	no	—
126	C₂×D₂₆₉	—	4	16	no	—
127	C₂×D₂₆₉	C₂²×Q₈	1	8	yes	—
128	C₄×C₂₆₉	—	2	2	no	—
129	C₄×C₂₆₉	—	2	2	no	—
130	C₄×C₂₆₉	—	2	4	no	—
131	C₄×C₂₆₉	—	2	4	no	—
132	Dic₂₆₉	—	2	4	no	—
133	Dic₂₆₉	—	2	4	no	—
134	Dic₂₆₉	—	2	4	no	—
135	C₂×C₂₆₉	—	2	4	no	—
136	C₂×C₂₆₉	#543	1	4	yes	complemented
137	C₂×C₂₆₉	C₂³×Q₈	1	1	yes	characteristic
138	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	D₂₆₉	—	2	8	no	—
141	C₂₆₉	C₂×#543	1	1	yes	characteristic, complemented, Sylow
142	C₂×#543	—	269	1	no	maximal, Sylow
143	#543	—	269	16	no	—
144	C₂×C₂²⋊Q₈	—	269	2	no	—
145	C₂×C₂²⋊Q₈	—	269	1	no	—
146	C₂×C₂²⋊Q₈	—	269	2	no	—
147	C₂×C₂²⋊Q₈	—	269	2	no	—
148	C₂×C₂²⋊Q₈	—	269	2	no	—
149	C₂×C₂²⋊Q₈	—	269	2	no	—
150	C₂×C₂²⋊Q₈	D₂₆₉	1	1	yes	characteristic, complemented
151	C₂²×C₂²⋊C₄	—	269	1	no	—
152	C₂²×C₂²⋊C₄	—	269	2	no	—
153	C₂⁵	—	269	1	no	—
154	C₂²×Q₈	—	269	1	no	—
155	C₂²×Q₈	—	269	2	no	—
156	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
157	C₂³×C₄	—	269	1	no	—
158	C₂³×C₄	—	269	2	no	—
159	C₂³×C₄	—	269	2	no	—
160	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
161	C₂²⋊Q₈	—	269	16	no	—
162	C₂²⋊Q₈	—	269	8	no	—
163	C₂²⋊Q₈	—	269	16	no	—
164	C₂²⋊Q₈	—	269	16	no	—
165	C₂²⋊Q₈	—	269	16	no	—
166	C₂²⋊Q₈	—	269	16	no	—
167	C₂²⋊Q₈	C₂×D₂₆₉	1	8	yes	complemented
168	C₂×C₄⋊C₄	—	269	2	no	—
169	C₂×C₄⋊C₄	—	269	2	no	—
170	C₂×C₄⋊C₄	—	269	2	no	—
171	C₂×C₄⋊C₄	—	269	1	no	—
172	C₂×C₄⋊C₄	—	269	2	no	—
173	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic, complemented
174	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	complemented
175	C₂×C₂²⋊C₄	—	269	2	no	—
176	C₂×C₂²⋊C₄	—	269	2	no	—
177	C₂×C₂²⋊C₄	—	269	2	no	—
178	C₂×C₂²⋊C₄	—	269	2	no	—
179	C₂×C₂²⋊C₄	—	269	2	no	—
180	C₂×C₂²⋊C₄	—	269	8	no	—
181	C₂×C₂²⋊C₄	—	269	16	no	—
182	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
183	C₂⁴	—	269	2	no	—
184	C₂⁴	—	269	8	no	—
185	C₂⁴	—	269	4	no	—
186	C₂⁴	—	1076	2	no	—
187	C₂⁴	—	1076	2	no	—
188	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	269	4	no	—
191	C₂×Q₈	—	269	8	no	—
192	C₂×Q₈	—	1076	2	no	—
193	C₂×Q₈	—	1076	4	no	—
194	C₂×Q₈	C₂²×D₂₆₉	1	4	yes	complemented
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	269	2	no	—
197	C₂²×C₄	—	269	2	no	—
198	C₂²×C₄	—	269	2	no	—
199	C₂²×C₄	—	269	4	no	—
200	C₂²×C₄	—	269	8	no	—
201	C₂²×C₄	—	269	8	no	—
202	C₂²×C₄	—	1076	4	no	—
203	C₂²×C₄	—	1076	4	no	—
204	C₂²×C₄	—	1076	2	no	—
205	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
208	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
209	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
210	C₄⋊C₄	—	269	8	no	—
211	C₄⋊C₄	—	269	8	no	—
212	C₄⋊C₄	—	269	8	no	—
213	C₄⋊C₄	—	269	4	no	—
214	C₄⋊C₄	—	269	8	no	—
215	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	complemented
216	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	complemented
217	C₂²⋊C₄	—	269	8	no	—
218	C₂²⋊C₄	—	269	8	no	—
219	C₂²⋊C₄	—	269	8	no	—
220	C₂²⋊C₄	—	269	8	no	—
221	C₂²⋊C₄	—	269	8	no	—
222	C₂²⋊C₄	C₂²×D₂₆₉	1	8	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	269	8	no	—
226	C₂³	—	269	4	no	—
227	C₂³	—	269	16	no	—
228	C₂³	—	538	4	no	—
229	C₂³	—	538	4	no	—
230	C₂³	—	1076	4	no	—
231	C₂³	—	1076	4	no	—
232	C₂³	—	1076	16	no	—
233	C₂³	Q₈×D₂₆₉	1	4	yes	—
234	C₂³	C₂³×D₂₆₉	1	2	yes	—
235	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	1076	4	no	—
238	Q₈	—	1076	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	269	4	no	—
245	C₂×C₄	—	269	4	no	—
246	C₂×C₄	—	269	4	no	—
247	C₂×C₄	—	538	4	no	—
248	C₂×C₄	—	538	4	no	—
249	C₂×C₄	—	538	4	no	—
250	C₂×C₄	—	1076	8	no	—
251	C₂×C₄	—	1076	8	no	—
252	C₂×C₄	—	1076	4	no	—
253	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
254	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
255	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
256	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	269	8	no	—
261	C₂²	—	538	4	no	—
262	C₂²	—	538	4	no	—
263	C₂²	—	538	16	no	—
264	C₂²	—	1076	16	no	—
265	C₂²	C₂×Q₈×D₂₆₉	1	4	yes	—
266	C₂²	C₂×Q₈×D₂₆₉	1	2	yes	—
267	C₂²	#22575498	1	2	yes	—
268	C₂²	#22575499	1	2	yes	—
269	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	538	4	no	—
275	C₄	—	538	4	no	—
276	C₄	—	538	4	no	—
277	C₂	—	2	4	no	—
278	C₂	—	538	8	no	—
279	C₂	#31771230	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₂₆₉	1	1	yes	characteristic
281	C₂	C₂×#22575498	1	1	yes	characteristic
282	C₂	C₂×#22575499	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×C₂²⋊Q₈×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂⁴×D₂₆₉	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	C₂×#543
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #31771230	C₂	direct (N × H)
2. C₂×#22575433	C₂	semidirect
3. C₂×#22575432	C₂	semidirect
4. C₂×#22575431	C₂	semidirect
5. C₂×#22575431	C₂	semidirect
6. C₂×#22575430	C₂	semidirect
7. C₂×#22575429	C₂	semidirect
10. C₂×C₂²⋊Q₈×C₂₆₉	C₂	semidirect
12. C₂²×Dic₅₃₈	C₂²	semidirect
13. C₂²×Dic₅₃₈	C₂²	semidirect
17. #22575433	C₂²	semidirect
18. #22575432	C₂²	semidirect
19. #22575431	C₂²	semidirect
20. #22575431	C₂²	semidirect
21. #22575430	C₂²	semidirect
22. #22575429	C₂²	semidirect
23. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
24. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
25. C₂×#16864381	C₂²	semidirect
26. C₂×#16864381	C₂²	semidirect
27. C₂×#16864381	C₂²	semidirect
35. C₂²×Q₈×C₂₆₉	C₂²	semidirect
37. C₂²⋊Q₈×C₂₆₉	C₂²	semidirect
38. C₂×C₄⋊C₄×C₂₆₉	C₂²	semidirect
39. C₂×C₄⋊C₄×C₂₆₉	C₂²	semidirect
43. C₂³×D₂₆₉	Q₈	semidirect
48. C₂×Dic₅₃₈	C₂³	semidirect
49. C₂×Dic₅₃₈	C₂³	semidirect
59. C₄⋊Dic₂₆₉	C₂³	semidirect
60. C₄⋊Dic₂₆₉	C₂³	semidirect
61. #16864381	C₂³	semidirect
62. #16864381	C₂³	semidirect
63. #16864381	C₂³	semidirect
71. C₂×Q₈×C₂₆₉	C₂³	semidirect
78. C₄⋊C₄×C₂₆₉	C₂³	semidirect
79. C₄⋊C₄×C₂₆₉	C₂³	semidirect
87. C₂²×D₂₆₉	C₂×Q₈	semidirect
136. C₂×C₂₆₉	#543	semidirect
141. C₂₆₉	C₂×#543	semidirect
150. C₂×C₂²⋊Q₈	D₂₆₉	semidirect
156. C₂²×Q₈	C₂×D₂₆₉	semidirect
167. C₂²⋊Q₈	C₂×D₂₆₉	semidirect
173. C₂×C₄⋊C₄	C₂×D₂₆₉	semidirect
174. C₂×C₄⋊C₄	C₂×D₂₆₉	semidirect
194. C₂×Q₈	C₂²×D₂₆₉	semidirect
215. C₄⋊C₄	C₂²×D₂₆₉	semidirect
216. C₄⋊C₄	C₂²×D₂₆₉	semidirect
279. C₂	#31771230	direct (N × H)

Maximal subgroups

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

subgroup	index
#31771230	2
C₂×#22575433	2
C₂×#22575432	2
C₂×#22575431	2
C₂×#22575430	2
C₂×#22575429	2
C₂²×#16864379	2
C₂×C₂²⋊C₄×D₂₆₉	2
C₂×C₂²⋊Q₈×C₂₆₉	2
C₂×#543	269

Subgroups — C2×#31771230 #46824279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771230 #46824279