Subgroups — C₂×#31771285 #46824386

Order 34432.

1328 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771285	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22575446	C₂	1	2	yes	maximal, complemented
3	C₂×#22575435	C₂	1	2	yes	maximal, complemented
4	C₂×#22575434	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#22575429	C₂	1	2	yes	maximal, complemented
6	C₂×#22575421	C₂	1	1	yes	maximal, characteristic
7	C₂×#22575413	C₂	1	2	yes	maximal, complemented
8	C₂×#22575410	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₄×D₂₆₉	C₂	1	2	yes	maximal, complemented
10	C₂×#124×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₂₆₉	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₂₆₉	C₂²	1	2	yes	—
13	C₂³×Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×D₂₆₉	C₂²	1	2	yes	—
15	C₂²×C₄×D₂₆₉	C₂²	1	2	yes	—
16	#22575446	C₂²	1	16	yes	complemented
17	#22575435	C₂²	1	16	yes	complemented
18	#22575434	C₂²	1	8	yes	complemented
19	#22575429	C₂²	1	16	yes	complemented
20	#22575421	C₂²	1	8	yes	—
21	#22575413	C₂²	1	16	yes	complemented
22	#22575410	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	complemented
24	C₂×#16864381	C₂²	1	2	yes	complemented
25	C₂×#16864381	C₂²	1	2	yes	complemented
26	C₂×#16864381	C₂²	1	2	yes	complemented
27	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
28	C₂×C₂²⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₂₆₉	C₂²	1	1	yes	characteristic, complemented
30	C₂×#16864379	C₂²	1	2	yes	—
31	C₂×#16864379	C₂²	1	2	yes	—
32	C₂×#16864379	C₂²	1	2	yes	—
33	C₂²⋊C₄×D₂₆₉	C₂²	1	16	yes	complemented
34	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	complemented
35	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
36	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
37	#124×C₂₆₉	C₂²	1	8	yes	complemented
38	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	complemented
40	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₂₆₉	—	2	8	no	—
42	C₂³×D₂₆₉	—	4	2	no	—
43	C₂³×D₂₆₉	C₂³	1	4	yes	—
44	C₂³×D₂₆₉	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₂₆₉	—	2	8	no	—
46	C₂×C₂²⋊D₂₆₉	C₂³	1	8	yes	—
47	C₂²×Dic₂₆₉	—	4	2	no	—
48	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
49	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
50	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
51	C₂²×Dic₂₆₉	C₂³	1	4	yes	complemented
52	C₂×C₄×D₂₆₉	—	2	8	no	—
53	C₂×C₄×D₂₆₉	—	4	4	no	—
54	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
55	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
56	C₄⋊Dic₂₆₉	C₂³	1	8	yes	complemented
57	#16864381	C₂³	1	8	yes	complemented
58	#16864381	C₂³	1	8	yes	complemented
59	#16864381	C₂³	1	8	yes	complemented
60	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
61	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	—
62	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	complemented
63	#16864379	C₂³	1	8	yes	—
64	#16864379	C₂³	1	8	yes	—
65	#16864379	C₂³	1	8	yes	—
66	C₄×Dic₂₆₉	C₂³	1	8	yes	complemented
67	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
68	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
69	C₂×Q₈×C₂₆₉	—	4	2	no	—
70	C₂×Q₈×C₂₆₉	C₂³	1	4	yes	complemented
71	C₂×D₄×C₂₆₉	—	4	2	no	—
72	C₂×D₄×C₂₆₉	C₂³	1	4	yes	—
73	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
74	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
76	C₂²⋊C₄×C₂₆₉	C₂³	1	8	yes	—
77	C₂²⋊C₄×C₂₆₉	C₂³	1	8	yes	complemented
78	C₄²×C₂₆₉	C₂³	1	4	yes	complemented
79	C₂²×D₂₆₉	—	2	4	no	—
80	C₂²×D₂₆₉	—	2	4	no	—
81	C₂²×D₂₆₉	—	2	16	no	—
82	C₂²×D₂₆₉	—	4	4	no	—
83	C₂²×D₂₆₉	—	4	16	no	—
84	C₂²×D₂₆₉	#41	1	8	yes	—
85	C₂²×D₂₆₉	C₂⁴	1	4	yes	—
86	C₂²⋊D₂₆₉	—	2	16	no	—
87	C₂×Dic₂₆₉	—	2	4	no	—
88	C₂×Dic₂₆₉	—	2	4	no	—
89	C₂×Dic₂₆₉	—	2	4	no	—
90	C₂×Dic₂₆₉	—	4	4	no	—
91	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
92	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
93	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
94	C₄×D₂₆₉	—	2	16	no	—
95	C₄×D₂₆₉	—	4	8	no	—
96	C₂³×C₂₆₉	—	2	4	no	—
97	C₂³×C₂₆₉	—	2	4	no	—
98	C₂³×C₂₆₉	—	2	2	no	—
99	C₂³×C₂₆₉	—	2	2	no	—
100	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
101	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
102	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
103	Q₈×C₂₆₉	—	4	4	no	—
104	D₄×C₂₆₉	—	4	4	no	—
105	C₂×C₄×C₂₆₉	—	2	4	no	—
106	C₂×C₄×C₂₆₉	—	2	2	no	—
107	C₂×C₄×C₂₆₉	—	2	4	no	—
108	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
109	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
110	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
111	C₂²×C₂₆₉	—	2	4	no	—
112	C₂²×C₂₆₉	—	2	4	no	—
113	C₂²×C₂₆₉	—	2	2	no	—
114	C₂²×C₂₆₉	—	2	2	no	—
115	C₂²×C₂₆₉	—	4	4	no	—
116	C₂²×C₂₆₉	—	4	4	no	—
117	C₂²×C₂₆₉	C₂×#41	1	4	yes	—
118	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
120	C₂×D₂₆₉	—	2	16	no	—
121	C₂×D₂₆₉	—	2	4	no	—
122	C₂×D₂₆₉	—	2	4	no	—
123	C₂×D₂₆₉	—	4	16	no	—
124	C₂×D₂₆₉	C₂×#41	1	8	yes	—
125	C₄×C₂₆₉	—	2	4	no	—
126	C₄×C₂₆₉	—	2	2	no	—
127	C₄×C₂₆₉	—	2	4	no	—
128	Dic₂₆₉	—	2	4	no	—
129	Dic₂₆₉	—	2	4	no	—
130	Dic₂₆₉	—	2	4	no	—
131	C₂×C₂₆₉	—	4	2	no	—
132	C₂×C₂₆₉	—	4	2	no	—
133	C₂×C₂₆₉	#551	1	4	yes	complemented
134	C₂×C₂₆₉	C₂²×#41	1	2	yes	—
135	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
136	D₂₆₉	—	2	8	no	—
137	C₂₆₉	C₂×#551	1	1	yes	characteristic, complemented, Sylow
138	C₂×#551	—	269	1	no	maximal, Sylow
139	#551	—	269	16	no	—
140	C₂×#126	—	269	2	no	—
141	C₂×#124	D₂₆₉	1	1	yes	characteristic, complemented
142	C₂×#123	—	269	1	no	—
143	C₂×#123	—	269	2	no	—
144	C₂×C₂²⋊Q₈	—	269	2	no	—
145	C₂×#120	—	269	1	no	—
146	C₂×C₄×D₄	—	269	2	no	—
147	C₂×#117	—	269	2	no	—
148	C₂²×C₂²⋊C₄	—	269	2	no	—
149	C₂⁵	—	269	1	no	—
150	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic, complemented
151	C₂²×D₄	—	269	2	no	—
152	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
153	C₂³×C₄	—	269	1	no	—
154	C₂³×C₄	—	269	2	no	—
155	C₂³×C₄	—	269	2	no	—
156	#126	—	269	16	no	—
157	#124	C₂×D₂₆₉	1	8	yes	complemented
158	#123	—	269	8	no	—
159	#123	—	269	16	no	—
160	C₂²⋊Q₈	—	269	16	no	—
161	#120	—	269	8	no	—
162	C₄×D₄	—	269	16	no	—
163	#117	—	269	16	no	—
164	C₂×C₄⋊C₄	—	269	2	no	—
165	C₂×C₄⋊C₄	—	269	2	no	—
166	C₂×C₄⋊C₄	—	269	2	no	—
167	C₂×C₄⋊C₄	—	269	2	no	—
168	C₂×C₂²⋊C₄	—	269	2	no	—
169	C₂×C₂²⋊C₄	—	269	1	no	—
170	C₂×C₂²⋊C₄	—	269	1	no	—
171	C₂×C₂²⋊C₄	—	269	2	no	—
172	C₂×C₂²⋊C₄	—	269	2	no	—
173	C₂×C₂²⋊C₄	—	269	2	no	—
174	C₂×C₂²⋊C₄	—	269	16	no	—
175	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	complemented
177	C₂×C₄²	—	269	2	no	—
178	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic, complemented
179	C₂⁴	—	269	2	no	—
180	C₂⁴	—	269	4	no	—
181	C₂⁴	—	538	8	no	—
182	C₂⁴	—	1076	2	no	—
183	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
184	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	C₂²×D₂₆₉	1	4	yes	complemented
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	269	8	no	—
189	C₂×D₄	—	538	8	no	—
190	C₂×D₄	C₂²×D₂₆₉	1	4	yes	—
191	C₂²×C₄	—	269	2	no	—
192	C₂²×C₄	—	269	2	no	—
193	C₂²×C₄	—	269	2	no	—
194	C₂²×C₄	—	269	4	no	—
195	C₂²×C₄	—	269	8	no	—
196	C₂²×C₄	—	269	8	no	—
197	C₂²×C₄	—	538	8	no	—
198	C₂²×C₄	—	1076	4	no	—
199	C₂²×C₄	—	1076	2	no	—
200	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
201	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
203	C₄⋊C₄	—	269	8	no	—
204	C₄⋊C₄	—	269	8	no	—
205	C₄⋊C₄	—	269	8	no	—
206	C₄⋊C₄	—	269	8	no	—
207	C₂²⋊C₄	—	269	8	no	—
208	C₂²⋊C₄	—	269	4	no	—
209	C₂²⋊C₄	—	269	4	no	—
210	C₂²⋊C₄	—	269	8	no	—
211	C₂²⋊C₄	—	269	8	no	—
212	C₂²⋊C₄	—	269	8	no	—
213	C₂²⋊C₄	C₂²×D₂₆₉	1	8	yes	—
214	C₂²⋊C₄	C₂²×D₂₆₉	1	8	yes	complemented
215	C₄²	—	269	8	no	—
216	C₄²	C₂²×D₂₆₉	1	4	yes	complemented
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	269	8	no	—
222	C₂³	—	269	4	no	—
223	C₂³	—	538	16	no	—
224	C₂³	—	538	4	no	—
225	C₂³	—	538	4	no	—
226	C₂³	—	1076	4	no	—
227	C₂³	—	1076	16	no	—
228	C₂³	C₂³×D₂₆₉	1	2	yes	—
229	C₂³	C₂³×D₂₆₉	1	2	yes	—
230	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
231	Q₈	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	538	16	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	269	4	no	—
238	C₂×C₄	—	269	4	no	—
239	C₂×C₄	—	269	4	no	—
240	C₂×C₄	—	538	16	no	—
241	C₂×C₄	—	538	4	no	—
242	C₂×C₄	—	538	4	no	—
243	C₂×C₄	—	538	4	no	—
244	C₂×C₄	—	1076	8	no	—
245	C₂×C₄	—	1076	4	no	—
246	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
247	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
248	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	269	8	no	—
256	C₂²	—	538	4	no	—
257	C₂²	—	538	4	no	—
258	C₂²	—	538	16	no	—
259	C₂²	—	1076	16	no	—
260	C₂²	#41×D₂₆₉	1	4	yes	—
261	C₂²	#22575498	1	2	yes	—
262	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	538	4	no	—
267	C₄	—	538	4	no	—
268	C₄	—	538	4	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	2	no	—
271	C₂	—	538	8	no	—
272	C₂	#31771285	1	4	yes	direct product factor
273	C₂	C₂×#41×D₂₆₉	1	2	yes	—
274	C₂	C₂×#22575498	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×#124×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂×#22575421	C₂
hypercenter	C₂×#124	D₂₆₉
hypocenter	C₂₆₉	C₂×#551
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. #31771285	C₂	direct (N × H)
2. C₂×#22575446	C₂	semidirect
3. C₂×#22575435	C₂	semidirect
4. C₂×#22575434	C₂	semidirect
5. C₂×#22575429	C₂	semidirect
7. C₂×#22575413	C₂	semidirect
8. C₂×#22575410	C₂	semidirect
9. C₂×C₂²⋊C₄×D₂₆₉	C₂	semidirect
10. C₂×#124×C₂₆₉	C₂	semidirect
13. C₂³×Dic₂₆₉	C₂²	semidirect
16. #22575446	C₂²	semidirect
17. #22575435	C₂²	semidirect
18. #22575434	C₂²	semidirect
19. #22575429	C₂²	semidirect
21. #22575413	C₂²	semidirect
22. #22575410	C₂²	semidirect
23. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
24. C₂×#16864381	C₂²	semidirect
25. C₂×#16864381	C₂²	semidirect
26. C₂×#16864381	C₂²	semidirect
29. C₂×C₂²⋊Dic₂₆₉	C₂²	semidirect
33. C₂²⋊C₄×D₂₆₉	C₂²	semidirect
34. C₂×C₄×Dic₂₆₉	C₂²	semidirect
35. C₂²×Q₈×C₂₆₉	C₂²	semidirect
37. #124×C₂₆₉	C₂²	semidirect
39. C₂×C₂²⋊C₄×C₂₆₉	C₂²	semidirect
40. C₂×C₄²×C₂₆₉	C₂²	semidirect
51. C₂²×Dic₂₆₉	C₂³	semidirect
56. C₄⋊Dic₂₆₉	C₂³	semidirect
57. #16864381	C₂³	semidirect
58. #16864381	C₂³	semidirect
59. #16864381	C₂³	semidirect
62. C₂²⋊Dic₂₆₉	C₂³	semidirect
66. C₄×Dic₂₆₉	C₂³	semidirect
70. C₂×Q₈×C₂₆₉	C₂³	semidirect
77. C₂²⋊C₄×C₂₆₉	C₂³	semidirect
78. C₄²×C₂₆₉	C₂³	semidirect
133. C₂×C₂₆₉	#551	semidirect
137. C₂₆₉	C₂×#551	semidirect
141. C₂×#124	D₂₆₉	semidirect
150. C₂²×Q₈	C₂×D₂₆₉	semidirect
157. #124	C₂×D₂₆₉	semidirect
176. C₂×C₂²⋊C₄	C₂×D₂₆₉	semidirect
178. C₂×C₄²	C₂×D₂₆₉	semidirect
186. C₂×Q₈	C₂²×D₂₆₉	semidirect
214. C₂²⋊C₄	C₂²×D₂₆₉	semidirect
216. C₄²	C₂²×D₂₆₉	semidirect
272. C₂	#31771285	direct (N × H)

Maximal subgroups

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

subgroup	index
#31771285	2
C₂×#22575446	2
C₂×#22575435	2
C₂×#22575434	2
C₂×#22575429	2
C₂×#22575421	2
C₂×#22575413	2
C₂×#22575410	2
C₂×C₂²⋊C₄×D₂₆₉	2
C₂×#124×C₂₆₉	2
C₂×#551	269

Subgroups — C2×#31771285 #46824386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771285 #46824386