Subgroups — C₂×#31771408 #46825636

Order 34432.

1512 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22575499	C₂	1	1	yes	maximal, characteristic, complemented
2	#31771408	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22575479	C₂	1	3	yes	maximal, complemented
4	C₂×#22575478	C₂	1	3	yes	maximal, complemented
5	C₂×#22575473	C₂	1	3	yes	maximal, complemented
6	C₂²×#16864387	C₂	1	3	yes	maximal, complemented
7	C₂×#22575460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#22575499	—	2	4	no	—
10	#22575499	C₂²	1	8	yes	complemented
11	C₂×#16864403	—	2	1	no	—
12	C₂×#16864402	C₂²	1	3	yes	complemented
13	C₂²×D₁₀₇₆	C₂²	1	3	yes	complemented
14	C₂×D₄×D₂₆₉	—	2	3	no	—
15	#22575479	C₂²	1	24	yes	complemented
16	#22575478	C₂²	1	24	yes	complemented
17	#22575473	C₂²	1	24	yes	complemented
18	C₂×#16864392	C₂²	1	1	yes	characteristic, complemented
19	C₂×#16864390	C₂²	1	3	yes	complemented
20	C₂×#16864389	C₂²	1	3	yes	complemented
21	C₂×#16864387	C₂²	1	6	yes	complemented
22	C₂×#16864387	C₂²	1	3	yes	complemented
23	C₂×#16864387	C₂²	1	24	yes	complemented
24	#22575460	C₂²	1	8	yes	complemented
25	C₂×#16864386	C₂²	1	3	yes	complemented
26	C₂²×#14524130	C₂²	1	3	yes	complemented
27	H(5,𝔽₂)×C₂₆₉	—	2	4	no	—
28	H(5,𝔽₂)×C₂₆₉	C₂²	1	8	yes	complemented
29	C₂×#41×C₂₆₉	—	2	1	no	—
30	C₂×#41×C₂₆₉	C₂²	1	1	yes	characteristic
31	C₂×#41×C₂₆₉	C₂²	1	3	yes	complemented
32	C₂²×D₄×C₂₆₉	—	2	3	no	—
33	C₂²×D₄×C₂₆₉	C₂²	1	3	yes	complemented
34	C₂³×D₂₆₉	—	2	3	no	—
35	#16864403	—	2	8	no	—
36	#16864402	—	2	6	no	—
37	#16864402	C₂³	1	12	yes	complemented
38	C₂×Dic₅₃₈	C₂³	1	1	yes	characteristic
39	C₂×D₁₀₇₆	—	2	3	no	—
40	C₂×D₁₀₇₆	—	2	6	no	—
41	C₂×D₁₀₇₆	C₂³	1	3	yes	—
42	C₂×D₁₀₇₆	C₂³	1	12	yes	complemented
43	C₂×C₂²⋊D₂₆₉	—	2	3	no	—
44	D₄×D₂₆₉	—	2	24	no	—
45	C₂×C₄×D₂₆₉	—	2	3	no	—
46	#16864392	C₂³	1	4	yes	complemented
47	#16864390	C₂³	1	12	yes	complemented
48	#16864389	C₂³	1	12	yes	complemented
49	#16864387	C₂³	1	12	yes	complemented
50	#16864387	C₂³	1	24	yes	complemented
51	#16864386	C₂³	1	12	yes	complemented
52	C₂×#14524130	C₂³	1	1	yes	characteristic
53	C₂×#14524130	C₂³	1	3	yes	—
54	C₂×#14524130	C₂³	1	12	yes	complemented
55	C₂⁴×C₂₆₉	—	2	3	no	—
56	#41×C₂₆₉	—	2	8	no	—
57	#41×C₂₆₉	—	2	6	no	—
58	#41×C₂₆₉	D₄	1	4	yes	—
59	#41×C₂₆₉	C₂³	1	4	yes	—
60	#41×C₂₆₉	C₂³	1	12	yes	complemented
61	C₂×Q₈×C₂₆₉	D₄	1	1	yes	characteristic
62	C₂×Q₈×C₂₆₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₂₆₉	—	2	3	no	—
64	C₂×D₄×C₂₆₉	—	2	24	no	—
65	C₂×D₄×C₂₆₉	—	2	3	no	—
66	C₂×D₄×C₂₆₉	—	2	6	no	—
67	C₂×D₄×C₂₆₉	D₄	1	3	yes	—
68	C₂×D₄×C₂₆₉	C₂³	1	3	yes	—
69	C₂×D₄×C₂₆₉	C₂³	1	12	yes	complemented
70	C₂²×C₄×C₂₆₉	—	2	3	no	—
71	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
72	C₂²×D₂₆₉	—	2	3	no	—
73	C₂²×D₂₆₉	—	2	12	no	—
74	C₂²×D₂₆₉	—	8	6	no	—
75	Dic₅₃₈	—	2	1	no	—
76	Dic₅₃₈	C₂⁴	1	2	yes	—
77	D₁₀₇₆	—	2	12	no	—
78	D₁₀₇₆	—	2	3	no	—
79	D₁₀₇₆	C₂⁴	1	6	yes	—
80	C₂²⋊D₂₆₉	—	2	12	no	—
81	C₂×Dic₂₆₉	—	2	1	no	—
82	C₄×D₂₆₉	—	2	12	no	—
83	#14524130	C₂⁴	1	2	yes	—
84	#14524130	C₂⁴	1	6	yes	—
85	C₂³×C₂₆₉	—	2	3	no	—
86	C₂³×C₂₆₉	—	2	12	no	—
87	C₂³×C₂₆₉	—	8	6	no	—
88	C₂³×C₂₆₉	C₂×D₄	1	3	yes	—
89	Q₈×C₂₆₉	—	2	1	no	—
90	Q₈×C₂₆₉	C₂×D₄	1	4	yes	—
91	Q₈×C₂₆₉	C₂⁴	1	2	yes	—
92	D₄×C₂₆₉	—	2	12	no	—
93	D₄×C₂₆₉	—	2	3	no	—
94	D₄×C₂₆₉	—	2	12	no	—
95	D₄×C₂₆₉	C₂×D₄	1	12	yes	—
96	D₄×C₂₆₉	C₂⁴	1	6	yes	—
97	C₂×C₄×C₂₆₉	—	2	1	no	—
98	C₂×C₄×C₂₆₉	—	2	12	no	—
99	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
100	C₂×C₄×C₂₆₉	C₂×D₄	1	6	yes	—
101	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
102	C₂×C₄×C₂₆₉	C₂⁴	1	6	yes	—
103	C₂²×C₂₆₉	—	2	6	no	—
104	C₂²×C₂₆₉	—	2	6	no	—
105	C₂²×C₂₆₉	—	4	6	no	—
106	C₂²×C₂₆₉	—	8	12	no	—
107	C₂²×C₂₆₉	C₂²×D₄	1	6	yes	—
108	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×D₂₆₉	—	2	6	no	—
110	C₂×D₂₆₉	—	4	6	no	—
111	C₂×D₂₆₉	—	8	12	no	—
112	C₄×C₂₆₉	—	2	2	no	—
113	C₄×C₂₆₉	C₂²×D₄	1	6	yes	—
114	C₄×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₂₆₉	C₂⁵	1	1	yes	characteristic
116	Dic₂₆₉	—	2	2	no	—
117	C₂×C₂₆₉	—	2	6	no	—
118	C₂×C₂₆₉	—	4	6	no	—
119	C₂×C₂₆₉	#576	1	2	yes	complemented
120	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
121	D₂₆₉	—	4	6	no	—
122	C₂₆₉	C₂×#576	1	1	yes	characteristic, complemented, Sylow
123	C₂×#576	—	269	1	no	maximal, Sylow
124	C₂×H(5,𝔽₂)	—	269	1	no	—
125	C₂×H(5,𝔽₂)	D₂₆₉	1	1	yes	characteristic, complemented
126	#576	—	269	16	no	—
127	C₂×AGL(1,ℤ/8)	—	269	3	no	—
128	C₂×AGL(1,ℤ/8)	—	269	3	no	—
129	C₂×#135	—	269	3	no	—
130	C₂²×D₈	—	269	3	no	—
131	C₂×#131	—	269	1	no	—
132	H(5,𝔽₂)	—	2	4	no	—
133	H(5,𝔽₂)	—	269	8	no	—
134	H(5,𝔽₂)	—	538	4	no	—
135	H(5,𝔽₂)	C₂×D₂₆₉	1	8	yes	complemented
136	C₂×#41	—	2	1	no	—
137	C₂×#41	—	269	3	no	—
138	C₂×#41	—	538	1	no	—
139	C₂×#41	C₂×D₂₆₉	1	1	yes	characteristic
140	C₂×#41	C₂×D₂₆₉	1	3	yes	complemented
141	C₂²×D₄	—	2	3	no	—
142	C₂²×D₄	—	269	3	no	—
143	C₂²×D₄	—	538	3	no	—
144	C₂²×D₄	C₂×D₂₆₉	1	3	yes	complemented
145	AGL(1,ℤ/8)	—	269	24	no	—
146	AGL(1,ℤ/8)	—	269	24	no	—
147	#135	—	269	24	no	—
148	C₂×Q₁₆	—	269	1	no	—
149	C₂×SD₁₆	—	269	3	no	—
150	C₂×SD₁₆	—	269	3	no	—
151	C₂×D₈	—	269	3	no	—
152	C₂×D₈	—	269	6	no	—
153	C₂×D₈	—	269	24	no	—
154	#131	—	269	8	no	—
155	C₂×M₁₆	—	269	3	no	—
156	C₂²×C₈	—	269	3	no	—
157	C₂⁴	—	2	3	no	—
158	C₂⁴	—	538	3	no	—
159	#41	—	2	8	no	—
160	#41	—	2	6	no	—
161	#41	—	269	12	no	—
162	#41	—	538	8	no	—
163	#41	—	538	6	no	—
164	#41	C₂²⋊D₂₆₉	1	4	yes	—
165	#41	C₂²×D₂₆₉	1	4	yes	—
166	#41	C₂²×D₂₆₉	1	12	yes	complemented
167	C₂×Q₈	—	269	1	no	—
168	C₂×Q₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₂₆₉	1	1	yes	characteristic
170	C₂×D₄	—	2	3	no	—
171	C₂×D₄	—	2	24	no	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	269	3	no	—
175	C₂×D₄	—	269	12	no	—
176	C₂×D₄	—	538	3	no	—
177	C₂×D₄	—	538	24	no	—
178	C₂×D₄	—	538	3	no	—
179	C₂×D₄	—	538	6	no	—
180	C₂×D₄	C₂²⋊D₂₆₉	1	3	yes	—
181	C₂×D₄	C₂²×D₂₆₉	1	3	yes	—
182	C₂×D₄	C₂²×D₂₆₉	1	12	yes	complemented
183	C₂²×C₄	—	2	3	no	—
184	C₂²×C₄	—	538	3	no	—
185	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
186	Q₁₆	—	269	4	no	—
187	SD₁₆	—	269	12	no	—
188	SD₁₆	—	269	12	no	—
189	D₈	—	269	12	no	—
190	D₈	—	269	24	no	—
191	M₁₆	—	269	12	no	—
192	C₂×C₈	—	269	1	no	—
193	C₂×C₈	—	269	3	no	—
194	C₂×C₈	—	269	12	no	—
195	C₂³	—	2	3	no	—
196	C₂³	—	2	12	no	—
197	C₂³	—	8	6	no	—
198	C₂³	—	538	3	no	—
199	C₂³	—	538	12	no	—
200	C₂³	—	2152	6	no	—
201	C₂³	C₂×C₂²⋊D₂₆₉	1	3	yes	—
202	Q₈	—	2	1	no	—
203	Q₈	—	269	2	no	—
204	Q₈	—	538	1	no	—
205	Q₈	C₂×C₂²⋊D₂₆₉	1	4	yes	—
206	Q₈	C₂³×D₂₆₉	1	2	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	2	3	no	—
209	D₄	—	2	12	no	—
210	D₄	—	269	6	no	—
211	D₄	—	538	12	no	—
212	D₄	—	538	12	no	—
213	D₄	—	538	3	no	—
214	D₄	C₂×C₂²⋊D₂₆₉	1	12	yes	—
215	D₄	C₂³×D₂₆₉	1	6	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	538	1	no	—
219	C₂×C₄	—	538	12	no	—
220	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	6	yes	—
221	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	3	yes	—
222	C₂×C₄	C₂³×D₂₆₉	1	6	yes	—
223	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
224	C₈	—	269	2	no	—
225	C₈	—	269	6	no	—
226	C₂²	—	2	6	no	—
227	C₂²	—	2	6	no	—
228	C₂²	—	4	6	no	—
229	C₂²	—	8	12	no	—
230	C₂²	—	538	6	no	—
231	C₂²	—	1076	6	no	—
232	C₂²	—	2152	12	no	—
233	C₂²	C₂²×C₂²⋊D₂₆₉	1	6	yes	—
234	C₂²	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	538	2	no	—
237	C₄	C₂²×C₂²⋊D₂₆₉	1	6	yes	—
238	C₄	C₂²×C₂²⋊D₂₆₉	1	1	yes	characteristic
239	C₄	C₂⁴×D₂₆₉	1	1	yes	characteristic
240	C₂	—	2	6	no	—
241	C₂	—	4	6	no	—
242	C₂	—	1076	6	no	—
243	C₂	#31771408	1	2	yes	direct product factor
244	C₂	C₂³×C₂²⋊D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₆₉
commutator, cosocle	C₄×C₂₆₉	C₂⁵
Frattini	C₄	C₂⁴×D₂₆₉
Fitting	C₂×H(5,𝔽₂)×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₂₆₉
hypocenter	C₂₆₉	C₂×#576
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#22575499	C₂	semidirect
2. #31771408	C₂	direct (N × H)
3. C₂×#22575479	C₂	semidirect
4. C₂×#22575478	C₂	semidirect
5. C₂×#22575473	C₂	semidirect
6. C₂²×#16864387	C₂	semidirect
7. C₂×#22575460	C₂	semidirect
8. C₂×H(5,𝔽₂)×C₂₆₉	C₂	semidirect
10. #22575499	C₂²	semidirect
12. C₂×#16864402	C₂²	semidirect
13. C₂²×D₁₀₇₆	C₂²	semidirect
15. #22575479	C₂²	semidirect
16. #22575478	C₂²	semidirect
17. #22575473	C₂²	semidirect
18. C₂×#16864392	C₂²	semidirect
19. C₂×#16864390	C₂²	semidirect
20. C₂×#16864389	C₂²	semidirect
21. C₂×#16864387	C₂²	semidirect
22. C₂×#16864387	C₂²	semidirect
23. C₂×#16864387	C₂²	semidirect
24. #22575460	C₂²	semidirect
25. C₂×#16864386	C₂²	semidirect
26. C₂²×#14524130	C₂²	semidirect
28. H(5,𝔽₂)×C₂₆₉	C₂²	semidirect
31. C₂×#41×C₂₆₉	C₂²	semidirect
33. C₂²×D₄×C₂₆₉	C₂²	semidirect
37. #16864402	C₂³	semidirect
42. C₂×D₁₀₇₆	C₂³	semidirect
46. #16864392	C₂³	semidirect
47. #16864390	C₂³	semidirect
48. #16864389	C₂³	semidirect
49. #16864387	C₂³	semidirect
50. #16864387	C₂³	semidirect
51. #16864386	C₂³	semidirect
54. C₂×#14524130	C₂³	semidirect
60. #41×C₂₆₉	C₂³	semidirect
69. C₂×D₄×C₂₆₉	C₂³	semidirect
119. C₂×C₂₆₉	#576	semidirect
122. C₂₆₉	C₂×#576	semidirect
125. C₂×H(5,𝔽₂)	D₂₆₉	semidirect
135. H(5,𝔽₂)	C₂×D₂₆₉	semidirect
140. C₂×#41	C₂×D₂₆₉	semidirect
144. C₂²×D₄	C₂×D₂₆₉	semidirect
166. #41	C₂²×D₂₆₉	semidirect
182. C₂×D₄	C₂²×D₂₆₉	semidirect
243. C₂	#31771408	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×#22575499	2
#31771408	2
C₂×#22575479	2
C₂×#22575478	2
C₂×#22575473	2
C₂²×#16864387	2
C₂×#22575460	2
C₂×H(5,𝔽₂)×C₂₆₉	2
C₂×#576	269

Subgroups — C2×#31771408 #46825636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771408 #46825636