Subgroups — C₂×#31771079 #46824071

Order 34432.

1920 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₀₇₆	C₂	1	1	yes	maximal, characteristic
2	#31771079	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×D₁₀₇₆	C₂	1	4	yes	maximal, complemented
4	C₂×#22575415	C₂	1	4	yes	maximal, complemented
5	C₂×#22575412	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊C₄×D₂₆₉	C₂	1	4	yes	maximal, complemented
7	C₂×#117×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×D₂₆₉	C₂²	1	2	yes	—
9	C₂²×D₁₀₇₆	C₄	1	16	yes	complemented
10	C₂²×D₁₀₇₆	C₂²	1	2	yes	—
11	C₂²×D₁₀₇₆	C₂²	1	2	yes	—
12	C₂²×D₁₀₇₆	C₂²	1	8	yes	—
13	C₂²×C₄×D₂₆₉	C₂²	1	8	yes	complemented
14	C₄×D₁₀₇₆	C₂²	1	32	yes	complemented
15	#22575415	C₂²	1	32	yes	complemented
16	#22575412	C₂²	1	8	yes	complemented
17	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	complemented
18	C₂×C₂²⋊Dic₂₆₉	C₂²	1	2	yes	—
19	C₂×#16864379	C₂²	1	8	yes	complemented
20	C₂²⋊C₄×D₂₆₉	C₂²	1	32	yes	complemented
21	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	complemented
22	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
23	#117×C₂₆₉	C₂²	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	complemented
25	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₆₉	C₂²	1	2	yes	complemented
27	C₂³×D₂₆₉	—	4	8	no	—
28	C₂³×D₂₆₉	C₂×C₄	1	16	yes	complemented
29	C₂³×D₂₆₉	C₂³	1	4	yes	—
30	C₂³×D₂₆₉	C₂³	1	8	yes	—
31	C₂×D₁₀₇₆	C₂×C₄	1	16	yes	complemented
32	C₂×D₁₀₇₆	C₂×C₄	1	16	yes	complemented
33	C₂×D₁₀₇₆	C₂×C₄	1	64	yes	complemented
34	C₂×D₁₀₇₆	C₂³	1	8	yes	—
35	C₂×D₁₀₇₆	C₂³	1	8	yes	—
36	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
37	C₂×C₄×D₂₆₉	—	4	16	no	—
38	C₂×C₄×D₂₆₉	C₂³	1	32	yes	complemented
39	C₄⋊Dic₂₆₉	C₂³	1	8	yes	complemented
40	C₂²⋊Dic₂₆₉	C₂³	1	8	yes	—
41	#16864379	C₂³	1	32	yes	complemented
42	C₄×Dic₂₆₉	C₂³	1	8	yes	complemented
43	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₆₉	C₂×C₄	1	8	yes	complemented
45	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
46	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
49	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	complemented
50	C₂²⋊C₄×C₂₆₉	C₂³	1	8	yes	—
51	C₄²×C₂₆₉	C₂³	1	8	yes	complemented
52	C₂²×D₂₆₉	—	2	16	no	—
53	C₂²×D₂₆₉	—	4	32	no	—
54	C₂²×D₂₆₉	—	4	16	no	—
55	C₂²×D₂₆₉	C₂²×C₄	1	16	yes	—
56	C₂²×D₂₆₉	C₂²×C₄	1	32	yes	complemented
57	C₂²×D₂₆₉	C₂⁴	1	8	yes	—
58	D₁₀₇₆	C₂²×C₄	1	32	yes	complemented
59	D₁₀₇₆	C₂²×C₄	1	32	yes	complemented
60	C₂×Dic₂₆₉	—	2	8	no	—
61	C₂×Dic₂₆₉	C₂⁴	1	8	yes	—
62	C₄×D₂₆₉	—	4	32	no	—
63	C₂³×C₂₆₉	—	2	4	no	—
64	C₂³×C₂₆₉	C₂²×C₄	1	4	yes	—
65	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
66	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
67	C₂×C₄×C₂₆₉	—	2	8	no	—
68	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₂₆₉	C₂²×C₄	1	16	yes	complemented
71	C₂×C₄×C₂₆₉	C₂⁴	1	8	yes	—
72	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
73	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
74	C₂²×C₂₆₉	—	2	8	no	—
75	C₂²×C₂₆₉	—	2	4	no	—
76	C₂²×C₂₆₉	C₂³×C₄	1	4	yes	—
77	C₂²×C₂₆₉	C₂³×C₄	1	2	yes	—
78	C₂²×C₂₆₉	H(5,𝔽₂)	1	4	yes	—
79	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
80	C₂×D₂₆₉	—	2	32	no	—
81	C₂×D₂₆₉	—	2	16	no	—
82	C₂×D₂₆₉	—	4	32	no	—
83	C₂×D₂₆₉	C₂³×C₄	1	16	yes	—
84	C₄×C₂₆₉	—	2	8	no	—
85	C₄×C₂₆₉	C₂³×C₄	1	4	yes	—
86	C₄×C₂₆₉	C₂³×C₄	1	4	yes	—
87	Dic₂₆₉	—	2	8	no	—
88	C₂×C₂₆₉	—	2	4	no	—
89	C₂×C₂₆₉	#518	1	4	yes	complemented
90	C₂×C₂₆₉	C₂⁴×C₄	1	1	yes	characteristic
91	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	2	yes	—
92	D₂₆₉	—	2	16	no	—
93	C₂₆₉	C₂×#518	1	1	yes	characteristic, complemented, Sylow
94	C₂×#518	—	269	1	no	maximal, Sylow
95	C₂³×D₄	—	269	1	no	—
96	#518	—	269	16	no	—
97	C₂×C₄×D₄	—	269	4	no	—
98	C₂×C₄×D₄	—	269	4	no	—
99	C₂×#117	—	269	1	no	—
100	C₂×#117	D₂₆₉	1	1	yes	characteristic, complemented
101	C₂²×C₂²⋊C₄	—	269	4	no	—
102	C₂⁵	—	269	2	no	—
103	C₂²×D₄	—	269	16	no	—
104	C₂²×D₄	—	269	2	no	—
105	C₂²×D₄	—	269	2	no	—
106	C₂²×D₄	—	269	8	no	—
107	C₂³×C₄	—	269	8	no	—
108	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
109	C₄×D₄	—	269	32	no	—
110	C₄×D₄	—	269	32	no	—
111	#117	—	269	8	no	—
112	#117	C₂×D₂₆₉	1	8	yes	complemented
113	C₂×C₄⋊C₄	—	269	2	no	—
114	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	complemented
115	C₂×C₂²⋊C₄	—	269	2	no	—
116	C₂×C₂²⋊C₄	—	269	8	no	—
117	C₂×C₂²⋊C₄	—	269	32	no	—
118	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
119	C₂×C₄²	—	269	2	no	—
120	C₂×C₄²	C₂×D₂₆₉	1	2	yes	complemented
121	C₂⁴	—	269	4	no	—
122	C₂⁴	—	269	16	no	—
123	C₂⁴	—	269	8	no	—
124	C₂⁴	—	1076	8	no	—
125	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
126	C₂×D₄	—	269	16	no	—
127	C₂×D₄	—	269	16	no	—
128	C₂×D₄	—	269	64	no	—
129	C₂×D₄	—	269	8	no	—
130	C₂×D₄	—	269	8	no	—
131	C₂²×C₄	—	269	4	no	—
132	C₂²×C₄	—	269	32	no	—
133	C₂²×C₄	—	1076	16	no	—
134	C₂²×C₄	C₄×D₂₆₉	1	8	yes	complemented
135	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
136	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
139	C₄⋊C₄	—	269	8	no	—
140	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	complemented
141	C₂²⋊C₄	—	269	8	no	—
142	C₂²⋊C₄	—	269	32	no	—
143	C₂²⋊C₄	C₂²×D₂₆₉	1	8	yes	—
144	C₄²	—	269	8	no	—
145	C₄²	C₂²×D₂₆₉	1	8	yes	complemented
146	C₂³	—	2	4	no	—
147	C₂³	—	269	16	no	—
148	C₂³	—	269	8	no	—
149	C₂³	—	269	32	no	—
150	C₂³	—	538	16	no	—
151	C₂³	—	1076	16	no	—
152	C₂³	—	1076	32	no	—
153	C₂³	C₂×C₄×D₂₆₉	1	4	yes	—
154	C₂³	C₂³×D₂₆₉	1	2	yes	—
155	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
156	D₄	—	269	32	no	—
157	D₄	—	269	32	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	269	8	no	—
160	C₂×C₄	—	538	8	no	—
161	C₂×C₄	—	1076	32	no	—
162	C₂×C₄	C₂×C₄×D₂₆₉	1	4	yes	—
163	C₂×C₄	C₂×C₄×D₂₆₉	1	4	yes	—
164	C₂×C₄	C₂×C₄×D₂₆₉	1	16	yes	complemented
165	C₂×C₄	C₂³×D₂₆₉	1	8	yes	—
166	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
167	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
168	C₂²	—	2	4	no	—
169	C₂²	—	2	8	no	—
170	C₂²	—	269	16	no	—
171	C₂²	—	538	16	no	—
172	C₂²	—	538	32	no	—
173	C₂²	—	1076	32	no	—
174	C₂²	C₂²×C₄×D₂₆₉	1	4	yes	—
175	C₂²	C₂²×C₄×D₂₆₉	1	2	yes	—
176	C₂²	#22575499	1	4	yes	—
177	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
178	C₄	—	2	8	no	—
179	C₄	—	538	8	no	—
180	C₄	C₂²×C₄×D₂₆₉	1	4	yes	—
181	C₄	C₂²×C₄×D₂₆₉	1	4	yes	—
182	C₂	—	2	4	no	—
183	C₂	—	538	16	no	—
184	C₂	#31771079	1	4	yes	direct product factor
185	C₂	C₂³×C₄×D₂₆₉	1	1	yes	characteristic
186	C₂	C₂×#22575499	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator	C₂×C₂₆₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×#117×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂³×D₁₀₇₆	C₂
cosocle	C₂²×C₂₆₉	C₂⁵
hypercenter	C₂×#117	D₂₆₉
hypocenter	C₂₆₉	C₂×#518
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #31771079	C₂	direct (N × H)
3. C₂×C₄×D₁₀₇₆	C₂	semidirect
4. C₂×#22575415	C₂	semidirect
5. C₂×#22575412	C₂	semidirect
6. C₂×C₂²⋊C₄×D₂₆₉	C₂	semidirect
7. C₂×#117×C₂₆₉	C₂	semidirect
9. C₂²×D₁₀₇₆	C₄	semidirect
13. C₂²×C₄×D₂₆₉	C₂²	semidirect
14. C₄×D₁₀₇₆	C₂²	semidirect
15. #22575415	C₂²	semidirect
16. #22575412	C₂²	semidirect
17. C₂×C₄⋊Dic₂₆₉	C₂²	semidirect
19. C₂×#16864379	C₂²	semidirect
20. C₂²⋊C₄×D₂₆₉	C₂²	semidirect
21. C₂×C₄×Dic₂₆₉	C₂²	semidirect
23. #117×C₂₆₉	C₂²	semidirect
24. C₂×C₄⋊C₄×C₂₆₉	C₂²	semidirect
26. C₂×C₄²×C₂₆₉	C₂²	semidirect
28. C₂³×D₂₆₉	C₂×C₄	semidirect
31. C₂×D₁₀₇₆	C₂×C₄	semidirect
32. C₂×D₁₀₇₆	C₂×C₄	semidirect
33. C₂×D₁₀₇₆	C₂×C₄	semidirect
38. C₂×C₄×D₂₆₉	C₂³	semidirect
39. C₄⋊Dic₂₆₉	C₂³	semidirect
41. #16864379	C₂³	semidirect
42. C₄×Dic₂₆₉	C₂³	semidirect
44. C₂²×C₄×C₂₆₉	C₂×C₄	semidirect
49. C₄⋊C₄×C₂₆₉	C₂³	semidirect
51. C₄²×C₂₆₉	C₂³	semidirect
56. C₂²×D₂₆₉	C₂²×C₄	semidirect
58. D₁₀₇₆	C₂²×C₄	semidirect
59. D₁₀₇₆	C₂²×C₄	semidirect
70. C₂×C₄×C₂₆₉	C₂²×C₄	semidirect
89. C₂×C₂₆₉	#518	semidirect
93. C₂₆₉	C₂×#518	semidirect
100. C₂×#117	D₂₆₉	semidirect
112. #117	C₂×D₂₆₉	semidirect
114. C₂×C₄⋊C₄	C₂×D₂₆₉	semidirect
120. C₂×C₄²	C₂×D₂₆₉	semidirect
134. C₂²×C₄	C₄×D₂₆₉	semidirect
140. C₄⋊C₄	C₂²×D₂₆₉	semidirect
145. C₄²	C₂²×D₂₆₉	semidirect
164. C₂×C₄	C₂×C₄×D₂₆₉	semidirect
184. C₂	#31771079	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₁₀₇₆	2
#31771079	2
C₂×C₄×D₁₀₇₆	2
C₂×#22575415	2
C₂×#22575412	2
C₂×C₂²⋊C₄×D₂₆₉	2
C₂×#117×C₂₆₉	2
C₂×#518	269

Subgroups — C2×#31771079 #46824071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771079 #46824071