Subgroups — C₂×#31771073 #46824064

Order 34432.

1728 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16864402	C₂	1	1	yes	maximal, characteristic
2	#31771073	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22575418	C₂	1	2	yes	maximal, complemented
4	C₂×#22575415	C₂	1	2	yes	maximal, complemented
5	C₂×#22575413	C₂	1	4	yes	maximal, complemented
6	C₂×#22575410	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₂₆₉	C₂	1	2	yes	maximal, complemented
9	C₂×#117×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16864402	C₄	1	16	yes	complemented
11	C₂×#16864402	C₂²	1	8	yes	—
12	C₂²×Dic₅₃₈	C₂²	1	1	yes	characteristic
13	C₂²×D₁₀₇₆	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₂₆₉	C₂²	1	2	yes	—
15	C₂³×Dic₂₆₉	C₂²	1	2	yes	—
16	C₂²×C₄×D₂₆₉	C₂²	1	4	yes	—
17	C₂²×C₄×D₂₆₉	C₂²	1	2	yes	—
18	#22575418	C₂²	1	16	yes	complemented
19	#22575415	C₂²	1	16	yes	complemented
20	#22575413	C₂²	1	32	yes	complemented
21	#22575410	C₂²	1	16	yes	complemented
22	C₂×#16864381	C₂²	1	4	yes	—
23	C₂×#16864379	C₂²	1	4	yes	—
24	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₆₉	C₂²	1	4	yes	—
26	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
28	C₂×C₄×Dic₂₆₉	C₂²	1	8	yes	complemented
29	C₄²×D₂₆₉	C₂²	1	16	yes	complemented
30	C₂³×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
31	#117×C₂₆₉	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
34	C₂×C₄²×C₂₆₉	C₂²	1	2	yes	—
35	C₂³×D₂₆₉	C₂³	1	2	yes	—
36	#16864402	C₂×C₄	1	64	yes	complemented
37	C₂×Dic₅₃₈	C₂×C₄	1	8	yes	complemented
38	C₂×Dic₅₃₈	C₂³	1	4	yes	—
39	C₂×D₁₀₇₆	C₂×C₄	1	8	yes	complemented
40	C₂×D₁₀₇₆	C₂³	1	4	yes	—
41	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	16	yes	complemented
42	C₂×C₂²⋊D₂₆₉	C₂³	1	8	yes	—
43	C₂²×Dic₂₆₉	—	2	8	no	—
44	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
45	C₂²×Dic₂₆₉	C₂³	1	8	yes	—
46	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
47	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
48	C₂×C₄×D₂₆₉	—	2	16	no	—
49	C₂×C₄×D₂₆₉	C₂×C₄	1	16	yes	complemented
50	C₂×C₄×D₂₆₉	C₂³	1	16	yes	—
51	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
52	#16864381	C₂³	1	16	yes	—
53	#16864379	C₂³	1	16	yes	—
54	C₄×Dic₂₆₉	C₂³	1	4	yes	—
55	C₄×Dic₂₆₉	C₂³	1	16	yes	—
56	C₄×Dic₂₆₉	C₂³	1	4	yes	—
57	C₄×Dic₂₆₉	C₂³	1	8	yes	—
58	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₆₉	C₂×C₄	1	8	yes	complemented
60	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
61	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
63	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	—
65	C₂²⋊C₄×C₂₆₉	C₂³	1	8	yes	—
66	C₄²×C₂₆₉	C₂³	1	8	yes	—
67	C₂²×D₂₆₉	—	2	8	no	—
68	C₂²×D₂₆₉	C₂²×C₄	1	8	yes	—
69	C₂²×D₂₆₉	C₂⁴	1	4	yes	—
70	Dic₅₃₈	C₂²×C₄	1	16	yes	complemented
71	D₁₀₇₆	C₂²×C₄	1	16	yes	complemented
72	C₂²⋊D₂₆₉	C₂²×C₄	1	32	yes	complemented
73	C₂×Dic₂₆₉	—	2	4	no	—
74	C₂×Dic₂₆₉	—	2	16	no	—
75	C₂×Dic₂₆₉	C₂²×C₄	1	8	yes	—
76	C₂×Dic₂₆₉	#41	1	8	yes	complemented
77	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
78	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
79	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
80	C₄×D₂₆₉	—	2	32	no	—
81	C₄×D₂₆₉	C₂²×C₄	1	32	yes	complemented
82	C₂³×C₂₆₉	—	2	4	no	—
83	C₂³×C₂₆₉	C₂²×C₄	1	4	yes	—
84	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
85	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
86	C₂×C₄×C₂₆₉	—	2	8	no	—
87	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
88	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₂₆₉	C₂²×C₄	1	16	yes	complemented
90	C₂×C₄×C₂₆₉	C₂⁴	1	8	yes	—
91	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
92	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
93	C₂²×C₂₆₉	—	2	8	no	—
94	C₂²×C₂₆₉	—	2	4	no	—
95	C₂²×C₂₆₉	C₂³×C₄	1	4	yes	—
96	C₂²×C₂₆₉	C₂³×C₄	1	2	yes	—
97	C₂²×C₂₆₉	C₂×#41	1	4	yes	—
98	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
99	C₂×D₂₆₉	—	2	16	no	—
100	C₂×D₂₆₉	—	2	8	no	—
101	C₂×D₂₆₉	C₂³×C₄	1	8	yes	—
102	C₄×C₂₆₉	—	2	8	no	—
103	C₄×C₂₆₉	C₂³×C₄	1	4	yes	—
104	C₄×C₂₆₉	C₂³×C₄	1	4	yes	—
105	Dic₂₆₉	—	2	4	no	—
106	Dic₂₆₉	C₂³×C₄	1	8	yes	—
107	Dic₂₆₉	C₂×#41	1	8	yes	complemented
108	C₂×C₂₆₉	—	2	4	no	—
109	C₂×C₂₆₉	C₄×#41	1	4	yes	complemented
110	C₂×C₂₆₉	C₂⁴×C₄	1	1	yes	characteristic
111	C₂×C₂₆₉	C₂²×#41	1	2	yes	—
112	D₂₆₉	—	2	8	no	—
113	C₂₆₉	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄×#41	—	269	1	no	maximal, Sylow
115	C₂²×#41	—	269	1	no	—
116	C₄×#41	—	269	16	no	—
117	C₂×C₄×Q₈	—	269	2	no	—
118	C₂×C₄×D₄	—	269	2	no	—
119	C₂×C₄×D₄	—	269	4	no	—
120	C₂×#117	—	269	2	no	—
121	C₂×#117	D₂₆₉	1	1	yes	characteristic, complemented
122	C₂²×C₄²	—	269	1	no	—
123	C₂²×C₄²	—	269	2	no	—
124	C₂×#41	—	269	16	no	—
125	C₂×#41	—	269	8	no	—
126	C₂²×Q₈	—	269	1	no	—
127	C₂²×D₄	—	269	2	no	—
128	C₂²×D₄	—	269	1	no	—
129	C₂³×C₄	—	269	2	no	—
130	C₂³×C₄	—	269	2	no	—
131	C₂³×C₄	—	269	4	no	—
132	C₂³×C₄	C₂×D₂₆₉	1	1	yes	characteristic
133	C₄×Q₈	—	269	16	no	—
134	C₄×D₄	—	269	16	no	—
135	C₄×D₄	—	269	32	no	—
136	#117	—	269	16	no	—
137	#117	C₂×D₂₆₉	1	8	yes	complemented
138	C₂×C₄⋊C₄	—	269	4	no	—
139	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
140	C₂×C₂²⋊C₄	—	269	4	no	—
141	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
142	C₂×C₄²	—	269	4	no	—
143	C₂×C₄²	—	269	2	no	—
144	C₂×C₄²	—	269	8	no	—
145	C₂×C₄²	—	269	1	no	—
146	C₂×C₄²	—	269	1	no	—
147	C₂×C₄²	—	269	16	no	—
148	C₂×C₄²	C₂×D₂₆₉	1	2	yes	—
149	C₂⁴	—	269	2	no	—
150	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
151	#41	—	269	64	no	—
152	C₂×Q₈	—	269	8	no	—
153	C₂×Q₈	—	269	4	no	—
154	C₂×D₄	—	269	16	no	—
155	C₂×D₄	—	269	8	no	—
156	C₂×D₄	—	269	8	no	—
157	C₂×D₄	—	269	4	no	—
158	C₂²×C₄	—	269	2	no	—
159	C₂²×C₄	—	269	2	no	—
160	C₂²×C₄	—	269	16	no	—
161	C₂²×C₄	—	269	8	no	—
162	C₂²×C₄	—	269	2	no	—
163	C₂²×C₄	—	269	8	no	—
164	C₂²×C₄	—	269	16	no	—
165	C₂²×C₄	—	538	16	no	—
166	C₂²×C₄	—	538	8	no	—
167	C₂²×C₄	C₄×D₂₆₉	1	8	yes	complemented
168	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
169	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
171	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
172	C₄⋊C₄	—	269	16	no	—
173	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	—
174	C₂²⋊C₄	—	269	16	no	—
175	C₂²⋊C₄	C₂²×D₂₆₉	1	8	yes	—
176	C₄²	—	269	4	no	—
177	C₄²	—	269	4	no	—
178	C₄²	—	269	16	no	—
179	C₄²	—	269	8	no	—
180	C₄²	C₂²×D₂₆₉	1	8	yes	—
181	C₂³	—	2	4	no	—
182	C₂³	—	269	8	no	—
183	C₂³	—	269	4	no	—
184	C₂³	—	538	8	no	—
185	C₂³	C₂×C₄×D₂₆₉	1	4	yes	—
186	C₂³	C₂³×D₂₆₉	1	2	yes	—
187	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
188	Q₈	—	269	16	no	—
189	D₄	—	269	32	no	—
190	D₄	—	269	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	269	8	no	—
193	C₂×C₄	—	269	8	no	—
194	C₂×C₄	—	269	4	no	—
195	C₂×C₄	—	269	4	no	—
196	C₂×C₄	—	269	32	no	—
197	C₂×C₄	—	269	4	no	—
198	C₂×C₄	—	538	32	no	—
199	C₂×C₄	—	538	16	no	—
200	C₂×C₄	—	538	4	no	—
201	C₂×C₄	C₂×C₄×D₂₆₉	1	4	yes	—
202	C₂×C₄	C₂×C₄×D₂₆₉	1	4	yes	—
203	C₂×C₄	C₂×C₄×D₂₆₉	1	16	yes	complemented
204	C₂×C₄	C₂³×D₂₆₉	1	8	yes	—
205	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
206	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	269	8	no	—
210	C₂²	—	538	8	no	—
211	C₂²	—	538	16	no	—
212	C₂²	C₂²×C₄×D₂₆₉	1	4	yes	—
213	C₂²	C₂²×C₄×D₂₆₉	1	2	yes	—
214	C₂²	#41×D₂₆₉	1	4	yes	—
215	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
216	C₄	—	2	8	no	—
217	C₄	—	269	8	no	—
218	C₄	—	269	8	no	—
219	C₄	—	538	4	no	—
220	C₄	C₂²×C₄×D₂₆₉	1	4	yes	—
221	C₄	C₂²×C₄×D₂₆₉	1	4	yes	—
222	C₂	—	2	4	no	—
223	C₂	—	538	8	no	—
224	C₂	#31771073	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₂₆₉	1	1	yes	characteristic
226	C₂	C₂×#41×D₂₆₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₆₉
commutator	C₂×C₂₆₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×#117×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂²×#16864402	C₂
cosocle	C₂²×C₂₆₉	C₂⁵
hypercenter	C₂×#117	D₂₆₉
hypocenter	C₂₆₉	C₂×C₄×#41
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #31771073	C₂	direct (N × H)
3. C₂×#22575418	C₂	semidirect
4. C₂×#22575415	C₂	semidirect
5. C₂×#22575413	C₂	semidirect
6. C₂×#22575410	C₂	semidirect
7. C₂²×C₄×Dic₂₆₉	C₂	semidirect
8. C₂×C₄²×D₂₆₉	C₂	semidirect
9. C₂×#117×C₂₆₉	C₂	semidirect
10. C₂×#16864402	C₄	semidirect
18. #22575418	C₂²	semidirect
19. #22575415	C₂²	semidirect
20. #22575413	C₂²	semidirect
21. #22575410	C₂²	semidirect
28. C₂×C₄×Dic₂₆₉	C₂²	semidirect
29. C₄²×D₂₆₉	C₂²	semidirect
31. #117×C₂₆₉	C₂²	semidirect
36. #16864402	C₂×C₄	semidirect
37. C₂×Dic₅₃₈	C₂×C₄	semidirect
39. C₂×D₁₀₇₆	C₂×C₄	semidirect
41. C₂×C₂²⋊D₂₆₉	C₂×C₄	semidirect
49. C₂×C₄×D₂₆₉	C₂×C₄	semidirect
59. C₂²×C₄×C₂₆₉	C₂×C₄	semidirect
70. Dic₅₃₈	C₂²×C₄	semidirect
71. D₁₀₇₆	C₂²×C₄	semidirect
72. C₂²⋊D₂₆₉	C₂²×C₄	semidirect
76. C₂×Dic₂₆₉	#41	semidirect
81. C₄×D₂₆₉	C₂²×C₄	semidirect
89. C₂×C₄×C₂₆₉	C₂²×C₄	semidirect
107. Dic₂₆₉	C₂×#41	semidirect
109. C₂×C₂₆₉	C₄×#41	semidirect
113. C₂₆₉	C₂×C₄×#41	semidirect
121. C₂×#117	D₂₆₉	semidirect
137. #117	C₂×D₂₆₉	semidirect
167. C₂²×C₄	C₄×D₂₆₉	semidirect
203. C₂×C₄	C₂×C₄×D₂₆₉	semidirect
224. C₂	#31771073	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16864402	2
#31771073	2
C₂×#22575418	2
C₂×#22575415	2
C₂×#22575413	2
C₂×#22575410	2
C₂²×C₄×Dic₂₆₉	2
C₂×C₄²×D₂₆₉	2
C₂×#117×C₂₆₉	2
C₂×C₄×#41	269

Subgroups — C2×#31771073 #46824064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771073 #46824064