Subgroups — C₂×#31771266 #46824353

Order 34432.

1620 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16864403	C₂	1	1	yes	maximal, characteristic, complemented
2	#31771266	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22575442	C₂	1	3	yes	maximal, complemented
4	C₂×#22575429	C₂	1	3	yes	maximal, complemented
5	C₂×#22575427	C₂	1	3	yes	maximal, complemented
6	C₂×Q₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₂²⋊D₂₆₉	C₂	1	3	yes	maximal, complemented
8	C₂³×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16864403	—	2	8	no	—
10	C₂×#16864403	C₂²	1	8	yes	complemented
11	C₂²×D₁₀₇₆	C₂²	1	3	yes	—
12	C₂²×C₂²⋊D₂₆₉	C₂²	1	3	yes	—
13	C₂²×C₄×D₂₆₉	C₂²	1	3	yes	—
14	#22575442	C₂²	1	24	yes	complemented
15	#22575429	C₂²	1	24	yes	complemented
16	#22575427	C₂²	1	24	yes	complemented
17	Q₈×Dic₂₆₉	C₂²	1	8	yes	complemented
18	C₄×C₂²⋊D₂₆₉	C₂²	1	24	yes	complemented
19	C₂×C₄⋊Dic₂₆₉	C₂²	1	3	yes	—
20	C₂×#16864381	C₂²	1	3	yes	—
21	C₂×C₂²⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
22	C₂×#16864379	C₂²	1	3	yes	—
23	C₂×#16864379	C₂²	1	6	yes	—
24	C₂×C₄×Dic₂₆₉	C₂²	1	3	yes	—
25	C₂²×Q₈×C₂₆₉	—	2	8	no	—
26	C₂²×Q₈×C₂₆₉	C₂²	1	3	yes	—
27	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
28	C₂²×Q₈×C₂₆₉	C₂²	1	8	yes	complemented
29	C₂³×C₄×C₂₆₉	C₂²	1	3	yes	—
30	C₂³×D₂₆₉	C₂³	1	3	yes	—
31	#16864403	—	2	32	no	—
32	C₂×D₁₀₇₆	—	2	12	no	—
33	C₂×D₁₀₇₆	C₂³	1	12	yes	—
34	C₂×C₂²⋊D₂₆₉	—	2	12	no	—
35	C₂×C₂²⋊D₂₆₉	C₂³	1	12	yes	—
36	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₆₉	C₂³	1	3	yes	—
38	C₂²×Dic₂₆₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×D₂₆₉	—	2	12	no	—
40	C₂×C₄×D₂₆₉	C₂³	1	12	yes	—
41	C₄⋊Dic₂₆₉	C₂³	1	12	yes	—
42	#16864381	C₂³	1	12	yes	—
43	C₂²⋊Dic₂₆₉	C₂³	1	4	yes	—
44	#16864379	C₂³	1	12	yes	—
45	#16864379	C₂³	1	24	yes	—
46	C₄×Dic₂₆₉	C₂³	1	12	yes	—
47	C₂⁴×C₂₆₉	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₂₆₉	—	2	32	no	—
49	C₂×Q₈×C₂₆₉	—	2	12	no	—
50	C₂×Q₈×C₂₆₉	D₄	1	8	yes	complemented
51	C₂×Q₈×C₂₆₉	C₂³	1	12	yes	—
52	C₂×Q₈×C₂₆₉	C₂³	1	4	yes	—
53	C₂²×C₄×C₂₆₉	—	2	12	no	—
54	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
55	C₂²×C₄×C₂₆₉	C₂³	1	12	yes	—
56	C₂²×C₄×C₂₆₉	C₂³	1	3	yes	—
57	C₂²×D₂₆₉	—	2	6	no	—
58	C₂²×D₂₆₉	—	2	6	no	—
59	C₂²×D₂₆₉	—	2	6	no	—
60	C₂²×D₂₆₉	C₂⁴	1	6	yes	—
61	D₁₀₇₆	—	2	24	no	—
62	C₂²⋊D₂₆₉	—	2	24	no	—
63	C₂×Dic₂₆₉	—	2	2	no	—
64	C₂×Dic₂₆₉	—	2	6	no	—
65	C₂×Dic₂₆₉	—	2	2	no	—
66	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
67	C₂×Dic₂₆₉	C₂⁴	1	6	yes	—
68	C₂×Dic₂₆₉	C₂⁴	1	2	yes	—
69	C₄×D₂₆₉	—	2	24	no	—
70	C₂³×C₂₆₉	—	2	2	no	—
71	C₂³×C₂₆₉	—	2	2	no	—
72	C₂³×C₂₆₉	#41	1	4	yes	complemented
73	C₂³×C₂₆₉	C₂⁴	1	2	yes	—
74	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
75	Q₈×C₂₆₉	—	2	24	no	—
76	Q₈×C₂₆₉	C₂×D₄	1	16	yes	complemented
77	C₂×C₄×C₂₆₉	—	2	6	no	—
78	C₂×C₄×C₂₆₉	—	2	24	no	—
79	C₂×C₄×C₂₆₉	C₂×D₄	1	12	yes	—
80	C₂×C₄×C₂₆₉	C₂⁴	1	6	yes	—
81	C₂×C₄×C₂₆₉	C₂⁴	1	6	yes	—
82	C₂²×C₂₆₉	—	2	8	no	—
83	C₂²×C₂₆₉	—	2	2	no	—
84	C₂²×C₂₆₉	—	2	2	no	—
85	C₂²×C₂₆₉	C₂²×D₄	1	2	yes	—
86	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
87	C₂²×C₂₆₉	C₂×#41	1	4	yes	complemented
88	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
89	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
90	C₂×D₂₆₉	—	2	6	no	—
91	C₂×D₂₆₉	—	2	6	no	—
92	C₂×D₂₆₉	—	2	6	no	—
93	C₂×D₂₆₉	—	4	12	no	—
94	C₄×C₂₆₉	—	2	6	no	—
95	C₄×C₂₆₉	C₂²×D₄	1	12	yes	—
96	Dic₂₆₉	—	2	2	no	—
97	Dic₂₆₉	—	2	6	no	—
98	Dic₂₆₉	—	2	2	no	—
99	C₂×C₂₆₉	—	2	4	no	—
100	C₂×C₂₆₉	#548	1	4	yes	complemented
101	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
102	C₂×C₂₆₉	C₂²×#41	1	1	yes	characteristic
103	C₂×C₂₆₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
104	D₂₆₉	—	4	6	no	—
105	C₂₆₉	C₂×#548	1	1	yes	characteristic, complemented, Sylow
106	C₂×#548	—	269	1	no	maximal, Sylow
107	C₂²×#41	—	269	1	no	—
108	C₂³×Q₈	D₂₆₉	1	1	yes	characteristic, complemented
109	#548	—	269	16	no	—
110	C₂×#124	—	269	3	no	—
111	C₂×C₂²⋊Q₈	—	269	3	no	—
112	C₂×#121	—	269	3	no	—
113	C₂×C₄×Q₈	—	269	1	no	—
114	C₂×C₄×D₄	—	269	3	no	—
115	C₂×#41	—	269	8	no	—
116	C₂×#41	—	538	8	no	—
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	C₂×D₂₆₉	1	3	yes	—
119	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
120	C₂²×Q₈	C₂×D₂₆₉	1	8	yes	complemented
121	C₂²×D₄	—	269	3	no	—
122	C₂²×D₄	—	269	3	no	—
123	C₂³×C₄	—	269	3	no	—
124	C₂³×C₄	C₂×D₂₆₉	1	3	yes	—
125	#124	—	269	24	no	—
126	C₂²⋊Q₈	—	269	24	no	—
127	#121	—	269	24	no	—
128	C₄×Q₈	—	269	8	no	—
129	C₄×D₄	—	269	24	no	—
130	C₂×C₄⋊C₄	—	269	3	no	—
131	C₂×C₄⋊C₄	—	269	3	no	—
132	C₂×C₂²⋊C₄	—	269	1	no	—
133	C₂×C₂²⋊C₄	—	269	3	no	—
134	C₂×C₂²⋊C₄	—	269	6	no	—
135	C₂×C₄²	—	269	3	no	—
136	C₂⁴	—	269	3	no	—
137	C₂⁴	C₂²×D₂₆₉	1	1	yes	characteristic
138	#41	—	538	32	no	—
139	C₂×Q₈	—	2	32	no	—
140	C₂×Q₈	—	2	12	no	—
141	C₂×Q₈	C₂²⋊D₂₆₉	1	8	yes	complemented
142	C₂×Q₈	C₂²×D₂₆₉	1	12	yes	—
143	C₂×Q₈	C₂²×D₂₆₉	1	4	yes	—
144	C₂×D₄	—	269	12	no	—
145	C₂×D₄	—	269	12	no	—
146	C₂×D₄	—	538	12	no	—
147	C₂×D₄	—	538	12	no	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	269	1	no	—
150	C₂²×C₄	—	269	3	no	—
151	C₂²×C₄	—	269	1	no	—
152	C₂²×C₄	—	269	12	no	—
153	C₂²×C₄	—	538	12	no	—
154	C₂²×C₄	C₂²×D₂₆₉	1	12	yes	—
155	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
156	C₂²×C₄	C₂²×D₂₆₉	1	3	yes	—
157	C₄⋊C₄	—	269	12	no	—
158	C₄⋊C₄	—	269	12	no	—
159	C₂²⋊C₄	—	269	4	no	—
160	C₂²⋊C₄	—	269	12	no	—
161	C₂²⋊C₄	—	269	24	no	—
162	C₄²	—	269	12	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	269	6	no	—
166	C₂³	—	538	6	no	—
167	C₂³	—	538	6	no	—
168	C₂³	—	538	6	no	—
169	C₂³	#16864403	1	4	yes	complemented
170	C₂³	C₂³×D₂₆₉	1	2	yes	—
171	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
172	Q₈	—	2	24	no	—
173	Q₈	C₂×C₂²⋊D₂₆₉	1	16	yes	complemented
174	D₄	—	538	24	no	—
175	D₄	—	538	24	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	24	no	—
178	C₂×C₄	—	269	2	no	—
179	C₂×C₄	—	269	6	no	—
180	C₂×C₄	—	269	2	no	—
181	C₂×C₄	—	538	2	no	—
182	C₂×C₄	—	538	6	no	—
183	C₂×C₄	—	538	2	no	—
184	C₂×C₄	—	538	24	no	—
185	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	12	yes	—
186	C₂×C₄	C₂³×D₂₆₉	1	6	yes	—
187	C₂×C₄	C₂³×D₂₆₉	1	6	yes	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	8	no	—
191	C₂²	—	538	6	no	—
192	C₂²	—	538	6	no	—
193	C₂²	—	538	6	no	—
194	C₂²	—	1076	12	no	—
195	C₂²	C₂²×C₂²⋊D₂₆₉	1	2	yes	—
196	C₂²	C₂×#16864403	1	2	yes	—
197	C₂²	C₂×#16864403	1	4	yes	complemented
198	C₂²	#22575501	1	2	yes	—
199	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
200	C₄	—	2	6	no	—
201	C₄	—	538	2	no	—
202	C₄	—	538	6	no	—
203	C₄	—	538	2	no	—
204	C₄	C₂²×C₂²⋊D₂₆₉	1	12	yes	—
205	C₂	—	2	4	no	—
206	C₂	—	1076	6	no	—
207	C₂	#31771266	1	4	yes	direct product factor
208	C₂	C₂³×C₂²⋊D₂₆₉	1	1	yes	characteristic
209	C₂	C₂²×#16864403	1	1	yes	characteristic
210	C₂	C₂×#22575501	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂³×Q₈×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
hypercenter	C₂³×Q₈	D₂₆₉
hypocenter	C₂₆₉	C₂×#548
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#16864403	C₂	semidirect
2. #31771266	C₂	direct (N × H)
3. C₂×#22575442	C₂	semidirect
4. C₂×#22575429	C₂	semidirect
5. C₂×#22575427	C₂	semidirect
6. C₂×Q₈×Dic₂₆₉	C₂	semidirect
7. C₂×C₄×C₂²⋊D₂₆₉	C₂	semidirect
8. C₂³×Q₈×C₂₆₉	C₂	semidirect
10. C₂×#16864403	C₂²	semidirect
14. #22575442	C₂²	semidirect
15. #22575429	C₂²	semidirect
16. #22575427	C₂²	semidirect
17. Q₈×Dic₂₆₉	C₂²	semidirect
18. C₄×C₂²⋊D₂₆₉	C₂²	semidirect
28. C₂²×Q₈×C₂₆₉	C₂²	semidirect
50. C₂×Q₈×C₂₆₉	D₄	semidirect
72. C₂³×C₂₆₉	#41	semidirect
76. Q₈×C₂₆₉	C₂×D₄	semidirect
87. C₂²×C₂₆₉	C₂×#41	semidirect
100. C₂×C₂₆₉	#548	semidirect
105. C₂₆₉	C₂×#548	semidirect
108. C₂³×Q₈	D₂₆₉	semidirect
120. C₂²×Q₈	C₂×D₂₆₉	semidirect
141. C₂×Q₈	C₂²⋊D₂₆₉	semidirect
169. C₂³	#16864403	semidirect
173. Q₈	C₂×C₂²⋊D₂₆₉	semidirect
197. C₂²	C₂×#16864403	semidirect
207. C₂	#31771266	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×#16864403	2
#31771266	2
C₂×#22575442	2
C₂×#22575429	2
C₂×#22575427	2
C₂×Q₈×Dic₂₆₉	2
C₂×C₄×C₂²⋊D₂₆₉	2
C₂³×Q₈×C₂₆₉	2
C₂×#548	269

Subgroups — C2×#31771266 #46824353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771266 #46824353