Subgroups — C₂×#31771340 #46824473

Order 34432.

1240 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771340	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22575450	C₂	1	1	yes	maximal, characteristic
3	C₂×#22575442	C₂	1	2	yes	maximal, complemented
4	C₂×#22575441	C₂	1	2	yes	maximal, complemented
5	C₂×#22575438	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₁₀₇₆	C₂	1	2	yes	maximal, complemented
8	C₂×#22575415	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₅₃₈	C₂²	1	1	yes	characteristic
11	C₂²×D₁₀₇₆	C₂²	1	2	yes	—
12	C₂²×D₁₀₇₆	C₂²	1	2	yes	—
13	C₂²×D₁₀₇₆	C₂²	1	2	yes	—
14	C₂²×C₄×D₂₆₉	C₂²	1	4	yes	—
15	#22575450	C₂²	1	8	yes	—
16	#22575442	C₂²	1	16	yes	complemented
17	#22575441	C₂²	1	16	yes	complemented
18	#22575438	C₂²	1	32	yes	complemented
19	C₄×Dic₅₃₈	C₂²	1	8	yes	complemented
20	C₄×D₁₀₇₆	C₂²	1	16	yes	complemented
21	#22575415	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
23	C₂×#16864381	C₂²	1	2	yes	—
24	C₂×#16864379	C₂²	1	4	yes	—
25	C₂×#16864379	C₂²	1	4	yes	—
26	C₂×#16864379	C₂²	1	4	yes	—
27	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
28	C₂²×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₂₆₉	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
32	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₂₆₉	C₂²	1	2	yes	—
34	C₂³×D₂₆₉	C₂³	1	4	yes	—
35	C₂×Dic₅₃₈	—	2	4	no	—
36	C₂×Dic₅₃₈	C₂³	1	4	yes	—
37	C₂×D₁₀₇₆	—	2	8	no	—
38	C₂×D₁₀₇₆	—	2	8	no	—
39	C₂×D₁₀₇₆	—	4	4	no	—
40	C₂×D₁₀₇₆	C₂³	1	8	yes	—
41	C₂×D₁₀₇₆	C₂³	1	8	yes	—
42	C₂×D₁₀₇₆	C₂³	1	8	yes	—
43	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
44	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
45	C₂×C₄×D₂₆₉	—	4	8	no	—
46	C₂×C₄×D₂₆₉	C₂³	1	16	yes	—
47	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
48	#16864381	C₂³	1	8	yes	—
49	#16864379	C₂³	1	16	yes	—
50	#16864379	C₂³	1	16	yes	—
51	#16864379	C₂³	1	16	yes	—
52	C₄×Dic₂₆₉	C₂³	1	8	yes	—
53	C₂×Q₈×C₂₆₉	—	2	4	no	—
54	C₂×Q₈×C₂₆₉	C₂³	1	4	yes	—
55	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₆₉	C₂³	1	4	yes	—
61	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	—
62	C₄²×C₂₆₉	C₂³	1	8	yes	—
63	C₄²×C₂₆₉	C₂³	1	4	yes	—
64	C₂²×D₂₆₉	—	2	8	no	—
65	C₂²×D₂₆₉	—	2	8	no	—
66	C₂²×D₂₆₉	—	2	8	no	—
67	C₂²×D₂₆₉	C₂⁴	1	8	yes	—
68	Dic₅₃₈	—	2	8	no	—
69	D₁₀₇₆	—	2	16	no	—
70	D₁₀₇₆	—	2	16	no	—
71	D₁₀₇₆	—	4	8	no	—
72	C₂×Dic₂₆₉	—	2	4	no	—
73	C₂×Dic₂₆₉	—	2	4	no	—
74	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
75	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
76	C₄×D₂₆₉	—	4	16	no	—
77	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
78	Q₈×C₂₆₉	—	2	8	no	—
79	C₂×C₄×C₂₆₉	—	2	4	no	—
80	C₂×C₄×C₂₆₉	—	2	4	no	—
81	C₂×C₄×C₂₆₉	—	2	2	no	—
82	C₂×C₄×C₂₆₉	#41	1	4	yes	—
83	C₂×C₄×C₂₆₉	#41	1	4	yes	—
84	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
85	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
86	C₂×C₄×C₂₆₉	C₂⁴	1	4	yes	—
87	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
88	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
89	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
90	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
91	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
92	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
93	C₂×D₂₆₉	—	2	8	no	—
94	C₂×D₂₆₉	—	2	8	no	—
95	C₂×D₂₆₉	—	2	8	no	—
96	C₂×D₂₆₉	—	4	16	no	—
97	C₄×C₂₆₉	—	2	4	no	—
98	C₄×C₂₆₉	—	2	4	no	—
99	C₄×C₂₆₉	—	2	2	no	—
100	C₄×C₂₆₉	C₂×#41	1	4	yes	—
101	C₄×C₂₆₉	C₂×#41	1	4	yes	—
102	Dic₂₆₉	—	2	4	no	—
103	Dic₂₆₉	—	2	4	no	—
104	C₂×C₂₆₉	#559	1	4	yes	complemented
105	C₂×C₂₆₉	C₂²×#41	1	1	yes	characteristic
106	C₂×C₂₆₉	C₂²×#41	1	1	yes	characteristic
107	C₂×C₂₆₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
108	D₂₆₉	—	4	8	no	—
109	C₂₆₉	C₂×#559	1	1	yes	characteristic, complemented, Sylow
110	C₂×#559	—	269	1	no	maximal, Sylow
111	#559	—	269	16	no	—
112	C₂×#127	—	269	1	no	—
113	C₂×#124	—	269	2	no	—
114	C₂×#124	—	269	2	no	—
115	C₂×#123	—	269	4	no	—
116	C₂×C₄×Q₈	—	269	1	no	—
117	C₂×C₄×Q₈	D₂₆₉	1	1	yes	characteristic, complemented
118	C₂×C₄×D₄	—	269	2	no	—
119	C₂×C₄×D₄	—	269	2	no	—
120	C₂²×Q₈	—	269	1	no	—
121	C₂²×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
122	C₂²×D₄	—	269	2	no	—
123	C₂²×D₄	—	269	2	no	—
124	C₂²×D₄	—	269	2	no	—
125	C₂³×C₄	—	269	4	no	—
126	#127	—	269	8	no	—
127	#124	—	269	16	no	—
128	#124	—	269	16	no	—
129	#123	—	269	32	no	—
130	C₄×Q₈	—	269	8	no	—
131	C₄×Q₈	C₂×D₂₆₉	1	8	yes	complemented
132	C₄×D₄	—	269	16	no	—
133	C₄×D₄	—	269	16	no	—
134	C₂×C₄⋊C₄	—	269	1	no	—
135	C₂×C₄⋊C₄	—	269	2	no	—
136	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
138	C₂×C₂²⋊C₄	—	269	4	no	—
139	C₂×C₂²⋊C₄	—	269	4	no	—
140	C₂×C₂²⋊C₄	—	269	4	no	—
141	C₂×C₄²	—	269	2	no	—
142	C₂×C₄²	C₂×D₂₆₉	1	2	yes	—
143	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
144	C₂⁴	—	269	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	269	4	no	—
147	C₂×Q₈	—	538	4	no	—
148	C₂×Q₈	C₂²×D₂₆₉	1	4	yes	—
149	C₂×D₄	—	269	8	no	—
150	C₂×D₄	—	269	8	no	—
151	C₂×D₄	—	269	8	no	—
152	C₂×D₄	—	538	8	no	—
153	C₂×D₄	—	538	8	no	—
154	C₂×D₄	—	1076	4	no	—
155	C₂²×C₄	—	269	2	no	—
156	C₂²×C₄	—	269	2	no	—
157	C₂²×C₄	—	269	16	no	—
158	C₂²×C₄	—	1076	8	no	—
159	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
160	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
161	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
164	C₄⋊C₄	—	269	4	no	—
165	C₄⋊C₄	—	269	8	no	—
166	C₄⋊C₄	C₂²×D₂₆₉	1	4	yes	—
167	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	—
168	C₂²⋊C₄	—	269	16	no	—
169	C₂²⋊C₄	—	269	16	no	—
170	C₂²⋊C₄	—	269	16	no	—
171	C₄²	—	269	8	no	—
172	C₄²	C₂²×D₂₆₉	1	8	yes	—
173	C₄²	C₂²×D₂₆₉	1	4	yes	—
174	C₂³	—	269	8	no	—
175	C₂³	—	538	8	no	—
176	C₂³	—	538	8	no	—
177	C₂³	—	538	8	no	—
178	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
179	Q₈	—	2	8	no	—
180	Q₈	—	538	8	no	—
181	D₄	—	538	16	no	—
182	D₄	—	538	16	no	—
183	D₄	—	1076	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	269	4	no	—
188	C₂×C₄	—	269	4	no	—
189	C₂×C₄	—	538	4	no	—
190	C₂×C₄	—	538	4	no	—
191	C₂×C₄	—	1076	16	no	—
192	C₂×C₄	#16864402	1	4	yes	—
193	C₂×C₄	#16864403	1	4	yes	—
194	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
195	C₂×C₄	C₂³×D₂₆₉	1	4	yes	—
196	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
197	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
198	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
199	C₂²	—	538	8	no	—
200	C₂²	—	538	8	no	—
201	C₂²	—	538	8	no	—
202	C₂²	—	1076	16	no	—
203	C₂²	C₂×#16864402	1	2	yes	—
204	C₂²	C₂×#16864403	1	2	yes	—
205	C₂²	#22575499	1	2	yes	—
206	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	538	4	no	—
211	C₄	—	538	4	no	—
212	C₄	C₂×#16864402	1	4	yes	—
213	C₄	C₂×#16864403	1	4	yes	—
214	C₂	—	1076	8	no	—
215	C₂	#31771340	1	4	yes	direct product factor
216	C₂	C₂²×#16864402	1	1	yes	characteristic
217	C₂	C₂²×#16864403	1	1	yes	characteristic
218	C₂	C₂×#22575499	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₆₉
commutator, cosocle	C₂²×C₂₆₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₆₉
Fitting	C₂×C₄×Q₈×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂×#22575450	C₂
hypercenter	C₂×C₄×Q₈	D₂₆₉
hypocenter	C₂₆₉	C₂×#559
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #31771340	C₂	direct (N × H)
3. C₂×#22575442	C₂	semidirect
4. C₂×#22575441	C₂	semidirect
5. C₂×#22575438	C₂	semidirect
6. C₂×C₄×Dic₅₃₈	C₂	semidirect
7. C₂×C₄×D₁₀₇₆	C₂	semidirect
8. C₂×#22575415	C₂	semidirect
9. C₂×C₄×Q₈×C₂₆₉	C₂	semidirect
16. #22575442	C₂²	semidirect
17. #22575441	C₂²	semidirect
18. #22575438	C₂²	semidirect
19. C₄×Dic₅₃₈	C₂²	semidirect
20. C₄×D₁₀₇₆	C₂²	semidirect
21. #22575415	C₂²	semidirect
29. C₄×Q₈×C₂₆₉	C₂²	semidirect
104. C₂×C₂₆₉	#559	semidirect
109. C₂₆₉	C₂×#559	semidirect
117. C₂×C₄×Q₈	D₂₆₉	semidirect
131. C₄×Q₈	C₂×D₂₆₉	semidirect
215. C₂	#31771340	direct (N × H)

Maximal subgroups

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

subgroup	index
#31771340	2
C₂×#22575450	2
C₂×#22575442	2
C₂×#22575441	2
C₂×#22575438	2
C₂×C₄×Dic₅₃₈	2
C₂×C₄×D₁₀₇₆	2
C₂×#22575415	2
C₂×C₄×Q₈×C₂₆₉	2
C₂×#559	269

Subgroups — C2×#31771340 #46824473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771340 #46824473