Subgroups — C₂×#31771270 #46824358

Order 34432.

1568 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₂₆₉	C₂	1	2	yes	maximal, complemented
2	#31771270	C₂	1	16	yes	maximal, direct product factor
3	C₂×#22575452	C₂	1	2	yes	maximal, complemented
4	C₂×#22575431	C₂	1	4	yes	maximal, complemented
5	C₂×#22575429	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₁₀₇₆	C₂	1	1	yes	maximal, characteristic
8	C₂×#22575415	C₂	1	2	yes	maximal
9	C₂×C₄⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₅₃₈	C₂²	1	4	yes	—
11	C₂²×Dic₅₃₈	C₂²	1	1	yes	characteristic
12	C₂×Q₈×D₂₆₉	—	2	16	no	—
13	C₂×Q₈×D₂₆₉	C₂²	1	16	yes	complemented
14	C₂²×D₁₀₇₆	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₂₆₉	C₂²	1	4	yes	—
16	C₂²×C₄×D₂₆₉	C₂²	1	2	yes	—
17	#22575452	C₂²	1	16	yes	complemented
18	#22575431	C₂²	1	32	yes	complemented
19	#22575429	C₂²	1	16	yes	complemented
20	C₄×Dic₅₃₈	C₂²	1	8	yes	complemented
21	C₄×D₁₀₇₆	C₂²	1	8	yes	—
22	#22575415	C₂²	1	16	yes	—
23	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×#16864381	C₂²	1	4	yes	—
26	C₂×#16864381	C₂²	1	2	yes	—
27	C₂×#16864379	C₂²	1	4	yes	—
28	C₂×#16864379	C₂²	1	2	yes	—
29	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
30	C₂²×Q₈×C₂₆₉	C₂²	1	2	yes	—
31	C₄⋊Q₈×C₂₆₉	C₂²	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
34	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
35	C₂³×D₂₆₉	C₂³	1	2	yes	—
36	C₂×Dic₅₃₈	—	2	16	no	—
37	C₂×Dic₅₃₈	D₄	1	8	yes	complemented
38	C₂×Dic₅₃₈	C₂³	1	16	yes	—
39	C₂×Dic₅₃₈	C₂³	1	4	yes	—
40	Q₈×D₂₆₉	—	2	64	no	—
41	C₂×D₁₀₇₆	Q₈	1	8	yes	complemented
42	C₂×D₁₀₇₆	C₂³	1	4	yes	—
43	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
44	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
45	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
46	C₂×C₄×D₂₆₉	—	2	16	no	—
47	C₂×C₄×D₂₆₉	—	2	8	no	—
48	C₂×C₄×D₂₆₉	C₂³	1	16	yes	—
49	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
50	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
51	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
52	#16864381	C₂³	1	8	yes	—
53	#16864381	C₂³	1	16	yes	—
54	#16864379	C₂³	1	16	yes	—
55	#16864379	C₂³	1	8	yes	—
56	C₄×Dic₂₆₉	C₂³	1	8	yes	—
57	C₂×Q₈×C₂₆₉	—	2	8	no	—
58	C₂×Q₈×C₂₆₉	C₂³	1	8	yes	—
59	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
61	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	—
64	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	—
65	C₄²×C₂₆₉	C₂³	1	4	yes	—
66	C₂²×D₂₆₉	—	2	4	no	—
67	C₂²×D₂₆₉	—	2	4	no	—
68	C₂²×D₂₆₉	C₂×Q₈	1	8	yes	complemented
69	C₂²×D₂₆₉	C₂⁴	1	4	yes	—
70	Dic₅₃₈	—	2	32	no	—
71	Dic₅₃₈	C₂×D₄	1	16	yes	complemented
72	D₁₀₇₆	C₂×Q₈	1	16	yes	complemented
73	C₂×Dic₂₆₉	—	2	4	no	—
74	C₂×Dic₂₆₉	—	2	4	no	—
75	C₂×Dic₂₆₉	C₂×D₄	1	8	yes	—
76	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
77	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
78	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
79	C₄×D₂₆₉	—	2	16	no	—
80	C₄×D₂₆₉	—	2	32	no	—
81	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
82	Q₈×C₂₆₉	—	2	16	no	—
83	C₂×C₄×C₂₆₉	—	2	8	no	—
84	C₂×C₄×C₂₆₉	—	2	2	no	—
85	C₂×C₄×C₂₆₉	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₂₆₉	C₂×Q₈	1	4	yes	complemented
87	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
88	C₂×C₄×C₂₆₉	C₂⁴	1	8	yes	—
89	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
90	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
91	C₂²×C₂₆₉	C₂²×D₄	1	2	yes	—
92	C₂²×C₂₆₉	C₂²×Q₈	1	2	yes	—
93	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
94	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
95	C₂×D₂₆₉	—	2	16	no	—
96	C₂×D₂₆₉	—	2	4	no	—
97	C₂×D₂₆₉	—	2	4	no	—
98	C₂×D₂₆₉	C₂²×Q₈	1	8	yes	complemented
99	C₄×C₂₆₉	—	2	8	no	—
100	C₄×C₂₆₉	—	2	2	no	—
101	C₄×C₂₆₉	C₂²×D₄	1	4	yes	—
102	C₄×C₂₆₉	C₂²×Q₈	1	4	yes	complemented
103	Dic₂₆₉	—	2	4	no	—
104	Dic₂₆₉	—	2	4	no	—
105	Dic₂₆₉	C₂²×D₄	1	8	yes	—
106	C₂×C₂₆₉	D₄×Q₈	1	4	yes	complemented
107	C₂×C₂₆₉	C₂³×D₄	1	1	yes	characteristic
108	C₂×C₂₆₉	C₂³×Q₈	1	1	yes	characteristic
109	C₂×C₂₆₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
110	D₂₆₉	—	2	8	no	—
111	C₂₆₉	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
112	C₂×D₄×Q₈	—	269	1	no	maximal, Sylow
113	C₂³×Q₈	—	269	2	no	—
114	D₄×Q₈	—	269	16	no	—
115	C₂×C₄⋊Q₈	—	269	2	no	—
116	C₂×C₄⋊Q₈	D₂₆₉	1	1	yes	characteristic, complemented
117	C₂×C₂²⋊Q₈	—	269	2	no	—
118	C₂×C₂²⋊Q₈	—	269	4	no	—
119	C₂×C₄×Q₈	—	269	1	no	—
120	C₂×C₄×D₄	—	269	2	no	—
121	C₂×C₄×D₄	—	269	1	no	—
122	C₂²×Q₈	—	269	16	no	—
123	C₂²×Q₈	—	269	1	no	—
124	C₂²×Q₈	—	269	4	no	—
125	C₂²×Q₈	—	538	16	no	—
126	C₂²×Q₈	C₂×D₂₆₉	1	2	yes	—
127	C₂²×D₄	—	269	1	no	—
128	C₂³×C₄	—	269	4	no	—
129	C₂³×C₄	—	269	2	no	—
130	C₄⋊Q₈	—	269	16	no	—
131	C₄⋊Q₈	C₂×D₂₆₉	1	8	yes	complemented
132	C₂²⋊Q₈	—	269	16	no	—
133	C₂²⋊Q₈	—	269	32	no	—
134	C₄×Q₈	—	269	8	no	—
135	C₄×D₄	—	269	16	no	—
136	C₄×D₄	—	269	8	no	—
137	C₂×C₄⋊C₄	—	269	1	no	—
138	C₂×C₄⋊C₄	—	269	2	no	—
139	C₂×C₄⋊C₄	—	269	1	no	—
140	C₂×C₄⋊C₄	—	269	4	no	—
141	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
143	C₂×C₂²⋊C₄	—	269	4	no	—
144	C₂×C₂²⋊C₄	—	269	2	no	—
145	C₂×C₄²	—	269	2	no	—
146	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
147	C₂⁴	—	269	2	no	—
148	C₂×Q₈	—	2	8	no	—
149	C₂×Q₈	—	269	8	no	—
150	C₂×Q₈	—	269	4	no	—
151	C₂×Q₈	—	269	16	no	—
152	C₂×Q₈	—	538	16	no	—
153	C₂×Q₈	—	538	64	no	—
154	C₂×Q₈	C₂²×D₂₆₉	1	8	yes	—
155	C₂×D₄	—	269	8	no	—
156	C₂×D₄	—	269	4	no	—
157	C₂²×C₄	—	269	2	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	—	538	16	no	—
163	C₂²×C₄	—	538	8	no	—
164	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
166	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
168	C₄⋊C₄	—	269	4	no	—
169	C₄⋊C₄	—	269	8	no	—
170	C₄⋊C₄	—	269	4	no	—
171	C₄⋊C₄	—	269	16	no	—
172	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	—
173	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	—
174	C₂²⋊C₄	—	269	16	no	—
175	C₂²⋊C₄	—	269	8	no	—
176	C₄²	—	269	8	no	—
177	C₄²	C₂²×D₂₆₉	1	4	yes	—
178	C₂³	—	269	8	no	—
179	C₂³	—	269	4	no	—
180	C₂³	—	538	4	no	—
181	C₂³	—	538	4	no	—
182	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
183	Q₈	—	2	16	no	—
184	Q₈	—	269	16	no	—
185	Q₈	—	538	32	no	—
186	D₄	—	269	16	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	269	8	no	—
190	C₂×C₄	—	269	4	no	—
191	C₂×C₄	—	269	4	no	—
192	C₂×C₄	—	269	4	no	—
193	C₂×C₄	—	538	32	no	—
194	C₂×C₄	—	538	4	no	—
195	C₂×C₄	—	538	4	no	—
196	C₂×C₄	—	538	16	no	—
197	C₂×C₄	D₄×D₂₆₉	1	4	yes	—
198	C₂×C₄	Q₈×D₂₆₉	1	4	yes	complemented
199	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
200	C₂×C₄	C₂³×D₂₆₉	1	8	yes	—
201	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
202	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
203	C₂²	—	269	8	no	—
204	C₂²	—	538	4	no	—
205	C₂²	—	538	4	no	—
206	C₂²	—	538	16	no	—
207	C₂²	C₂×D₄×D₂₆₉	1	2	yes	—
208	C₂²	C₂×Q₈×D₂₆₉	1	2	yes	—
209	C₂²	#22575501	1	2	yes	—
210	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
211	C₄	—	2	8	no	—
212	C₄	—	2	2	no	—
213	C₄	—	269	8	no	—
214	C₄	—	538	4	no	—
215	C₄	—	538	4	no	—
216	C₄	C₂×D₄×D₂₆₉	1	4	yes	—
217	C₄	C₂×Q₈×D₂₆₉	1	4	yes	complemented
218	C₂	—	538	8	no	—
219	C₂	#31771270	1	4	yes	direct product factor
220	C₂	C₂²×D₄×D₂₆₉	1	1	yes	characteristic
221	C₂	C₂²×Q₈×D₂₆₉	1	1	yes	characteristic
222	C₂	C₂×#22575501	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	C₂²×D₁₀₇₆	C₂²
hypercenter	C₂×C₄⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	C₂×D₄×Q₈
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. C₂²×Q₈×D₂₆₉	C₂	semidirect
2. #31771270	C₂	direct (N × H)
3. C₂×#22575452	C₂	semidirect
4. C₂×#22575431	C₂	semidirect
5. C₂×#22575429	C₂	semidirect
6. C₂×C₄×Dic₅₃₈	C₂	semidirect
9. C₂×C₄⋊Q₈×C₂₆₉	C₂	semidirect
13. C₂×Q₈×D₂₆₉	C₂²	semidirect
17. #22575452	C₂²	semidirect
18. #22575431	C₂²	semidirect
19. #22575429	C₂²	semidirect
20. C₄×Dic₅₃₈	C₂²	semidirect
31. C₄⋊Q₈×C₂₆₉	C₂²	semidirect
37. C₂×Dic₅₃₈	D₄	semidirect
41. C₂×D₁₀₇₆	Q₈	semidirect
68. C₂²×D₂₆₉	C₂×Q₈	semidirect
71. Dic₅₃₈	C₂×D₄	semidirect
72. D₁₀₇₆	C₂×Q₈	semidirect
86. C₂×C₄×C₂₆₉	C₂×Q₈	semidirect
98. C₂×D₂₆₉	C₂²×Q₈	semidirect
102. C₄×C₂₆₉	C₂²×Q₈	semidirect
106. C₂×C₂₆₉	D₄×Q₈	semidirect
111. C₂₆₉	C₂×D₄×Q₈	semidirect
116. C₂×C₄⋊Q₈	D₂₆₉	semidirect
131. C₄⋊Q₈	C₂×D₂₆₉	semidirect
198. C₂×C₄	Q₈×D₂₆₉	semidirect
217. C₄	C₂×Q₈×D₂₆₉	semidirect
219. C₂	#31771270	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×Q₈×D₂₆₉	2
#31771270	2
C₂×#22575452	2
C₂×#22575431	2
C₂×#22575429	2
C₂×C₄×Dic₅₃₈	2
C₂×C₄×D₁₀₇₆	2
C₂×#22575415	2
C₂×C₄⋊Q₈×C₂₆₉	2
C₂×D₄×Q₈	269

Subgroups — C2×#31771270 #46824358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771270 #46824358