Subgroups — C₂×#31771329 #46824435

Order 34432.

1112 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	#31771329	C₂	1	16	yes	maximal, direct product factor
2	C₂×#22575448	C₂	1	4	yes	maximal, complemented
3	C₂×#22575442	C₂	1	2	yes	maximal
4	C₂×#22575433	C₂	1	2	yes	maximal, complemented
5	C₂×Q₈×Dic₂₆₉	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×#22575411	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	1	yes	characteristic
12	C₂²×C₄×D₂₆₉	C₂²	1	2	yes	—
13	#22575448	C₂²	1	32	yes	complemented
14	#22575442	C₂²	1	16	yes	—
15	#22575433	C₂²	1	16	yes	complemented
16	Q₈×Dic₂₆₉	C₂²	1	16	yes	complemented
17	C₄×Dic₅₃₈	C₂²	1	8	yes	complemented
18	C₄×D₁₀₇₆	C₂²	1	8	yes	—
19	#22575411	C₂²	1	16	yes	complemented
20	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₂₆₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₂₆₉	C₂²	1	4	yes	—
23	C₂×#16864381	C₂²	1	2	yes	—
24	C₂×#16864379	C₂²	1	4	yes	—
25	C₂×#16864379	C₂²	1	2	yes	—
26	C₂×#16864379	C₂²	1	4	yes	—
27	C₂×C₄×Dic₂₆₉	C₂²	1	4	yes	—
28	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
29	C₂²×Q₈×C₂₆₉	C₂²	1	2	yes	—
30	C₄⋊Q₈×C₂₆₉	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
34	C₂³×D₂₆₉	C₂³	1	2	yes	—
35	C₂×Dic₅₃₈	—	2	4	no	—
36	C₂×Dic₅₃₈	C₂³	1	4	yes	—
37	C₂×D₁₀₇₆	—	2	4	no	—
38	C₂×D₁₀₇₆	C₂³	1	4	yes	—
39	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
40	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
41	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
42	C₂×C₄×D₂₆₉	—	2	8	no	—
43	C₂×C₄×D₂₆₉	C₂³	1	8	yes	—
44	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
45	C₄⋊Dic₂₆₉	C₂³	1	4	yes	—
46	C₄⋊Dic₂₆₉	C₂³	1	16	yes	—
47	#16864381	C₂³	1	8	yes	—
48	#16864379	C₂³	1	16	yes	—
49	#16864379	C₂³	1	8	yes	—
50	#16864379	C₂³	1	16	yes	—
51	C₄×Dic₂₆₉	C₂³	1	16	yes	—
52	C₄×Dic₂₆₉	C₂³	1	8	yes	—
53	C₂×Q₈×C₂₆₉	—	2	8	no	—
54	C₂×Q₈×C₂₆₉	C₂³	1	8	yes	—
55	C₂²×C₄×C₂₆₉	C₂³	1	4	yes	—
56	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	—
60	C₄⋊C₄×C₂₆₉	C₂³	1	8	yes	—
61	C₄²×C₂₆₉	C₂³	1	4	yes	—
62	C₂²×D₂₆₉	—	2	4	no	—
63	C₂²×D₂₆₉	—	2	4	no	—
64	C₂²×D₂₆₉	—	2	4	no	—
65	C₂²×D₂₆₉	C₂⁴	1	4	yes	—
66	Dic₅₃₈	—	2	8	no	—
67	D₁₀₇₆	—	2	8	no	—
68	C₂×Dic₂₆₉	—	2	4	no	—
69	C₂×Dic₂₆₉	—	2	4	no	—
70	C₂×Dic₂₆₉	—	2	4	no	—
71	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
72	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
73	C₂×Dic₂₆₉	C₂⁴	1	4	yes	—
74	C₄×D₂₆₉	—	2	16	no	—
75	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
76	Q₈×C₂₆₉	—	2	16	no	—
77	C₂×C₄×C₂₆₉	—	2	8	no	—
78	C₂×C₄×C₂₆₉	—	2	2	no	—
79	C₂×C₄×C₂₆₉	#41	1	4	yes	—
80	C₂×C₄×C₂₆₉	#41	1	4	yes	—
81	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
82	C₂×C₄×C₂₆₉	C₂⁴	1	8	yes	—
83	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
84	C₂×C₄×C₂₆₉	C₂⁴	1	2	yes	—
85	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
86	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
87	C₂²×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
88	C₂²×C₂₆₉	C₂⁵	1	1	yes	characteristic
89	C₂×D₂₆₉	—	2	4	no	—
90	C₂×D₂₆₉	—	2	4	no	—
91	C₂×D₂₆₉	—	2	4	no	—
92	C₂×D₂₆₉	—	4	8	no	—
93	C₄×C₂₆₉	—	2	8	no	—
94	C₄×C₂₆₉	—	2	2	no	—
95	C₄×C₂₆₉	C₂×#41	1	4	yes	—
96	C₄×C₂₆₉	C₂×#41	1	4	yes	—
97	Dic₂₆₉	—	2	4	no	—
98	Dic₂₆₉	—	2	4	no	—
99	Dic₂₆₉	—	2	4	no	—
100	C₂×C₂₆₉	#556	1	4	yes	complemented
101	C₂×C₂₆₉	C₂²×#41	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	4	no	—
105	C₂₆₉	C₂×#556	1	1	yes	characteristic, complemented, Sylow
106	C₂×#556	—	269	1	no	maximal, Sylow
107	#556	—	269	16	no	—
108	C₂×C₄⋊Q₈	D₂₆₉	1	1	yes	characteristic, complemented
109	C₂×#126	—	269	4	no	—
110	C₂×#124	—	269	2	no	—
111	C₂×C₂²⋊Q₈	—	269	2	no	—
112	C₂×C₄×Q₈	—	269	2	no	—
113	C₂×C₄×Q₈	—	269	1	no	—
114	C₂×C₄×D₄	—	269	1	no	—
115	C₂×#117	—	269	2	no	—
116	C₂²×Q₈	—	269	1	no	—
117	C₂²×Q₈	C₂×D₂₆₉	1	2	yes	—
118	C₂²×D₄	—	269	1	no	—
119	C₂³×C₄	—	269	2	no	—
120	C₄⋊Q₈	C₂×D₂₆₉	1	8	yes	complemented
121	#126	—	269	32	no	—
122	#124	—	269	16	no	—
123	C₂²⋊Q₈	—	269	16	no	—
124	C₄×Q₈	—	269	16	no	—
125	C₄×Q₈	—	269	8	no	—
126	C₄×D₄	—	269	8	no	—
127	#117	—	269	16	no	—
128	C₂×C₄⋊C₄	—	269	1	no	—
129	C₂×C₄⋊C₄	—	269	1	no	—
130	C₂×C₄⋊C₄	—	269	4	no	—
131	C₂×C₄⋊C₄	—	269	2	no	—
132	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
133	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
134	C₂×C₂²⋊C₄	—	269	4	no	—
135	C₂×C₂²⋊C₄	—	269	2	no	—
136	C₂×C₂²⋊C₄	—	269	4	no	—
137	C₂×C₄²	—	269	4	no	—
138	C₂×C₄²	—	269	2	no	—
139	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
140	C₂⁴	—	269	2	no	—
141	C₂×Q₈	—	2	8	no	—
142	C₂×Q₈	—	269	4	no	—
143	C₂×Q₈	—	538	4	no	—
144	C₂×Q₈	C₂²×D₂₆₉	1	8	yes	—
145	C₂×D₄	—	269	4	no	—
146	C₂×D₄	—	538	4	no	—
147	C₂²×C₄	—	269	2	no	—
148	C₂²×C₄	—	269	2	no	—
149	C₂²×C₄	—	269	2	no	—
150	C₂²×C₄	—	269	8	no	—
151	C₂²×C₄	—	538	8	no	—
152	C₂²×C₄	C₂²×D₂₆₉	1	4	yes	—
153	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
156	C₄⋊C₄	—	269	4	no	—
157	C₄⋊C₄	—	269	4	no	—
158	C₄⋊C₄	—	269	16	no	—
159	C₄⋊C₄	—	269	8	no	—
160	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	—
161	C₄⋊C₄	C₂²×D₂₆₉	1	8	yes	—
162	C₂²⋊C₄	—	269	16	no	—
163	C₂²⋊C₄	—	269	8	no	—
164	C₂²⋊C₄	—	269	16	no	—
165	C₄²	—	269	16	no	—
166	C₄²	—	269	8	no	—
167	C₄²	C₂²×D₂₆₉	1	4	yes	—
168	C₂³	—	269	4	no	—
169	C₂³	—	538	4	no	—
170	C₂³	—	538	4	no	—
171	C₂³	—	538	4	no	—
172	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
173	Q₈	—	2	16	no	—
174	Q₈	—	538	8	no	—
175	D₄	—	538	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	269	4	no	—
179	C₂×C₄	—	269	4	no	—
180	C₂×C₄	—	269	4	no	—
181	C₂×C₄	—	538	4	no	—
182	C₂×C₄	—	538	4	no	—
183	C₂×C₄	—	538	4	no	—
184	C₂×C₄	—	538	16	no	—
185	C₂×C₄	#16864401	1	4	yes	—
186	C₂×C₄	#16864403	1	4	yes	—
187	C₂×C₄	C₂³×D₂₆₉	1	8	yes	—
188	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
189	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
190	C₂×C₄	C₂³×D₂₆₉	1	2	yes	—
191	C₂²	—	538	4	no	—
192	C₂²	—	538	4	no	—
193	C₂²	—	538	4	no	—
194	C₂²	—	1076	8	no	—
195	C₂²	C₂×#16864401	1	2	yes	—
196	C₂²	C₂×#16864403	1	2	yes	—
197	C₂²	#22575501	1	2	yes	—
198	C₂²	C₂⁴×D₂₆₉	1	1	yes	characteristic
199	C₄	—	2	8	no	—
200	C₄	—	2	2	no	—
201	C₄	—	538	4	no	—
202	C₄	—	538	4	no	—
203	C₄	—	538	4	no	—
204	C₄	C₂×#16864401	1	4	yes	—
205	C₄	C₂×#16864403	1	4	yes	—
206	C₂	—	1076	4	no	—
207	C₂	#31771329	1	4	yes	direct product factor
208	C₂	C₂²×#16864401	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	C₂×C₄⋊Q₈×C₂₆₉	C₂
socle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂²×D₁₀₇₆	C₂²
hypercenter	C₂×C₄⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	C₂×#556
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. #31771329	C₂	direct (N × H)
2. C₂×#22575448	C₂	semidirect
4. C₂×#22575433	C₂	semidirect
5. C₂×Q₈×Dic₂₆₉	C₂	semidirect
6. C₂×C₄×Dic₅₃₈	C₂	semidirect
8. C₂×#22575411	C₂	semidirect
9. C₂×C₄⋊Q₈×C₂₆₉	C₂	semidirect
13. #22575448	C₂²	semidirect
15. #22575433	C₂²	semidirect
16. Q₈×Dic₂₆₉	C₂²	semidirect
17. C₄×Dic₅₃₈	C₂²	semidirect
19. #22575411	C₂²	semidirect
30. C₄⋊Q₈×C₂₆₉	C₂²	semidirect
100. C₂×C₂₆₉	#556	semidirect
105. C₂₆₉	C₂×#556	semidirect
108. C₂×C₄⋊Q₈	D₂₆₉	semidirect
120. C₄⋊Q₈	C₂×D₂₆₉	semidirect
207. C₂	#31771329	direct (N × H)

Maximal subgroups

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

subgroup	index
#31771329	2
C₂×#22575448	2
C₂×#22575442	2
C₂×#22575433	2
C₂×Q₈×Dic₂₆₉	2
C₂×C₄×Dic₅₃₈	2
C₂×C₄×D₁₀₇₆	2
C₂×#22575411	2
C₂×C₄⋊Q₈×C₂₆₉	2
C₂×#556	269

Subgroups — C2×#31771329 #46824435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31771329 #46824435