Subgroups — #46813795

Order 34432.

704 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₂₆₉	C₂	1	2	yes	maximal
2	#31770504	C₂	1	2	yes	maximal, complemented
3	#31770480	C₂	1	4	yes	maximal, complemented
4	#31770471	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770465	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#22575360	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₂₆₉	C₂²	1	4	yes	—
10	D₄×Dic₂₆₉	—	2	8	no	—
11	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₆₉	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₂₆₉	C₂²	1	4	yes	—
15	C₂×C₄×Dic₂₆₉	C₂²	1	2	yes	—
16	#22575360	C₂²	1	1	yes	characteristic
17	#22575360	C₂²	1	2	yes	—
18	#22575360	C₂²	1	4	yes	—
19	#22575360	C₂²	1	1	yes	characteristic
20	#22575360	C₂²	1	2	yes	—
21	C₂²×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₂₆₉	C₂²	1	2	yes	—
23	C₄×D₄×C₂₆₉	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₆₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
27	C₂²×Dic₂₆₉	—	2	8	no	—
28	C₂²×Dic₂₆₉	—	2	8	no	—
29	C₂²×Dic₂₆₉	—	4	4	no	—
30	C₂²×Dic₂₆₉	C₂³	1	4	yes	—
31	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
32	C₂²×Dic₂₆₉	C₂³	1	2	yes	—
33	C₄⋊Dic₂₆₉	—	2	4	no	—
34	C₄⋊Dic₂₆₉	—	2	4	no	—
35	C₄⋊Dic₂₆₉	—	4	4	no	—
36	C₂²⋊Dic₂₆₉	—	2	8	no	—
37	C₄×Dic₂₆₉	—	2	4	no	—
38	C₂⁴×C₂₆₉	C₂³	1	2	yes	—
39	C₂×D₄×C₂₆₉	C₂×C₄	1	4	yes	—
40	C₂×D₄×C₂₆₉	D₄	1	4	yes	—
41	C₂×D₄×C₂₆₉	Q₈	1	4	yes	—
42	C₂²×C₄×C₂₆₉	—	4	4	no	—
43	C₂²×C₄×C₂₆₉	C₂×C₄	1	8	yes	complemented
44	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₂₆₉	C₂×C₄	1	4	yes	complemented
49	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	8	yes	complemented
50	C₄²×C₂₆₉	C₂×C₄	1	4	yes	complemented
51	C₂×Dic₂₆₉	—	2	2	no	—
52	C₂×Dic₂₆₉	—	2	2	no	—
53	C₂×Dic₂₆₉	—	2	2	no	—
54	C₂×Dic₂₆₉	—	2	4	no	—
55	C₂×Dic₂₆₉	—	2	2	no	—
56	C₂×Dic₂₆₉	—	2	4	no	—
57	C₂×Dic₂₆₉	—	2	2	no	—
58	C₂×Dic₂₆₉	—	2	4	no	—
59	C₂×Dic₂₆₉	—	4	16	no	—
60	C₂×Dic₂₆₉	#41	1	4	yes	—
61	C₂³×C₂₆₉	—	2	4	no	—
62	C₂³×C₂₆₉	—	2	4	no	—
63	C₂³×C₂₆₉	C₂²×C₄	1	4	yes	—
64	C₂³×C₂₆₉	C₂×D₄	1	4	yes	—
65	C₂³×C₂₆₉	C₂×Q₈	1	4	yes	—
66	C₂³×C₂₆₉	C₂⁴	1	1	yes	characteristic
67	D₄×C₂₆₉	C₄⋊C₄	1	16	yes	complemented
68	C₂×C₄×C₂₆₉	—	2	2	no	—
69	C₂×C₄×C₂₆₉	—	2	2	no	—
70	C₂×C₄×C₂₆₉	—	2	4	no	—
71	C₂×C₄×C₂₆₉	—	4	8	no	—
72	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₂₆₉	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
78	C₂²×C₂₆₉	—	2	4	no	—
79	C₂²×C₂₆₉	—	2	4	no	—
80	C₂²×C₂₆₉	—	2	8	no	—
81	C₂²×C₂₆₉	—	2	8	no	—
82	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	8	yes	—
83	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₂₆₉	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₂₆₉	C₂×#41	1	2	yes	—
87	C₂²×C₂₆₉	H(5,𝔽₂)	1	2	yes	—
88	C₄×C₂₆₉	—	2	2	no	—
89	C₄×C₂₆₉	—	2	2	no	—
90	C₄×C₂₆₉	—	2	4	no	—
91	C₄×C₂₆₉	C₂×C₄⋊C₄	1	4	yes	—
92	Dic₂₆₉	—	2	4	no	—
93	Dic₂₆₉	—	4	2	no	—
94	Dic₂₆₉	—	4	4	no	—
95	C₂×C₂₆₉	—	2	8	no	—
96	C₂×C₂₆₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₂₆₉	C₄×#41	1	1	yes	characteristic
98	C₂×C₂₆₉	#518	1	1	yes	characteristic
99	C₂×C₂₆₉	#546	1	2	yes	—
100	C₂×C₂₆₉	#554	1	2	yes	—
101	C₂₆₉	#2349	1	1	yes	characteristic, complemented, Sylow
102	#2349	—	269	1	no	maximal, Sylow
103	C₂×C₄×D₄	—	269	2	no	—
104	C₂×C₄×D₄	D₂₆₉	1	1	yes	characteristic
105	#389	—	269	2	no	—
106	#385	—	269	4	no	—
107	#382	—	269	1	no	—
108	#380	—	269	2	no	—
109	C₄×C₄⋊C₄	—	269	1	no	—
110	C₂×#95	—	269	2	no	—
111	C₂²×D₄	C₂×D₂₆₉	1	1	yes	characteristic
112	C₂³×C₄	—	269	4	no	—
113	C₂³×C₄	C₂×D₂₆₉	1	2	yes	—
114	C₄×D₄	—	538	8	no	—
115	C₄×D₄	Dic₂₆₉	1	8	yes	complemented
116	C₂×C₄⋊C₄	—	269	2	no	—
117	C₂×C₄⋊C₄	—	269	2	no	—
118	C₂×C₄⋊C₄	—	269	4	no	—
119	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
120	C₂×C₂²⋊C₄	—	269	4	no	—
121	C₂×C₂²⋊C₄	C₂×D₂₆₉	1	2	yes	—
122	C₂×C₄²	—	269	2	no	—
123	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
124	#95	—	269	1	no	—
125	#95	—	269	2	no	—
126	#95	—	269	4	no	—
127	#95	—	269	1	no	—
128	#95	—	269	2	no	—
129	C₂⁴	C₂²×D₂₆₉	1	2	yes	—
130	C₂×D₄	C₂×Dic₂₆₉	1	4	yes	—
131	C₂×D₄	D₁₀₇₆	1	4	yes	—
132	C₂×D₄	Dic₅₃₈	1	4	yes	—
133	C₂²×C₄	—	4	4	no	—
134	C₂²×C₄	—	269	2	no	—
135	C₂²×C₄	—	269	4	no	—
136	C₂²×C₄	—	269	2	no	—
137	C₂²×C₄	—	538	8	no	—
138	C₂²×C₄	—	538	8	no	—
139	C₂²×C₄	—	1076	4	no	—
140	C₂²×C₄	C₂×Dic₂₆₉	1	8	yes	complemented
141	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
144	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
145	C₄⋊C₄	—	538	4	no	—
146	C₄⋊C₄	—	538	4	no	—
147	C₄⋊C₄	—	1076	4	no	—
148	C₄⋊C₄	C₂×Dic₂₆₉	1	4	yes	complemented
149	C₂²⋊C₄	—	538	8	no	—
150	C₂²⋊C₄	C₂×Dic₂₆₉	1	8	yes	complemented
151	C₄²	—	538	4	no	—
152	C₄²	C₂×Dic₂₆₉	1	4	yes	complemented
153	C₂³	—	2	4	no	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂²×Dic₂₆₉	1	4	yes	—
156	C₂³	C₂×D₁₀₇₆	1	4	yes	—
157	C₂³	C₂×Dic₅₃₈	1	4	yes	—
158	C₂³	C₂³×D₂₆₉	1	1	yes	characteristic
159	D₄	C₄⋊Dic₂₆₉	1	16	yes	complemented
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	269	4	no	—
165	C₂×C₄	—	538	2	no	—
166	C₂×C₄	—	538	2	no	—
167	C₂×C₄	—	538	2	no	—
168	C₂×C₄	—	538	4	no	—
169	C₂×C₄	—	538	2	no	—
170	C₂×C₄	—	538	4	no	—
171	C₂×C₄	—	538	2	no	—
172	C₂×C₄	—	538	4	no	—
173	C₂×C₄	—	1076	16	no	—
174	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
175	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
176	C₂×C₄	C₂²×Dic₂₆₉	1	4	yes	—
177	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
178	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
179	C₂×C₄	C₂×Dic₅₃₈	1	2	yes	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	C₂×C₄⋊Dic₂₆₉	1	8	yes	—
185	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
186	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
187	C₂²	C₂²×Dic₅₃₈	1	1	yes	characteristic
188	C₂²	#41×D₂₆₉	1	2	yes	—
189	C₂²	#22575498	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	4	no	—
193	C₄	—	538	4	no	—
194	C₄	—	1076	4	no	—
195	C₄	—	1076	2	no	—
196	C₄	C₂×C₄⋊Dic₂₆₉	1	4	yes	—
197	C₂	—	2	8	no	—
198	C₂	C₂²×C₄⋊Dic₂₆₉	1	1	yes	characteristic
199	C₂	#41×Dic₂₆₉	1	1	yes	characteristic
200	C₂	#31771080	1	1	yes	characteristic
201	C₂	#31771250	1	2	yes	—
202	C₂	#31771314	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₆₉
commutator	C₂²×C₂₆₉	C₂³×C₄
Fitting	C₂×C₄×D₄×C₂₆₉	C₂
socle, cosocle	C₂³×C₂₆₉	C₂⁴
prime radical	C₂²×D₄×C₂₆₉	C₂²
hypercenter	C₂×C₄×D₄	D₂₆₉
hypocenter	C₂₆₉	#2349
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #31770504	C₂	semidirect
3. #31770480	C₂	semidirect
4. #31770471	C₂	semidirect
5. #31770465	C₂	semidirect
6. C₄×C₄⋊Dic₂₆₉	C₂	semidirect
7. C₂×#22575360	C₂	semidirect
23. C₄×D₄×C₂₆₉	C₄	semidirect
43. C₂²×C₄×C₂₆₉	C₂×C₄	semidirect
48. C₄⋊C₄×C₂₆₉	C₂×C₄	semidirect
49. C₂²⋊C₄×C₂₆₉	C₂×C₄	semidirect
50. C₄²×C₂₆₉	C₂×C₄	semidirect
67. D₄×C₂₆₉	C₄⋊C₄	semidirect
101. C₂₆₉	#2349	semidirect
115. C₄×D₄	Dic₂₆₉	semidirect
140. C₂²×C₄	C₂×Dic₂₆₉	semidirect
148. C₄⋊C₄	C₂×Dic₂₆₉	semidirect
150. C₂²⋊C₄	C₂×Dic₂₆₉	semidirect
152. C₄²	C₂×Dic₂₆₉	semidirect
159. D₄	C₄⋊Dic₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₂₆₉	2
#31770504	2
#31770480	2
#31770471	2
#31770465	2
C₄×C₄⋊Dic₂₆₉	2
C₂×#22575360	2
C₂×C₄×D₄×C₂₆₉	2
#2349	269