Subgroups — #46820024

Order 34432.

356 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770736	C₂	1	2	yes	maximal, complemented
2	#31770677	C₂	1	2	yes	maximal, complemented
3	#31770677	C₂	1	2	yes	maximal, complemented
4	D₄×#14524130	C₂	1	2	yes	maximal, complemented
5	#31770666	C₂	1	2	yes	maximal, complemented
6	#31770662	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770632	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊#14524130	C₂	1	1	yes	maximal, characteristic, complemented
9	#529×C₂₆₉	C₂	1	1	yes	maximal, characteristic
10	C₂×#16864386	C₂²	1	2	yes	—
11	C₂²×#14524130	C₂²	1	1	yes	characteristic
12	C₂²×#14524130	C₂²	1	1	yes	characteristic
13	C₂²×#14524130	C₂²	1	2	yes	—
14	C₄⋊#14524130	C₂²	1	2	yes	—
15	C₄⋊#14524130	C₂²	1	2	yes	—
16	C₄⋊#14524130	C₂²	1	1	yes	characteristic
17	C₄⋊#14524130	C₂²	1	2	yes	—
18	C₄⋊#14524130	C₂²	1	2	yes	—
19	C₄⋊#14524130	C₂²	1	1	yes	characteristic
20	C₂²⋊#14524130	C₂²	1	2	yes	—
21	C₂²⋊#14524130	C₂²	1	2	yes	—
22	C₂²⋊#14524130	C₂²	1	2	yes	—
23	#22575364	C₂²	1	2	yes	—
24	#22575364	C₂²	1	2	yes	—
25	C₄×#14524130	C₂²	1	2	yes	—
26	#126×C₂₆₉	C₄	1	2	yes	—
27	#125×C₂₆₉	C₄	1	1	yes	characteristic
28	#124×C₂₆₉	C₄	1	1	yes	characteristic
29	#123×C₂₆₉	C₄	1	2	yes	—
30	C₂²⋊Q₈×C₂₆₉	C₄	1	1	yes	characteristic
31	#121×C₂₆₉	C₄	1	1	yes	characteristic
32	C₄×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₂₆₉	C₂²	1	2	yes	—
34	C₄×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
35	#117×C₂₆₉	C₂²	1	2	yes	—
36	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
37	#16864386	—	4	2	no	—
38	C₂×#14524130	—	4	1	no	—
39	C₂×#14524130	—	4	1	no	—
40	C₂×#14524130	—	4	2	no	—
41	C₂×#14524130	C₂³	1	2	yes	—
42	C₂×#14524130	C₂³	1	2	yes	—
43	C₂×#14524130	C₂³	1	2	yes	—
44	C₂×#14524130	C₂³	1	2	yes	—
45	C₂×Q₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₂₆₉	C₂×C₄	1	2	yes	—
47	C₂×D₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
52	C₄⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
62	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
63	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₂₆₉	C₂³	1	2	yes	—
66	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
67	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
68	#14524130	—	2	2	no	—
69	#14524130	—	2	2	no	—
70	#14524130	—	2	2	no	—
71	#14524130	—	2	2	no	—
72	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
73	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₂₆₉	—	4	1	no	—
75	D₄×C₂₆₉	—	2	4	no	—
76	D₄×C₂₆₉	—	4	1	no	—
77	C₂×C₄×C₂₆₉	—	2	2	no	—
78	C₂×C₄×C₂₆₉	—	2	1	no	—
79	C₂×C₄×C₂₆₉	—	2	2	no	—
80	C₂×C₄×C₂₆₉	—	2	1	no	—
81	C₂×C₄×C₂₆₉	—	2	2	no	—
82	C₂×C₄×C₂₆₉	—	4	1	no	—
83	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
92	C₂²×C₂₆₉	—	2	1	no	—
93	C₂²×C₂₆₉	—	2	1	no	—
94	C₂²×C₂₆₉	—	2	2	no	—
95	C₂²×C₂₆₉	—	2	2	no	—
96	C₂²×C₂₆₉	—	2	2	no	—
97	C₂²×C₂₆₉	#131	1	2	yes	—
98	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
99	C₄×C₂₆₉	—	2	1	no	—
100	C₄×C₂₆₉	—	2	2	no	—
101	C₄×C₂₆₉	—	2	1	no	—
102	C₄×C₂₆₉	—	2	1	no	—
103	C₄×C₂₆₉	—	2	2	no	—
104	C₄×C₂₆₉	—	2	1	no	—
105	C₄×C₂₆₉	—	2	1	no	—
106	C₄×C₂₆₉	—	2	2	no	—
107	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
108	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₂₆₉	—	2	2	no	—
110	C₂×C₂₆₉	—	4	2	no	—
111	C₂×C₂₆₉	#520	1	1	yes	characteristic
112	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
113	C₂×C₂₆₉	#568	1	1	yes	characteristic
114	C₂₆₉	#2992	1	1	yes	characteristic, complemented, Sylow
115	#2992	—	269	1	no	maximal, Sylow
116	#529	D₂₆₉	1	1	yes	characteristic
117	#446	—	269	2	no	—
118	#435	—	269	2	no	—
119	#435	—	269	2	no	—
120	C₈×D₄	—	269	2	no	—
121	#433	—	269	2	no	—
122	#432	—	269	1	no	—
123	#424	—	269	2	no	—
124	C₂×C₄⋊C₈	—	269	1	no	—
125	C₂×M₁₆	—	269	2	no	—
126	C₂²×C₈	—	269	1	no	—
127	C₂²×C₈	—	269	1	no	—
128	C₂²×C₈	—	269	2	no	—
129	#126	Dic₂₆₉	1	2	yes	—
130	#125	Dic₂₆₉	1	1	yes	characteristic
131	#124	Dic₂₆₉	1	1	yes	characteristic
132	#123	Dic₂₆₉	1	2	yes	—
133	C₂²⋊Q₈	Dic₂₆₉	1	1	yes	characteristic
134	#121	Dic₂₆₉	1	1	yes	characteristic
135	C₄×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
136	C₄×D₄	C₂×D₂₆₉	1	2	yes	—
137	C₄×D₄	C₂×D₂₆₉	1	1	yes	characteristic
138	#117	C₂×D₂₆₉	1	2	yes	—
139	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
140	C₄⋊C₈	—	269	2	no	—
141	C₄⋊C₈	—	269	2	no	—
142	C₄⋊C₈	—	269	1	no	—
143	C₄⋊C₈	—	269	2	no	—
144	C₄⋊C₈	—	269	2	no	—
145	C₄⋊C₈	—	269	1	no	—
146	C₂²⋊C₈	—	269	2	no	—
147	C₂²⋊C₈	—	269	2	no	—
148	C₂²⋊C₈	—	269	2	no	—
149	#97	—	269	2	no	—
150	#97	—	269	2	no	—
151	C₄×C₈	—	269	2	no	—
152	C₂×Q₈	C₂×Dic₂₆₉	1	1	yes	characteristic
153	C₂×D₄	C₂×Dic₂₆₉	1	2	yes	—
154	C₂×D₄	C₂×Dic₂₆₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
159	M₁₆	—	1076	2	no	—
160	C₂×C₈	—	269	2	no	—
161	C₂×C₈	—	269	2	no	—
162	C₂×C₈	—	269	2	no	—
163	C₂×C₈	—	269	2	no	—
164	C₂×C₈	—	1076	1	no	—
165	C₂×C₈	—	1076	2	no	—
166	C₂×C₈	—	1076	1	no	—
167	C₄⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
168	C₄⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
169	C₄⋊C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
170	C₄⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
171	C₄⋊C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
172	C₄⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
173	C₂²⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
174	C₂²⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
175	C₂²⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
176	C₂²⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₂₆₉	1	2	yes	—
178	C₄²	C₂×Dic₂₆₉	1	1	yes	characteristic
179	C₄²	C₂×Dic₂₆₉	1	1	yes	characteristic
180	C₄²	C₂²×D₂₆₉	1	2	yes	—
181	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
182	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
183	C₂³	C₂²×Dic₂₆₉	1	2	yes	—
184	C₂³	C₂²×Dic₂₆₉	1	1	yes	characteristic
185	Q₈	—	4	1	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
198	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
199	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
200	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
201	C₂×C₄	C₂²×Dic₂₆₉	1	2	yes	—
202	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
203	C₈	—	538	2	no	—
204	C₈	—	538	2	no	—
205	C₈	—	538	2	no	—
206	C₈	—	538	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	#22575460	1	2	yes	—
213	C₂²	C₂³×Dic₂₆₉	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	2	no	—
222	C₄	#22575499	1	1	yes	characteristic
223	C₄	#22575500	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	2	no	—
226	C₂	#31771088	1	1	yes	characteristic
227	C₂	C₂×#22575460	1	1	yes	characteristic
228	C₂	#31771372	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting	#529×C₂₆₉	C₂
prime radical	#121×C₂₆₉	C₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#529	D₂₆₉
hypocenter	C₂₆₉	#2992
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #31770736	C₂	semidirect
2. #31770677	C₂	semidirect
3. #31770677	C₂	semidirect
4. D₄×#14524130	C₂	semidirect
5. #31770666	C₂	semidirect
6. #31770662	C₂	semidirect
7. #31770632	C₂	semidirect
8. C₂×C₄⋊#14524130	C₂	semidirect
114. C₂₆₉	#2992	semidirect

Maximal subgroups

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

subgroup	index
#31770736	2
#31770677	2
D₄×#14524130	2
#31770666	2
#31770662	2
#31770632	2
C₂×C₄⋊#14524130	2
#529×C₂₆₉	2
#2992	269