Subgroups — #46819962

Order 34432.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771152	C₂	1	1	yes	maximal, characteristic
2	#31770684	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770674	C₂	1	2	yes	maximal, complemented
4	#31770673	C₂	1	2	yes	maximal, complemented
5	D₄×#14524130	C₂	1	1	yes	maximal, characteristic, complemented
6	#31770670	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770661	C₂	1	1	yes	maximal, characteristic, complemented
8	#31770656	C₂	1	1	yes	maximal, characteristic, complemented
9	#31770573	C₂	1	2	yes	maximal, complemented
10	#31770570	C₂	1	2	yes	maximal, complemented
11	#436×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16864402	C₂²	1	2	yes	—
13	C₂×#16864385	C₂²	1	2	yes	—
14	C₂²×#14524130	C₂²	1	2	yes	—
15	C₂×C₈×D₂₆₉	C₂²	1	2	yes	—
16	#22575452	C₄	1	1	yes	characteristic
17	#22575449	C₄	1	1	yes	characteristic
18	#22575439	C₄	1	2	yes	—
19	#22575428	C₄	1	1	yes	characteristic
20	#22575426	C₄	1	2	yes	—
21	#22575424	C₄	1	1	yes	characteristic
22	C₄×D₁₀₇₆	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₂₆₉	C₂²	1	2	yes	—
24	C₄²×D₂₆₉	C₂²	1	1	yes	characteristic
25	C₄⋊#14524130	C₂²	1	1	yes	characteristic
26	#22575382	C₂²	1	1	yes	characteristic
27	#22575382	C₂²	1	1	yes	characteristic
28	C₂²⋊#14524130	C₂²	1	2	yes	—
29	#22575365	C₂²	1	1	yes	characteristic
30	#22575365	C₂²	1	1	yes	characteristic
31	#22575365	C₂²	1	2	yes	—
32	#22575365	C₂²	1	2	yes	—
33	#22575365	C₂²	1	2	yes	—
34	#22575363	C₂²	1	2	yes	—
35	C₄×#14524130	C₂²	1	1	yes	characteristic
36	C₂×M₁₆×C₂₆₉	C₂²	1	2	yes	—
37	C₄×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₂₆₉	C₂²	1	2	yes	—
40	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
41	#16864402	—	4	2	no	—
42	#16864402	—	4	2	no	—
43	C₂×Dic₅₃₈	C₂×C₄	1	2	yes	—
44	C₂×D₁₀₇₆	C₂×C₄	1	1	yes	characteristic
45	C₂×D₁₀₇₆	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₂₆₉	C₂³	1	2	yes	—
53	#16864385	—	4	2	no	—
54	C₂×#14524130	—	4	2	no	—
55	C₂×#14524130	C₂³	1	1	yes	characteristic
56	C₂×#14524130	C₂³	1	1	yes	characteristic
57	C₂×#14524130	C₂³	1	2	yes	—
58	C₈×D₂₆₉	—	4	2	no	—
59	C₄⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
60	#16864381	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₂₆₉	C₂×C₄	1	2	yes	—
62	#16864379	C₂×C₄	1	2	yes	—
63	#16864379	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₆₉	C₂³	1	2	yes	—
69	M₁₆×C₂₆₉	—	4	2	no	—
70	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
71	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	2	yes	—
75	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
76	C₂²×D₂₆₉	C₂²×C₄	1	2	yes	—
77	C₂²×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
78	Dic₅₃₈	—	4	2	no	—
79	D₁₀₇₆	—	2	2	no	—
80	D₁₀₇₆	—	4	2	no	—
81	C₂²⋊D₂₆₉	—	4	2	no	—
82	C₂²⋊D₂₆₉	—	4	2	no	—
83	C₂²⋊D₂₆₉	—	4	2	no	—
84	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₂₆₉	C₂²×C₄	1	2	yes	—
88	C₄×D₂₆₉	—	2	2	no	—
89	C₄×D₂₆₉	—	2	2	no	—
90	C₄×D₂₆₉	—	2	2	no	—
91	C₄×D₂₆₉	—	2	1	no	—
92	C₄×D₂₆₉	—	2	1	no	—
93	C₄×D₂₆₉	—	4	1	no	—
94	#14524130	—	2	1	no	—
95	#14524130	—	2	1	no	—
96	#14524130	—	2	2	no	—
97	C₂³×C₂₆₉	C₂²×C₄	1	2	yes	—
98	D₄×C₂₆₉	—	2	2	no	—
99	C₂×C₄×C₂₆₉	—	2	2	no	—
100	C₂×C₄×C₂₆₉	—	2	2	no	—
101	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
105	C₈×C₂₆₉	—	2	1	no	—
106	C₈×C₂₆₉	—	2	2	no	—
107	C₈×C₂₆₉	—	2	1	no	—
108	C₂²×C₂₆₉	—	2	2	no	—
109	C₂²×C₂₆₉	—	2	2	no	—
110	C₂²×C₂₆₉	—	2	2	no	—
111	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₂₆₉	—	2	1	no	—
113	C₂×D₂₆₉	—	2	2	no	—
114	C₂×D₂₆₉	—	2	1	no	—
115	C₂×D₂₆₉	—	2	2	no	—
116	C₂×D₂₆₉	—	2	1	no	—
117	C₂×D₂₆₉	—	2	2	no	—
118	C₄×C₂₆₉	—	2	2	no	—
119	C₄×C₂₆₉	—	2	1	no	—
120	C₄×C₂₆₉	#131	1	2	yes	—
121	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₂₆₉	—	2	1	no	—
124	Dic₂₆₉	—	2	2	no	—
125	Dic₂₆₉	—	2	1	no	—
126	Dic₂₆₉	—	2	1	no	—
127	C₂×C₂₆₉	—	4	2	no	—
128	C₂×C₂₆₉	#518	1	1	yes	characteristic
129	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
130	C₂×C₂₆₉	#568	1	1	yes	characteristic
131	D₂₆₉	—	4	1	no	—
132	D₂₆₉	—	4	2	no	—
133	C₂₆₉	#2981	1	1	yes	characteristic, complemented, Sylow
134	#2981	—	269	1	no	maximal, Sylow
135	#532	—	269	1	no	—
136	#436	—	269	1	no	—
137	#436	D₂₆₉	1	1	yes	characteristic, complemented
138	#435	—	269	2	no	—
139	#435	—	269	2	no	—
140	C₈×D₄	—	269	1	no	—
141	C₈×D₄	—	269	1	no	—
142	#432	—	269	1	no	—
143	#431	—	269	1	no	—
144	#408	—	269	2	no	—
145	#408	—	269	2	no	—
146	C₂×#41	—	269	2	no	—
147	C₂×M₁₆	—	269	2	no	—
148	C₂×M₁₆	C₂×D₂₆₉	1	2	yes	—
149	C₂²×C₈	—	269	2	no	—
150	C₂²×C₈	—	269	2	no	—
151	C₄⋊Q₈	—	269	1	no	—
152	#127	—	269	1	no	—
153	#124	—	269	2	no	—
154	#121	—	269	1	no	—
155	#121	—	269	1	no	—
156	#121	—	269	2	no	—
157	C₄×D₄	—	269	2	no	—
158	C₄×D₄	—	269	1	no	—
159	C₄×D₄	C₂×D₂₆₉	1	1	yes	characteristic
160	C₂×C₄²	—	269	1	no	—
161	C₄⋊C₈	—	269	1	no	—
162	C₄⋊C₈	—	269	1	no	—
163	C₄⋊C₈	—	269	1	no	—
164	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
165	C₂²⋊C₈	—	269	2	no	—
166	C₂²⋊C₈	—	269	1	no	—
167	C₂²⋊C₈	—	269	1	no	—
168	C₂²⋊C₈	—	269	2	no	—
169	C₂²⋊C₈	—	269	2	no	—
170	C₂²⋊C₈	—	269	2	no	—
171	C₂²⋊C₈	C₂×D₂₆₉	1	2	yes	—
172	#97	—	269	2	no	—
173	C₄×C₈	—	269	1	no	—
174	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
175	#41	—	1076	2	no	—
176	#41	—	1076	2	no	—
177	C₂×Q₈	—	269	2	no	—
178	C₂×D₄	—	269	2	no	—
179	C₂×D₄	—	269	2	no	—
180	C₂×D₄	—	269	2	no	—
181	C₂×D₄	—	269	1	no	—
182	C₂×D₄	—	269	2	no	—
183	C₂×D₄	C₄×D₂₆₉	1	1	yes	characteristic
184	C₂²×C₄	—	269	1	no	—
185	C₂²×C₄	—	269	1	no	—
186	C₂²×C₄	—	269	2	no	—
187	C₂²×C₄	—	269	1	no	—
188	C₂²×C₄	C₂²×D₂₆₉	1	2	yes	—
189	M₁₆	—	4	2	no	—
190	M₁₆	—	1076	2	no	—
191	C₂×C₈	—	269	1	no	—
192	C₂×C₈	—	269	1	no	—
193	C₂×C₈	—	269	2	no	—
194	C₂×C₈	—	1076	2	no	—
195	C₂×C₈	—	1076	2	no	—
196	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
199	C₄⋊C₄	—	269	2	no	—
200	C₄⋊C₄	—	269	1	no	—
201	C₄⋊C₄	C₄×D₂₆₉	1	1	yes	characteristic
202	C₂²⋊C₄	—	269	2	no	—
203	C₂²⋊C₄	—	269	2	no	—
204	C₂²⋊C₄	—	269	2	no	—
205	C₂²⋊C₄	C₄×D₂₆₉	1	2	yes	—
206	C₄²	—	269	1	no	—
207	C₄²	—	269	1	no	—
208	C₄²	—	269	1	no	—
209	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
210	C₂³	—	269	2	no	—
211	C₂³	—	269	1	no	—
212	C₂³	C₂×C₄×D₂₆₉	1	2	yes	—
213	Q₈	—	1076	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	538	2	no	—
216	D₄	—	1076	2	no	—
217	D₄	—	1076	2	no	—
218	D₄	—	1076	2	no	—
219	D₄	—	1076	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	269	1	no	—
223	C₂×C₄	—	269	1	no	—
224	C₂×C₄	—	269	1	no	—
225	C₂×C₄	—	269	2	no	—
226	C₂×C₄	—	538	2	no	—
227	C₂×C₄	—	538	2	no	—
228	C₂×C₄	—	538	1	no	—
229	C₂×C₄	—	538	2	no	—
230	C₂×C₄	—	538	1	no	—
231	C₂×C₄	—	1076	1	no	—
232	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₂₆₉	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	538	1	no	—
240	C₈	—	538	1	no	—
241	C₈	—	538	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	538	1	no	—
246	C₂²	—	538	2	no	—
247	C₂²	—	538	1	no	—
248	C₂²	—	538	2	no	—
249	C₂²	—	538	1	no	—
250	C₂²	—	538	2	no	—
251	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	538	1	no	—
255	C₄	—	538	2	no	—
256	C₄	—	538	1	no	—
257	C₄	—	538	1	no	—
258	C₄	#22575461	1	2	yes	—
259	C₄	#22575498	1	1	yes	characteristic
260	C₄	#22575499	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	1076	1	no	—
263	C₂	—	1076	2	no	—
264	C₂	#31771077	1	1	yes	characteristic
265	C₂	C₂×#22575461	1	1	yes	characteristic
266	C₂	#31771370	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting	#436×C₂₆₉	C₂
prime radical	#31771152	C₂
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#436	D₂₆₉
hypocenter	C₂₆₉	#2981
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #31770684	C₂	semidirect
3. #31770674	C₂	semidirect
4. #31770673	C₂	semidirect
5. D₄×#14524130	C₂	semidirect
6. #31770670	C₂	semidirect
7. #31770661	C₂	semidirect
8. #31770656	C₂	semidirect
9. #31770573	C₂	semidirect
10. #31770570	C₂	semidirect
11. #436×C₂₆₉	C₂	semidirect
133. C₂₆₉	#2981	semidirect
137. #436	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771152	2
#31770684	2
#31770674	2
#31770673	2
D₄×#14524130	2
#31770670	2
#31770661	2
#31770656	2
#31770573	2
#31770570	2
#436×C₂₆₉	2
#2981	269