Subgroups — #46820012

Order 34432.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#16864402	C₂	1	1	yes	maximal, characteristic
2	#31770733	C₂	1	2	yes	maximal, complemented
3	#31770678	C₂	1	2	yes	maximal, complemented
4	#31770673	C₂	1	2	yes	maximal, complemented
5	#31770668	C₂	1	2	yes	maximal, complemented
6	#31770661	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770657	C₂	1	1	yes	maximal, characteristic, complemented
8	#31770627	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊#14524130	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#432×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16864402	C₄	1	1	yes	characteristic
13	C₂×#16864385	C₂²	1	2	yes	—
14	C₂²×#14524130	C₂²	1	2	yes	—
15	C₂×C₈×D₂₆₉	C₂²	1	2	yes	—
16	C₄×Dic₅₃₈	C₄	1	1	yes	characteristic
17	C₄×Dic₅₃₈	C₂²	1	1	yes	characteristic
18	C₄×D₁₀₇₆	C₄	1	1	yes	characteristic
19	C₄×D₁₀₇₆	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₂₆₉	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₂₆₉	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic
24	#22575412	C₄	1	1	yes	characteristic
25	#22575410	C₄	1	1	yes	characteristic
26	#22575410	C₄	1	1	yes	characteristic
27	C₄²×D₂₆₉	C₂²	1	1	yes	characteristic
28	C₄²×D₂₆₉	C₂²	1	1	yes	characteristic
29	C₄⋊#14524130	C₂²	1	1	yes	characteristic
30	C₄⋊#14524130	C₂²	1	1	yes	characteristic
31	C₄⋊#14524130	C₂²	1	2	yes	—
32	#22575382	C₂²	1	2	yes	—
33	#22575382	C₂²	1	2	yes	—
34	#22575365	C₂²	1	2	yes	—
35	#22575365	C₂²	1	2	yes	—
36	#22575363	C₂²	1	2	yes	—
37	C₈×Dic₂₆₉	C₂²	1	2	yes	—
38	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₂₆₉	C₂²	1	2	yes	—
42	#97×C₂₆₉	C₂²	1	2	yes	—
43	#16864402	—	2	4	no	—
44	C₂×Dic₅₃₈	C₂×C₄	1	1	yes	characteristic
45	C₂×D₁₀₇₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
54	#16864385	—	4	2	no	—
55	C₂×#14524130	—	4	2	no	—
56	C₂×#14524130	C₂³	1	2	yes	—
57	C₂×#14524130	C₂³	1	2	yes	—
58	C₈×D₂₆₉	—	4	2	no	—
59	C₄⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
61	#16864381	C₂×C₄	1	1	yes	characteristic
62	#16864381	C₂×C₄	1	1	yes	characteristic
63	#16864381	C₂×C₄	1	1	yes	characteristic
64	#16864381	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
67	#16864379	C₂×C₄	1	1	yes	characteristic
68	#16864379	C₂×C₄	1	1	yes	characteristic
69	#16864379	C₂×C₄	1	1	yes	characteristic
70	#16864379	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
76	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
81	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
82	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
85	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
86	C₂²×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
88	Dic₅₃₈	—	2	2	no	—
89	D₁₀₇₆	—	2	2	no	—
90	C₂²⋊D₂₆₉	—	2	2	no	—
91	C₂²⋊D₂₆₉	M₁₆	1	4	yes	complemented
92	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
98	C₄×D₂₆₉	—	2	2	no	—
99	C₄×D₂₆₉	—	2	2	no	—
100	C₄×D₂₆₉	—	2	2	no	—
101	C₄×D₂₆₉	—	2	1	no	—
102	C₄×D₂₆₉	—	2	1	no	—
103	C₄×D₂₆₉	—	2	1	no	—
104	C₄×D₂₆₉	—	2	1	no	—
105	C₄×D₂₆₉	—	2	2	no	—
106	#14524130	—	2	2	no	—
107	#14524130	—	2	2	no	—
108	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₆₉	—	2	2	no	—
110	C₂×C₄×C₂₆₉	—	2	2	no	—
111	C₂×C₄×C₂₆₉	—	2	1	no	—
112	C₂×C₄×C₂₆₉	—	2	1	no	—
113	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
119	C₈×C₂₆₉	—	2	2	no	—
120	C₈×C₂₆₉	—	2	2	no	—
121	C₂²×C₂₆₉	—	2	1	no	—
122	C₂²×C₂₆₉	—	2	1	no	—
123	C₂²×C₂₆₉	C₂×M₁₆	1	2	yes	—
124	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₂₆₉	—	2	1	no	—
126	C₂×D₂₆₉	—	2	1	no	—
127	C₂×D₂₆₉	—	2	1	no	—
128	C₂×D₂₆₉	—	2	1	no	—
129	C₂×D₂₆₉	—	2	1	no	—
130	C₂×D₂₆₉	C₂×M₁₆	1	2	yes	—
131	C₄×C₂₆₉	—	2	1	no	—
132	C₄×C₂₆₉	—	2	1	no	—
133	C₄×C₂₆₉	—	2	1	no	—
134	C₄×C₂₆₉	—	2	1	no	—
135	C₄×C₂₆₉	#131	1	2	yes	—
136	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
137	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
138	Dic₂₆₉	—	2	1	no	—
139	Dic₂₆₉	—	2	1	no	—
140	Dic₂₆₉	—	2	1	no	—
141	Dic₂₆₉	—	2	1	no	—
142	Dic₂₆₉	—	2	1	no	—
143	Dic₂₆₉	C₂×M₁₆	1	2	yes	—
144	C₂×C₂₆₉	—	2	2	no	—
145	C₂×C₂₆₉	#520	1	1	yes	characteristic
146	C₂×C₂₆₉	C₂²×M₁₆	1	1	yes	characteristic
147	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
148	D₂₆₉	—	2	2	no	—
149	D₂₆₉	—	4	1	no	—
150	C₂₆₉	#2990	1	1	yes	characteristic, complemented, Sylow
151	#2990	—	269	1	no	maximal, Sylow
152	C₄×#41	—	269	1	no	—
153	#446	—	269	2	no	—
154	#435	—	269	2	no	—
155	#435	—	269	2	no	—
156	C₈×D₄	—	269	2	no	—
157	#432	—	269	1	no	—
158	#432	D₂₆₉	1	1	yes	characteristic, complemented
159	#431	—	269	1	no	—
160	#423	—	269	1	no	—
161	C₂×C₄⋊C₈	—	269	1	no	—
162	C₂×C₄⋊C₈	—	269	1	no	—
163	C₂×#41	—	269	1	no	—
164	C₂×M₁₆	—	269	2	no	—
165	C₂²×C₈	—	269	2	no	—
166	C₂²×C₈	—	269	2	no	—
167	C₄×Q₈	—	269	1	no	—
168	C₄×Q₈	—	269	1	no	—
169	C₄×D₄	—	269	1	no	—
170	C₄×D₄	—	269	1	no	—
171	C₄×D₄	—	269	1	no	—
172	C₄×D₄	—	269	1	no	—
173	C₄×D₄	—	269	1	no	—
174	C₄×D₄	—	269	1	no	—
175	#117	—	269	1	no	—
176	#117	—	269	1	no	—
177	#117	—	269	1	no	—
178	C₂×C₄²	—	269	1	no	—
179	C₂×C₄²	—	269	1	no	—
180	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
181	C₄⋊C₈	—	269	2	no	—
182	C₄⋊C₈	—	269	2	no	—
183	C₄⋊C₈	—	269	1	no	—
184	C₄⋊C₈	—	269	1	no	—
185	C₄⋊C₈	—	269	2	no	—
186	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
187	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
188	C₂²⋊C₈	—	269	2	no	—
189	C₂²⋊C₈	—	269	2	no	—
190	C₂²⋊C₈	C₂×D₂₆₉	1	2	yes	—
191	#97	—	269	2	no	—
192	#97	C₂×D₂₆₉	1	2	yes	—
193	C₄×C₈	—	269	2	no	—
194	#41	—	538	4	no	—
195	C₂×Q₈	—	269	1	no	—
196	C₂×D₄	—	269	1	no	—
197	C₂×D₄	—	269	1	no	—
198	C₂×D₄	—	269	1	no	—
199	C₂²×C₄	—	269	1	no	—
200	C₂²×C₄	—	269	1	no	—
201	C₂²×C₄	—	269	1	no	—
202	C₂²×C₄	—	269	1	no	—
203	C₂²×C₄	—	269	1	no	—
204	C₂²×C₄	—	269	1	no	—
205	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₂₆₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
208	M₁₆	—	1076	2	no	—
209	C₂×C₈	—	269	2	no	—
210	C₂×C₈	—	269	2	no	—
211	C₂×C₈	—	1076	2	no	—
212	C₂×C₈	—	1076	2	no	—
213	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
214	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
215	C₄⋊C₄	—	269	1	no	—
216	C₄⋊C₄	—	269	1	no	—
217	C₄⋊C₄	—	269	1	no	—
218	C₄⋊C₄	—	269	1	no	—
219	C₄⋊C₄	—	269	1	no	—
220	C₄⋊C₄	—	269	1	no	—
221	C₂²⋊C₄	—	269	1	no	—
222	C₂²⋊C₄	—	269	1	no	—
223	C₂²⋊C₄	—	269	1	no	—
224	C₂²⋊C₄	—	269	1	no	—
225	C₂²⋊C₄	—	269	1	no	—
226	C₂²⋊C₄	—	269	1	no	—
227	C₄²	—	269	1	no	—
228	C₄²	—	269	1	no	—
229	C₄²	—	269	1	no	—
230	C₄²	—	269	1	no	—
231	C₄²	—	269	1	no	—
232	C₄²	—	269	1	no	—
233	C₄²	C₄×D₂₆₉	1	1	yes	characteristic
234	C₄²	C₄×D₂₆₉	1	1	yes	characteristic
235	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
236	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
237	C₂³	—	269	1	no	—
238	C₂³	—	269	1	no	—
239	C₂³	C₂×C₄×D₂₆₉	1	1	yes	characteristic
240	Q₈	—	538	2	no	—
241	D₄	—	269	4	no	—
242	D₄	—	538	2	no	—
243	D₄	—	538	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	269	1	no	—
249	C₂×C₄	—	269	1	no	—
250	C₂×C₄	—	269	1	no	—
251	C₂×C₄	—	269	1	no	—
252	C₂×C₄	—	269	1	no	—
253	C₂×C₄	—	269	1	no	—
254	C₂×C₄	—	538	2	no	—
255	C₂×C₄	—	538	2	no	—
256	C₂×C₄	—	538	2	no	—
257	C₂×C₄	—	538	2	no	—
258	C₂×C₄	—	538	1	no	—
259	C₂×C₄	—	538	1	no	—
260	C₂×C₄	—	538	1	no	—
261	C₂×C₄	—	538	1	no	—
262	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
267	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	538	2	no	—
271	C₈	—	538	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	269	2	no	—
275	C₂²	—	538	1	no	—
276	C₂²	—	538	1	no	—
277	C₂²	—	538	1	no	—
278	C₂²	—	538	1	no	—
279	C₂²	—	538	1	no	—
280	C₂²	M₁₆×D₂₆₉	1	2	yes	—
281	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	269	2	no	—
287	C₄	—	538	1	no	—
288	C₄	—	538	1	no	—
289	C₄	—	538	1	no	—
290	C₄	—	538	1	no	—
291	C₄	—	538	1	no	—
292	C₄	#22575461	1	2	yes	—
293	C₄	#22575499	1	1	yes	characteristic
294	C₄	#22575500	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	538	2	no	—
297	C₂	—	1076	1	no	—
298	C₂	#31771085	1	1	yes	characteristic
299	C₂	C₂×M₁₆×D₂₆₉	1	1	yes	characteristic
300	C₂	C₂×#22575461	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting	#432×C₂₆₉	C₂
prime radical	C₂×#16864402	C₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#432	D₂₆₉
hypocenter	C₂₆₉	#2990
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #31770733	C₂	semidirect
3. #31770678	C₂	semidirect
4. #31770673	C₂	semidirect
5. #31770668	C₂	semidirect
6. #31770661	C₂	semidirect
7. #31770657	C₂	semidirect
8. #31770627	C₂	semidirect
9. C₂×C₄⋊#14524130	C₂	semidirect
10. C₄⋊C₈×D₂₆₉	C₂	semidirect
11. #432×C₂₆₉	C₂	semidirect
91. C₂²⋊D₂₆₉	M₁₆	semidirect
150. C₂₆₉	#2990	semidirect
158. #432	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#16864402	2
#31770733	2
#31770678	2
#31770673	2
#31770668	2
#31770661	2
#31770657	2
#31770627	2
C₂×C₄⋊#14524130	2
C₄⋊C₈×D₂₆₉	2
#432×C₂₆₉	2
#2990	269