Subgroups — #46820043

Order 34432.

364 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771133	C₂	1	1	yes	maximal, characteristic
2	#31770734	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770729	C₂	1	1	yes	maximal, characteristic, complemented
4	#31770681	C₂	1	1	yes	maximal, characteristic, complemented
5	#31770675	C₂	1	1	yes	maximal, characteristic, complemented
6	#31770674	C₂	1	1	yes	maximal, characteristic, complemented
7	#31770670	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₂²⋊D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₂²⋊D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#31770666	C₂	1	1	yes	maximal, characteristic, complemented
11	#31770665	C₂	1	1	yes	maximal, characteristic, complemented
12	#31770657	C₂	1	1	yes	maximal, characteristic, complemented
13	#31770630	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#424×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#16864385	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₆₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₂₆₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₂₆₉	C₂²	1	1	yes	characteristic
20	#22575448	C₄	1	1	yes	characteristic
21	#22575446	C₄	1	1	yes	characteristic
22	#22575443	C₄	1	1	yes	characteristic
23	#22575439	C₄	1	1	yes	characteristic
24	#22575438	C₄	1	1	yes	characteristic
25	#22575435	C₄	1	1	yes	characteristic
26	#22575431	C₄	1	1	yes	characteristic
27	#22575426	C₄	1	1	yes	characteristic
28	C₄×Dic₅₃₈	C₂²	1	1	yes	characteristic
29	C₄×D₁₀₇₆	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₂₆₉	C₂²	1	1	yes	characteristic
32	#22575410	C₂²	1	1	yes	characteristic
33	C₄²×D₂₆₉	C₂²	1	1	yes	characteristic
34	C₄⋊#14524130	C₂²	1	1	yes	characteristic
35	C₄⋊#14524130	C₂²	1	1	yes	characteristic
36	#22575382	C₂²	1	1	yes	characteristic
37	#22575382	C₂²	1	1	yes	characteristic
38	#22575382	C₂²	1	1	yes	characteristic
39	#22575382	C₂²	1	1	yes	characteristic
40	C₂²⋊#14524130	C₂²	1	1	yes	characteristic
41	C₂²⋊#14524130	C₂²	1	1	yes	characteristic
42	#22575365	C₂²	1	1	yes	characteristic
43	#22575365	C₂²	1	1	yes	characteristic
44	#22575365	C₂²	1	1	yes	characteristic
45	#22575365	C₂²	1	1	yes	characteristic
46	#22575364	C₂²	1	1	yes	characteristic
47	#22575363	C₂²	1	1	yes	characteristic
48	C₄×#14524130	C₂²	1	1	yes	characteristic
49	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic
50	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
54	#117×C₂₆₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
57	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₂₆₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₅₃₈	C₂×C₄	1	1	yes	characteristic
60	C₂×D₁₀₇₆	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₂₆₉	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₂₆₉	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₂₆₉	C₂³	1	1	yes	characteristic
67	#16864385	—	4	1	no	—
68	C₂×#14524130	C₂³	1	1	yes	characteristic
69	C₂×#14524130	C₂³	1	1	yes	characteristic
70	C₂×#14524130	C₂³	1	1	yes	characteristic
71	C₂×#14524130	C₂³	1	1	yes	characteristic
72	C₈×D₂₆₉	—	4	1	no	—
73	C₈×D₂₆₉	—	4	1	no	—
74	C₈×D₂₆₉	—	4	1	no	—
75	C₄⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
76	C₄⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
77	#16864381	C₂×C₄	1	1	yes	characteristic
78	#16864381	C₂×C₄	1	1	yes	characteristic
79	#16864381	C₂×C₄	1	1	yes	characteristic
80	#16864381	C₂×C₄	1	1	yes	characteristic
81	#16864381	C₂×C₄	1	1	yes	characteristic
82	#16864381	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
85	#16864379	C₂×C₄	1	1	yes	characteristic
86	#16864379	C₂×C₄	1	1	yes	characteristic
87	#16864379	C₂×C₄	1	1	yes	characteristic
88	#16864379	C₂×C₄	1	1	yes	characteristic
89	#16864379	C₂×C₄	1	1	yes	characteristic
90	#16864379	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
94	C₄×Dic₂₆₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
96	M₁₆×C₂₆₉	—	4	1	no	—
97	C₂×C₈×C₂₆₉	—	4	1	no	—
98	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
106	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
107	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
108	C₂²×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₂₆₉	C₂²×C₄	1	1	yes	characteristic
110	Dic₅₃₈	—	2	2	no	—
111	D₁₀₇₆	—	2	2	no	—
112	C₂²⋊D₂₆₉	—	2	2	no	—
113	C₂²⋊D₂₆₉	—	2	2	no	—
114	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
120	C₄×D₂₆₉	—	2	2	no	—
121	C₄×D₂₆₉	—	2	1	no	—
122	C₄×D₂₆₉	—	2	1	no	—
123	C₄×D₂₆₉	—	2	1	no	—
124	C₄×D₂₆₉	—	2	1	no	—
125	C₄×D₂₆₉	—	2	2	no	—
126	#14524130	—	2	1	no	—
127	#14524130	—	2	1	no	—
128	#14524130	—	2	1	no	—
129	#14524130	—	2	1	no	—
130	C₂³×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×C₂₆₉	—	2	1	no	—
132	C₂×C₄×C₂₆₉	—	2	1	no	—
133	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
139	C₈×C₂₆₉	—	2	1	no	—
140	C₈×C₂₆₉	—	2	1	no	—
141	C₈×C₂₆₉	—	2	1	no	—
142	C₈×C₂₆₉	—	2	1	no	—
143	C₂²×C₂₆₉	—	2	1	no	—
144	C₂²×C₂₆₉	—	2	1	no	—
145	C₂²×C₂₆₉	—	2	1	no	—
146	C₂²×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
147	C₂×D₂₆₉	—	2	1	no	—
148	C₂×D₂₆₉	—	2	1	no	—
149	C₂×D₂₆₉	—	2	1	no	—
150	C₂×D₂₆₉	—	2	1	no	—
151	C₂×D₂₆₉	—	2	1	no	—
152	C₂×D₂₆₉	#131	1	2	yes	—
153	C₄×C₂₆₉	—	2	1	no	—
154	C₄×C₂₆₉	—	2	1	no	—
155	C₄×C₂₆₉	—	2	1	no	—
156	C₄×C₂₆₉	—	2	1	no	—
157	C₄×C₂₆₉	—	2	1	no	—
158	C₄×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₂₆₉	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₂₆₉	—	2	1	no	—
161	Dic₂₆₉	—	2	1	no	—
162	Dic₂₆₉	—	2	1	no	—
163	Dic₂₆₉	—	2	1	no	—
164	Dic₂₆₉	—	2	1	no	—
165	Dic₂₆₉	#131	1	2	yes	—
166	C₂×C₂₆₉	—	4	1	no	—
167	C₂×C₂₆₉	#520	1	1	yes	characteristic
168	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
169	C₂×C₂₆₉	C₂×#131	1	1	yes	characteristic
170	D₂₆₉	—	2	2	no	—
171	D₂₆₉	—	4	1	no	—
172	C₂₆₉	#2994	1	1	yes	characteristic, complemented, Sylow
173	#2994	—	269	1	no	maximal, Sylow
174	#529	—	269	1	no	—
175	#446	—	269	1	no	—
176	C₈×Q₈	—	269	1	no	—
177	#436	—	269	1	no	—
178	#435	—	269	1	no	—
179	#435	—	269	1	no	—
180	C₈×D₄	—	269	1	no	—
181	C₈×D₄	—	269	1	no	—
182	C₈×D₄	—	269	1	no	—
183	#433	—	269	1	no	—
184	#433	—	269	1	no	—
185	#431	—	269	1	no	—
186	#424	—	269	1	no	—
187	#424	D₂₆₉	1	1	yes	characteristic, complemented
188	C₂×C₄⋊C₈	—	269	1	no	—
189	C₂×M₁₆	—	269	1	no	—
190	C₂×M₁₆	C₂×D₂₆₉	1	1	yes	characteristic
191	C₂²×C₈	—	269	1	no	—
192	C₂²×C₈	—	269	1	no	—
193	C₂²×C₈	—	269	1	no	—
194	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
195	#126	—	269	1	no	—
196	#126	—	269	1	no	—
197	#125	—	269	1	no	—
198	#124	—	269	1	no	—
199	#123	—	269	1	no	—
200	#123	—	269	1	no	—
201	C₂²⋊Q₈	—	269	1	no	—
202	#121	—	269	1	no	—
203	C₄×Q₈	—	269	1	no	—
204	C₄×D₄	—	269	1	no	—
205	C₄×D₄	—	269	1	no	—
206	C₄×D₄	—	269	1	no	—
207	#117	—	269	1	no	—
208	#117	C₂×D₂₆₉	1	1	yes	characteristic
209	C₂×C₄²	—	269	1	no	—
210	C₄⋊C₈	—	269	1	no	—
211	C₄⋊C₈	—	269	1	no	—
212	C₄⋊C₈	—	269	1	no	—
213	C₄⋊C₈	—	269	1	no	—
214	C₄⋊C₈	—	269	1	no	—
215	C₄⋊C₈	—	269	1	no	—
216	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
217	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
218	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
219	C₄⋊C₈	C₂×D₂₆₉	1	1	yes	characteristic
220	C₂²⋊C₈	—	269	1	no	—
221	C₂²⋊C₈	—	269	1	no	—
222	C₂²⋊C₈	—	269	1	no	—
223	C₂²⋊C₈	—	269	1	no	—
224	C₂²⋊C₈	—	269	1	no	—
225	C₂²⋊C₈	—	269	1	no	—
226	#97	—	269	1	no	—
227	#97	—	269	1	no	—
228	C₄×C₈	—	269	1	no	—
229	C₄×C₈	—	269	1	no	—
230	C₄×C₈	—	269	1	no	—
231	C₄×C₈	—	269	1	no	—
232	C₂×Q₈	—	269	1	no	—
233	C₂×D₄	—	269	1	no	—
234	C₂×D₄	—	269	1	no	—
235	C₂×D₄	—	269	1	no	—
236	C₂²×C₄	—	269	1	no	—
237	C₂²×C₄	—	269	1	no	—
238	C₂²×C₄	—	269	1	no	—
239	C₂²×C₄	—	269	1	no	—
240	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
241	M₁₆	—	4	1	no	—
242	M₁₆	—	1076	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	269	1	no	—
245	C₂×C₈	—	269	1	no	—
246	C₂×C₈	—	269	1	no	—
247	C₂×C₈	—	269	1	no	—
248	C₂×C₈	—	1076	1	no	—
249	C₂×C₈	—	1076	1	no	—
250	C₂×C₈	—	1076	1	no	—
251	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
255	C₄⋊C₄	—	269	1	no	—
256	C₄⋊C₄	—	269	1	no	—
257	C₄⋊C₄	—	269	1	no	—
258	C₄⋊C₄	—	269	1	no	—
259	C₄⋊C₄	—	269	1	no	—
260	C₄⋊C₄	—	269	1	no	—
261	C₄⋊C₄	—	269	1	no	—
262	C₄⋊C₄	—	269	1	no	—
263	C₄⋊C₄	C₄×D₂₆₉	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₂₆₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	269	1	no	—
266	C₂²⋊C₄	—	269	1	no	—
267	C₂²⋊C₄	—	269	1	no	—
268	C₂²⋊C₄	—	269	1	no	—
269	C₂²⋊C₄	—	269	1	no	—
270	C₂²⋊C₄	—	269	1	no	—
271	C₂²⋊C₄	—	269	1	no	—
272	C₂²⋊C₄	—	269	1	no	—
273	C₂²⋊C₄	C₄×D₂₆₉	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₂₆₉	1	1	yes	characteristic
275	C₄²	—	269	1	no	—
276	C₄²	—	269	1	no	—
277	C₄²	—	269	1	no	—
278	C₄²	—	269	1	no	—
279	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
280	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
281	C₂³	—	269	1	no	—
282	C₂³	—	269	1	no	—
283	C₂³	C₂×C₄×D₂₆₉	1	1	yes	characteristic
284	Q₈	—	538	2	no	—
285	D₄	—	538	2	no	—
286	D₄	—	538	2	no	—
287	D₄	—	538	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	269	1	no	—
291	C₂×C₄	—	269	1	no	—
292	C₂×C₄	—	269	1	no	—
293	C₂×C₄	—	269	1	no	—
294	C₂×C₄	—	269	1	no	—
295	C₂×C₄	—	269	1	no	—
296	C₂×C₄	—	538	2	no	—
297	C₂×C₄	—	538	2	no	—
298	C₂×C₄	—	538	1	no	—
299	C₂×C₄	—	538	1	no	—
300	C₂×C₄	—	538	1	no	—
301	C₂×C₄	—	538	1	no	—
302	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	538	1	no	—
313	C₈	—	538	1	no	—
314	C₈	—	538	1	no	—
315	C₈	—	538	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	269	2	no	—
320	C₂²	—	538	1	no	—
321	C₂²	—	538	1	no	—
322	C₂²	—	538	1	no	—
323	C₂²	—	538	1	no	—
324	C₂²	—	538	1	no	—
325	C₂²	C₂²×C₄×D₂₆₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	269	2	no	—
332	C₄	—	538	1	no	—
333	C₄	—	538	1	no	—
334	C₄	—	538	1	no	—
335	C₄	—	538	1	no	—
336	C₄	—	538	1	no	—
337	C₄	#22575498	1	1	yes	characteristic
338	C₄	#22575501	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	538	2	no	—
341	C₂	—	1076	1	no	—
342	C₂	#31771086	1	1	yes	characteristic
343	C₂	#131×D₂₆₉	1	1	yes	characteristic
344	C₂	#131×D₂₆₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₆₉
commutator, socle	C₂²×C₂₆₉	C₂³×C₄
Fitting	#424×C₂₆₉	C₂
prime radical	#22575426	C₄
cosocle	C₂×C₄×C₂₆₉	C₂⁴
hypercenter	#424	D₂₆₉
hypocenter	C₂₆₉	#2994
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #31770734	C₂	semidirect
3. #31770729	C₂	semidirect
4. #31770681	C₂	semidirect
5. #31770675	C₂	semidirect
6. #31770674	C₂	semidirect
7. #31770670	C₂	semidirect
8. C₈×C₂²⋊D₂₆₉	C₂	semidirect
9. C₈×C₂²⋊D₂₆₉	C₂	semidirect
10. #31770666	C₂	semidirect
11. #31770665	C₂	semidirect
12. #31770657	C₂	semidirect
13. #31770630	C₂	semidirect
14. C₄⋊C₈×D₂₆₉	C₂	semidirect
15. #424×C₂₆₉	C₂	semidirect
172. C₂₆₉	#2994	semidirect
187. #424	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31771133	2
#31770734	2
#31770729	2
#31770681	2
#31770675	2
#31770674	2
#31770670	2
C₈×C₂²⋊D₂₆₉	2
#31770666	2
#31770665	2
#31770657	2
#31770630	2
C₄⋊C₈×D₂₆₉	2
#424×C₂₆₉	2
#2994	269