Subgroups — #44402349

Order 29056.

600 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300246	C₂	1	1	yes	maximal, characteristic
2	C₂×#20300245	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₂₂₇	C₂	1	1	yes	maximal, characteristic
4	#28408931	C₂	1	1	yes	maximal, characteristic, complemented
5	#28408930	C₂	1	1	yes	maximal, characteristic, complemented
6	#28408929	C₂	1	2	yes	maximal, complemented
7	#28408928	C₂	1	1	yes	maximal, characteristic, complemented
8	#28408927	C₂	1	2	yes	maximal, complemented
9	#28408891	C₂	1	2	yes	maximal
10	#28408889	C₂	1	2	yes	maximal
11	#389×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₂₇	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₂₇	C₂²	1	1	yes	characteristic
14	#20300246	—	2	4	no	—
15	#20300245	—	2	4	no	—
16	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₂₇⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₂₇⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
24	C₄⋊C₄×D₂₂₇	C₄	1	8	yes	complemented
25	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	2	yes	—
27	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
29	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
30	C₂₂₇⋊#95	C₂²	1	2	yes	—
31	C₂₂₇⋊#95	C₂²	1	2	yes	—
32	C₂₂₇⋊#95	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
37	#95×C₂₂₇	C₂²	1	2	yes	—
38	C₂³×D₂₂₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
42	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
43	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₂₂₇	—	4	2	no	—
45	C₂×C₄×D₂₂₇	—	4	2	no	—
46	C₂×C₄×D₂₂₇	C₂×C₄	1	8	yes	—
47	C₂×C₄×D₂₂₇	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₂₂₇	D₄	1	4	yes	—
49	C₂×C₄×D₂₂₇	Q₈	1	4	yes	—
50	C₄⋊Dic₂₂₇	—	2	2	no	—
51	C₄⋊Dic₂₂₇	—	4	2	no	—
52	C₄⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
53	Dic₂₂₇⋊C₄	—	2	2	no	—
54	Dic₂₂₇⋊C₄	—	2	2	no	—
55	Dic₂₂₇⋊C₄	—	4	2	no	—
56	Dic₂₂₇⋊C₄	—	4	2	no	—
57	Dic₂₂₇⋊C₄	C₂×C₄	1	8	yes	—
58	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
59	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
60	(C₂×D₂₂₇)⋊C₄	—	2	4	no	—
61	C₄×Dic₂₂₇	—	2	4	no	—
62	C₄×Dic₂₂₇	—	2	2	no	—
63	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
65	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₂₇	—	2	2	no	—
69	C₄⋊C₄×C₂₂₇	—	4	2	no	—
70	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
71	C₄²×C₂₂₇	—	2	2	no	—
72	C₂²×D₂₂₇	—	2	1	no	—
73	C₂²×D₂₂₇	—	2	1	no	—
74	C₂²×D₂₂₇	—	2	1	no	—
75	C₂²×D₂₂₇	—	2	1	no	—
76	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
77	C₂²×D₂₂₇	C₂×D₄	1	2	yes	—
78	C₂²×D₂₂₇	C₂×Q₈	1	2	yes	—
79	C₂×Dic₂₂₇	—	2	2	no	—
80	C₂×Dic₂₂₇	—	2	2	no	—
81	C₂×Dic₂₂₇	—	2	1	no	—
82	C₂×Dic₂₂₇	—	2	1	no	—
83	C₂×Dic₂₂₇	—	2	2	no	—
84	C₂×Dic₂₂₇	—	2	1	no	—
85	C₂×Dic₂₂₇	—	2	2	no	—
86	C₂×Dic₂₂₇	—	2	1	no	—
87	C₂×Dic₂₂₇	—	2	1	no	—
88	C₂×Dic₂₂₇	—	2	1	no	—
89	C₂×Dic₂₂₇	—	2	2	no	—
90	C₂×Dic₂₂₇	C₂²×C₄	1	4	yes	—
91	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₂₂₇	C₂×D₄	1	2	yes	—
93	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
94	C₄×D₂₂₇	—	4	8	no	—
95	C₄×D₂₂₇	C₄⋊C₄	1	16	yes	complemented
96	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
97	C₂×C₄×C₂₂₇	—	2	2	no	—
98	C₂×C₄×C₂₂₇	—	2	2	no	—
99	C₂×C₄×C₂₂₇	—	2	1	no	—
100	C₂×C₄×C₂₂₇	—	2	1	no	—
101	C₂×C₄×C₂₂₇	—	2	2	no	—
102	C₂×C₄×C₂₂₇	—	2	1	no	—
103	C₂×C₄×C₂₂₇	—	2	2	no	—
104	C₂×C₄×C₂₂₇	—	2	1	no	—
105	C₂×C₄×C₂₂₇	—	2	2	no	—
106	C₂×C₄×C₂₂₇	—	2	1	no	—
107	C₂×C₄×C₂₂₇	—	2	1	no	—
108	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
109	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
112	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
113	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
114	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×D₂₂₇	—	2	2	no	—
120	C₂×D₂₂₇	—	2	2	no	—
121	C₂×D₂₂₇	—	2	2	no	—
122	C₂×D₂₂₇	—	2	2	no	—
123	C₂×D₂₂₇	—	2	2	no	—
124	C₂×D₂₂₇	—	2	2	no	—
125	C₂×D₂₂₇	C₂×C₄⋊C₄	1	4	yes	—
126	C₄×C₂₂₇	—	2	4	no	—
127	C₄×C₂₂₇	—	4	1	no	—
128	C₄×C₂₂₇	—	4	2	no	—
129	C₄×C₂₂₇	—	4	1	no	—
130	C₄×C₂₂₇	C₂×C₄⋊C₄	1	4	yes	—
131	Dic₂₂₇	—	2	4	no	—
132	Dic₂₂₇	—	4	1	no	—
133	Dic₂₂₇	—	4	1	no	—
134	Dic₂₂₇	—	4	2	no	—
135	Dic₂₂₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×C₂₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₂₂₇	#518	1	1	yes	characteristic
138	C₂×C₂₂₇	#519	1	1	yes	characteristic
139	C₂×C₂₂₇	#537	1	1	yes	characteristic
140	C₂×C₂₂₇	#537	1	1	yes	characteristic
141	C₂×C₂₂₇	#544	1	1	yes	characteristic
142	C₂×C₂₂₇	#544	1	1	yes	characteristic
143	D₂₂₇	—	2	4	no	—
144	C₂₂₇	#2316	1	1	yes	characteristic, complemented, Sylow
145	#2316	—	227	1	no	maximal, Sylow
146	C₂×#117	—	227	1	no	—
147	C₂×#117	—	227	1	no	—
148	C₂²×C₄⋊C₄	—	227	1	no	—
149	#389	—	227	2	no	—
150	#389	—	227	1	no	—
151	#389	—	227	1	no	—
152	#389	—	227	1	no	—
153	#389	—	227	2	no	—
154	#389	D₂₂₇	1	1	yes	characteristic, complemented
155	#380	—	227	2	no	—
156	#380	—	227	2	no	—
157	C₂³×C₄	—	227	2	no	—
158	C₂³×C₄	—	227	1	no	—
159	#117	—	454	4	no	—
160	#117	—	454	4	no	—
161	C₂×C₄⋊C₄	—	227	2	no	—
162	C₂×C₄⋊C₄	—	227	2	no	—
163	C₂×C₄⋊C₄	—	227	2	no	—
164	C₂×C₄⋊C₄	—	227	1	no	—
165	C₂×C₄⋊C₄	—	227	1	no	—
166	C₂×C₄⋊C₄	—	227	1	no	—
167	C₂×C₄⋊C₄	—	227	2	no	—
168	C₂×C₄⋊C₄	—	227	1	no	—
169	C₂×C₄⋊C₄	—	227	8	no	—
170	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×D₂₂₇	1	2	yes	—
172	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	227	1	no	—
174	C₂×C₂²⋊C₄	—	227	2	no	—
175	C₂×C₂²⋊C₄	—	227	1	no	—
176	C₂×C₄²	—	227	2	no	—
177	C₂×C₄²	—	227	1	no	—
178	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
179	#95	—	227	2	no	—
180	#95	—	227	2	no	—
181	#95	—	227	2	no	—
182	#95	C₂×D₂₂₇	1	2	yes	—
183	C₂⁴	—	227	1	no	—
184	C₂²×C₄	—	908	2	no	—
185	C₂²×C₄	—	908	2	no	—
186	C₂²×C₄	—	227	8	no	—
187	C₂²×C₄	—	227	1	no	—
188	C₂²×C₄	—	227	2	no	—
189	C₂²×C₄	—	227	1	no	—
190	C₂²×C₄	—	227	2	no	—
191	C₂²×C₄	—	227	1	no	—
192	C₂²×C₄	—	227	4	no	—
193	C₂²×C₄	—	227	4	no	—
194	C₂²×C₄	—	227	4	no	—
195	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
197	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
199	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	908	2	no	—
203	C₄⋊C₄	—	908	2	no	—
204	C₄⋊C₄	—	908	2	no	—
205	C₄⋊C₄	—	454	2	no	—
206	C₄⋊C₄	—	454	2	no	—
207	C₄⋊C₄	—	454	2	no	—
208	C₄⋊C₄	—	227	8	no	—
209	C₄⋊C₄	—	227	4	no	—
210	C₄⋊C₄	C₄×D₂₂₇	1	4	yes	—
211	C₂²⋊C₄	—	454	2	no	—
212	C₂²⋊C₄	—	454	2	no	—
213	C₂²⋊C₄	—	454	4	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	454	4	no	—
216	C₄²	—	454	2	no	—
217	C₂³	—	454	1	no	—
218	C₂³	—	454	1	no	—
219	C₂³	—	454	1	no	—
220	C₂³	—	454	1	no	—
221	C₂³	—	227	2	no	—
222	C₂³	—	227	2	no	—
223	C₂³	—	227	2	no	—
224	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	908	8	no	—
237	C₂×C₄	—	454	1	no	—
238	C₂×C₄	—	454	2	no	—
239	C₂×C₄	—	454	1	no	—
240	C₂×C₄	—	454	1	no	—
241	C₂×C₄	—	454	2	no	—
242	C₂×C₄	—	454	1	no	—
243	C₂×C₄	—	454	1	no	—
244	C₂×C₄	—	454	2	no	—
245	C₂×C₄	—	454	1	no	—
246	C₂×C₄	—	454	2	no	—
247	C₂×C₄	—	454	2	no	—
248	C₂×C₄	—	227	4	no	—
249	C₂×C₄	—	227	2	no	—
250	C₂×C₄	—	227	2	no	—
251	C₂×C₄	—	227	2	no	—
252	C₂×C₄	—	227	16	no	—
253	C₂×C₄	C₂×C₄×D₂₂₇	1	4	yes	—
254	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
255	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
256	C₂×C₄	Q₈×D₂₂₇	1	2	yes	—
257	C₂²	—	454	2	no	—
258	C₂²	—	454	2	no	—
259	C₂²	—	454	2	no	—
260	C₂²	—	454	2	no	—
261	C₂²	—	454	2	no	—
262	C₂²	—	454	2	no	—
263	C₂²	—	227	4	no	—
264	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
265	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
266	C₂²	C₂×Q₈×D₂₂₇	1	1	yes	characteristic
267	C₂²	#20300333	1	1	yes	characteristic
268	C₂²	#20300334	1	1	yes	characteristic
269	C₂²	#20300335	1	1	yes	characteristic
270	C₂²	#20300336	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	908	1	no	—
276	C₄	—	908	1	no	—
277	C₄	—	908	2	no	—
278	C₄	—	454	4	no	—
279	C₄	—	227	4	no	—
280	C₄	C₄⋊C₄×D₂₂₇	1	4	yes	—
281	C₂	—	454	4	no	—
282	C₂	C₂×C₄⋊C₄×D₂₂₇	1	1	yes	characteristic
283	C₂	#28409504	1	1	yes	characteristic
284	C₂	#28409508	1	1	yes	characteristic
285	C₂	#28409611	1	1	yes	characteristic
286	C₂	#28409612	1	1	yes	characteristic
287	C₂	#28409659	1	1	yes	characteristic
288	C₂	#28409662	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Fitting	#389×C₂₂₇	C₂
socle, cosocle	C₂³×C₂₂₇	C₂⁴
prime radical	C₂³×D₂₂₇	C₂³
hypercenter	#389	D₂₂₇
hypocenter	C₂₂₇	#2316
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
4. #28408931	C₂	semidirect
5. #28408930	C₂	semidirect
6. #28408929	C₂	semidirect
7. #28408928	C₂	semidirect
8. #28408927	C₂	semidirect
11. #389×C₂₂₇	C₂	semidirect
24. C₄⋊C₄×D₂₂₇	C₄	semidirect
95. C₄×D₂₂₇	C₄⋊C₄	semidirect
144. C₂₂₇	#2316	semidirect
154. #389	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300246	2
C₂×#20300245	2
C₂×C₄⋊C₄×D₂₂₇	2
#28408931	2
#28408930	2
#28408929	2
#28408928	2
#28408927	2
#28408891	2
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