Subgroups — C₄×#20300264 #44402263

Order 29056.

656 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300264	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×D₂₂₇	C₂	1	1	yes	maximal, characteristic
3	#28408939	C₂	1	1	yes	maximal, characteristic, complemented
4	#28408928	C₂	1	1	yes	maximal, characteristic, complemented
5	#28408913	C₂	1	2	yes	maximal, complemented
6	#28408903	C₂	1	2	yes	maximal
7	#28408889	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₂₂₇⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×(C₂×D₂₂₇)⋊C₄	C₂	1	2	yes	maximal
11	C₂×C₄×Q₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₂₂₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₂₇	C₂²	1	1	yes	characteristic
15	#20300264	C₄	1	8	yes	direct product factor
16	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
26	C₄²×D₂₂₇	—	2	4	no	—
27	C₂₂₇⋊#95	C₂²	1	2	yes	—
28	C₂₂₇⋊#95	C₂²	1	2	yes	—
29	C₂₂₇⋊#95	C₂²	1	2	yes	—
30	C₂₂₇⋊#95	C₂²	1	2	yes	—
31	C₂₂₇⋊#95	C₂²	1	2	yes	—
32	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₂₂₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₂₂₇	C₂²	1	2	yes	—
37	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
38	C₂³×D₂₂₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
40	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
42	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₂₂₇	—	2	2	no	—
45	C₂×C₄×D₂₂₇	—	2	2	no	—
46	C₂×C₄×D₂₂₇	—	2	2	no	—
47	C₂×C₄×D₂₂₇	—	2	2	no	—
48	C₂×C₄×D₂₂₇	—	2	2	no	—
49	C₂×C₄×D₂₂₇	—	2	2	no	—
50	C₂×C₄×D₂₂₇	—	2	2	no	—
51	C₂×C₄×D₂₂₇	C₂×C₄	1	4	yes	—
52	C₂×C₄×D₂₂₇	Q₈	1	4	yes	complemented
53	C₄⋊Dic₂₂₇	—	2	2	no	—
54	C₄⋊Dic₂₂₇	C₂×C₄	1	4	yes	complemented
55	Dic₂₂₇⋊C₄	—	2	4	no	—
56	Dic₂₂₇⋊C₄	C₂×C₄	1	8	yes	complemented
57	(C₂×D₂₂₇)⋊C₄	—	2	4	no	—
58	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	8	yes	—
59	C₄×Dic₂₂₇	—	2	2	no	—
60	C₄×Dic₂₂₇	—	2	2	no	—
61	C₄×Dic₂₂₇	—	2	2	no	—
62	C₄×Dic₂₂₇	—	2	4	no	—
63	C₂×Q₈×C₂₂₇	—	2	2	no	—
64	C₂×Q₈×C₂₂₇	—	2	2	no	—
65	C₂×Q₈×C₂₂₇	C₂×C₄	1	4	yes	complemented
66	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
68	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₂₂₇	—	2	4	no	—
72	C₄⋊C₄×C₂₂₇	—	2	2	no	—
73	C₄²×C₂₂₇	—	2	4	no	—
74	C₄²×C₂₂₇	D₄	1	4	yes	—
75	C₂²×D₂₂₇	—	2	1	no	—
76	C₂²×D₂₂₇	—	2	1	no	—
77	C₂²×D₂₂₇	—	2	1	no	—
78	C₂²×D₂₂₇	—	2	1	no	—
79	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
80	C₂²×D₂₂₇	C₂×Q₈	1	2	yes	—
81	C₂²×D₂₂₇	#41	1	2	yes	—
82	C₂×Dic₂₂₇	—	2	1	no	—
83	C₂×Dic₂₂₇	—	2	1	no	—
84	C₂×Dic₂₂₇	—	2	1	no	—
85	C₂×Dic₂₂₇	—	2	2	no	—
86	C₂×Dic₂₂₇	—	2	2	no	—
87	C₂×Dic₂₂₇	—	2	1	no	—
88	C₂×Dic₂₂₇	—	2	1	no	—
89	C₂×Dic₂₂₇	—	2	1	no	—
90	C₂×Dic₂₂₇	—	2	2	no	—
91	C₂×Dic₂₂₇	—	2	2	no	—
92	C₂×Dic₂₂₇	—	2	2	no	—
93	C₂×Dic₂₂₇	C₂²×C₄	1	4	yes	—
94	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
95	C₂×Dic₂₂₇	C₂×Q₈	1	2	yes	—
96	C₂×Dic₂₂₇	#41	1	2	yes	—
97	C₄×D₂₂₇	—	2	8	no	—
98	C₄×D₂₂₇	—	2	8	no	—
99	C₄×D₂₂₇	—	2	8	no	—
100	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
101	Q₈×C₂₂₇	—	2	8	no	—
102	C₂×C₄×C₂₂₇	—	2	2	no	—
103	C₂×C₄×C₂₂₇	—	2	2	no	—
104	C₂×C₄×C₂₂₇	—	2	1	no	—
105	C₂×C₄×C₂₂₇	—	2	1	no	—
106	C₂×C₄×C₂₂₇	—	2	2	no	—
107	C₂×C₄×C₂₂₇	—	2	2	no	—
108	C₂×C₄×C₂₂₇	—	2	2	no	—
109	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
110	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
111	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
115	C₂×C₄×C₂₂₇	#41	1	2	yes	—
116	C₂×C₄×C₂₂₇	#41	1	2	yes	—
117	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
118	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
121	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
122	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
123	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
124	C₂×D₂₂₇	—	2	2	no	—
125	C₂×D₂₂₇	—	2	2	no	—
126	C₂×D₂₂₇	—	2	2	no	—
127	C₂×D₂₂₇	—	2	2	no	—
128	C₂×D₂₂₇	—	2	2	no	—
129	C₂×D₂₂₇	—	2	2	no	—
130	C₂×D₂₂₇	C₄×Q₈	1	4	yes	complemented
131	C₄×C₂₂₇	—	2	4	no	—
132	C₄×C₂₂₇	—	2	2	no	—
133	C₄×C₂₂₇	—	4	2	no	—
134	C₄×C₂₂₇	C₄×D₄	1	4	yes	—
135	C₄×C₂₂₇	C₂²⋊Q₈	1	4	yes	complemented
136	Dic₂₂₇	—	2	2	no	—
137	Dic₂₂₇	—	2	4	no	—
138	Dic₂₂₇	—	2	2	no	—
139	Dic₂₂₇	—	2	2	no	—
140	Dic₂₂₇	—	4	2	no	—
141	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂×C₂₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₂×C₂₂₇	C₄×#41	1	1	yes	characteristic
144	C₂×C₂₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₂₂₇	#525	1	1	yes	characteristic
146	C₂×C₂₂₇	#529	1	1	yes	characteristic
147	C₂×C₂₂₇	#533	1	1	yes	characteristic
148	D₂₂₇	—	2	4	no	—
149	C₂₂₇	C₄×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
150	C₄×C₂²⋊Q₈	—	227	1	no	maximal, Sylow
151	C₂×C₂²⋊Q₈	—	227	1	no	—
152	C₂×C₄×Q₈	D₂₂₇	1	1	yes	characteristic, complemented
153	C₂²×C₄²	—	227	1	no	—
154	#391	—	227	1	no	—
155	#389	—	227	1	no	—
156	#387	—	227	2	no	—
157	#385	—	227	2	no	—
158	#380	—	227	1	no	—
159	C₄×C₄⋊C₄	—	227	1	no	—
160	C₄×C₄⋊C₄	—	227	2	no	—
161	C₄×C₂²⋊C₄	—	227	2	no	—
162	C₂²×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
163	C₂³×C₄	—	227	1	no	—
164	C₂³×C₄	—	227	1	no	—
165	C₂³×C₄	—	227	1	no	—
166	C₂²⋊Q₈	—	227	8	no	—
167	C₄×Q₈	—	2	4	no	—
168	C₂×C₄⋊C₄	—	227	1	no	—
169	C₂×C₄⋊C₄	—	227	2	no	—
170	C₂×C₄⋊C₄	—	227	2	no	—
171	C₂×C₄⋊C₄	—	227	1	no	—
172	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×D₂₂₇	1	2	yes	—
174	C₂×C₂²⋊C₄	—	227	2	no	—
175	C₂×C₂²⋊C₄	—	227	2	no	—
176	C₂×C₄²	—	454	4	no	—
177	C₂×C₄²	—	227	1	no	—
178	C₂×C₄²	—	227	1	no	—
179	C₂×C₄²	—	227	2	no	—
180	C₂×C₄²	—	227	1	no	—
181	C₂×C₄²	C₂×D₂₂₇	1	2	yes	—
182	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
183	#95	—	227	2	no	—
184	#95	—	227	2	no	—
185	#95	—	227	2	no	—
186	#95	—	227	2	no	—
187	#95	—	227	2	no	—
188	C₂⁴	—	227	1	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	C₄×D₂₂₇	1	4	yes	complemented
192	C₂²×C₄	—	454	2	no	—
193	C₂²×C₄	—	454	2	no	—
194	C₂²×C₄	—	454	2	no	—
195	C₂²×C₄	—	454	2	no	—
196	C₂²×C₄	—	454	2	no	—
197	C₂²×C₄	—	454	2	no	—
198	C₂²×C₄	—	454	2	no	—
199	C₂²×C₄	—	227	2	no	—
200	C₂²×C₄	—	227	1	no	—
201	C₂²×C₄	—	227	1	no	—
202	C₂²×C₄	—	227	1	no	—
203	C₂²×C₄	—	227	2	no	—
204	C₂²×C₄	—	227	4	no	—
205	C₂²×C₄	—	227	4	no	—
206	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
207	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
208	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	454	4	no	—
214	C₄⋊C₄	—	454	2	no	—
215	C₄⋊C₄	—	227	8	no	—
216	C₄⋊C₄	—	227	4	no	—
217	C₂²⋊C₄	—	454	4	no	—
218	C₂²⋊C₄	—	227	8	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	454	2	no	—
221	C₄²	—	454	2	no	—
222	C₄²	—	454	2	no	—
223	C₄²	—	454	4	no	—
224	C₄²	C₂²⋊D₂₂₇	1	4	yes	—
225	C₂³	—	454	1	no	—
226	C₂³	—	454	1	no	—
227	C₂³	—	454	1	no	—
228	C₂³	—	454	1	no	—
229	C₂³	—	227	2	no	—
230	C₂³	—	227	2	no	—
231	C₂³	—	227	2	no	—
232	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
233	Q₈	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	454	8	no	—
242	C₂×C₄	—	454	2	no	—
243	C₂×C₄	—	454	2	no	—
244	C₂×C₄	—	454	2	no	—
245	C₂×C₄	—	454	1	no	—
246	C₂×C₄	—	454	1	no	—
247	C₂×C₄	—	454	1	no	—
248	C₂×C₄	—	454	2	no	—
249	C₂×C₄	—	454	1	no	—
250	C₂×C₄	—	454	1	no	—
251	C₂×C₄	—	454	2	no	—
252	C₂×C₄	—	454	1	no	—
253	C₂×C₄	—	454	8	no	—
254	C₂×C₄	—	454	8	no	—
255	C₂×C₄	—	227	4	no	—
256	C₂×C₄	—	227	2	no	—
257	C₂×C₄	—	227	2	no	—
258	C₂×C₄	—	227	2	no	—
259	C₂×C₄	C₂×C₄×D₂₂₇	1	4	yes	—
260	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	2	yes	—
262	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	2	yes	—
264	C₂×C₄	Q₈×D₂₂₇	1	2	yes	—
265	C₂×C₄	#16813884	1	2	yes	—
266	C₂×C₄	#16813885	1	2	yes	—
267	C₂²	—	454	2	no	—
268	C₂²	—	454	2	no	—
269	C₂²	—	454	2	no	—
270	C₂²	—	454	2	no	—
271	C₂²	—	454	2	no	—
272	C₂²	—	454	2	no	—
273	C₂²	—	227	4	no	—
274	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
275	C₂²	C₂²×C₂²⋊D₂₂₇	1	1	yes	characteristic
276	C₂²	C₂×Q₈×D₂₂₇	1	1	yes	characteristic
277	C₂²	#41×D₂₂₇	1	1	yes	characteristic
278	C₂²	#41×D₂₂₇	1	1	yes	characteristic
279	C₂²	C₂×#16813884	1	1	yes	characteristic
280	C₂²	C₂×#16813885	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	908	2	no	—
285	C₄	—	454	2	no	—
286	C₄	—	454	4	no	—
287	C₄	—	454	2	no	—
288	C₄	—	454	2	no	—
289	C₄	C₄×C₂²⋊D₂₂₇	1	4	yes	—
290	C₄	#20300264	1	4	yes	direct product factor
291	C₂	—	454	4	no	—
292	C₂	C₂×C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
293	C₂	C₄×Q₈×D₂₂₇	1	1	yes	characteristic
294	C₂	C₄×#16813885	1	1	yes	characteristic
295	C₂	C₂×#20300264	1	1	yes	characteristic
296	C₂	#28409539	1	1	yes	characteristic
297	C₂	#28409569	1	1	yes	characteristic
298	C₂	#28409586	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Frattini	C₂³	C₂³×D₂₂₇
Fitting	C₂×C₄×Q₈×C₂₂₇	C₂
socle, cosocle	C₂³×C₂₂₇	C₂⁴
prime radical	C₂³×D₂₂₇	C₂³
hypercenter	C₂×C₄×Q₈	D₂₂₇
hypocenter	C₂₂₇	C₄×C₂²⋊Q₈
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #28408939	C₂	semidirect
4. #28408928	C₂	semidirect
5. #28408913	C₂	semidirect
8. C₄×Dic₂₂₇⋊C₄	C₂	semidirect
9. C₄×C₄⋊Dic₂₂₇	C₂	semidirect
11. C₂×C₄×Q₈×C₂₂₇	C₂	semidirect
15. #20300264	C₄	direct (N × H)
52. C₂×C₄×D₂₂₇	Q₈	semidirect
54. C₄⋊Dic₂₂₇	C₂×C₄	semidirect
56. Dic₂₂₇⋊C₄	C₂×C₄	semidirect
65. C₂×Q₈×C₂₂₇	C₂×C₄	semidirect
130. C₂×D₂₂₇	C₄×Q₈	semidirect
135. C₄×C₂₂₇	C₂²⋊Q₈	semidirect
149. C₂₂₇	C₄×C₂²⋊Q₈	semidirect
152. C₂×C₄×Q₈	D₂₂₇	semidirect
191. C₂×Q₈	C₄×D₂₂₇	semidirect
290. C₄	#20300264	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20300264	2
C₂×C₄²×D₂₂₇	2
#28408939	2
#28408928	2
#28408913	2
#28408903	2
#28408889	2
C₄×Dic₂₂₇⋊C₄	2
C₄×C₄⋊Dic₂₂₇	2
C₄×(C₂×D₂₂₇)⋊C₄	2
C₂×C₄×Q₈×C₂₂₇	2
C₄×C₂²⋊Q₈	227

Subgroups — C4×#20300264 #44402263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#20300264 #44402263