Subgroups — #44409191

Order 29056.

496 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	C₂×#20300310	C₂	1	2	yes	maximal, complemented
2	#28409582	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#16813884	C₂	1	1	yes	maximal, characteristic, complemented
4	#28409255	C₂	1	2	yes	maximal, complemented
5	#28409248	C₂	1	2	yes	maximal, complemented
6	#28409243	C₂	1	1	yes	maximal, characteristic, complemented
7	#28409238	C₂	1	1	yes	maximal, characteristic, complemented
8	#28409233	C₂	1	1	yes	maximal, characteristic, complemented
9	#28409228	C₂	1	1	yes	maximal, characteristic, complemented
10	#28409029	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16813884	C₂²	1	2	yes	—
13	C₂×#16813884	C₂²	1	1	yes	characteristic
14	#20300310	—	2	8	no	—
15	C₂×Dic₉₀₈	C₂²	1	2	yes	—
16	C₂×#16813873	C₂²	1	2	yes	—
17	C₂×#16813873	C₂²	1	2	yes	—
18	C₂×D₁₈₁₆	C₂²	1	2	yes	—
19	#20300288	C₂²	1	1	yes	characteristic
20	#20300285	C₂²	1	1	yes	characteristic
21	#20300276	C₂²	1	2	yes	—
22	#20300262	—	2	1	no	—
23	#20300262	—	2	1	no	—
24	C₄×Dic₄₅₄	C₂²	1	1	yes	characteristic
25	C₄×Dic₄₅₄	C₂²	1	1	yes	characteristic
26	C₄×D₉₀₈	—	2	1	no	—
27	C₄×D₉₀₈	—	2	1	no	—
28	C₄×D₉₀₈	C₂²	1	1	yes	characteristic
29	C₄×D₉₀₈	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₂₂₇	—	2	1	no	—
31	C₄×C₂²⋊D₂₂₇	—	2	1	no	—
32	#20300247	C₂²	1	1	yes	characteristic
33	#20300245	—	2	1	no	—
34	C₄²×D₂₂₇	—	2	1	no	—
35	#20300213	C₂²	1	2	yes	—
36	#20300213	C₂²	1	2	yes	—
37	#20300210	C₂²	1	2	yes	—
38	#20300210	C₂²	1	2	yes	—
39	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
40	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
42	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
43	C₄⋊C₈×C₂₂₇	C₂²	1	2	yes	—
44	#16813884	—	2	4	no	—
45	#16813884	—	2	2	no	—
46	#16813884	—	2	4	no	—
47	#16813884	D₄	1	4	yes	complemented
48	C₂×Dic₄₅₄	C₂³	1	2	yes	—
49	C₂×Dic₄₅₄	C₂³	1	1	yes	characteristic
50	C₂×D₉₀₈	—	2	1	no	—
51	C₂×D₉₀₈	—	2	1	no	—
52	C₂×D₉₀₈	C₂³	1	2	yes	—
53	C₂×D₉₀₈	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₂₂₇	—	2	1	no	—
55	C₂×C₂²⋊D₂₂₇	—	2	2	no	—
56	C₂×C₄×D₂₂₇	—	2	1	no	—
57	C₂×C₄×D₂₂₇	—	2	2	no	—
58	C₂×C₄×D₂₂₇	—	2	1	no	—
59	C₂×C₄×D₂₂₇	—	2	1	no	—
60	Dic₉₀₈	—	2	4	no	—
61	#16813873	—	2	4	no	—
62	#16813873	—	2	4	no	—
63	D₁₈₁₆	—	2	4	no	—
64	C₄⋊Dic₂₂₇	—	2	1	no	—
65	C₄⋊Dic₂₂₇	—	2	1	no	—
66	C₄⋊Dic₂₂₇	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₂₂₇	C₂³	1	1	yes	characteristic
68	Dic₂₂₇⋊C₄	—	2	1	no	—
69	Dic₂₂₇⋊C₄	—	2	1	no	—
70	C₂²⋊Dic₂₂₇	—	2	1	no	—
71	(C₂×D₂₂₇)⋊C₄	—	2	1	no	—
72	(C₂×D₂₂₇)⋊C₄	—	2	1	no	—
73	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
74	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
75	C₄×Dic₂₂₇	—	2	1	no	—
76	C₄×Dic₂₂₇	—	2	1	no	—
77	C₄×Dic₂₂₇	—	2	1	no	—
78	C₂²×C₄×C₂₂₇	D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₂₇	D₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₂₂₇	—	2	4	no	—
82	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
83	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
84	C₄²×C₂₂₇	D₄	1	1	yes	characteristic
85	C₄²×C₂₂₇	D₄	1	1	yes	characteristic
86	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
87	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
88	C₂²×D₂₂₇	—	2	1	no	—
89	C₂²×D₂₂₇	—	2	2	no	—
90	Dic₄₅₄	—	2	1	no	—
91	Dic₄₅₄	—	2	2	no	—
92	Dic₄₅₄	—	2	2	no	—
93	Dic₄₅₄	C₂×D₄	1	2	yes	—
94	D₉₀₈	—	2	2	no	—
95	D₉₀₈	—	2	1	no	—
96	D₉₀₈	—	2	2	no	—
97	D₉₀₈	—	4	2	no	—
98	D₉₀₈	—	4	2	no	—
99	D₉₀₈	C₂×D₄	1	2	yes	—
100	C₂²⋊D₂₂₇	—	2	4	no	—
101	C₂²⋊D₂₂₇	—	4	4	no	—
102	C₂×Dic₂₂₇	—	2	1	no	—
103	C₂×Dic₂₂₇	—	2	1	no	—
104	C₂×Dic₂₂₇	—	2	1	no	—
105	C₂×Dic₂₂₇	—	2	2	no	—
106	C₄×D₂₂₇	—	2	4	no	—
107	C₄×D₂₂₇	—	4	2	no	—
108	C₄×D₂₂₇	—	4	4	no	—
109	C₄×D₂₂₇	—	4	2	no	—
110	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₂₇	—	2	2	no	—
112	C₂×C₄×C₂₂₇	—	4	1	no	—
113	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₂₇	#41	1	2	yes	—
120	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
121	C₈×C₂₂₇	—	2	2	no	—
122	C₈×C₂₂₇	—	2	2	no	—
123	C₂²×C₂₂₇	—	2	1	no	—
124	C₂²×C₂₂₇	—	2	1	no	—
125	C₂²×C₂₂₇	#121	1	2	yes	—
126	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×D₂₂₇	—	2	2	no	—
128	C₂×D₂₂₇	—	4	1	no	—
129	C₂×D₂₂₇	—	4	1	no	—
130	C₂×D₂₂₇	—	4	2	no	—
131	C₂×D₂₂₇	—	4	2	no	—
132	C₂×D₂₂₇	—	4	2	no	—
133	C₄×C₂₂₇	—	2	1	no	—
134	C₄×C₂₂₇	—	2	1	no	—
135	C₄×C₂₂₇	—	2	1	no	—
136	C₄×C₂₂₇	#121	1	2	yes	—
137	C₄×C₂₂₇	#135	1	2	yes	complemented
138	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
140	Dic₂₂₇	—	2	2	no	—
141	Dic₂₂₇	—	4	2	no	—
142	Dic₂₂₇	—	4	1	no	—
143	Dic₂₂₇	—	4	1	no	—
144	C₂×C₂₂₇	—	2	2	no	—
145	C₂×C₂₂₇	C₂×#121	1	1	yes	characteristic
146	C₂×C₂₂₇	C₂×#135	1	1	yes	characteristic
147	C₂×C₂₂₇	#575	1	1	yes	characteristic
148	D₂₂₇	—	4	2	no	—
149	D₂₂₇	—	8	2	no	—
150	C₂₂₇	#3035	1	1	yes	characteristic, complemented, Sylow
151	#3035	—	227	1	no	maximal, Sylow
152	C₂×#135	—	227	2	no	—
153	#532	—	227	1	no	—
154	C₄×#41	—	227	1	no	—
155	#464	—	227	2	no	—
156	#463	—	227	2	no	—
157	#462	—	227	1	no	—
158	#461	—	227	1	no	—
159	#460	—	227	1	no	—
160	#459	—	227	1	no	—
161	C₂×C₄⋊C₈	D₂₂₇	1	1	yes	characteristic, complemented
162	#416	—	227	2	no	—
163	C₂×#41	—	227	2	no	—
164	C₂×#41	—	227	1	no	—
165	#135	—	454	8	no	—
166	C₂×Q₁₆	—	227	2	no	—
167	C₂×SD₁₆	—	227	2	no	—
168	C₂×SD₁₆	—	227	2	no	—
169	C₂×D₈	—	227	2	no	—
170	C₂²×C₈	C₂×D₂₂₇	1	2	yes	—
171	C₄⋊Q₈	—	227	1	no	—
172	#127	—	227	1	no	—
173	#124	—	227	2	no	—
174	#121	—	454	1	no	—
175	#121	—	454	1	no	—
176	C₄×Q₈	—	227	1	no	—
177	C₄×Q₈	—	227	1	no	—
178	C₄×D₄	—	454	1	no	—
179	C₄×D₄	—	454	1	no	—
180	C₄×D₄	—	454	1	no	—
181	C₄×D₄	—	454	1	no	—
182	C₄×D₄	—	227	1	no	—
183	C₄×D₄	—	227	1	no	—
184	#117	—	454	1	no	—
185	#117	—	227	1	no	—
186	C₂×C₄²	—	454	1	no	—
187	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
188	C₄⋊C₈	C₂×D₂₂₇	1	1	yes	characteristic
189	C₄⋊C₈	C₂×D₂₂₇	1	1	yes	characteristic
190	C₄⋊C₈	C₂×D₂₂₇	1	2	yes	—
191	#103	—	227	2	no	—
192	#103	—	227	2	no	—
193	#102	—	227	2	no	—
194	#102	—	227	2	no	—
195	#41	—	454	4	no	—
196	#41	—	454	4	no	—
197	#41	—	454	2	no	—
198	#41	—	227	4	no	—
199	C₂×Q₈	—	227	2	no	—
200	C₂×Q₈	—	227	1	no	—
201	C₂×D₄	—	454	2	no	—
202	C₂×D₄	—	454	1	no	—
203	C₂×D₄	—	454	1	no	—
204	C₂×D₄	—	454	1	no	—
205	C₂×D₄	—	227	2	no	—
206	C₂×D₄	—	227	1	no	—
207	C₂²×C₄	—	454	1	no	—
208	C₂²×C₄	—	454	1	no	—
209	C₂²×C₄	—	454	2	no	—
210	C₂²×C₄	—	454	1	no	—
211	C₂²×C₄	D₉₀₈	1	1	yes	characteristic
212	C₂²×C₄	D₉₀₈	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
214	Q₁₆	—	454	4	no	—
215	SD₁₆	—	454	4	no	—
216	SD₁₆	—	454	4	no	—
217	D₈	—	454	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
220	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
221	C₄⋊C₄	—	454	1	no	—
222	C₄⋊C₄	—	454	1	no	—
223	C₄⋊C₄	—	454	1	no	—
224	C₄⋊C₄	—	454	1	no	—
225	C₄⋊C₄	—	227	1	no	—
226	C₄⋊C₄	—	227	1	no	—
227	C₂²⋊C₄	—	454	1	no	—
228	C₂²⋊C₄	—	454	1	no	—
229	C₂²⋊C₄	—	454	1	no	—
230	C₂²⋊C₄	—	454	2	no	—
231	C₂²⋊C₄	—	454	2	no	—
232	C₄²	—	454	1	no	—
233	C₄²	—	454	1	no	—
234	C₄²	—	454	1	no	—
235	C₄²	D₉₀₈	1	1	yes	characteristic
236	C₄²	D₉₀₈	1	1	yes	characteristic
237	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
238	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
239	C₂³	—	454	2	no	—
240	C₂³	—	454	1	no	—
241	C₂³	C₂×D₉₀₈	1	1	yes	characteristic
242	Q₈	—	454	2	no	—
243	Q₈	—	454	2	no	—
244	Q₈	—	454	1	no	—
245	Q₈	—	227	2	no	—
246	D₄	—	908	2	no	—
247	D₄	—	908	4	no	—
248	D₄	—	908	2	no	—
249	D₄	—	454	4	no	—
250	D₄	—	454	2	no	—
251	D₄	—	454	2	no	—
252	D₄	—	454	1	no	—
253	D₄	—	227	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	908	2	no	—
257	C₂×C₄	—	908	2	no	—
258	C₂×C₄	—	908	4	no	—
259	C₂×C₄	—	454	1	no	—
260	C₂×C₄	—	454	1	no	—
261	C₂×C₄	—	454	2	no	—
262	C₂×C₄	—	454	1	no	—
263	C₂×C₄	—	454	4	no	—
264	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
265	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
266	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
267	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
268	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
269	C₂×C₄	C₂×D₉₀₈	1	1	yes	characteristic
270	C₂×C₄	#16813885	1	2	yes	—
271	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	908	2	no	—
277	C₂²	—	908	2	no	—
278	C₂²	—	908	2	no	—
279	C₂²	—	908	1	no	—
280	C₂²	—	908	1	no	—
281	C₂²	—	454	2	no	—
282	C₂²	#20300263	1	2	yes	—
283	C₂²	C₂²×D₉₀₈	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	908	2	no	—
288	C₄	—	908	1	no	—
289	C₄	—	908	1	no	—
290	C₄	—	454	2	no	—
291	C₄	#20300263	1	2	yes	—
292	C₄	#20300310	1	2	yes	complemented
293	C₄	C₂×D₄×D₂₂₇	1	1	yes	characteristic
294	C₄	C₂×#16813885	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	1816	2	no	—
297	C₂	—	908	2	no	—
298	C₂	C₂×#20300263	1	1	yes	characteristic
299	C₂	C₂×#20300310	1	1	yes	characteristic
300	C₂	#28409832	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₉₀₈
commutator, cosocle	C₂×C₄×C₂₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	C₂×C₄⋊C₈×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂²×D₄
hypercenter	C₂×C₄⋊C₈	D₂₂₇
hypocenter	C₂₂₇	#3035
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#20300310	C₂	semidirect
2. #28409582	C₂	semidirect
3. C₄×#16813884	C₂	semidirect
4. #28409255	C₂	semidirect
5. #28409248	C₂	semidirect
6. #28409243	C₂	semidirect
7. #28409238	C₂	semidirect
8. #28409233	C₂	semidirect
9. #28409228	C₂	semidirect
10. #28409029	C₂	semidirect
11. C₂×C₄⋊C₈×C₂₂₇	C₂	semidirect
47. #16813884	D₄	semidirect
137. C₄×C₂₂₇	#135	semidirect
150. C₂₂₇	#3035	semidirect
161. C₂×C₄⋊C₈	D₂₂₇	semidirect
292. C₄	#20300310	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300310	2
#28409582	2
C₄×#16813884	2
#28409255	2
#28409248	2
#28409243	2
#28409238	2
#28409233	2
#28409228	2
#28409029	2
C₂×C₄⋊C₈×C₂₂₇	2
#3035	227