Subgroups — #30647394

Order 16256.

392 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550344	C₂	1	1	yes	maximal, characteristic
2	#22549938	C₂	1	2	yes	maximal, complemented
3	#22549935	C₂	1	2	yes	maximal, complemented
4	#22549931	C₂	1	2	yes	maximal, complemented
5	#22549927	C₂	1	2	yes	maximal, complemented
6	#22549899	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549844	C₂	1	1	yes	maximal, characteristic
9	#22549842	C₂	1	1	yes	maximal, characteristic
10	#22549832	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	C₂×#14523232	C₂²	1	1	yes	characteristic
14	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	#16861174	C₄	1	1	yes	characteristic
16	#16861174	C₄	1	1	yes	characteristic
17	#16861171	C₄	1	1	yes	characteristic
18	#16861171	C₄	1	1	yes	characteristic
19	(C₂²⋊D₁₂₇)⋊C₄	C₄	1	2	yes	—
20	(C₂²⋊D₁₂₇)⋊C₄	C₄	1	2	yes	—
21	#16861166	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
23	C₄²×D₁₂₇	C₂²	1	2	yes	—
24	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
29	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
34	#16861120	C₂²	1	2	yes	—
35	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
36	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
38	#117×C₁₂₇	C₂²	1	1	yes	characteristic
39	#97×C₁₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
41	#14523248	—	2	2	no	—
42	#14523248	—	2	2	no	—
43	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
44	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
46	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
51	#14523232	—	2	2	no	—
52	C₂×C₁₂₇⋊C₈	—	2	2	no	—
53	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
56	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₂₇	—	2	2	no	—
68	C₂×C₈×C₁₂₇	—	2	2	no	—
69	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
71	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
75	C₄²×C₁₂₇	C₂³	1	2	yes	—
76	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
77	Dic₂₅₄	—	2	2	no	—
78	D₅₀₈	—	2	2	no	—
79	C₂²⋊D₁₂₇	—	2	4	no	—
80	C₂×Dic₁₂₇	—	2	2	no	—
81	C₂×Dic₁₂₇	—	4	1	no	—
82	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₄×D₁₂₇	—	2	2	no	—
88	C₄×D₁₂₇	—	2	2	no	—
89	C₄×D₁₂₇	—	2	4	no	—
90	C₄×D₁₂₇	—	4	2	no	—
91	C₁₂₇⋊C₈	—	2	1	no	—
92	C₁₂₇⋊C₈	—	2	1	no	—
93	C₁₂₇⋊C₈	—	2	2	no	—
94	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	#41	1	2	yes	—
99	C₂×C₄×C₁₂₇	#41	1	2	yes	—
100	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
101	C₈×C₁₂₇	—	2	1	no	—
102	C₈×C₁₂₇	—	2	1	no	—
103	C₈×C₁₂₇	—	2	2	no	—
104	C₂²×C₁₂₇	—	2	1	no	—
105	C₂²×C₁₂₇	—	2	1	no	—
106	C₂²×C₁₂₇	#117	1	2	yes	—
107	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₁₂₇	—	2	2	no	—
109	C₂×D₁₂₇	—	2	2	no	—
110	C₂×D₁₂₇	—	2	2	no	—
111	C₄×C₁₂₇	—	2	2	no	—
112	C₄×C₁₂₇	—	2	2	no	—
113	C₄×C₁₂₇	#117	1	2	yes	—
114	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
115	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
116	Dic₁₂₇	—	2	2	no	—
117	Dic₁₂₇	—	2	1	no	—
118	Dic₁₂₇	—	2	1	no	—
119	Dic₁₂₇	—	2	1	no	—
120	Dic₁₂₇	#131	1	2	yes	—
121	C₂×C₁₂₇	—	2	2	no	—
122	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
123	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
124	C₂×C₁₂₇	#568	1	1	yes	characteristic
125	D₁₂₇	—	4	2	no	—
126	#2925	—	127	1	no	maximal, Sylow
127	C₁₂₇	#2925	1	1	yes	characteristic, complemented, Sylow
128	C₄×#41	—	127	1	no	—
129	#433	—	127	2	no	—
130	#433	—	127	2	no	—
131	#432	—	127	2	no	—
132	#431	—	127	2	no	—
133	#423	—	127	1	no	—
134	C₂×C₄⋊C₈	—	127	1	no	—
135	#408	—	127	1	no	—
136	#408	—	127	1	no	—
137	#405	—	127	1	no	—
138	#405	D₁₂₇	1	1	yes	characteristic, complemented
139	C₂×#41	—	127	1	no	—
140	C₂×M₁₆	—	127	1	no	—
141	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
142	C₂²×C₈	—	127	1	no	—
143	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
144	C₄×Q₈	—	127	1	no	—
145	C₄×Q₈	—	127	1	no	—
146	C₄×D₄	—	127	1	no	—
147	C₄×D₄	—	127	1	no	—
148	C₄×D₄	—	127	2	no	—
149	C₄×D₄	—	127	2	no	—
150	#117	—	127	2	no	—
151	#117	C₂×D₁₂₇	1	1	yes	characteristic
152	C₂×C₄²	—	127	1	no	—
153	C₂×C₄²	—	127	2	no	—
154	C₄⋊C₈	—	127	1	no	—
155	C₄⋊C₈	—	127	1	no	—
156	C₄⋊C₈	—	127	1	no	—
157	C₄⋊C₈	—	127	1	no	—
158	C₄⋊C₈	—	127	2	no	—
159	C₄⋊C₈	—	127	2	no	—
160	C₂²⋊C₈	—	127	2	no	—
161	C₂²⋊C₈	—	127	2	no	—
162	C₂²⋊C₈	—	127	2	no	—
163	C₂²⋊C₈	—	127	2	no	—
164	#97	—	127	2	no	—
165	#97	C₂×D₁₂₇	1	2	yes	—
166	C₄×C₈	—	127	2	no	—
167	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
168	#41	—	254	2	no	—
169	#41	—	254	2	no	—
170	C₂×Q₈	—	127	1	no	—
171	C₂×D₄	—	127	2	no	—
172	C₂×D₄	—	127	1	no	—
173	C₂²×C₄	—	127	1	no	—
174	C₂²×C₄	—	127	1	no	—
175	C₂²×C₄	—	127	2	no	—
176	C₂²×C₄	—	127	2	no	—
177	C₂²×C₄	—	127	2	no	—
178	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
179	M₁₆	—	2	2	no	—
180	M₁₆	—	254	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	127	1	no	—
183	C₂×C₈	—	127	2	no	—
184	C₂×C₈	—	127	1	no	—
185	C₂×C₈	—	254	2	no	—
186	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
187	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
188	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
189	C₄⋊C₄	—	127	2	no	—
190	C₄⋊C₄	—	127	2	no	—
191	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
192	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
193	C₂²⋊C₄	—	127	2	no	—
194	C₂²⋊C₄	—	127	2	no	—
195	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
196	C₄²	—	127	1	no	—
197	C₄²	—	127	2	no	—
198	C₄²	—	127	1	no	—
199	C₄²	—	127	2	no	—
200	C₄²	—	127	1	no	—
201	C₄²	—	127	1	no	—
202	C₄²	C₂²×D₁₂₇	1	2	yes	—
203	C₂³	—	127	2	no	—
204	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
205	Q₈	—	254	2	no	—
206	D₄	—	254	4	no	—
207	D₄	—	254	2	no	—
208	C₂×C₄	—	127	1	no	—
209	C₂×C₄	—	127	1	no	—
210	C₂×C₄	—	127	1	no	—
211	C₂×C₄	—	127	1	no	—
212	C₂×C₄	—	127	2	no	—
213	C₂×C₄	—	508	2	no	—
214	C₂×C₄	—	508	1	no	—
215	C₂×C₄	—	254	4	no	—
216	C₂×C₄	—	254	2	no	—
217	C₂×C₄	—	254	2	no	—
218	C₂×C₄	—	254	2	no	—
219	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
221	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
222	C₂×C₄	#14523248	1	2	yes	—
223	C₂×C₄	#14523248	1	2	yes	—
224	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	2	no	—
228	C₈	—	254	1	no	—
229	C₈	—	254	1	no	—
230	C₈	—	254	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	254	2	no	—
234	C₂²	—	254	2	no	—
235	C₂²	—	254	2	no	—
236	C₂²	#16861166	1	2	yes	—
237	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	127	2	no	—
241	C₄	—	254	2	no	—
242	C₄	—	254	1	no	—
243	C₄	—	254	1	no	—
244	C₄	—	254	1	no	—
245	C₄	#16861166	1	2	yes	—
246	C₄	C₂×#14523248	1	1	yes	characteristic
247	C₄	C₂×#14523248	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	508	2	no	—
250	C₂	C₂×#16861166	1	1	yes	characteristic
251	C₂	#131×D₁₂₇	1	1	yes	characteristic
252	C₂	#22550641	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#405×C₁₂₇	C₂
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#405	D₁₂₇
hypocenter	C₁₂₇	#2925
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22549938	C₂	semidirect
3. #22549935	C₂	semidirect
4. #22549931	C₂	semidirect
5. #22549927	C₂	semidirect
6. #22549899	C₂	semidirect
7. C₂×Dic₁₂₇⋊C₈	C₂	semidirect
10. #22549832	C₂	semidirect
11. #405×C₁₂₇	C₂	semidirect
127. C₁₂₇	#2925	semidirect
138. #405	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550344	2
#22549938	2
#22549935	2
#22549931	2
#22549927	2
#22549899	2
C₂×Dic₁₂₇⋊C₈	2
#22549844	2
#22549842	2
#22549832	2
#405×C₁₂₇	2
#2925	127