Subgroups — C₂×#22549991 #30647618

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861229	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861215	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549991	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16861144	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16861136	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16861135	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523248	C₂²	1	1	yes	characteristic
10	#16861229	C₄	1	4	yes	complemented
11	#16861229	C₂²	1	4	yes	—
12	C₂×#14523238	C₄	1	1	yes	characteristic, complemented
13	C₂×#14523236	C₄	1	1	yes	characteristic, complemented
14	C₂×#14523235	C₄	1	1	yes	characteristic, complemented
15	C₂×#14523233	C₄	1	1	yes	characteristic, complemented
16	#16861215	—	2	2	no	—
17	#16861215	C₂²	1	4	yes	complemented
18	C₂×#14523232	C₂²	1	1	yes	characteristic
19	C₂×#14523231	—	2	1	no	—
20	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₂₇	—	2	1	no	—
22	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
23	#16861144	C₂²	1	4	yes	complemented
24	#16861136	C₂²	1	4	yes	complemented
25	#16861135	C₂²	1	4	yes	complemented
26	C₈×Dic₁₂₇	C₂²	1	4	yes	complemented
27	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
28	#131×C₁₂₇	—	2	2	no	—
29	#131×C₁₂₇	C₂²	1	4	yes	complemented
30	C₂×M₁₆×C₁₂₇	—	2	1	no	—
31	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₂₇	—	2	1	no	—
33	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
34	#14523248	—	2	1	no	—
35	#14523248	—	2	1	no	—
36	#14523248	C₂×C₄	1	2	yes	—
37	#14523248	C₂³	1	2	yes	—
38	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
39	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
41	C₂²×Dic₁₂₇	—	2	1	no	—
42	C₂×C₄×D₁₂₇	—	2	1	no	—
43	#14523238	C₂×C₄	1	4	yes	complemented
44	#14523236	C₂×C₄	1	4	yes	complemented
45	#14523235	C₂×C₄	1	4	yes	complemented
46	#14523233	C₂×C₄	1	4	yes	complemented
47	#14523232	—	2	1	no	—
48	#14523232	C₂³	1	2	yes	—
49	#14523231	—	2	4	no	—
50	C₂×C₁₂₇⋊C₈	—	2	1	no	—
51	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
54	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
55	C₈×D₁₂₇	—	2	4	no	—
56	C₄×Dic₁₂₇	—	2	1	no	—
57	C₄×Dic₁₂₇	C₂³	1	2	yes	—
58	#41×C₁₂₇	—	2	1	no	—
59	#41×C₁₂₇	—	2	1	no	—
60	#41×C₁₂₇	C₂×C₄	1	2	yes	—
61	#41×C₁₂₇	C₂³	1	2	yes	—
62	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₂×D₄×C₁₂₇	—	2	1	no	—
64	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₂₇	—	2	1	no	—
66	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₂₇	—	2	1	no	—
68	M₁₆×C₁₂₇	—	2	4	no	—
69	M₁₆×C₁₂₇	C₂³	1	2	yes	—
70	C₂×C₈×C₁₂₇	—	2	1	no	—
71	C₂×C₈×C₁₂₇	—	2	4	no	—
72	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
74	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
76	C₂²×D₁₂₇	—	2	1	no	—
77	Dic₂₅₄	—	2	1	no	—
78	Dic₂₅₄	C₂²×C₄	1	2	yes	—
79	D₅₀₈	—	2	1	no	—
80	D₅₀₈	C₂²×C₄	1	2	yes	—
81	C₂²⋊D₁₂₇	—	2	2	no	—
82	C₂²⋊D₁₂₇	—	2	2	no	—
83	C₂×Dic₁₂₇	—	2	4	no	—
84	C₂×Dic₁₂₇	—	2	1	no	—
85	C₂×Dic₁₂₇	—	2	2	no	—
86	C₄×D₁₂₇	—	2	2	no	—
87	C₄×D₁₂₇	—	2	2	no	—
88	C₁₂₇⋊C₈	—	2	2	no	—
89	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
90	C₁₂₇⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂³×C₁₂₇	—	2	1	no	—
92	C₂³×C₁₂₇	#41	1	1	yes	characteristic
93	Q₈×C₁₂₇	—	2	1	no	—
94	Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
95	D₄×C₁₂₇	—	2	2	no	—
96	D₄×C₁₂₇	—	2	2	no	—
97	D₄×C₁₂₇	—	2	1	no	—
98	D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₂₇	—	2	1	no	—
100	C₂×C₄×C₁₂₇	—	2	2	no	—
101	C₂×C₄×C₁₂₇	—	2	2	no	—
102	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₂₇	—	2	2	no	—
109	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
111	C₂²×C₁₂₇	—	2	2	no	—
112	C₂²×C₁₂₇	—	2	2	no	—
113	C₂²×C₁₂₇	—	4	2	no	—
114	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
117	C₂×D₁₂₇	—	2	2	no	—
118	C₂×D₁₂₇	—	4	2	no	—
119	C₄×C₁₂₇	—	2	2	no	—
120	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
121	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
122	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
124	Dic₁₂₇	—	2	4	no	—
125	Dic₁₂₇	—	2	2	no	—
126	C₂×C₁₂₇	—	2	1	no	—
127	C₂×C₁₂₇	—	2	1	no	—
128	C₂×C₁₂₇	—	4	2	no	—
129	C₂×C₁₂₇	#443	1	2	yes	complemented
130	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
131	D₁₂₇	—	4	2	no	—
132	C₂×#443	—	127	1	no	maximal, Sylow
133	C₁₂₇	C₂×#443	1	1	yes	characteristic, complemented, Sylow
134	C₂×#135	—	127	1	no	—
135	C₂×#131	—	127	1	no	—
136	C₂×#131	D₁₂₇	1	1	yes	characteristic, complemented
137	#443	—	127	8	no	—
138	C₂×#108	—	127	1	no	—
139	C₂×#104	—	127	1	no	—
140	C₂×#104	—	127	1	no	—
141	C₂×C₄×C₈	—	127	1	no	—
142	C₂×#41	—	127	1	no	—
143	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
144	#135	—	127	4	no	—
145	#135	—	127	4	no	—
146	C₂×Q₁₆	—	127	1	no	—
147	C₂×SD₁₆	—	127	1	no	—
148	C₂×SD₁₆	—	127	1	no	—
149	C₂×D₈	—	127	1	no	—
150	#131	—	2	2	no	—
151	#131	—	127	4	no	—
152	#131	—	254	2	no	—
153	#131	C₂×D₁₂₇	1	4	yes	complemented
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	127	1	no	—
156	C₂×M₁₆	—	254	1	no	—
157	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	127	1	no	—
160	C₂²×C₈	—	254	1	no	—
161	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
162	C₂×C₄²	—	127	1	no	—
163	#108	—	127	4	no	—
164	#104	—	127	4	no	—
165	#104	—	127	4	no	—
166	C₄×C₈	—	127	4	no	—
167	#41	—	2	1	no	—
168	#41	—	2	1	no	—
169	#41	—	127	2	no	—
170	#41	—	127	2	no	—
171	#41	—	254	1	no	—
172	#41	—	254	1	no	—
173	#41	C₄×D₁₂₇	1	2	yes	—
174	#41	C₂²×D₁₂₇	1	2	yes	—
175	C₂×Q₈	—	127	1	no	—
176	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	127	1	no	—
179	C₂×D₄	—	254	1	no	—
180	C₂×D₄	C₄×D₁₂₇	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	254	1	no	—
183	C₂²×C₄	—	254	1	no	—
184	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
185	Q₁₆	—	127	4	no	—
186	SD₁₆	—	127	4	no	—
187	SD₁₆	—	127	4	no	—
188	D₈	—	127	4	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	127	2	no	—
192	M₁₆	—	254	1	no	—
193	M₁₆	—	254	4	no	—
194	M₁₆	C₂²×D₁₂₇	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	127	1	no	—
198	C₂×C₈	—	127	1	no	—
199	C₂×C₈	—	127	2	no	—
200	C₂×C₈	—	127	2	no	—
201	C₂×C₈	—	254	1	no	—
202	C₂×C₈	—	254	4	no	—
203	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
205	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
206	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
207	C₄²	—	127	2	no	—
208	C₄²	—	254	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	254	1	no	—
211	C₂³	#14523248	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	127	2	no	—
214	Q₈	—	254	1	no	—
215	Q₈	C₂×C₄×D₁₂₇	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	127	2	no	—
220	D₄	—	254	2	no	—
221	D₄	—	254	2	no	—
222	D₄	—	254	1	no	—
223	D₄	C₂×C₄×D₁₂₇	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	254	4	no	—
228	C₂×C₄	—	254	2	no	—
229	C₂×C₄	—	254	1	no	—
230	C₂×C₄	—	254	2	no	—
231	C₂×C₄	—	254	2	no	—
232	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
236	C₂×C₄	#14523248	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	127	2	no	—
240	C₈	—	127	2	no	—
241	C₈	—	254	2	no	—
242	C₈	C₂×C₂²⋊D₁₂₇	1	2	yes	—
243	C₈	C₂×C₂²⋊D₁₂₇	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	508	2	no	—
248	C₂²	—	254	2	no	—
249	C₂²	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
250	C₂²	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
251	C₂²	C₂×#14523248	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	254	4	no	—
254	C₄	—	254	2	no	—
255	C₄	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	508	2	no	—
263	C₂	#22549991	1	2	yes	direct product factor
264	C₂	C₂×C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#131×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₄×D₄
prime radical	C₂×#16861229	C₂
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#131	D₁₂₇
hypocenter	C₁₂₇	C₂×#443
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#16861215	C₂	semidirect
3. #22549991	C₂	direct (N × H)
4. C₂×#16861144	C₂	semidirect
5. C₂×#16861136	C₂	semidirect
6. C₂×#16861135	C₂	semidirect
7. C₂×C₈×Dic₁₂₇	C₂	semidirect
8. C₂×#131×C₁₂₇	C₂	semidirect
10. #16861229	C₄	semidirect
12. C₂×#14523238	C₄	semidirect
13. C₂×#14523236	C₄	semidirect
14. C₂×#14523235	C₄	semidirect
15. C₂×#14523233	C₄	semidirect
17. #16861215	C₂²	semidirect
23. #16861144	C₂²	semidirect
24. #16861136	C₂²	semidirect
25. #16861135	C₂²	semidirect
26. C₈×Dic₁₂₇	C₂²	semidirect
29. #131×C₁₂₇	C₂²	semidirect
43. #14523238	C₂×C₄	semidirect
44. #14523236	C₂×C₄	semidirect
45. #14523235	C₂×C₄	semidirect
46. #14523233	C₂×C₄	semidirect
129. C₂×C₁₂₇	#443	semidirect
133. C₁₂₇	C₂×#443	semidirect
136. C₂×#131	D₁₂₇	semidirect
153. #131	C₂×D₁₂₇	semidirect
263. C₂	#22549991	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×#16861229	2
C₂×#16861215	2
#22549991	2
C₂×#16861144	2
C₂×#16861136	2
C₂×#16861135	2
C₂×C₈×Dic₁₂₇	2
C₂×#131×C₁₂₇	2
C₂×#443	127

Subgroups — C2×#22549991 #30647618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549991 #30647618