Subgroups — C₂×#471×C₁₂₇ #30636016

Order 16256.

468 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×AGL(1,ℤ/8)×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#471×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#101×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#100×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	#137×C₁₂₇	—	2	2	no	—
13	#137×C₁₂₇	C₂²	1	4	yes	complemented
14	AGL(1,ℤ/8)×C₁₂₇	—	2	2	no	—
15	AGL(1,ℤ/8)×C₁₂₇	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
17	C₂×SD₁₆×C₁₂₇	—	2	1	no	—
18	C₂×SD₁₆×C₁₂₇	—	2	1	no	—
19	C₂×SD₁₆×C₁₂₇	—	2	1	no	—
20	C₂×SD₁₆×C₁₂₇	—	2	1	no	—
21	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
22	C₂×SD₁₆×C₁₂₇	C₂²	1	4	yes	complemented
23	C₂×D₈×C₁₂₇	—	2	1	no	—
24	#131×C₁₂₇	—	2	2	no	—
25	#131×C₁₂₇	C₂²	1	4	yes	complemented
26	C₂×M₁₆×C₁₂₇	—	2	1	no	—
27	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₁₂₇	—	2	1	no	—
31	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
32	#108×C₁₂₇	C₂²	1	4	yes	complemented
33	#101×C₁₂₇	C₂²	1	4	yes	complemented
34	#100×C₁₂₇	C₂²	1	4	yes	complemented
35	C₂⁴×C₁₂₇	—	2	1	no	—
36	#41×C₁₂₇	—	2	1	no	—
37	#41×C₁₂₇	—	2	1	no	—
38	#41×C₁₂₇	D₄	1	2	yes	—
39	#41×C₁₂₇	C₂³	1	2	yes	—
40	C₂×Q₈×C₁₂₇	—	2	1	no	—
41	C₂×Q₈×C₁₂₇	—	2	1	no	—
42	C₂×Q₈×C₁₂₇	—	2	1	no	—
43	C₂×Q₈×C₁₂₇	—	2	1	no	—
44	C₂×Q₈×C₁₂₇	—	2	1	no	—
45	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
46	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
47	C₂×D₄×C₁₂₇	—	2	1	no	—
48	C₂×D₄×C₁₂₇	—	2	1	no	—
49	C₂×D₄×C₁₂₇	—	2	1	no	—
50	C₂×D₄×C₁₂₇	—	2	1	no	—
51	C₂×D₄×C₁₂₇	—	2	1	no	—
52	C₂×D₄×C₁₂₇	—	2	1	no	—
53	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
54	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₂₇	—	2	1	no	—
56	C₂²×C₄×C₁₂₇	—	2	1	no	—
57	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
58	Q₁₆×C₁₂₇	—	2	4	no	—
59	SD₁₆×C₁₂₇	—	2	4	no	—
60	SD₁₆×C₁₂₇	—	2	4	no	—
61	SD₁₆×C₁₂₇	—	2	4	no	—
62	SD₁₆×C₁₂₇	—	2	4	no	—
63	D₈×C₁₂₇	—	2	4	no	—
64	M₁₆×C₁₂₇	—	2	1	no	—
65	M₁₆×C₁₂₇	—	2	1	no	—
66	M₁₆×C₁₂₇	—	2	1	no	—
67	M₁₆×C₁₂₇	—	2	4	no	—
68	M₁₆×C₁₂₇	C₂³	1	2	yes	—
69	M₁₆×C₁₂₇	C₂³	1	2	yes	—
70	M₁₆×C₁₂₇	C₂³	1	2	yes	—
71	C₂×C₈×C₁₂₇	—	2	1	no	—
72	C₂×C₈×C₁₂₇	—	2	1	no	—
73	C₂×C₈×C₁₂₇	—	2	1	no	—
74	C₂×C₈×C₁₂₇	—	2	4	no	—
75	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
78	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
79	C₂³×C₁₂₇	—	2	1	no	—
80	C₂³×C₁₂₇	—	2	2	no	—
81	C₂³×C₁₂₇	—	4	1	no	—
82	C₂³×C₁₂₇	—	4	1	no	—
83	C₂³×C₁₂₇	—	8	2	no	—
84	C₂³×C₁₂₇	#41	1	1	yes	characteristic
85	Q₈×C₁₂₇	—	2	1	no	—
86	Q₈×C₁₂₇	—	2	2	no	—
87	Q₈×C₁₂₇	—	2	2	no	—
88	Q₈×C₁₂₇	—	4	1	no	—
89	Q₈×C₁₂₇	—	4	1	no	—
90	Q₈×C₁₂₇	C₂×D₄	1	2	yes	—
91	D₄×C₁₂₇	—	2	2	no	—
92	D₄×C₁₂₇	—	2	2	no	—
93	D₄×C₁₂₇	—	2	1	no	—
94	D₄×C₁₂₇	—	2	2	no	—
95	D₄×C₁₂₇	—	2	2	no	—
96	D₄×C₁₂₇	—	4	1	no	—
97	D₄×C₁₂₇	—	4	1	no	—
98	D₄×C₁₂₇	C₂×D₄	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	—	2	2	no	—
103	C₂×C₄×C₁₂₇	—	4	1	no	—
104	C₂×C₄×C₁₂₇	—	4	1	no	—
105	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
111	C₈×C₁₂₇	—	2	2	no	—
112	C₈×C₁₂₇	—	2	2	no	—
113	C₈×C₁₂₇	—	2	2	no	—
114	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
115	C₈×C₁₂₇	C₂×D₄	1	2	yes	—
116	C₂²×C₁₂₇	—	2	2	no	—
117	C₂²×C₁₂₇	—	2	2	no	—
118	C₂²×C₁₂₇	—	4	2	no	—
119	C₂²×C₁₂₇	—	4	2	no	—
120	C₂²×C₁₂₇	—	8	2	no	—
121	C₂²×C₁₂₇	—	8	2	no	—
122	C₂²×C₁₂₇	—	8	2	no	—
123	C₂²×C₁₂₇	#121	1	1	yes	characteristic
124	C₂²×C₁₂₇	#121	1	1	yes	characteristic
125	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
126	C₄×C₁₂₇	—	2	2	no	—
127	C₄×C₁₂₇	—	4	2	no	—
128	C₄×C₁₂₇	#121	1	1	yes	characteristic
129	C₄×C₁₂₇	#121	1	1	yes	characteristic
130	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₁₂₇	—	2	1	no	—
133	C₂×C₁₂₇	—	2	1	no	—
134	C₂×C₁₂₇	—	4	2	no	—
135	C₂×C₁₂₇	—	8	2	no	—
136	C₂×C₁₂₇	#471	1	2	yes	direct product factor
137	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
138	C₂×#471	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₁₂₇	C₂×#471	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×#137	C₂×C₁₂₇	1	1	yes	characteristic, complemented
141	C₂×AGL(1,ℤ/8)	C₂×C₁₂₇	1	1	yes	characteristic
142	C₂²×SD₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
143	C₂×#131	C₂×C₁₂₇	1	1	yes	characteristic, complemented
144	#471	C₂×C₁₂₇	1	8	yes	direct product factor
145	C₂×#108	C₂×C₁₂₇	1	1	yes	characteristic, complemented
146	C₂×#101	C₂×C₁₂₇	1	1	yes	characteristic, complemented
147	C₂×#100	C₂×C₁₂₇	1	1	yes	characteristic, complemented
148	C₂×#41	C₂²×C₁₂₇	1	1	yes	characteristic
149	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
150	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
151	#137	—	2	2	no	—
152	#137	C₂²×C₁₂₇	1	4	yes	complemented
153	AGL(1,ℤ/8)	—	2	2	no	—
154	AGL(1,ℤ/8)	C₂²×C₁₂₇	1	4	yes	—
155	C₂×Q₁₆	—	2	1	no	—
156	C₂×SD₁₆	—	2	1	no	—
157	C₂×SD₁₆	—	2	1	no	—
158	C₂×SD₁₆	—	2	1	no	—
159	C₂×SD₁₆	—	2	1	no	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	C₂²×C₁₂₇	1	4	yes	complemented
162	C₂×D₈	—	2	1	no	—
163	#131	—	2	2	no	—
164	#131	C₂²×C₁₂₇	1	4	yes	complemented
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
167	C₂×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
168	C₂×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
171	#108	C₂²×C₁₂₇	1	4	yes	complemented
172	#101	C₂²×C₁₂₇	1	4	yes	complemented
173	#100	C₂²×C₁₂₇	1	4	yes	complemented
174	C₂⁴	—	2	1	no	—
175	#41	—	2	1	no	—
176	#41	—	2	1	no	—
177	#41	D₄×C₁₂₇	1	2	yes	—
178	#41	C₂³×C₁₂₇	1	2	yes	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	—	2	1	no	—
184	C₂×Q₈	D₄×C₁₂₇	1	1	yes	characteristic
185	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	D₄×C₁₂₇	1	1	yes	characteristic
193	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
197	Q₁₆	—	2	4	no	—
198	SD₁₆	—	2	4	no	—
199	SD₁₆	—	2	4	no	—
200	SD₁₆	—	2	4	no	—
201	SD₁₆	—	2	4	no	—
202	D₈	—	2	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	C₂³×C₁₂₇	1	2	yes	—
208	M₁₆	C₂³×C₁₂₇	1	2	yes	—
209	M₁₆	C₂³×C₁₂₇	1	2	yes	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
215	C₂×C₈	D₄×C₁₂₇	1	1	yes	characteristic
216	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
217	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	4	1	no	—
221	C₂³	—	4	1	no	—
222	C₂³	—	8	2	no	—
223	C₂³	#41×C₁₂₇	1	1	yes	characteristic
224	Q₈	—	2	1	no	—
225	Q₈	—	2	2	no	—
226	Q₈	—	2	2	no	—
227	Q₈	—	4	1	no	—
228	Q₈	—	4	1	no	—
229	Q₈	C₂×D₄×C₁₂₇	1	2	yes	—
230	D₄	—	2	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	2	no	—
235	D₄	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	C₂×D₄×C₁₂₇	1	2	yes	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
248	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
249	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
254	C₈	C₂×D₄×C₁₂₇	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	#121×C₁₂₇	1	1	yes	characteristic
263	C₂²	#121×C₁₂₇	1	1	yes	characteristic
264	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	4	2	no	—
267	C₄	#121×C₁₂₇	1	1	yes	characteristic
268	C₄	#121×C₁₂₇	1	1	yes	characteristic
269	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
270	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	8	2	no	—
275	C₂	#471×C₁₂₇	1	2	yes	direct product factor
276	C₂	C₂×#121×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
prime radical	C₂×AGL(1,ℤ/8)×C₁₂₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#137×C₁₂₇	C₂	semidirect
3. C₂²×SD₁₆×C₁₂₇	C₂	semidirect
4. C₂×#131×C₁₂₇	C₂	semidirect
5. #471×C₁₂₇	C₂	direct (N × H)
6. C₂×#108×C₁₂₇	C₂	semidirect
7. C₂×#101×C₁₂₇	C₂	semidirect
8. C₂×#100×C₁₂₇	C₂	semidirect
13. #137×C₁₂₇	C₂²	semidirect
22. C₂×SD₁₆×C₁₂₇	C₂²	semidirect
25. #131×C₁₂₇	C₂²	semidirect
32. #108×C₁₂₇	C₂²	semidirect
33. #101×C₁₂₇	C₂²	semidirect
34. #100×C₁₂₇	C₂²	semidirect
136. C₂×C₁₂₇	#471	direct (N × H)
138. C₂×#471	C₁₂₇	direct (N × H)
139. C₁₂₇	C₂×#471	direct (N × H)
140. C₂×#137	C₂×C₁₂₇	semidirect
142. C₂²×SD₁₆	C₂×C₁₂₇	semidirect
143. C₂×#131	C₂×C₁₂₇	semidirect
144. #471	C₂×C₁₂₇	direct (N × H)
145. C₂×#108	C₂×C₁₂₇	semidirect
146. C₂×#101	C₂×C₁₂₇	semidirect
147. C₂×#100	C₂×C₁₂₇	semidirect
152. #137	C₂²×C₁₂₇	semidirect
161. C₂×SD₁₆	C₂²×C₁₂₇	semidirect
164. #131	C₂²×C₁₂₇	semidirect
171. #108	C₂²×C₁₂₇	semidirect
172. #101	C₂²×C₁₂₇	semidirect
173. #100	C₂²×C₁₂₇	semidirect
275. C₂	#471×C₁₂₇	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×#137×C₁₂₇	2
C₂×AGL(1,ℤ/8)×C₁₂₇	2
C₂²×SD₁₆×C₁₂₇	2
C₂×#131×C₁₂₇	2
#471×C₁₂₇	2
C₂×#108×C₁₂₇	2
C₂×#101×C₁₂₇	2
C₂×#100×C₁₂₇	2
C₂×#471	127

Subgroups — C2×#471×C127 #30636016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#471×C₁₂₇ #30636016