Subgroups — C₂×#417×C₁₂₇ #30635847

Order 16256.

816 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#417×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#103×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×#102×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂×#41×C₁₂₇	—	2	4	no	—
8	C₂×#41×C₁₂₇	—	2	2	no	—
9	C₂×#41×C₁₂₇	C₄	1	8	yes	complemented
10	C₂×#41×C₁₂₇	C₂²	1	4	yes	—
11	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₂₇	—	2	1	no	—
13	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₂₇	—	2	1	no	—
15	C₂³×C₄×C₁₂₇	—	2	1	no	—
16	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₂₇	—	2	4	no	—
18	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	complemented
19	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
22	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	complemented
24	#103×C₁₂₇	C₂²	1	8	yes	complemented
25	#102×C₁₂₇	C₂²	1	8	yes	complemented
26	C₂⁴×C₁₂₇	—	2	1	no	—
27	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
28	#41×C₁₂₇	—	2	16	no	—
29	#41×C₁₂₇	—	2	8	no	—
30	#41×C₁₂₇	C₂×C₄	1	16	yes	complemented
31	C₂×Q₈×C₁₂₇	—	2	2	no	—
32	C₂×Q₈×C₁₂₇	—	2	1	no	—
33	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
34	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
35	C₂×D₄×C₁₂₇	—	2	8	no	—
36	C₂×D₄×C₁₂₇	—	2	4	no	—
37	C₂×D₄×C₁₂₇	—	2	1	no	—
38	C₂×D₄×C₁₂₇	—	2	2	no	—
39	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
40	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₂₇	—	2	1	no	—
42	C₂²×C₄×C₁₂₇	—	2	8	no	—
43	C₂²×C₄×C₁₂₇	—	2	4	no	—
44	C₂²×C₄×C₁₂₇	—	2	4	no	—
45	C₂²×C₄×C₁₂₇	—	2	2	no	—
46	C₂²×C₄×C₁₂₇	—	4	4	no	—
47	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
49	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
51	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
53	M₁₆×C₁₂₇	—	2	8	no	—
54	C₂×C₈×C₁₂₇	—	2	4	no	—
55	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
56	C₄⋊C₄×C₁₂₇	—	2	2	no	—
57	C₄⋊C₄×C₁₂₇	—	2	4	no	—
58	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
59	C₂³×C₁₂₇	—	2	2	no	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	—	2	4	no	—
62	C₂³×C₁₂₇	—	2	2	no	—
63	C₂³×C₁₂₇	—	4	2	no	—
64	C₂³×C₁₂₇	—	4	2	no	—
65	C₂³×C₁₂₇	C₂²⋊C₄	1	4	yes	—
66	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
67	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₁₂₇	—	2	4	no	—
69	Q₈×C₁₂₇	—	2	2	no	—
70	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	complemented
71	D₄×C₁₂₇	—	2	16	no	—
72	D₄×C₁₂₇	—	2	4	no	—
73	D₄×C₁₂₇	—	2	2	no	—
74	D₄×C₁₂₇	C₂²×C₄	1	4	yes	complemented
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	—	2	16	no	—
78	C₂×C₄×C₁₂₇	—	2	4	no	—
79	C₂×C₄×C₁₂₇	—	4	8	no	—
80	C₂×C₄×C₁₂₇	—	4	4	no	—
81	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
82	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
83	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
89	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₂₇	—	2	4	no	—
91	C₂²×C₁₂₇	—	2	2	no	—
92	C₂²×C₁₂₇	—	2	2	no	—
93	C₂²×C₁₂₇	—	2	8	no	—
94	C₂²×C₁₂₇	—	2	4	no	—
95	C₂²×C₁₂₇	—	4	2	no	—
96	C₂²×C₁₂₇	—	4	2	no	—
97	C₂²×C₁₂₇	—	4	8	no	—
98	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
101	C₂²×C₁₂₇	#137	1	2	yes	—
102	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₁₂₇	—	2	4	no	—
104	C₄×C₁₂₇	—	4	4	no	—
105	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
106	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
107	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₁₂₇	—	2	4	no	—
110	C₂×C₁₂₇	—	4	4	no	—
111	C₂×C₁₂₇	#417	1	4	yes	direct product factor
112	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₁₂₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₁₂₇	C₂×#137	1	1	yes	characteristic
115	C₂×#417	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₁₂₇	C₂×#417	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×#41	C₂×C₁₂₇	1	1	yes	characteristic
118	C₂²×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
119	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
120	#417	C₂×C₁₂₇	1	8	yes	direct product factor
121	C₂×#103	C₂×C₁₂₇	1	2	yes	complemented
122	C₂×#102	C₂×C₁₂₇	1	2	yes	complemented
123	C₂×#41	—	2	4	no	—
124	C₂×#41	—	2	2	no	—
125	C₂×#41	C₄×C₁₂₇	1	8	yes	complemented
126	C₂×#41	C₂²×C₁₂₇	1	4	yes	—
127	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₂²×D₄	—	2	1	no	—
129	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
130	C₂³×C₄	—	2	1	no	—
131	C₂³×C₄	—	2	1	no	—
132	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₂×M₁₆	—	2	4	no	—
134	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	complemented
135	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	complemented
140	#103	C₂²×C₁₂₇	1	8	yes	complemented
141	#102	C₂²×C₁₂₇	1	8	yes	complemented
142	C₂⁴	—	2	1	no	—
143	C₂⁴	D₄×C₁₂₇	1	1	yes	characteristic
144	#41	—	2	16	no	—
145	#41	—	2	8	no	—
146	#41	C₂×C₄×C₁₂₇	1	16	yes	complemented
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
150	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
151	C₂×D₄	—	2	8	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	complemented
156	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
164	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
165	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
166	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
167	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
169	M₁₆	—	2	8	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	4	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	C₂²⋊C₄×C₁₂₇	1	4	yes	—
182	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
183	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
184	Q₈	—	2	4	no	—
185	Q₈	—	2	2	no	—
186	Q₈	C₂²×C₄×C₁₂₇	1	4	yes	complemented
187	D₄	—	2	16	no	—
188	D₄	—	2	4	no	—
189	D₄	—	2	2	no	—
190	D₄	C₂²×C₄×C₁₂₇	1	4	yes	complemented
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
198	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
199	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
200	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
201	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
202	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
204	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
205	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
206	C₈	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	8	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	8	no	—
214	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
215	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
216	C₂²	AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
217	C₂²	#137×C₁₂₇	1	2	yes	—
218	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	4	4	no	—
221	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
222	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
223	C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
224	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
225	C₂	—	2	4	no	—
226	C₂	—	4	4	no	—
227	C₂	#417×C₁₂₇	1	4	yes	direct product factor
228	C₂	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
229	C₂	C₂×AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
230	C₂	C₂×#137×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂×D₄
commutator	C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
prime radical	C₂²×#41×C₁₂₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₁₂₇	C₂	semidirect
3. C₂²×C₄⋊C₄×C₁₂₇	C₂	semidirect
4. #417×C₁₂₇	C₂	direct (N × H)
5. C₂×#103×C₁₂₇	C₂	semidirect
6. C₂×#102×C₁₂₇	C₂	semidirect
9. C₂×#41×C₁₂₇	C₄	semidirect
18. C₂×M₁₆×C₁₂₇	C₂²	semidirect
23. C₂×C₄⋊C₄×C₁₂₇	C₂²	semidirect
24. #103×C₁₂₇	C₂²	semidirect
25. #102×C₁₂₇	C₂²	semidirect
30. #41×C₁₂₇	C₂×C₄	semidirect
33. C₂×Q₈×C₁₂₇	C₂×C₄	semidirect
39. C₂×D₄×C₁₂₇	C₂×C₄	semidirect
70. Q₈×C₁₂₇	C₂²×C₄	semidirect
74. D₄×C₁₂₇	C₂²×C₄	semidirect
111. C₂×C₁₂₇	#417	direct (N × H)
115. C₂×#417	C₁₂₇	direct (N × H)
116. C₁₂₇	C₂×#417	direct (N × H)
118. C₂²×M₁₆	C₂×C₁₂₇	semidirect
119. C₂²×C₄⋊C₄	C₂×C₁₂₇	semidirect
120. #417	C₂×C₁₂₇	direct (N × H)
121. C₂×#103	C₂×C₁₂₇	semidirect
122. C₂×#102	C₂×C₁₂₇	semidirect
125. C₂×#41	C₄×C₁₂₇	semidirect
134. C₂×M₁₆	C₂²×C₁₂₇	semidirect
139. C₂×C₄⋊C₄	C₂²×C₁₂₇	semidirect
140. #103	C₂²×C₁₂₇	semidirect
141. #102	C₂²×C₁₂₇	semidirect
146. #41	C₂×C₄×C₁₂₇	semidirect
149. C₂×Q₈	C₂×C₄×C₁₂₇	semidirect
155. C₂×D₄	C₂×C₄×C₁₂₇	semidirect
186. Q₈	C₂²×C₄×C₁₂₇	semidirect
190. D₄	C₂²×C₄×C₁₂₇	semidirect
227. C₂	#417×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
C₂²×#41×C₁₂₇	2
C₂²×M₁₆×C₁₂₇	2
C₂²×C₄⋊C₄×C₁₂₇	2
#417×C₁₂₇	2
C₂×#103×C₁₂₇	2
C₂×#102×C₁₂₇	2
C₂×#417	127

Subgroups — C2×#417×C127 #30635847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#417×C₁₂₇ #30635847