Subgroups — C₁₂₇×#2311 #30635263

Order 16256.

880 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	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#391×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#390×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#386×C₁₂₇	C₂	1	4	yes	maximal, complemented
7	#380×C₁₂₇	C₂	1	4	yes	maximal, complemented
8	C₂×#41×C₁₂₇	—	2	4	no	—
9	C₂×#41×C₁₂₇	—	2	4	no	—
10	C₂×#41×C₁₂₇	C₄	1	8	yes	complemented
11	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
16	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	#117×C₁₂₇	—	2	4	no	—
18	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	8	yes	—
24	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
25	#95×C₁₂₇	C₂²	1	8	yes	—
26	C₂⁴×C₁₂₇	C₂³	1	2	yes	—
27	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
28	#41×C₁₂₇	—	2	32	no	—
29	C₂×Q₈×C₁₂₇	—	2	2	no	—
30	C₂×Q₈×C₁₂₇	—	2	2	no	—
31	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
32	C₂×D₄×C₁₂₇	—	2	4	no	—
33	C₂×D₄×C₁₂₇	—	2	4	no	—
34	C₂×D₄×C₁₂₇	—	2	2	no	—
35	C₂×D₄×C₁₂₇	—	2	2	no	—
36	C₂×D₄×C₁₂₇	C₂×C₄	1	8	yes	—
37	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
38	C₂²×C₄×C₁₂₇	—	2	4	no	—
39	C₂²×C₄×C₁₂₇	—	2	4	no	—
40	C₂²×C₄×C₁₂₇	—	2	4	no	—
41	C₂²×C₄×C₁₂₇	—	4	4	no	—
42	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	complemented
44	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
45	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
46	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
47	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
48	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₂₇	—	2	4	no	—
52	C₄⋊C₄×C₁₂₇	—	2	4	no	—
53	C₄⋊C₄×C₁₂₇	—	2	4	no	—
54	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
55	C₂²⋊C₄×C₁₂₇	—	4	8	no	—
56	C₄²×C₁₂₇	—	2	4	no	—
57	C₂³×C₁₂₇	—	2	2	no	—
58	C₂³×C₁₂₇	—	2	2	no	—
59	C₂³×C₁₂₇	—	2	2	no	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	—	2	4	no	—
62	C₂³×C₁₂₇	—	2	4	no	—
63	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
64	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
65	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
66	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
67	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
68	Q₈×C₁₂₇	—	2	8	no	—
69	D₄×C₁₂₇	—	2	16	no	—
70	D₄×C₁₂₇	—	2	8	no	—
71	C₂×C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄×C₁₂₇	—	2	2	no	—
73	C₂×C₄×C₁₂₇	—	2	16	no	—
74	C₂×C₄×C₁₂₇	—	2	4	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	2	4	no	—
77	C₂×C₄×C₁₂₇	—	2	8	no	—
78	C₂×C₄×C₁₂₇	—	2	4	no	—
79	C₂×C₄×C₁₂₇	—	4	8	no	—
80	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	16	yes	complemented
81	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂²×C₁₂₇	—	2	4	no	—
89	C₂²×C₁₂₇	—	2	2	no	—
90	C₂²×C₁₂₇	—	2	2	no	—
91	C₂²×C₁₂₇	—	2	4	no	—
92	C₂²×C₁₂₇	—	2	4	no	—
93	C₂²×C₁₂₇	—	2	8	no	—
94	C₂²×C₁₂₇	—	4	2	no	—
95	C₂²×C₁₂₇	—	4	2	no	—
96	C₂²×C₁₂₇	—	4	4	no	—
97	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
103	C₄×C₁₂₇	—	2	4	no	—
104	C₄×C₁₂₇	—	4	4	no	—
105	C₄×C₁₂₇	—	4	4	no	—
106	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
107	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂×C₁₂₇	—	2	4	no	—
109	C₂×C₁₂₇	—	4	4	no	—
110	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₁₂₇	#520	1	2	yes	—
112	C₂×C₁₂₇	#535	1	1	yes	characteristic
113	C₂×C₁₂₇	#536	1	1	yes	characteristic
114	C₂×C₁₂₇	#537	1	2	yes	—
115	#2311	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₁₂₇	#2311	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×#41	C₂×C₁₂₇	1	1	yes	characteristic
118	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic, complemented
119	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
120	#391	C₂×C₁₂₇	1	2	yes	complemented
121	#390	C₂×C₁₂₇	1	2	yes	complemented
122	#386	C₂×C₁₂₇	1	4	yes	complemented
123	#380	C₂×C₁₂₇	1	4	yes	complemented
124	C₂×#41	—	2	4	no	—
125	C₂×#41	—	2	4	no	—
126	C₂×#41	C₄×C₁₂₇	1	8	yes	complemented
127	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₂²×D₄	C₂²×C₁₂₇	1	2	yes	—
129	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
131	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
132	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
133	#117	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	8	yes	—
140	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
141	#95	C₂²×C₁₂₇	1	8	yes	—
142	C₂⁴	C₂³×C₁₂₇	1	2	yes	—
143	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
144	#41	—	2	32	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
148	C₂×D₄	—	2	4	no	—
149	C₂×D₄	—	2	4	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	C₂×C₄×C₁₂₇	1	8	yes	—
153	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	complemented
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
159	C₂²×C₄	C₂×C₄×C₁₂₇	1	8	yes	complemented
160	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
161	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
162	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
163	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
164	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
165	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	4	8	no	—
172	C₄²	—	2	4	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	4	no	—
178	C₂³	—	2	4	no	—
179	C₂³	C₂²×C₄×C₁₂₇	1	4	yes	—
180	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
181	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
182	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
183	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	D₄	—	2	16	no	—
186	D₄	—	2	8	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	16	yes	complemented
197	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
198	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
199	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
200	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
201	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
202	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	4	no	—
213	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
214	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
215	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
216	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
217	C₂²	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
218	C₂²	2₋¹⁺⁴×C₁₂₇	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	4	4	no	—
221	C₄	—	4	4	no	—
222	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
223	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
224	C₂	—	2	4	no	—
225	C₂	—	4	4	no	—
226	C₂	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
227	C₂	#520×C₁₂₇	1	2	yes	—
228	C₂	#535×C₁₂₇	1	1	yes	characteristic
229	C₂	#536×C₁₂₇	1	1	yes	characteristic
230	C₂	#537×C₁₂₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#117×C₁₂₇	C₂	semidirect
3. C₂²×C₄⋊C₄×C₁₂₇	C₂	semidirect
4. #391×C₁₂₇	C₂	semidirect
5. #390×C₁₂₇	C₂	semidirect
6. #386×C₁₂₇	C₂	semidirect
7. #380×C₁₂₇	C₂	semidirect
10. C₂×#41×C₁₂₇	C₄	semidirect
31. C₂×Q₈×C₁₂₇	C₂×C₄	semidirect
37. C₂×D₄×C₁₂₇	C₂×C₄	semidirect
43. C₂²×C₄×C₁₂₇	C₂×C₄	semidirect
80. C₂×C₄×C₁₂₇	C₂²⋊C₄	semidirect
115. #2311	C₁₂₇	direct (N × H)
116. C₁₂₇	#2311	direct (N × H)
118. C₂×#117	C₂×C₁₂₇	semidirect
119. C₂²×C₄⋊C₄	C₂×C₁₂₇	semidirect
120. #391	C₂×C₁₂₇	semidirect
121. #390	C₂×C₁₂₇	semidirect
122. #386	C₂×C₁₂₇	semidirect
123. #380	C₂×C₁₂₇	semidirect
126. C₂×#41	C₄×C₁₂₇	semidirect
147. C₂×Q₈	C₂×C₄×C₁₂₇	semidirect
153. C₂×D₄	C₂×C₄×C₁₂₇	semidirect
159. C₂²×C₄	C₂×C₄×C₁₂₇	semidirect
196. C₂×C₄	C₂²⋊C₄×C₁₂₇	semidirect

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
C₂×#117×C₁₂₇	2
C₂²×C₄⋊C₄×C₁₂₇	2
#391×C₁₂₇	2
#390×C₁₂₇	2
#386×C₁₂₇	2
#380×C₁₂₇	2
#2311	127

Subgroups — C127×#2311 #30635263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2311 #30635263