Subgroups — C₁₂₇×#2628 #30635580

Order 16256.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#400×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#399×C₁₂₇	C₂	1	2	yes	maximal
5	#395×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#393×C₁₂₇	C₂	1	2	yes	maximal
7	#389×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#380×C₁₂₇	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
13	#117×C₁₂₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
25	#95×C₁₂₇	C₂²	1	4	yes	—
26	#95×C₁₂₇	C₂²	1	2	yes	—
27	#95×C₁₂₇	C₂²	1	2	yes	—
28	#95×C₁₂₇	C₂²	1	4	yes	—
29	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₂₇	—	2	4	no	—
31	C₂×Q₈×C₁₂₇	—	4	4	no	—
32	C₂²×C₄×C₁₂₇	—	2	2	no	—
33	C₂²×C₄×C₁₂₇	—	2	2	no	—
34	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	complemented
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₂₇	—	2	2	no	—
44	C₄⋊C₄×C₁₂₇	—	2	2	no	—
45	C₄⋊C₄×C₁₂₇	—	2	4	no	—
46	C₄⋊C₄×C₁₂₇	—	2	4	no	—
47	C₄⋊C₄×C₁₂₇	—	2	4	no	—
48	C₄⋊C₄×C₁₂₇	—	4	4	no	—
49	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
50	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
51	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
52	C₄²×C₁₂₇	—	2	2	no	—
53	C₄²×C₁₂₇	D₄	1	4	yes	—
54	C₂³×C₁₂₇	—	2	1	no	—
55	C₂³×C₁₂₇	—	2	1	no	—
56	C₂³×C₁₂₇	—	2	2	no	—
57	C₂³×C₁₂₇	—	2	2	no	—
58	C₂³×C₁₂₇	C₂×Q₈	1	2	yes	—
59	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
60	Q₈×C₁₂₇	—	4	8	no	—
61	C₂×C₄×C₁₂₇	—	2	1	no	—
62	C₂×C₄×C₁₂₇	—	2	1	no	—
63	C₂×C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄×C₁₂₇	—	2	8	no	—
66	C₂×C₄×C₁₂₇	—	2	1	no	—
67	C₂×C₄×C₁₂₇	—	2	1	no	—
68	C₂×C₄×C₁₂₇	—	2	4	no	—
69	C₂×C₄×C₁₂₇	—	2	4	no	—
70	C₂×C₄×C₁₂₇	—	2	4	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	2	2	no	—
73	C₂×C₄×C₁₂₇	—	2	4	no	—
74	C₂×C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₁₂₇	#41	1	2	yes	—
82	C₂²×C₁₂₇	—	2	2	no	—
83	C₂²×C₁₂₇	—	2	4	no	—
84	C₂²×C₁₂₇	—	4	1	no	—
85	C₂²×C₁₂₇	—	4	1	no	—
86	C₂²×C₁₂₇	—	4	2	no	—
87	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
90	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
91	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
92	C₄×C₁₂₇	—	2	2	no	—
93	C₄×C₁₂₇	—	2	2	no	—
94	C₄×C₁₂₇	—	2	2	no	—
95	C₄×C₁₂₇	—	4	1	no	—
96	C₄×C₁₂₇	—	4	4	no	—
97	C₄×C₁₂₇	—	4	1	no	—
98	C₄×C₁₂₇	—	4	4	no	—
99	C₄×C₁₂₇	C₂²⋊Q₈	1	4	yes	—
100	C₂×C₁₂₇	—	4	2	no	—
101	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₁₂₇	#535	1	1	yes	characteristic
103	C₂×C₁₂₇	#536	1	1	yes	characteristic
104	C₂×C₁₂₇	#542	1	2	yes	—
105	C₂×C₁₂₇	#544	1	2	yes	—
106	#2628	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₁₂₇	#2628	1	1	yes	characteristic, direct product factor, Sylow
108	C₂×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
109	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic
110	#400	C₂×C₁₂₇	1	2	yes	complemented
111	#399	C₂×C₁₂₇	1	2	yes	—
112	#395	C₂×C₁₂₇	1	2	yes	complemented
113	#393	C₂×C₁₂₇	1	2	yes	—
114	#389	C₂×C₁₂₇	1	2	yes	complemented
115	#382	C₂×C₁₂₇	1	1	yes	characteristic, complemented
116	#380	C₂×C₁₂₇	1	2	yes	—
117	C₂²×Q₈	C₂²×C₁₂₇	1	2	yes	—
118	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
119	C₄⋊Q₈	—	2	4	no	—
120	#117	—	2	4	no	—
121	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
130	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
131	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
132	#95	C₂²×C₁₂₇	1	4	yes	—
133	#95	C₂²×C₁₂₇	1	2	yes	—
134	#95	C₂²×C₁₂₇	1	2	yes	—
135	#95	C₂²×C₁₂₇	1	4	yes	—
136	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	Q₈×C₁₂₇	1	4	yes	complemented
142	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
144	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
145	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	4	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	D₄×C₁₂₇	1	4	yes	—
161	C₂³	—	2	1	no	—
162	C₂³	—	2	1	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₂×Q₈×C₁₂₇	1	2	yes	—
166	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
167	Q₈	—	4	8	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
184	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
186	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
187	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
188	C₂×C₄	#41×C₁₂₇	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	4	1	no	—
192	C₂²	—	4	1	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
195	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
196	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
197	C₂²	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
198	C₂²	2₋¹⁺⁴×C₁₂₇	1	2	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	4	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	1	no	—
205	C₄	—	4	4	no	—
206	C₄	C₂²⋊Q₈×C₁₂₇	1	4	yes	—
207	C₂	—	4	2	no	—
208	C₂	C₂×C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
209	C₂	#535×C₁₂₇	1	1	yes	characteristic
210	C₂	#536×C₁₂₇	1	1	yes	characteristic
211	C₂	#542×C₁₂₇	1	2	yes	—
212	C₂	#544×C₁₂₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Q₈×C₁₂₇	C₂	semidirect
3. #400×C₁₂₇	C₂	semidirect
5. #395×C₁₂₇	C₂	semidirect
7. #389×C₁₂₇	C₂	semidirect
8. #382×C₁₂₇	C₂	semidirect
34. C₂²×C₄×C₁₂₇	Q₈	semidirect
106. #2628	C₁₂₇	direct (N × H)
107. C₁₂₇	#2628	direct (N × H)
108. C₂×C₄⋊Q₈	C₂×C₁₂₇	semidirect
110. #400	C₂×C₁₂₇	semidirect
112. #395	C₂×C₁₂₇	semidirect
114. #389	C₂×C₁₂₇	semidirect
115. #382	C₂×C₁₂₇	semidirect
141. C₂²×C₄	Q₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×C₁₂₇	2
C₂×#117×C₁₂₇	2
#400×C₁₂₇	2
#399×C₁₂₇	2
#395×C₁₂₇	2
#393×C₁₂₇	2
#389×C₁₂₇	2
#382×C₁₂₇	2
#380×C₁₂₇	2
#2628	127

Subgroups — C127×#2628 #30635580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2628 #30635580