Subgroups — C₁₂₇×#13826 #44577499

Order 32512.

1606 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#391×C₁₂₇	C₂	1	3	yes	maximal
2	C₁₂₇×#1436	C₂	1	6	yes	maximal
3	C₄×#95×C₁₂₇	C₂	1	6	yes	maximal
4	C₂³×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
5	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
6	C₂²×C₄²×C₁₂₇	C₂²	1	6	yes	—
7	C₂²×C₄²×C₁₂₇	C₂²	1	3	yes	—
8	#391×C₁₂₇	C₄	1	24	yes	complemented
9	C₂×#95×C₁₂₇	C₂²	1	6	yes	—
10	C₂×#95×C₁₂₇	C₂²	1	2	yes	—
11	C₂×#95×C₁₂₇	C₂²	1	12	yes	—
12	C₂×#95×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×Q₈×C₁₂₇	—	4	3	no	—
14	C₂²×Q₈×C₁₂₇	—	4	1	no	—
15	C₂²×Q₈×C₁₂₇	C₂×C₄	1	12	yes	—
16	C₂³×C₄×C₁₂₇	C₂³	1	6	yes	—
17	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
18	C₂³×C₄×C₁₂₇	C₂³	1	6	yes	—
19	C₂×C₄⋊C₄×C₁₂₇	—	4	3	no	—
20	C₂×C₄⋊C₄×C₁₂₇	—	4	3	no	—
21	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	12	yes	—
22	C₂×C₄²×C₁₂₇	—	2	24	no	—
23	C₂×C₄²×C₁₂₇	—	4	6	no	—
24	C₂×C₄²×C₁₂₇	—	4	6	no	—
25	C₂×C₄²×C₁₂₇	C₂×C₄	1	12	yes	—
26	#95×C₁₂₇	—	2	12	no	—
27	#95×C₁₂₇	—	4	2	no	—
28	#95×C₁₂₇	C₂×C₄	1	48	yes	—
29	#95×C₁₂₇	Q₈	1	8	yes	complemented
30	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
31	C₂×Q₈×C₁₂₇	—	4	12	no	—
32	C₂×Q₈×C₁₂₇	—	4	12	no	—
33	C₂×Q₈×C₁₂₇	C₄²	1	16	yes	complemented
34	C₂²×C₄×C₁₂₇	—	2	12	no	—
35	C₂²×C₄×C₁₂₇	—	2	12	no	—
36	C₂²×C₄×C₁₂₇	—	2	6	no	—
37	C₂²×C₄×C₁₂₇	—	2	3	no	—
38	C₂²×C₄×C₁₂₇	—	2	3	no	—
39	C₂²×C₄×C₁₂₇	—	2	12	no	—
40	C₂²×C₄×C₁₂₇	—	2	12	no	—
41	C₂²×C₄×C₁₂₇	—	2	6	no	—
42	C₂²×C₄×C₁₂₇	—	2	6	no	—
43	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
44	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
45	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
46	C₂²×C₄×C₁₂₇	C₂²×C₄	1	12	yes	—
47	C₂²×C₄×C₁₂₇	C₂×Q₈	1	12	yes	—
48	C₂²×C₄×C₁₂₇	#41	1	12	yes	—
49	C₄⋊C₄×C₁₂₇	—	4	12	no	—
50	C₄²×C₁₂₇	—	4	12	no	—
51	C₄²×C₁₂₇	—	4	24	no	—
52	C₂³×C₁₂₇	C₂³×C₄	1	3	yes	—
53	C₂³×C₁₂₇	C₂²×Q₈	1	2	yes	—
54	C₂³×C₁₂₇	C₂×#41	1	6	yes	—
55	C₂³×C₁₂₇	H(5,𝔽₂)	1	3	yes	—
56	C₂³×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
57	Q₈×C₁₂₇	—	4	16	no	—
58	C₂×C₄×C₁₂₇	—	2	12	no	—
59	C₂×C₄×C₁₂₇	—	2	12	no	—
60	C₂×C₄×C₁₂₇	—	2	6	no	—
61	C₂×C₄×C₁₂₇	—	2	6	no	—
62	C₂×C₄×C₁₂₇	—	2	12	no	—
63	C₂×C₄×C₁₂₇	—	2	12	no	—
64	C₂×C₄×C₁₂₇	—	2	24	no	—
65	C₂×C₄×C₁₂₇	—	2	24	no	—
66	C₂×C₄×C₁₂₇	—	2	24	no	—
67	C₂×C₄×C₁₂₇	—	2	12	no	—
68	C₂×C₄×C₁₂₇	—	4	6	no	—
69	C₂×C₄×C₁₂₇	—	4	6	no	—
70	C₂×C₄×C₁₂₇	—	4	12	no	—
71	C₂×C₄×C₁₂₇	C₂×C₄²	1	12	yes	—
72	C₂×C₄×C₁₂₇	C₄×Q₈	1	24	yes	—
73	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
74	C₂²×C₁₂₇	C₂×C₄×Q₈	1	6	yes	—
75	C₂²×C₁₂₇	C₄×#41	1	6	yes	—
76	C₂²×C₁₂₇	#518	1	3	yes	—
77	C₂²×C₁₂₇	#518	1	3	yes	—
78	C₂²×C₁₂₇	#551	1	6	yes	—
79	C₂²×C₁₂₇	#554	1	6	yes	—
80	C₂²×C₁₂₇	#555	1	3	yes	—
81	C₂²×C₁₂₇	Q₈²	1	1	yes	characteristic
82	C₄×C₁₂₇	—	2	24	no	—
83	C₄×C₁₂₇	—	2	12	no	—
84	C₄×C₁₂₇	—	4	12	no	—
85	#13826	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₁₂₇	C₄²×Q₈	1	2	yes	—
87	C₂×C₁₂₇	#2273	1	1	yes	characteristic
88	C₂×C₁₂₇	#2368	1	6	yes	—
89	C₂×C₁₂₇	#2369	1	3	yes	—
90	C₂×C₁₂₇	#2372	1	3	yes	—
91	C₂×#391	C₂×C₁₂₇	1	3	yes	—
92	#1436	C₂×C₁₂₇	1	6	yes	—
93	C₄×#95	C₂×C₁₂₇	1	6	yes	—
94	C₁₂₇	#13826	1	1	yes	characteristic, direct product factor, Sylow
95	C₂³×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
96	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	3	yes	—
97	C₂²×C₄²	C₂²×C₁₂₇	1	6	yes	—
98	C₂²×C₄²	C₂²×C₁₂₇	1	3	yes	—
99	#391	C₄×C₁₂₇	1	24	yes	complemented
100	C₂×#95	C₂²×C₁₂₇	1	6	yes	—
101	C₂×#95	C₂²×C₁₂₇	1	2	yes	—
102	C₂×#95	C₂²×C₁₂₇	1	12	yes	—
103	C₂×#95	C₂²×C₁₂₇	1	2	yes	—
104	C₂²×Q₈	—	4	3	no	—
105	C₂²×Q₈	—	4	1	no	—
106	C₂²×Q₈	C₂×C₄×C₁₂₇	1	12	yes	—
107	C₂³×C₄	C₂³×C₁₂₇	1	6	yes	—
108	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
109	C₂³×C₄	C₂³×C₁₂₇	1	6	yes	—
110	C₂×C₄⋊C₄	—	4	3	no	—
111	C₂×C₄⋊C₄	—	4	3	no	—
112	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	12	yes	—
113	C₂×C₄²	—	2	24	no	—
114	C₂×C₄²	—	4	6	no	—
115	C₂×C₄²	—	4	6	no	—
116	C₂×C₄²	C₂×C₄×C₁₂₇	1	12	yes	—
117	#95	—	2	12	no	—
118	#95	—	4	2	no	—
119	#95	C₂×C₄×C₁₂₇	1	48	yes	—
120	#95	Q₈×C₁₂₇	1	8	yes	complemented
121	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
122	C₂×Q₈	—	4	12	no	—
123	C₂×Q₈	—	4	12	no	—
124	C₂×Q₈	C₄²×C₁₂₇	1	16	yes	complemented
125	C₂²×C₄	—	2	12	no	—
126	C₂²×C₄	—	2	12	no	—
127	C₂²×C₄	—	2	6	no	—
128	C₂²×C₄	—	2	3	no	—
129	C₂²×C₄	—	2	3	no	—
130	C₂²×C₄	—	2	12	no	—
131	C₂²×C₄	—	2	12	no	—
132	C₂²×C₄	—	2	6	no	—
133	C₂²×C₄	—	2	6	no	—
134	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
135	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
136	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
137	C₂²×C₄	C₂²×C₄×C₁₂₇	1	12	yes	—
138	C₂²×C₄	C₂×Q₈×C₁₂₇	1	12	yes	—
139	C₂²×C₄	#41×C₁₂₇	1	12	yes	—
140	C₄⋊C₄	—	4	12	no	—
141	C₄²	—	4	12	no	—
142	C₄²	—	4	24	no	—
143	C₂³	C₂³×C₄×C₁₂₇	1	3	yes	—
144	C₂³	C₂²×Q₈×C₁₂₇	1	2	yes	—
145	C₂³	C₂×#41×C₁₂₇	1	6	yes	—
146	C₂³	H(5,𝔽₂)×C₁₂₇	1	3	yes	—
147	C₂³	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
148	Q₈	—	4	16	no	—
149	C₂×C₄	—	2	12	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	—	2	12	no	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	2	12	no	—
155	C₂×C₄	—	2	24	no	—
156	C₂×C₄	—	2	24	no	—
157	C₂×C₄	—	2	24	no	—
158	C₂×C₄	—	2	12	no	—
159	C₂×C₄	—	4	6	no	—
160	C₂×C₄	—	4	6	no	—
161	C₂×C₄	—	4	12	no	—
162	C₂×C₄	C₂×C₄²×C₁₂₇	1	12	yes	—
163	C₂×C₄	C₄×Q₈×C₁₂₇	1	24	yes	—
164	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
165	C₂²	C₂×C₄×Q₈×C₁₂₇	1	6	yes	—
166	C₂²	C₄×#41×C₁₂₇	1	6	yes	—
167	C₂²	#518×C₁₂₇	1	3	yes	—
168	C₂²	#518×C₁₂₇	1	3	yes	—
169	C₂²	#551×C₁₂₇	1	6	yes	—
170	C₂²	#554×C₁₂₇	1	6	yes	—
171	C₂²	#555×C₁₂₇	1	3	yes	—
172	C₂²	Q₈²×C₁₂₇	1	1	yes	characteristic
173	C₄	—	2	24	no	—
174	C₄	—	2	12	no	—
175	C₄	—	4	12	no	—
176	C₂	C₄²×Q₈×C₁₂₇	1	2	yes	—
177	C₂	C₁₂₇×#2273	1	1	yes	characteristic
178	C₂	C₁₂₇×#2368	1	6	yes	—
179	C₂	C₁₂₇×#2369	1	3	yes	—
180	C₂	C₁₂₇×#2372	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₂₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂⁴	C₂⁴×C₁₂₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. #391×C₁₂₇	C₄	semidirect
29. #95×C₁₂₇	Q₈	semidirect
33. C₂×Q₈×C₁₂₇	C₄²	semidirect
85. #13826	C₁₂₇	direct (N × H)
94. C₁₂₇	#13826	direct (N × H)
99. #391	C₄×C₁₂₇	semidirect
120. #95	Q₈×C₁₂₇	semidirect
124. C₂×Q₈	C₄²×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#391×C₁₂₇	2
C₁₂₇×#1436	2
C₄×#95×C₁₂₇	2
#13826	127

Subgroups — C127×#13826 #44577499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13826 #44577499