Subgroups — C₁₂₇×#14161 #44577834

Order 32512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊C₄×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1446	C₂	1	4	yes	maximal
4	C₁₂₇×C₄⋊#95	C₂	1	8	yes	maximal
5	C₂³×Q₈×C₁₂₇	C₂²	1	2	yes	—
6	C₂×C₄⋊Q₈×C₁₂₇	—	2	8	no	—
7	C₂×C₄⋊Q₈×C₁₂₇	C₄	1	8	yes	complemented
8	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
10	C₂²×C₄²×C₁₂₇	C₂²	1	4	yes	—
11	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₂₇	—	2	8	no	—
13	C₂×#95×C₁₂₇	C₂²	1	4	yes	—
14	C₂×#95×C₁₂₇	C₂²	1	16	yes	—
15	C₂²×Q₈×C₁₂₇	—	2	8	no	—
16	C₂²×Q₈×C₁₂₇	—	4	8	no	—
17	C₂²×Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
18	C₂³×C₄×C₁₂₇	C₂³	1	4	yes	—
19	C₂³×C₄×C₁₂₇	C₂³	1	8	yes	—
20	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
22	C₄⋊Q₈×C₁₂₇	—	2	32	no	—
23	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
24	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
25	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
26	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
27	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	16	yes	—
28	C₂×C₄⋊C₄×C₁₂₇	Q₈	1	16	yes	complemented
29	C₂×C₄²×C₁₂₇	—	2	8	no	—
30	C₂×C₄²×C₁₂₇	—	2	16	no	—
31	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₂₇	D₄	1	8	yes	—
33	#95×C₁₂₇	—	2	8	no	—
34	#95×C₁₂₇	—	4	16	no	—
35	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
36	C₂×Q₈×C₁₂₇	—	2	16	no	—
37	C₂×Q₈×C₁₂₇	—	4	16	no	—
38	C₂×Q₈×C₁₂₇	—	4	32	no	—
39	C₂²×C₄×C₁₂₇	—	2	4	no	—
40	C₂²×C₄×C₁₂₇	—	2	8	no	—
41	C₂²×C₄×C₁₂₇	—	2	8	no	—
42	C₂²×C₄×C₁₂₇	—	2	16	no	—
43	C₂²×C₄×C₁₂₇	—	2	8	no	—
44	C₂²×C₄×C₁₂₇	—	2	16	no	—
45	C₂²×C₄×C₁₂₇	—	2	16	no	—
46	C₂²×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
47	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₁₂₇	C₂×Q₈	1	16	yes	—
53	C₂²×C₄×C₁₂₇	C₂×Q₈	1	8	yes	—
54	C₂²×C₄×C₁₂₇	#41	1	8	yes	—
55	C₄⋊C₄×C₁₂₇	—	2	32	no	—
56	C₄⋊C₄×C₁₂₇	—	2	32	no	—
57	C₄²×C₁₂₇	—	4	16	no	—
58	C₄²×C₁₂₇	C₂²⋊C₄	1	16	yes	—
59	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
60	C₂³×C₁₂₇	C₂²×D₄	1	2	yes	—
61	C₂³×C₁₂₇	C₂²×Q₈	1	4	yes	—
62	C₂³×C₁₂₇	C₂×#41	1	4	yes	—
63	C₂³×C₁₂₇	H(5,𝔽₂)	1	4	yes	—
64	Q₈×C₁₂₇	—	4	32	no	—
65	C₂×C₄×C₁₂₇	—	2	4	no	—
66	C₂×C₄×C₁₂₇	—	2	8	no	—
67	C₂×C₄×C₁₂₇	—	2	8	no	—
68	C₂×C₄×C₁₂₇	—	2	16	no	—
69	C₂×C₄×C₁₂₇	—	2	16	no	—
70	C₂×C₄×C₁₂₇	—	2	32	no	—
71	C₂×C₄×C₁₂₇	—	4	8	no	—
72	C₂×C₄×C₁₂₇	—	4	8	no	—
73	C₂×C₄×C₁₂₇	—	4	16	no	—
74	C₂×C₄×C₁₂₇	—	4	16	no	—
75	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
76	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	8	yes	—
77	C₂×C₄×C₁₂₇	C₄×Q₈	1	16	yes	—
78	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	16	yes	—
79	C₂×C₄×C₁₂₇	#124	1	8	yes	—
80	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	8	yes	—
81	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₁₂₇	C₂×C₄×Q₈	1	4	yes	—
83	C₂²×C₁₂₇	#518	1	2	yes	—
84	C₂²×C₁₂₇	C₂×C₂²⋊Q₈	1	4	yes	—
85	C₂²×C₁₂₇	C₂×#124	1	2	yes	—
86	C₂²×C₁₂₇	C₂×C₄⋊Q₈	1	2	yes	—
87	C₂²×C₁₂₇	#535	1	4	yes	—
88	C₂²×C₁₂₇	#554	1	8	yes	—
89	C₂²×C₁₂₇	#555	1	4	yes	—
90	C₂²×C₁₂₇	Q₈²	1	4	yes	—
91	C₄×C₁₂₇	—	2	4	no	—
92	C₄×C₁₂₇	—	4	16	no	—
93	C₄×C₁₂₇	—	4	8	no	—
94	C₄×C₁₂₇	#391	1	16	yes	—
95	#14161	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₁₂₇	C₂×#391	1	2	yes	—
97	C₂×C₁₂₇	#2309	1	1	yes	characteristic
98	C₂×C₁₂₇	#2369	1	4	yes	—
99	C₂×C₁₂₇	#2563	1	4	yes	—
100	C₂×C₁₂₇	#2567	1	4	yes	—
101	C₂²×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
102	C₂×C₄×C₄⋊C₄	C₂×C₁₂₇	1	2	yes	—
103	#1446	C₂×C₁₂₇	1	4	yes	—
104	C₄⋊#95	C₂×C₁₂₇	1	8	yes	—
105	C₁₂₇	#14161	1	1	yes	characteristic, direct product factor, Sylow
106	C₂³×Q₈	C₂²×C₁₂₇	1	2	yes	—
107	C₂×C₄⋊Q₈	—	2	8	no	—
108	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	8	yes	complemented
109	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
110	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
111	C₂²×C₄²	C₂²×C₁₂₇	1	4	yes	—
112	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
113	C₄×C₄⋊C₄	—	2	8	no	—
114	C₂×#95	C₂²×C₁₂₇	1	4	yes	—
115	C₂×#95	C₂²×C₁₂₇	1	16	yes	—
116	C₂²×Q₈	—	2	8	no	—
117	C₂²×Q₈	—	4	8	no	—
118	C₂²×Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
119	C₂³×C₄	C₂³×C₁₂₇	1	4	yes	—
120	C₂³×C₄	C₂³×C₁₂₇	1	8	yes	—
121	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
122	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
123	C₄⋊Q₈	—	2	32	no	—
124	C₂×C₄⋊C₄	—	2	8	no	—
125	C₂×C₄⋊C₄	—	2	8	no	—
126	C₂×C₄⋊C₄	—	2	8	no	—
127	C₂×C₄⋊C₄	—	2	8	no	—
128	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	16	yes	—
129	C₂×C₄⋊C₄	Q₈×C₁₂₇	1	16	yes	complemented
130	C₂×C₄²	—	2	8	no	—
131	C₂×C₄²	—	2	16	no	—
132	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
133	C₂×C₄²	D₄×C₁₂₇	1	8	yes	—
134	#95	—	2	8	no	—
135	#95	—	4	16	no	—
136	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
137	C₂×Q₈	—	2	16	no	—
138	C₂×Q₈	—	4	16	no	—
139	C₂×Q₈	—	4	32	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	16	no	—
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	16	no	—
146	C₂²×C₄	—	2	16	no	—
147	C₂²×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
148	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
149	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
150	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
151	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
152	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
153	C₂²×C₄	C₂×Q₈×C₁₂₇	1	16	yes	—
154	C₂²×C₄	C₂×Q₈×C₁₂₇	1	8	yes	—
155	C₂²×C₄	#41×C₁₂₇	1	8	yes	—
156	C₄⋊C₄	—	2	32	no	—
157	C₄⋊C₄	—	2	32	no	—
158	C₄²	—	4	16	no	—
159	C₄²	C₂²⋊C₄×C₁₂₇	1	16	yes	—
160	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
161	C₂³	C₂²×D₄×C₁₂₇	1	2	yes	—
162	C₂³	C₂²×Q₈×C₁₂₇	1	4	yes	—
163	C₂³	C₂×#41×C₁₂₇	1	4	yes	—
164	C₂³	H(5,𝔽₂)×C₁₂₇	1	4	yes	—
165	Q₈	—	4	32	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	—	2	32	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	—	4	16	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	—	4	16	no	—
176	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
177	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	8	yes	—
178	C₂×C₄	C₄×Q₈×C₁₂₇	1	16	yes	—
179	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	16	yes	—
180	C₂×C₄	#124×C₁₂₇	1	8	yes	—
181	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	8	yes	—
182	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
183	C₂²	C₂×C₄×Q₈×C₁₂₇	1	4	yes	—
184	C₂²	#518×C₁₂₇	1	2	yes	—
185	C₂²	C₂×C₂²⋊Q₈×C₁₂₇	1	4	yes	—
186	C₂²	C₂×#124×C₁₂₇	1	2	yes	—
187	C₂²	C₂×C₄⋊Q₈×C₁₂₇	1	2	yes	—
188	C₂²	#535×C₁₂₇	1	4	yes	—
189	C₂²	#554×C₁₂₇	1	8	yes	—
190	C₂²	#555×C₁₂₇	1	4	yes	—
191	C₂²	Q₈²×C₁₂₇	1	4	yes	—
192	C₄	—	2	4	no	—
193	C₄	—	4	16	no	—
194	C₄	—	4	8	no	—
195	C₄	#391×C₁₂₇	1	16	yes	—
196	C₂	C₂×#391×C₁₂₇	1	2	yes	—
197	C₂	C₁₂₇×#2309	1	1	yes	characteristic
198	C₂	C₁₂₇×#2369	1	4	yes	—
199	C₂	C₁₂₇×#2563	1	4	yes	—
200	C₂	C₁₂₇×#2567	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊Q₈×C₁₂₇	C₄	semidirect
28. C₂×C₄⋊C₄×C₁₂₇	Q₈	semidirect
95. #14161	C₁₂₇	direct (N × H)
105. C₁₂₇	#14161	direct (N × H)
108. C₂×C₄⋊Q₈	C₄×C₁₂₇	semidirect
129. C₂×C₄⋊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
C₂²×C₄⋊Q₈×C₁₂₇	2
C₂×C₄×C₄⋊C₄×C₁₂₇	2
C₁₂₇×#1446	2
C₁₂₇×C₄⋊#95	2
#14161	127

Subgroups — C127×#14161 #44577834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#14161 #44577834