Subgroups — C₁₂₇×#13172 #44576845

Order 32512.

274 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×#368×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#367×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2108	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1952	C₂	1	2	yes	maximal
5	C₁₂₇×#1387	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
9	#428×C₁₂₇	—	2	2	no	—
10	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
11	#405×C₁₂₇	C₂²	1	1	yes	characteristic
12	#368×C₁₂₇	C₄	1	4	yes	complemented
13	#367×C₁₂₇	—	2	1	no	—
14	#367×C₁₂₇	—	2	1	no	—
15	#346×C₁₂₇	—	2	1	no	—
16	C₄×C₁₆×C₁₂₇	—	2	1	no	—
17	#343×C₁₂₇	—	2	2	no	—
18	C₂×M₁₆×C₁₂₇	—	2	2	no	—
19	C₂×M₁₆×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
21	#117×C₁₂₇	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
23	M₃₂×C₁₂₇	—	2	1	no	—
24	M₃₂×C₁₂₇	Q₈	1	2	yes	—
25	C₂×C₁₆×C₁₂₇	—	2	1	no	—
26	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₂₇	Q₈	1	2	yes	—
30	#108×C₁₂₇	—	2	2	no	—
31	#108×C₁₂₇	C₂×C₄	1	4	yes	—
32	#107×C₁₂₇	—	2	2	no	—
33	#107×C₁₂₇	—	2	2	no	—
34	#107×C₁₂₇	—	4	2	no	—
35	#106×C₁₂₇	—	4	2	no	—
36	#97×C₁₂₇	—	2	1	no	—
37	C₄×C₈×C₁₂₇	—	2	1	no	—
38	C₂²×C₄×C₁₂₇	—	4	2	no	—
39	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₁₂₇	—	2	1	no	—
41	M₁₆×C₁₂₇	—	2	4	no	—
42	M₁₆×C₁₂₇	—	4	2	no	—
43	M₁₆×C₁₂₇	Q₁₆	1	2	yes	—
44	C₂×C₈×C₁₂₇	—	2	1	no	—
45	C₂×C₈×C₁₂₇	—	4	2	no	—
46	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
52	C₄⋊C₄×C₁₂₇	—	4	2	no	—
53	C₄⋊C₄×C₁₂₇	—	4	2	no	—
54	C₄⋊C₄×C₁₂₇	D₈	1	2	yes	—
55	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
56	C₄²×C₁₂₇	—	2	1	no	—
57	C₁₆×C₁₂₇	—	2	2	no	—
58	C₁₆×C₁₂₇	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₁₂₇	C₄⋊C₄	1	2	yes	—
60	C₂³×C₁₂₇	#135	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	8	2	no	—
63	C₂×C₄×C₁₂₇	—	16	2	no	—
64	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	#135	1	1	yes	characteristic
70	C₈×C₁₂₇	—	2	2	no	—
71	C₈×C₁₂₇	—	4	4	no	—
72	C₈×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
74	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
75	C₈×C₁₂₇	#125	1	1	yes	characteristic
76	C₂²×C₁₂₇	—	2	2	no	—
77	C₂²×C₁₂₇	C₄×Q₁₆	1	1	yes	characteristic
78	C₂²×C₁₂₇	#466	1	1	yes	characteristic
79	C₂²×C₁₂₇	#499	1	1	yes	characteristic
80	C₄×C₁₂₇	—	4	2	no	—
81	C₄×C₁₂₇	—	16	2	no	—
82	C₄×C₁₂₇	#389	1	1	yes	characteristic
83	C₄×C₁₂₇	C₄×D₈	1	1	yes	characteristic
84	C₄×C₁₂₇	#467	1	1	yes	characteristic
85	C₄×C₁₂₇	#500	1	1	yes	characteristic
86	#13172	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₁₂₇	—	2	1	no	—
88	C₂×C₁₂₇	—	2	1	no	—
89	C₂×C₁₂₇	#1942	1	1	yes	characteristic
90	C₂×C₁₂₇	#2223	1	2	yes	—
91	C₂×#368	C₂×C₁₂₇	1	1	yes	characteristic
92	C₂×#367	C₂×C₁₂₇	1	1	yes	characteristic
93	#2108	C₂×C₁₂₇	1	1	yes	characteristic
94	#1952	C₂×C₁₂₇	1	2	yes	—
95	#1387	C₂×C₁₂₇	1	2	yes	—
96	C₁₂₇	#13172	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
98	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
99	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
100	#428	—	2	2	no	—
101	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
102	#405	C₂²×C₁₂₇	1	1	yes	characteristic
103	#368	C₄×C₁₂₇	1	4	yes	complemented
104	#367	—	2	1	no	—
105	#367	—	2	1	no	—
106	#346	—	2	1	no	—
107	C₄×C₁₆	—	2	1	no	—
108	#343	—	2	2	no	—
109	C₂×M₁₆	—	2	2	no	—
110	C₂×M₁₆	D₄×C₁₂₇	1	1	yes	characteristic
111	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
112	#117	D₄×C₁₂₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	M₃₂	—	2	1	no	—
115	M₃₂	Q₈×C₁₂₇	1	2	yes	—
116	C₂×C₁₆	—	2	1	no	—
117	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
118	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
119	C₂×C₁₆	D₄×C₁₂₇	1	1	yes	characteristic
120	C₂×C₁₆	Q₈×C₁₂₇	1	2	yes	—
121	#108	—	2	2	no	—
122	#108	C₂×C₄×C₁₂₇	1	4	yes	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	4	2	no	—
126	#106	—	4	2	no	—
127	#97	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
131	M₁₆	—	2	1	no	—
132	M₁₆	—	2	4	no	—
133	M₁₆	—	4	2	no	—
134	M₁₆	Q₁₆×C₁₂₇	1	2	yes	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
138	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
140	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
141	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
142	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	D₈×C₁₂₇	1	2	yes	—
146	C₂²⋊C₄	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₄⋊C₄×C₁₂₇	1	2	yes	—
150	C₁₆	C₄⋊C₄×C₁₂₇	1	2	yes	—
151	C₂³	#135×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	16	2	no	—
155	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
157	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
158	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
159	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
160	C₂×C₄	#135×C₁₂₇	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	4	4	no	—
163	C₈	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
164	C₈	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
165	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
166	C₈	#125×C₁₂₇	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	C₄×Q₁₆×C₁₂₇	1	1	yes	characteristic
169	C₂²	#466×C₁₂₇	1	1	yes	characteristic
170	C₂²	#499×C₁₂₇	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	16	2	no	—
173	C₄	#389×C₁₂₇	1	1	yes	characteristic
174	C₄	C₄×D₈×C₁₂₇	1	1	yes	characteristic
175	C₄	#467×C₁₂₇	1	1	yes	characteristic
176	C₄	#500×C₁₂₇	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₁₂₇×#1942	1	1	yes	characteristic
180	C₂	C₁₂₇×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#466
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ #405×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. #368×C₁₂₇	C₄	semidirect
86. #13172	C₁₂₇	direct (N × H)
96. C₁₂₇	#13172	direct (N × H)
103. #368	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#368×C₁₂₇	2
C₂×#367×C₁₂₇	2
C₁₂₇×#2108	2
C₁₂₇×#1952	2
C₁₂₇×#1387	2
#13172	127

Subgroups — C127×#13172 #44576845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13172 #44576845