Subgroups — C₁₂₇×#3341 #30636293

Order 16256.

364 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#555×C₁₂₇	C₂	1	2	yes	maximal, complemented
2	#500×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#482×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	#476×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#466×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₄×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#427×C₁₂₇	C₂	1	2	yes	maximal, complemented
9	C₂×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
12	#124×C₁₂₇	—	2	4	no	—
13	#121×C₁₂₇	—	2	2	no	—
14	#121×C₁₂₇	C₂²	1	4	yes	—
15	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
16	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	#117×C₁₂₇	—	2	2	no	—
18	#117×C₁₂₇	C₂²	1	4	yes	—
19	#107×C₁₂₇	C₂²	1	2	yes	—
20	#107×C₁₂₇	C₂²	1	2	yes	—
21	#107×C₁₂₇	C₂²	1	1	yes	characteristic
22	#106×C₁₂₇	C₂²	1	4	yes	—
23	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	#102×C₁₂₇	C₂²	1	2	yes	—
25	#102×C₁₂₇	C₂²	1	4	yes	—
26	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
27	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₂₇	—	2	2	no	—
29	C₂×D₄×C₁₂₇	—	2	4	no	—
30	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₂₇	—	2	2	no	—
33	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
34	D₈×C₁₂₇	—	2	2	no	—
35	C₂×C₈×C₁₂₇	—	2	4	no	—
36	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
38	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₂₇	—	2	2	no	—
40	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
44	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
45	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
46	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
47	C₄²×C₁₂₇	—	2	4	no	—
48	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₂³×C₁₂₇	—	2	2	no	—
50	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
51	Q₈×C₁₂₇	—	8	2	no	—
52	D₄×C₁₂₇	—	2	2	no	—
53	D₄×C₁₂₇	—	2	2	no	—
54	D₄×C₁₂₇	—	2	2	no	—
55	D₄×C₁₂₇	—	8	4	no	—
56	C₂×C₄×C₁₂₇	—	2	2	no	—
57	C₂×C₄×C₁₂₇	—	2	2	no	—
58	C₂×C₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄×C₁₂₇	—	2	4	no	—
60	C₂×C₄×C₁₂₇	—	4	4	no	—
61	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
65	C₈×C₁₂₇	—	2	2	no	—
66	C₈×C₁₂₇	—	2	1	no	—
67	C₈×C₁₂₇	#41	1	2	yes	—
68	C₂²×C₁₂₇	—	2	2	no	—
69	C₂²×C₁₂₇	—	2	2	no	—
70	C₂²×C₁₂₇	—	2	2	no	—
71	C₂²×C₁₂₇	—	4	2	no	—
72	C₂²×C₁₂₇	—	4	2	no	—
73	C₂²×C₁₂₇	—	4	2	no	—
74	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₁₂₇	—	2	2	no	—
76	C₄×C₁₂₇	—	2	1	no	—
77	C₄×C₁₂₇	—	4	2	no	—
78	C₄×C₁₂₇	—	4	4	no	—
79	C₄×C₁₂₇	#135	1	2	yes	—
80	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
81	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
82	C₂×C₁₂₇	—	4	2	no	—
83	C₂×C₁₂₇	—	8	2	no	—
84	C₂×C₁₂₇	#547	1	1	yes	characteristic
85	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
86	C₂×C₁₂₇	#576	1	1	yes	characteristic
87	#3341	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₁₂₇	#3341	1	1	yes	characteristic, direct product factor, Sylow
89	#555	C₂×C₁₂₇	1	2	yes	complemented
90	#500	C₂×C₁₂₇	1	1	yes	characteristic, complemented
91	#482	C₂×C₁₂₇	1	4	yes	complemented
92	#476	C₂×C₁₂₇	1	2	yes	complemented
93	#466	C₂×C₁₂₇	1	2	yes	complemented
94	C₄×D₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
95	C₈×D₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
96	#427	C₂×C₁₂₇	1	2	yes	complemented
97	C₂×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
99	C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
100	#124	—	2	4	no	—
101	#121	—	2	2	no	—
102	#121	C₂²×C₁₂₇	1	4	yes	—
103	C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
104	C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	C₂²×C₁₂₇	1	4	yes	—
107	#107	C₂²×C₁₂₇	1	1	yes	characteristic
108	#107	C₂²×C₁₂₇	1	2	yes	—
109	#107	C₂²×C₁₂₇	1	2	yes	—
110	#106	C₂²×C₁₂₇	1	4	yes	—
111	C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
112	#102	C₂²×C₁₂₇	1	2	yes	—
113	#102	C₂²×C₁₂₇	1	4	yes	—
114	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
115	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
116	C₂×Q₈	—	2	2	no	—
117	C₂×D₄	—	2	4	no	—
118	C₂×D₄	D₄×C₁₂₇	1	1	yes	characteristic
119	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
122	D₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
126	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	D₄×C₁₂₇	1	1	yes	characteristic
129	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
130	C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	D₄×C₁₂₇	1	2	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
137	C₂³	—	2	2	no	—
138	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
139	Q₈	—	8	2	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	8	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	1	no	—
155	C₈	#41×C₁₂₇	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	4	no	—
167	C₄	#135×C₁₂₇	1	2	yes	—
168	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
169	C₄	2₋¹⁺⁴×C₁₂₇	1	1	yes	characteristic
170	C₂	—	4	2	no	—
171	C₂	—	8	2	no	—
172	C₂	#547×C₁₂₇	1	1	yes	characteristic
173	C₂	C₂×#135×C₁₂₇	1	1	yes	characteristic
174	C₂	#576×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄×D₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #555×C₁₂₇	C₂	semidirect
2. #500×C₁₂₇	C₂	semidirect
3. #482×C₁₂₇	C₂	semidirect
4. #476×C₁₂₇	C₂	semidirect
5. #466×C₁₂₇	C₂	semidirect
6. C₄×D₈×C₁₂₇	C₂	semidirect
7. C₈×D₄×C₁₂₇	C₂	semidirect
8. #427×C₁₂₇	C₂	semidirect
87. #3341	C₁₂₇	direct (N × H)
88. C₁₂₇	#3341	direct (N × H)
89. #555	C₂×C₁₂₇	semidirect
90. #500	C₂×C₁₂₇	semidirect
91. #482	C₂×C₁₂₇	semidirect
92. #476	C₂×C₁₂₇	semidirect
93. #466	C₂×C₁₂₇	semidirect
94. C₄×D₈	C₂×C₁₂₇	semidirect
95. C₈×D₄	C₂×C₁₂₇	semidirect
96. #427	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#555×C₁₂₇	2
#500×C₁₂₇	2
#482×C₁₂₇	2
#476×C₁₂₇	2
#466×C₁₂₇	2
C₄×D₈×C₁₂₇	2
C₈×D₄×C₁₂₇	2
#427×C₁₂₇	2
#3341	127

Subgroups — C127×#3341 #30636293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3341 #30636293