Subgroups — C₁₂₇×#3352 #30636304

Order 16256.

336 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#552×C₁₂₇	C₂	1	2	yes	maximal
2	#499×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#485×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#484×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	#479×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#470×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#436×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	#428×C₁₂₇	C₂	1	2	yes	maximal, complemented
10	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
12	#126×C₁₂₇	—	2	2	no	—
13	#125×C₁₂₇	—	2	2	no	—
14	#125×C₁₂₇	C₂²	1	2	yes	—
15	#123×C₁₂₇	—	2	2	no	—
16	C₂²⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂²⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
18	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
19	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	#117×C₁₂₇	—	2	2	no	—
21	#117×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
23	#107×C₁₂₇	C₂²	1	1	yes	characteristic
24	#107×C₁₂₇	C₂²	1	1	yes	characteristic
25	#107×C₁₂₇	C₂²	1	1	yes	characteristic
26	#107×C₁₂₇	C₂²	1	2	yes	—
27	#106×C₁₂₇	C₂²	1	2	yes	—
28	#106×C₁₂₇	C₂²	1	2	yes	—
29	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
30	#103×C₁₂₇	C₂²	1	2	yes	—
31	#103×C₁₂₇	C₂²	1	2	yes	—
32	#103×C₁₂₇	C₂²	1	2	yes	—
33	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
34	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
36	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	—	2	2	no	—
38	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
39	Q₁₆×C₁₂₇	—	2	2	no	—
40	M₁₆×C₁₂₇	—	2	4	no	—
41	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
43	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₂₇	—	2	2	no	—
45	C₄⋊C₄×C₁₂₇	—	2	2	no	—
46	C₄⋊C₄×C₁₂₇	—	2	2	no	—
47	C₄⋊C₄×C₁₂₇	—	2	2	no	—
48	C₄⋊C₄×C₁₂₇	—	2	2	no	—
49	C₄⋊C₄×C₁₂₇	—	2	2	no	—
50	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
51	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
55	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
56	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
57	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
58	C₄²×C₁₂₇	—	2	2	no	—
59	C₄²×C₁₂₇	—	2	2	no	—
60	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
61	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
62	Q₈×C₁₂₇	—	2	2	no	—
63	Q₈×C₁₂₇	—	2	2	no	—
64	D₄×C₁₂₇	—	4	1	no	—
65	C₂×C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄×C₁₂₇	—	2	2	no	—
67	C₂×C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₄×C₁₂₇	—	2	2	no	—
70	C₂×C₄×C₁₂₇	—	2	2	no	—
71	C₂×C₄×C₁₂₇	—	4	4	no	—
72	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
76	C₈×C₁₂₇	—	2	2	no	—
77	C₈×C₁₂₇	—	2	1	no	—
78	C₈×C₁₂₇	#41	1	2	yes	—
79	C₂²×C₁₂₇	—	2	2	no	—
80	C₂²×C₁₂₇	—	2	2	no	—
81	C₂²×C₁₂₇	—	2	2	no	—
82	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₁₂₇	—	2	2	no	—
84	C₄×C₁₂₇	—	2	1	no	—
85	C₄×C₁₂₇	—	2	1	no	—
86	C₄×C₁₂₇	—	4	2	no	—
87	C₄×C₁₂₇	—	4	2	no	—
88	C₄×C₁₂₇	—	4	2	no	—
89	C₄×C₁₂₇	—	4	2	no	—
90	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
91	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
92	C₂×C₁₂₇	—	4	2	no	—
93	C₂×C₁₂₇	#547	1	1	yes	characteristic
94	C₂×C₁₂₇	#575	1	1	yes	characteristic
95	C₂×C₁₂₇	#578	1	1	yes	characteristic
96	#3352	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₁₂₇	#3352	1	1	yes	characteristic, direct product factor, Sylow
98	#552	C₂×C₁₂₇	1	2	yes	—
99	#499	C₂×C₁₂₇	1	1	yes	characteristic, complemented
100	#485	C₂×C₁₂₇	1	2	yes	complemented
101	#484	C₂×C₁₂₇	1	2	yes	complemented
102	#479	C₂×C₁₂₇	1	2	yes	complemented
103	#470	C₂×C₁₂₇	1	2	yes	complemented
104	C₄×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
105	#436	C₂×C₁₂₇	1	1	yes	characteristic
106	#428	C₂×C₁₂₇	1	2	yes	complemented
107	C₂×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
108	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
109	#126	—	2	2	no	—
110	#125	—	2	2	no	—
111	#125	C₂²×C₁₂₇	1	2	yes	—
112	#123	—	2	2	no	—
113	C₂²⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
114	C₂²⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
115	C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
116	C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
117	#117	—	2	2	no	—
118	#117	C₂²×C₁₂₇	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
120	#107	C₂²×C₁₂₇	1	1	yes	characteristic
121	#107	C₂²×C₁₂₇	1	2	yes	—
122	#107	C₂²×C₁₂₇	1	1	yes	characteristic
123	#107	C₂²×C₁₂₇	1	1	yes	characteristic
124	#106	C₂²×C₁₂₇	1	2	yes	—
125	#106	C₂²×C₁₂₇	1	2	yes	—
126	C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
127	#103	C₂²×C₁₂₇	1	2	yes	—
128	#103	C₂²×C₁₂₇	1	2	yes	—
129	#103	C₂²×C₁₂₇	1	2	yes	—
130	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
131	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
132	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
133	C₂×D₄	D₄×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
136	Q₁₆	—	2	2	no	—
137	M₁₆	—	2	4	no	—
138	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
139	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
140	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	D₄×C₁₂₇	1	1	yes	characteristic
148	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
149	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
150	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	D₄×C₁₂₇	1	2	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
158	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
159	Q₈	—	2	2	no	—
160	Q₈	—	2	2	no	—
161	D₄	—	4	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
171	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	1	no	—
175	C₈	#41×C₁₂₇	1	2	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
188	C₄	2₋¹⁺⁴×C₁₂₇	1	1	yes	characteristic
189	C₂	—	4	2	no	—
190	C₂	#547×C₁₂₇	1	1	yes	characteristic
191	C₂	#575×C₁₂₇	1	1	yes	characteristic
192	C₂	#578×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
prime radical	C₂×D₄×C₁₂₇	D₄
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
2. #499×C₁₂₇	C₂	semidirect
3. #485×C₁₂₇	C₂	semidirect
4. #484×C₁₂₇	C₂	semidirect
5. #479×C₁₂₇	C₂	semidirect
6. #470×C₁₂₇	C₂	semidirect
7. C₄×Q₁₆×C₁₂₇	C₂	semidirect
9. #428×C₁₂₇	C₂	semidirect
96. #3352	C₁₂₇	direct (N × H)
97. C₁₂₇	#3352	direct (N × H)
99. #499	C₂×C₁₂₇	semidirect
100. #485	C₂×C₁₂₇	semidirect
101. #484	C₂×C₁₂₇	semidirect
102. #479	C₂×C₁₂₇	semidirect
103. #470	C₂×C₁₂₇	semidirect
104. C₄×Q₁₆	C₂×C₁₂₇	semidirect
106. #428	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#552×C₁₂₇	2
#499×C₁₂₇	2
#485×C₁₂₇	2
#484×C₁₂₇	2
#479×C₁₂₇	2
#470×C₁₂₇	2
C₄×Q₁₆×C₁₂₇	2
#436×C₁₂₇	2
#428×C₁₂₇	2
#3352	127

Subgroups — C127×#3352 #30636304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3352 #30636304