Subgroups — C₂×#419×C₁₂₇ #30635846

Order 16256.

752 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#419×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#103×C₁₂₇	C₂	1	4	yes	maximal, complemented
6	C₂²×Q₈×C₁₂₇	—	2	4	no	—
7	C₂²×Q₈×C₁₂₇	—	2	2	no	—
8	C₂²×Q₈×C₁₂₇	—	2	1	no	—
9	C₂²×Q₈×C₁₂₇	C₄	1	8	yes	complemented
10	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂²×Q₈×C₁₂₇	C₂²	1	4	yes	—
12	C₂³×C₄×C₁₂₇	—	2	1	no	—
13	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₂₇	—	2	4	no	—
15	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
16	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
17	#117×C₁₂₇	—	2	2	no	—
18	#117×C₁₂₇	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
21	C₂×C₄²×C₁₂₇	—	2	1	no	—
22	#103×C₁₂₇	C₂²	1	16	yes	complemented
23	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
24	C₂×Q₈×C₁₂₇	—	2	8	no	—
25	C₂×Q₈×C₁₂₇	—	2	16	no	—
26	C₂×Q₈×C₁₂₇	—	2	4	no	—
27	C₂×Q₈×C₁₂₇	—	2	8	no	—
28	C₂×Q₈×C₁₂₇	—	2	4	no	—
29	C₂×Q₈×C₁₂₇	—	2	2	no	—
30	C₂×Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
31	C₂×Q₈×C₁₂₇	C₂×C₄	1	16	yes	complemented
32	C₂×Q₈×C₁₂₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₂₇	—	2	8	no	—
34	C₂²×C₄×C₁₂₇	—	2	4	no	—
35	C₂²×C₄×C₁₂₇	—	2	2	no	—
36	C₂²×C₄×C₁₂₇	—	2	2	no	—
37	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
39	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
41	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	M₁₆×C₁₂₇	—	2	8	no	—
44	C₂×C₈×C₁₂₇	—	2	4	no	—
45	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
46	C₄⋊C₄×C₁₂₇	—	2	2	no	—
47	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
49	C₄²×C₁₂₇	—	2	4	no	—
50	C₂³×C₁₂₇	—	2	4	no	—
51	C₂³×C₁₂₇	C₂²⋊C₄	1	4	yes	—
52	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
53	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₁₂₇	—	2	8	no	—
55	Q₈×C₁₂₇	—	2	16	no	—
56	Q₈×C₁₂₇	—	2	4	no	—
57	Q₈×C₁₂₇	C₂²×C₄	1	8	yes	—
58	C₂×C₄×C₁₂₇	—	2	8	no	—
59	C₂×C₄×C₁₂₇	—	2	16	no	—
60	C₂×C₄×C₁₂₇	—	2	4	no	—
61	C₂×C₄×C₁₂₇	—	2	4	no	—
62	C₂×C₄×C₁₂₇	—	4	4	no	—
63	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
64	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
65	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
67	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
72	C₈×C₁₂₇	—	2	4	no	—
73	C₂²×C₁₂₇	—	2	4	no	—
74	C₂²×C₁₂₇	—	2	8	no	—
75	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
76	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂²×C₁₂₇	#137	1	4	yes	—
78	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₁₂₇	—	2	8	no	—
80	C₄×C₁₂₇	—	4	4	no	—
81	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
82	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
83	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₁₂₇	—	2	4	no	—
86	C₂×C₁₂₇	#419	1	4	yes	direct product factor
87	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₁₂₇	C₂×#137	1	2	yes	—
89	C₂×#419	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₁₂₇	C₂×#419	1	1	yes	characteristic, direct product factor, Sylow
91	C₂³×Q₈	C₂×C₁₂₇	1	1	yes	characteristic
92	C₂²×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
93	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic
94	#419	C₂×C₁₂₇	1	8	yes	direct product factor
95	C₂×#103	C₂×C₁₂₇	1	4	yes	complemented
96	C₂²×Q₈	—	2	4	no	—
97	C₂²×Q₈	—	2	1	no	—
98	C₂²×Q₈	—	2	2	no	—
99	C₂²×Q₈	C₄×C₁₂₇	1	8	yes	complemented
100	C₂²×Q₈	C₂²×C₁₂₇	1	2	yes	—
101	C₂²×Q₈	C₂²×C₁₂₇	1	4	yes	—
102	C₂³×C₄	—	2	1	no	—
103	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
104	C₂×M₁₆	—	2	4	no	—
105	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
106	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
107	#117	—	2	2	no	—
108	#117	C₂²×C₁₂₇	1	4	yes	—
109	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
110	C₂×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₄²	—	2	1	no	—
112	#103	C₂²×C₁₂₇	1	16	yes	complemented
113	C₂⁴	D₄×C₁₂₇	1	1	yes	characteristic
114	C₂×Q₈	—	2	8	no	—
115	C₂×Q₈	—	2	16	no	—
116	C₂×Q₈	—	2	4	no	—
117	C₂×Q₈	—	2	8	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
121	C₂×Q₈	C₂×C₄×C₁₂₇	1	16	yes	complemented
122	C₂×Q₈	C₂³×C₁₂₇	1	4	yes	—
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
128	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
129	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
130	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
131	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
133	M₁₆	—	2	8	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
138	C₂²⋊C₄	—	2	4	no	—
139	C₄²	—	2	4	no	—
140	C₂³	—	2	4	no	—
141	C₂³	C₂²⋊C₄×C₁₂₇	1	4	yes	—
142	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
143	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
144	Q₈	—	2	8	no	—
145	Q₈	—	2	16	no	—
146	Q₈	—	2	4	no	—
147	Q₈	C₂²×C₄×C₁₂₇	1	8	yes	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	16	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
154	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
155	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
156	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
157	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
158	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
159	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
160	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
161	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
162	C₈	—	2	4	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
166	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
167	C₂²	#137×C₁₂₇	1	4	yes	—
168	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
169	C₄	—	2	8	no	—
170	C₄	—	4	4	no	—
171	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
172	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
173	C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
174	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
175	C₂	—	2	4	no	—
176	C₂	#419×C₁₂₇	1	4	yes	direct product factor
177	C₂	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
178	C₂	C₂×#137×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂×D₄
commutator	C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
prime radical	C₂⁴×C₁₂₇	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #419×C₁₂₇	C₂	direct (N × H)
5. C₂×#103×C₁₂₇	C₂	semidirect
9. C₂²×Q₈×C₁₂₇	C₄	semidirect
22. #103×C₁₂₇	C₂²	semidirect
31. C₂×Q₈×C₁₂₇	C₂×C₄	semidirect
86. C₂×C₁₂₇	#419	direct (N × H)
89. C₂×#419	C₁₂₇	direct (N × H)
90. C₁₂₇	C₂×#419	direct (N × H)
94. #419	C₂×C₁₂₇	direct (N × H)
95. C₂×#103	C₂×C₁₂₇	semidirect
99. C₂²×Q₈	C₄×C₁₂₇	semidirect
112. #103	C₂²×C₁₂₇	semidirect
121. C₂×Q₈	C₂×C₄×C₁₂₇	semidirect
176. C₂	#419×C₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₁₂₇	2
C₂²×M₁₆×C₁₂₇	2
C₂×#117×C₁₂₇	2
#419×C₁₂₇	2
C₂×#103×C₁₂₇	2
C₂×#419	127

Subgroups — C2×#419×C127 #30635846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#419×C₁₂₇ #30635846