Subgroups — C₂×#413×C₁₂₇ #30635839

Order 16256.

756 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	#413×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×#101×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₁₂₇	—	2	2	no	—
7	C₂×#41×C₁₂₇	—	2	4	no	—
8	C₂×#41×C₁₂₇	—	2	2	no	—
9	C₂×#41×C₁₂₇	C₄	1	4	yes	—
10	C₂×#41×C₁₂₇	C₂²	1	4	yes	—
11	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₂₇	C₄	1	2	yes	—
13	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₂₇	C₄	1	2	yes	—
15	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₂₇	—	2	2	no	—
17	C₂×M₁₆×C₁₂₇	—	2	4	no	—
18	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₂₇	C₂²	1	8	yes	complemented
20	C₂²×C₈×C₁₂₇	—	2	2	no	—
21	#101×C₁₂₇	C₂²	1	8	yes	complemented
22	#100×C₁₂₇	C₂²	1	8	yes	complemented
23	C₂⁴×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	#41×C₁₂₇	—	2	8	no	—
26	#41×C₁₂₇	—	2	8	no	—
27	#41×C₁₂₇	—	2	8	no	—
28	#41×C₁₂₇	—	2	8	no	—
29	C₂×Q₈×C₁₂₇	—	2	2	no	—
30	C₂×Q₈×C₁₂₇	—	2	2	no	—
31	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
33	C₂×D₄×C₁₂₇	—	2	4	no	—
34	C₂×D₄×C₁₂₇	—	2	4	no	—
35	C₂×D₄×C₁₂₇	—	2	2	no	—
36	C₂×D₄×C₁₂₇	—	2	2	no	—
37	C₂×D₄×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₂₇	—	2	4	no	—
41	C₂²×C₄×C₁₂₇	—	2	2	no	—
42	C₂²×C₄×C₁₂₇	—	2	2	no	—
43	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
45	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
48	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
49	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
50	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
52	M₁₆×C₁₂₇	—	2	4	no	—
53	M₁₆×C₁₂₇	—	2	8	no	—
54	M₁₆×C₁₂₇	C₂³	1	8	yes	—
55	C₂×C₈×C₁₂₇	—	2	4	no	—
56	C₂×C₈×C₁₂₇	—	2	8	no	—
57	C₂³×C₁₂₇	—	2	2	no	—
58	C₂³×C₁₂₇	—	2	2	no	—
59	C₂³×C₁₂₇	—	4	2	no	—
60	C₂³×C₁₂₇	—	4	4	no	—
61	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
62	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
63	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
64	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
65	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
66	Q₈×C₁₂₇	—	2	4	no	—
67	Q₈×C₁₂₇	—	2	4	no	—
68	D₄×C₁₂₇	—	2	16	no	—
69	D₄×C₁₂₇	—	2	4	no	—
70	D₄×C₁₂₇	—	2	4	no	—
71	C₂×C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄×C₁₂₇	—	2	2	no	—
73	C₂×C₄×C₁₂₇	—	2	8	no	—
74	C₂×C₄×C₁₂₇	—	2	8	no	—
75	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
78	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
79	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
88	C₈×C₁₂₇	—	2	8	no	—
89	C₂²×C₁₂₇	—	2	2	no	—
90	C₂²×C₁₂₇	—	2	4	no	—
91	C₂²×C₁₂₇	—	2	4	no	—
92	C₂²×C₁₂₇	—	4	2	no	—
93	C₂²×C₁₂₇	—	4	2	no	—
94	C₂²×C₁₂₇	—	4	4	no	—
95	C₂²×C₁₂₇	—	4	4	no	—
96	C₂²×C₁₂₇	—	4	4	no	—
97	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₁₂₇	—	2	4	no	—
102	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
104	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
105	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
106	C₂×C₁₂₇	—	2	1	no	—
107	C₂×C₁₂₇	—	2	1	no	—
108	C₂×C₁₂₇	—	2	4	no	—
109	C₂×C₁₂₇	—	4	4	no	—
110	C₂×C₁₂₇	#413	1	2	yes	direct product factor
111	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#413	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₂₇	C₂×#413	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×#41	C₂×C₁₂₇	1	1	yes	characteristic
115	C₂²×M₁₆	C₂×C₁₂₇	1	2	yes	complemented
116	#413	C₂×C₁₂₇	1	8	yes	direct product factor
117	C₂×#101	C₂×C₁₂₇	1	2	yes	complemented
118	C₂×#100	C₂×C₁₂₇	1	2	yes	complemented
119	C₂×#41	—	2	2	no	—
120	C₂×#41	—	2	2	no	—
121	C₂×#41	—	2	4	no	—
122	C₂×#41	C₄×C₁₂₇	1	4	yes	—
123	C₂×#41	C₂²×C₁₂₇	1	4	yes	—
124	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
125	C₂²×D₄	C₄×C₁₂₇	1	2	yes	—
126	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
127	C₂³×C₄	C₄×C₁₂₇	1	2	yes	—
128	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	2	4	no	—
131	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
132	C₂×M₁₆	C₂²×C₁₂₇	1	8	yes	complemented
133	C₂²×C₈	—	2	2	no	—
134	#101	C₂²×C₁₂₇	1	8	yes	complemented
135	#100	C₂²×C₁₂₇	1	8	yes	complemented
136	C₂⁴	C₂×C₄×C₁₂₇	1	2	yes	—
137	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
138	#41	—	2	8	no	—
139	#41	—	2	8	no	—
140	#41	—	2	8	no	—
141	#41	—	2	8	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
145	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	C₂×C₄×C₁₂₇	1	8	yes	—
151	C₂×D₄	C₂×C₄×C₁₂₇	1	2	yes	—
152	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
157	C₂²×C₄	C₂×C₄×C₁₂₇	1	8	yes	—
158	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
160	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
161	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
162	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
163	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
165	M₁₆	—	2	4	no	—
166	M₁₆	—	2	8	no	—
167	M₁₆	C₂³×C₁₂₇	1	8	yes	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	2	no	—
173	C₂³	—	4	4	no	—
174	C₂³	C₂²⋊C₄×C₁₂₇	1	2	yes	—
175	C₂³	C₂²⋊C₄×C₁₂₇	1	2	yes	—
176	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
177	C₂³	C₂²×C₄×C₁₂₇	1	4	yes	—
178	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
179	Q₈	—	2	4	no	—
180	Q₈	—	2	4	no	—
181	D₄	—	2	16	no	—
182	D₄	—	2	4	no	—
183	D₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	2	yes	—
189	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
191	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
192	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
193	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
194	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
195	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
196	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
197	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
198	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
199	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
200	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
201	C₈	—	2	8	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	4	no	—
208	C₂²	—	4	4	no	—
209	C₂²	—	4	4	no	—
210	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
211	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
212	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
213	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
214	C₄	—	2	4	no	—
215	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
216	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
217	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
218	C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	—	2	4	no	—
222	C₂	—	4	4	no	—
223	C₂	#413×C₁₂₇	1	2	yes	direct product factor
224	C₂	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²⋊C₄
commutator	C₂²	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₄
prime radical	C₂²×#41×C₁₂₇	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₁₂₇	C₂	semidirect
3. #413×C₁₂₇	C₂	direct (N × H)
4. C₂×#101×C₁₂₇	C₂	semidirect
5. C₂×#100×C₁₂₇	C₂	semidirect
19. C₂×M₁₆×C₁₂₇	C₂²	semidirect
21. #101×C₁₂₇	C₂²	semidirect
22. #100×C₁₂₇	C₂²	semidirect
110. C₂×C₁₂₇	#413	direct (N × H)
112. C₂×#413	C₁₂₇	direct (N × H)
113. C₁₂₇	C₂×#413	direct (N × H)
115. C₂²×M₁₆	C₂×C₁₂₇	semidirect
116. #413	C₂×C₁₂₇	direct (N × H)
117. C₂×#101	C₂×C₁₂₇	semidirect
118. C₂×#100	C₂×C₁₂₇	semidirect
132. C₂×M₁₆	C₂²×C₁₂₇	semidirect
134. #101	C₂²×C₁₂₇	semidirect
135. #100	C₂²×C₁₂₇	semidirect
223. C₂	#413×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
C₂²×#41×C₁₂₇	2
C₂²×M₁₆×C₁₂₇	2
#413×C₁₂₇	2
C₂×#101×C₁₂₇	2
C₂×#100×C₁₂₇	2
C₂×#413	127

Subgroups — C2×#413×C127 #30635839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#413×C₁₂₇ #30635839