Subgroups — C₂×#22549638 #30638443

Order 16256.

512 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₂₇	—	2	1	no	—
6	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
7	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
8	C₂²×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
12	C₂×C₄⋊Dic₁₂₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₂₇	—	2	1	no	—
15	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
17	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₂₇	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	C₂²×Dic₁₂₇	—	2	4	no	—
21	C₂²×Dic₁₂₇	—	2	2	no	—
22	C₂²×Dic₁₂₇	—	4	2	no	—
23	C₂²×Dic₁₂₇	—	4	2	no	—
24	C₂×C₁₂₇⋊C₈	—	2	8	no	—
25	C₂×C₁₂₇⋊C₈	—	2	2	no	—
26	C₂×C₁₂₇⋊C₈	—	2	2	no	—
27	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	complemented
28	C₄⋊Dic₁₂₇	—	2	4	no	—
29	C₄⋊Dic₁₂₇	—	2	2	no	—
30	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	complemented
31	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	—	2	4	no	—
33	C₂²×C₄×C₁₂₇	—	2	2	no	—
34	C₂²×C₄×C₁₂₇	—	4	2	no	—
35	C₂²×C₄×C₁₂₇	—	4	2	no	—
36	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
41	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
42	C₂²×C₄×C₁₂₇	Q₈	1	2	yes	—
43	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₂₇	—	2	4	no	—
45	C₄⋊C₄×C₁₂₇	—	2	2	no	—
46	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₁₂₇	—	2	4	no	—
48	C₂×Dic₁₂₇	—	4	8	no	—
49	C₂×Dic₁₂₇	—	4	2	no	—
50	C₂×Dic₁₂₇	—	4	2	no	—
51	C₁₂₇⋊C₈	—	2	4	no	—
52	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₁₂₇	C₂²⋊C₄	1	2	yes	—
54	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
55	C₂³×C₁₂₇	D₈	1	2	yes	—
56	C₂³×C₁₂₇	SD₁₆	1	2	yes	—
57	C₂³×C₁₂₇	SD₁₆	1	2	yes	—
58	C₂³×C₁₂₇	Q₁₆	1	2	yes	—
59	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	—	2	4	no	—
61	C₂×C₄×C₁₂₇	—	4	8	no	—
62	C₂×C₄×C₁₂₇	—	4	2	no	—
63	C₂×C₄×C₁₂₇	—	4	2	no	—
64	C₂×C₄×C₁₂₇	C₄²	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	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
70	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
71	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₁₂₇	#95	1	4	yes	—
78	C₂²×C₁₂₇	#102	1	4	yes	—
79	C₂²×C₁₂₇	#102	1	4	yes	—
80	C₂²×C₁₂₇	#103	1	4	yes	—
81	C₂²×C₁₂₇	#103	1	4	yes	—
82	C₂²×C₁₂₇	#106	1	4	yes	—
83	C₂²×C₁₂₇	#107	1	4	yes	—
84	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
91	C₄×C₁₂₇	—	4	4	no	—
92	C₄×C₁₂₇	#95	1	4	yes	—
93	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
94	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₁₂₇	—	4	4	no	—
98	C₂×C₁₂₇	#340	1	8	yes	complemented
99	C₂×C₁₂₇	C₂×#95	1	1	yes	characteristic
100	C₂×C₁₂₇	C₂×#102	1	1	yes	characteristic
101	C₂×C₁₂₇	C₂×#102	1	1	yes	characteristic
102	C₂×C₁₂₇	C₂×#103	1	1	yes	characteristic
103	C₂×C₁₂₇	C₂×#103	1	1	yes	characteristic
104	C₂×C₁₂₇	C₂×#106	1	1	yes	characteristic
105	C₂×C₁₂₇	C₂×#107	1	1	yes	characteristic
106	C₂×#340	—	127	1	no	maximal, Sylow
107	C₁₂₇	C₂×#340	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₈	—	127	1	no	—
109	C₂²×C₄⋊C₄	—	127	1	no	—
110	C₂²×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic
111	#340	—	127	4	no	—
112	C₂³×C₄	—	2	1	no	—
113	C₂³×C₄	—	254	1	no	—
114	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
115	C₂²×C₈	—	127	4	no	—
116	C₂²×C₈	—	127	2	no	—
117	C₂²×C₈	—	254	2	no	—
118	C₂²×C₈	—	254	2	no	—
119	C₂×C₄⋊C₄	—	2	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	127	4	no	—
122	C₂×C₄⋊C₄	—	127	2	no	—
123	C₂×C₄⋊C₄	—	254	1	no	—
124	C₂×C₄⋊C₄	—	254	2	no	—
125	C₂×C₄⋊C₄	Dic₁₂₇	1	4	yes	complemented
126	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
127	C₂⁴	C₂²⋊D₁₂₇	1	1	yes	characteristic
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	508	2	no	—
133	C₂²×C₄	—	508	2	no	—
134	C₂²×C₄	—	254	4	no	—
135	C₂²×C₄	—	254	2	no	—
136	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
137	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
138	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
139	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
140	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
141	C₂²×C₄	D₅₀₈	1	2	yes	—
142	C₂²×C₄	Dic₂₅₄	1	2	yes	—
143	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
144	C₂×C₈	—	127	4	no	—
145	C₂×C₈	—	254	8	no	—
146	C₂×C₈	—	254	2	no	—
147	C₂×C₈	—	254	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	127	4	no	—
151	C₄⋊C₄	—	254	4	no	—
152	C₄⋊C₄	—	254	2	no	—
153	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	complemented
154	C₂³	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
155	C₂³	C₂²⋊Dic₁₂₇	1	2	yes	—
156	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
157	C₂³	#14523233	1	2	yes	—
158	C₂³	#14523235	1	2	yes	—
159	C₂³	#14523236	1	2	yes	—
160	C₂³	#14523238	1	2	yes	—
161	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	508	2	no	—
167	C₂×C₄	—	508	2	no	—
168	C₂×C₄	—	508	8	no	—
169	C₂×C₄	—	254	4	no	—
170	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
171	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
172	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
173	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
174	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
175	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
176	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
177	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
181	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
183	C₈	—	254	4	no	—
184	C₂²	C₁₂₇⋊#95	1	4	yes	—
185	C₂²	#16861129	1	4	yes	—
186	C₂²	#16861130	1	4	yes	—
187	C₂²	#16861132	1	4	yes	—
188	C₂²	#16861133	1	4	yes	—
189	C₂²	#16861140	1	4	yes	—
190	C₂²	#16861142	1	4	yes	—
191	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
192	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
193	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
194	C₂²	C₂×#14523233	1	1	yes	characteristic
195	C₂²	C₂×#14523235	1	1	yes	characteristic
196	C₂²	C₂×#14523236	1	1	yes	characteristic
197	C₂²	C₂×#14523238	1	1	yes	characteristic
198	C₄	—	4	4	no	—
199	C₄	—	508	4	no	—
200	C₄	C₁₂₇⋊#95	1	4	yes	—
201	C₄	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
202	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
203	C₄	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
204	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
205	C₂	#22549638	1	8	yes	direct product factor
206	C₂	C₂×C₁₂₇⋊#95	1	1	yes	characteristic
207	C₂	C₂×#16861129	1	1	yes	characteristic
208	C₂	C₂×#16861130	1	1	yes	characteristic
209	C₂	C₂×#16861132	1	1	yes	characteristic
210	C₂	C₂×#16861133	1	1	yes	characteristic
211	C₂	C₂×#16861140	1	1	yes	characteristic
212	C₂	C₂×#16861142	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂²×C₄⋊C₄×C₁₂₇	C₂
socle, prime radical	C₂⁴×C₁₂₇	D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₂₇
hypocenter	C₁₂₇	C₂×#340
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #22549638	C₂	direct (N × H)
8. C₂²×C₁₂₇⋊C₈	C₄	semidirect
12. C₂×C₄⋊Dic₁₂₇	C₄	semidirect
18. C₂×C₄⋊C₄×C₁₂₇	C₄	semidirect
27. C₂×C₁₂₇⋊C₈	C₂×C₄	semidirect
30. C₄⋊Dic₁₂₇	C₂×C₄	semidirect
46. C₄⋊C₄×C₁₂₇	C₂×C₄	semidirect
98. C₂×C₁₂₇	#340	semidirect
107. C₁₂₇	C₂×#340	semidirect
125. C₂×C₄⋊C₄	Dic₁₂₇	semidirect
153. C₄⋊C₄	C₂×Dic₁₂₇	semidirect
205. C₂	#22549638	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×C₁₂₇⋊C₈	2
C₂²×C₄⋊Dic₁₂₇	2
#22549638	2
C₂²×C₄⋊C₄×C₁₂₇	2
C₂×#340	127

Subgroups — C2×#22549638 #30638443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549638 #30638443