Subgroups — C₂×#22549993 #30647620

Order 16256.

472 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861216	C₂	1	2	yes	maximal, complemented
2	#22549993	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16861145	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16861136	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	#16861216	—	2	4	no	—
8	#16861216	C₂²	1	8	yes	complemented
9	C₂×#14523232	—	2	2	no	—
10	C₂×#14523232	C₂²	1	2	yes	—
11	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
12	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	#16861145	C₂²	1	4	yes	complemented
15	#16861136	C₂²	1	8	yes	complemented
16	C₈×Dic₁₂₇	C₂²	1	4	yes	complemented
17	C₂×#41×C₁₂₇	C₂²	1	2	yes	—
18	#135×C₁₂₇	C₄	1	4	yes	complemented
19	#135×C₁₂₇	C₂²	1	4	yes	—
20	C₂×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic, complemented
21	C₂×SD₁₆×C₁₂₇	C₄	1	2	yes	complemented
22	C₂×D₈×C₁₂₇	C₄	1	1	yes	characteristic, complemented
23	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₂₇	—	2	1	no	—
25	#14523232	—	2	8	no	—
26	#14523232	—	2	2	no	—
27	#14523232	C₂³	1	4	yes	—
28	C₂×C₁₂₇⋊C₈	—	2	8	no	—
29	C₂×C₁₂₇⋊C₈	—	2	2	no	—
30	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
33	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
34	C₄×Dic₁₂₇	—	2	1	no	—
35	C₄×Dic₁₂₇	C₂³	1	2	yes	—
36	#41×C₁₂₇	—	2	2	no	—
37	#41×C₁₂₇	—	2	2	no	—
38	#41×C₁₂₇	C₂×C₄	1	4	yes	—
39	#41×C₁₂₇	C₂³	1	4	yes	—
40	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
41	C₂×D₄×C₁₂₇	—	2	2	no	—
42	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₁₂₇	—	2	2	no	—
44	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
46	SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
47	D₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
48	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
51	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
52	C₂×Dic₁₂₇	—	2	4	no	—
53	C₂×Dic₁₂₇	—	2	2	no	—
54	C₁₂₇⋊C₈	—	2	4	no	—
55	C₁₂₇⋊C₈	C₂×D₄	1	2	yes	—
56	C₁₂₇⋊C₈	C₂×D₄	1	2	yes	—
57	C₂³×C₁₂₇	—	2	2	no	—
58	C₂³×C₁₂₇	#41	1	1	yes	characteristic
59	Q₈×C₁₂₇	—	2	2	no	—
60	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
61	D₄×C₁₂₇	—	2	4	no	—
62	D₄×C₁₂₇	—	2	4	no	—
63	D₄×C₁₂₇	—	2	2	no	—
64	D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄×C₁₂₇	—	2	4	no	—
67	C₂×C₄×C₁₂₇	—	2	4	no	—
68	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
73	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
74	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
75	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₁₂₇	—	2	2	no	—
77	C₂²×C₁₂₇	—	2	4	no	—
78	C₂²×C₁₂₇	—	4	4	no	—
79	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
80	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
81	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
82	C₄×C₁₂₇	—	2	4	no	—
83	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
84	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
85	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
86	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
87	Dic₁₂₇	—	2	4	no	—
88	C₂×C₁₂₇	—	2	1	no	—
89	C₂×C₁₂₇	—	2	1	no	—
90	C₂×C₁₂₇	—	4	4	no	—
91	C₂×C₁₂₇	#443	1	2	yes	complemented
92	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
93	C₂×#443	—	127	1	no	maximal, Sylow
94	C₁₂₇	C₂×#443	1	1	yes	characteristic, complemented, Sylow
95	C₂×#135	D₁₂₇	1	1	yes	characteristic
96	C₂×#131	—	127	2	no	—
97	#443	—	127	8	no	—
98	C₂×#108	—	127	1	no	—
99	C₂×#104	—	127	2	no	—
100	C₂×C₄×C₈	—	127	1	no	—
101	C₂×#41	C₂×D₁₂₇	1	2	yes	—
102	#135	Dic₁₂₇	1	4	yes	complemented
103	#135	C₂×D₁₂₇	1	4	yes	—
104	C₂×Q₁₆	Dic₁₂₇	1	1	yes	characteristic, complemented
105	C₂×SD₁₆	Dic₁₂₇	1	2	yes	complemented
106	C₂×D₈	Dic₁₂₇	1	1	yes	characteristic, complemented
107	#131	—	127	8	no	—
108	#131	—	254	4	no	—
109	C₂×M₁₆	—	127	2	no	—
110	C₂×M₁₆	—	254	2	no	—
111	C₂²×C₈	—	127	1	no	—
112	C₂²×C₈	—	254	2	no	—
113	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
114	C₂×C₄²	—	127	1	no	—
115	#108	—	127	4	no	—
116	#104	—	127	8	no	—
117	C₄×C₈	—	127	4	no	—
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	#41	C₂×Dic₁₂₇	1	4	yes	—
121	#41	C₂²×D₁₂₇	1	4	yes	—
122	C₂×Q₈	C₂×Dic₁₂₇	1	2	yes	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	254	1	no	—
127	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
128	Q₁₆	C₂×Dic₁₂₇	1	4	yes	complemented
129	SD₁₆	C₂×Dic₁₂₇	1	8	yes	complemented
130	D₈	C₂×Dic₁₂₇	1	4	yes	complemented
131	M₁₆	—	127	4	no	—
132	M₁₆	—	254	8	no	—
133	M₁₆	—	254	2	no	—
134	C₂×C₈	—	127	2	no	—
135	C₂×C₈	—	127	1	no	—
136	C₂×C₈	—	127	1	no	—
137	C₂×C₈	—	127	2	no	—
138	C₂×C₈	—	254	8	no	—
139	C₂×C₈	—	254	2	no	—
140	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
143	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
144	C₄²	—	127	2	no	—
145	C₄²	—	254	1	no	—
146	C₂³	—	2	2	no	—
147	C₂³	#14523247	1	1	yes	characteristic
148	Q₈	—	2	2	no	—
149	Q₈	C₂²×Dic₁₂₇	1	4	yes	—
150	D₄	—	2	4	no	—
151	D₄	—	2	4	no	—
152	D₄	—	2	2	no	—
153	D₄	C₂²×Dic₁₂₇	1	4	yes	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	254	4	no	—
158	C₂×C₄	—	254	2	no	—
159	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
161	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
162	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
163	C₂×C₄	#14523247	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
165	C₈	—	127	2	no	—
166	C₈	—	127	2	no	—
167	C₈	—	254	4	no	—
168	C₈	C₂²×Dic₁₂₇	1	2	yes	—
169	C₈	C₂²×Dic₁₂₇	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	4	4	no	—
173	C₂²	D₄×Dic₁₂₇	1	1	yes	characteristic
174	C₂²	D₄×Dic₁₂₇	1	1	yes	characteristic
175	C₂²	C₂×#14523247	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	254	4	no	—
178	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
179	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
180	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
181	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
182	C₂	—	2	1	no	—
183	C₂	—	2	1	no	—
184	C₂	—	4	4	no	—
185	C₂	#22549993	1	2	yes	direct product factor
186	C₂	C₂×D₄×Dic₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting, prime radical	C₂×#135×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₄×D₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#135	D₁₂₇
hypocenter	C₁₂₇	C₂×#443
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#16861216	C₂	semidirect
2. #22549993	C₂	direct (N × H)
3. C₂×#16861145	C₂	semidirect
4. C₂×#16861136	C₂	semidirect
5. C₂×C₈×Dic₁₂₇	C₂	semidirect
8. #16861216	C₂²	semidirect
14. #16861145	C₂²	semidirect
15. #16861136	C₂²	semidirect
16. C₈×Dic₁₂₇	C₂²	semidirect
18. #135×C₁₂₇	C₄	semidirect
20. C₂×Q₁₆×C₁₂₇	C₄	semidirect
21. C₂×SD₁₆×C₁₂₇	C₄	semidirect
22. C₂×D₈×C₁₂₇	C₄	semidirect
45. Q₁₆×C₁₂₇	C₂×C₄	semidirect
46. SD₁₆×C₁₂₇	C₂×C₄	semidirect
47. D₈×C₁₂₇	C₂×C₄	semidirect
91. C₂×C₁₂₇	#443	semidirect
94. C₁₂₇	C₂×#443	semidirect
102. #135	Dic₁₂₇	semidirect
104. C₂×Q₁₆	Dic₁₂₇	semidirect
105. C₂×SD₁₆	Dic₁₂₇	semidirect
106. C₂×D₈	Dic₁₂₇	semidirect
128. Q₁₆	C₂×Dic₁₂₇	semidirect
129. SD₁₆	C₂×Dic₁₂₇	semidirect
130. D₈	C₂×Dic₁₂₇	semidirect
185. C₂	#22549993	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16861216	2
#22549993	2
C₂×#16861145	2
C₂×#16861136	2
C₂×C₈×Dic₁₂₇	2
C₂×#135×C₁₂₇	2
C₂×#443	127

Subgroups — C2×#22549993 #30647620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549993 #30647620