Subgroups — C₂×#22549998 #30647626

Order 16256.

464 conjugacy classes of subgroups.

The subgroup lattice has 184 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	#22549998	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×#16861119	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	#16861119	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	2	yes	—
31	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
32	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
33	C₄×Dic₁₂₇	—	2	1	no	—
34	C₄×Dic₁₂₇	C₂³	1	2	yes	—
35	#41×C₁₂₇	—	2	2	no	—
36	#41×C₁₂₇	—	2	2	no	—
37	#41×C₁₂₇	C₂×C₄	1	4	yes	—
38	#41×C₁₂₇	C₂³	1	4	yes	—
39	C₂×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₂×D₄×C₁₂₇	—	2	2	no	—
41	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₂₇	—	2	2	no	—
43	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
45	SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
46	D₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
47	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
51	C₂×Dic₁₂₇	—	2	2	no	—
52	C₂×Dic₁₂₇	—	4	2	no	—
53	C₁₂₇⋊C₈	—	2	4	no	—
54	C₁₂₇⋊C₈	C₂×D₄	1	4	yes	—
55	C₂³×C₁₂₇	—	2	2	no	—
56	C₂³×C₁₂₇	#41	1	1	yes	characteristic
57	Q₈×C₁₂₇	—	2	2	no	—
58	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
59	D₄×C₁₂₇	—	2	4	no	—
60	D₄×C₁₂₇	—	2	4	no	—
61	D₄×C₁₂₇	—	2	2	no	—
62	D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄×C₁₂₇	—	2	4	no	—
65	C₂×C₄×C₁₂₇	—	2	4	no	—
66	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
71	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
72	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
73	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₁₂₇	—	2	2	no	—
75	C₂²×C₁₂₇	—	2	4	no	—
76	C₂²×C₁₂₇	—	4	4	no	—
77	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
78	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
79	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
80	C₄×C₁₂₇	—	2	4	no	—
81	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
82	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
83	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
85	Dic₁₂₇	—	4	2	no	—
86	C₂×C₁₂₇	—	2	1	no	—
87	C₂×C₁₂₇	—	2	1	no	—
88	C₂×C₁₂₇	—	4	4	no	—
89	C₂×C₁₂₇	#444	1	2	yes	complemented
90	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
91	C₂×#444	—	127	1	no	maximal, Sylow
92	C₁₂₇	C₂×#444	1	1	yes	characteristic, complemented, Sylow
93	C₂×#135	D₁₂₇	1	1	yes	characteristic
94	C₂×#131	—	127	2	no	—
95	#444	—	127	8	no	—
96	C₂×#108	—	127	1	no	—
97	C₂×#104	—	127	2	no	—
98	C₂×#97	—	127	1	no	—
99	C₂×#41	C₂×D₁₂₇	1	2	yes	—
100	#135	Dic₁₂₇	1	4	yes	complemented
101	#135	C₂×D₁₂₇	1	4	yes	—
102	C₂×Q₁₆	Dic₁₂₇	1	1	yes	characteristic, complemented
103	C₂×SD₁₆	Dic₁₂₇	1	2	yes	complemented
104	C₂×D₈	Dic₁₂₇	1	1	yes	characteristic, complemented
105	#131	—	127	8	no	—
106	#131	—	254	4	no	—
107	C₂×M₁₆	—	127	2	no	—
108	C₂×M₁₆	—	254	2	no	—
109	C₂²×C₈	—	127	1	no	—
110	C₂²×C₈	—	254	2	no	—
111	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
112	C₂×C₄²	—	127	1	no	—
113	#108	—	127	4	no	—
114	#104	—	127	8	no	—
115	#97	—	127	4	no	—
116	#41	—	2	2	no	—
117	#41	—	2	2	no	—
118	#41	C₂×Dic₁₂₇	1	4	yes	—
119	#41	C₂²×D₁₂₇	1	4	yes	—
120	C₂×Q₈	C₂×Dic₁₂₇	1	2	yes	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	254	1	no	—
125	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
126	Q₁₆	C₂×Dic₁₂₇	1	4	yes	complemented
127	SD₁₆	C₂×Dic₁₂₇	1	8	yes	complemented
128	D₈	C₂×Dic₁₂₇	1	4	yes	complemented
129	M₁₆	—	127	4	no	—
130	M₁₆	—	254	8	no	—
131	M₁₆	—	254	2	no	—
132	C₂×C₈	—	127	2	no	—
133	C₂×C₈	—	127	2	no	—
134	C₂×C₈	—	127	2	no	—
135	C₂×C₈	—	254	8	no	—
136	C₂×C₈	—	254	2	no	—
137	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
138	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
140	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
141	C₄²	—	127	2	no	—
142	C₄²	—	254	1	no	—
143	C₂³	—	2	2	no	—
144	C₂³	#14523247	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	Q₈	C₂²×Dic₁₂₇	1	4	yes	—
147	D₄	—	2	4	no	—
148	D₄	—	2	4	no	—
149	D₄	—	2	2	no	—
150	D₄	C₂²×Dic₁₂₇	1	4	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	508	2	no	—
155	C₂×C₄	—	254	2	no	—
156	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
158	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
159	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
160	C₂×C₄	#14523247	1	1	yes	characteristic
161	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
162	C₈	—	127	4	no	—
163	C₈	—	254	4	no	—
164	C₈	C₂²×Dic₁₂₇	1	2	yes	—
165	C₈	C₂²×Dic₁₂₇	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	4	4	no	—
169	C₂²	D₄×Dic₁₂₇	1	1	yes	characteristic
170	C₂²	D₄×Dic₁₂₇	1	1	yes	characteristic
171	C₂²	C₂×#14523247	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	508	2	no	—
174	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
175	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
176	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
177	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	4	4	no	—
181	C₂	#22549998	1	2	yes	direct product factor
182	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×#444
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. #22549998	C₂	direct (N × H)
3. C₂×#16861145	C₂	semidirect
4. C₂×#16861136	C₂	semidirect
5. C₂×#16861119	C₂	semidirect
8. #16861216	C₂²	semidirect
14. #16861145	C₂²	semidirect
15. #16861136	C₂²	semidirect
16. #16861119	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
44. Q₁₆×C₁₂₇	C₂×C₄	semidirect
45. SD₁₆×C₁₂₇	C₂×C₄	semidirect
46. D₈×C₁₂₇	C₂×C₄	semidirect
89. C₂×C₁₂₇	#444	semidirect
92. C₁₂₇	C₂×#444	semidirect
100. #135	Dic₁₂₇	semidirect
102. C₂×Q₁₆	Dic₁₂₇	semidirect
103. C₂×SD₁₆	Dic₁₂₇	semidirect
104. C₂×D₈	Dic₁₂₇	semidirect
126. Q₁₆	C₂×Dic₁₂₇	semidirect
127. SD₁₆	C₂×Dic₁₂₇	semidirect
128. D₈	C₂×Dic₁₂₇	semidirect
181. C₂	#22549998	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
#22549998	2
C₂×#16861145	2
C₂×#16861136	2
C₂×#16861119	2
C₂×#135×C₁₂₇	2
C₂×#444	127

Subgroups — C2×#22549998 #30647626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549998 #30647626