Subgroups — C₂×#22550504 #30652128

Order 16256.

1080 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550504	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16861209	C₂	1	2	yes	maximal, complemented
3	C₂×#16861208	C₂	1	2	yes	maximal, complemented
4	C₂×#16861201	C₂	1	2	yes	maximal, complemented
5	C₂×#16861200	C₂	1	2	yes	maximal, complemented
6	C₂×#16861186	C₂	1	4	yes	maximal
7	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₅₄	C₂²	1	4	yes	—
11	C₂³×Dic₁₂₇	C₂²	1	2	yes	—
12	#16861209	C₂²	1	16	yes	complemented
13	#16861208	C₂²	1	16	yes	complemented
14	#16861201	C₂²	1	16	yes	complemented
15	#16861200	C₂²	1	16	yes	complemented
16	#16861186	C₂²	1	32	yes	—
17	#16861168	C₂²	1	8	yes	—
18	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₂₇	C₂²	1	8	yes	—
21	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
24	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
25	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
26	C₂×C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
28	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
29	#117×C₁₂₇	C₂²	1	8	yes	—
30	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
34	C₂×Dic₂₅₄	—	4	8	no	—
35	C₂×Dic₂₅₄	C₂³	1	16	yes	—
36	C₂²×Dic₁₂₇	—	4	4	no	—
37	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
38	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
39	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
41	C₄⋊Dic₁₂₇	C₂³	1	16	yes	—
42	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
43	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
44	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
45	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
46	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
47	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
48	C₂²⋊Dic₁₂₇	C₂³	1	8	yes	—
49	C₄×Dic₁₂₇	C₂³	1	8	yes	—
50	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₂₇	Q₈	1	8	yes	complemented
52	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
57	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
58	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
59	C₂²⋊C₄×C₁₂₇	C₂³	1	8	yes	—
60	C₄²×C₁₂₇	C₂³	1	8	yes	—
61	Dic₂₅₄	—	4	16	no	—
62	C₂×Dic₁₂₇	—	2	8	no	—
63	C₂×Dic₁₂₇	—	2	4	no	—
64	C₂×Dic₁₂₇	—	2	4	no	—
65	C₂×Dic₁₂₇	—	4	8	no	—
66	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
67	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
68	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
69	C₂³×C₁₂₇	—	2	2	no	—
70	C₂³×C₁₂₇	—	2	2	no	—
71	C₂³×C₁₂₇	C₂×Q₈	1	4	yes	—
72	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
73	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₄×C₁₂₇	—	2	4	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
77	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
78	C₂×C₄×C₁₂₇	C₂×Q₈	1	16	yes	complemented
79	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
80	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
81	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
83	C₂²×C₁₂₇	—	2	8	no	—
84	C₂²×C₁₂₇	—	2	2	no	—
85	C₂²×C₁₂₇	—	2	2	no	—
86	C₂²×C₁₂₇	C₂²×Q₈	1	4	yes	—
87	C₂²×C₁₂₇	C₂²×Q₈	1	2	yes	—
88	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
89	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
90	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
91	C₄×C₁₂₇	—	2	4	no	—
92	C₄×C₁₂₇	—	2	4	no	—
93	C₄×C₁₂₇	C₂²×Q₈	1	4	yes	—
94	C₄×C₁₂₇	C₂²×Q₈	1	4	yes	—
95	Dic₁₂₇	—	2	8	no	—
96	Dic₁₂₇	—	2	4	no	—
97	Dic₁₂₇	—	2	4	no	—
98	C₂×C₁₂₇	—	2	4	no	—
99	C₂×C₁₂₇	#544	1	4	yes	complemented
100	C₂×C₁₂₇	C₂³×Q₈	1	1	yes	characteristic
101	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
102	C₂×C₁₂₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×#544	—	127	1	no	maximal, Sylow
104	C₁₂₇	C₂×#544	1	1	yes	characteristic, complemented, Sylow
105	#544	—	127	16	no	—
106	C₂×C₄⋊Q₈	—	127	2	no	—
107	C₂×C₄⋊Q₈	—	127	2	no	—
108	C₂×#125	—	127	2	no	—
109	C₂×#125	—	127	2	no	—
110	C₂×C₂²⋊Q₈	—	127	4	no	—
111	C₂×#117	—	127	1	no	—
112	C₂×#117	D₁₂₇	1	1	yes	characteristic
113	C₂²×C₄⋊C₄	—	127	1	no	—
114	C₂²×Q₈	—	127	4	no	—
115	C₂³×C₄	—	127	2	no	—
116	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
117	C₄⋊Q₈	—	127	16	no	—
118	C₄⋊Q₈	—	127	16	no	—
119	#125	—	127	16	no	—
120	#125	—	127	16	no	—
121	C₂²⋊Q₈	—	127	32	no	—
122	#117	—	127	8	no	—
123	#117	C₂×D₁₂₇	1	8	yes	—
124	C₂×C₄⋊C₄	—	127	2	no	—
125	C₂×C₄⋊C₄	—	127	2	no	—
126	C₂×C₄⋊C₄	—	127	8	no	—
127	C₂×C₄⋊C₄	—	127	1	no	—
128	C₂×C₄⋊C₄	—	127	1	no	—
129	C₂×C₄⋊C₄	—	127	4	no	—
130	C₂×C₄⋊C₄	—	127	4	no	—
131	C₂×C₄⋊C₄	—	127	4	no	—
132	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
133	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
134	C₂×C₂²⋊C₄	—	127	2	no	—
135	C₂×C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
136	C₂×C₄²	—	127	2	no	—
137	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
138	C₂⁴	C₂²×D₁₂₇	1	1	yes	characteristic
139	C₂×Q₈	—	127	16	no	—
140	C₂×Q₈	—	508	8	no	—
141	C₂²×C₄	—	127	8	no	—
142	C₂²×C₄	—	127	2	no	—
143	C₂²×C₄	—	127	2	no	—
144	C₂²×C₄	—	127	4	no	—
145	C₂²×C₄	—	508	4	no	—
146	C₂²×C₄	Dic₂₅₄	1	8	yes	complemented
147	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
148	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
149	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
152	C₄⋊C₄	—	127	16	no	—
153	C₄⋊C₄	—	127	16	no	—
154	C₄⋊C₄	—	127	8	no	—
155	C₄⋊C₄	—	127	8	no	—
156	C₄⋊C₄	—	127	4	no	—
157	C₄⋊C₄	—	127	4	no	—
158	C₄⋊C₄	—	127	16	no	—
159	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
160	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
161	C₂²⋊C₄	—	127	8	no	—
162	C₂²⋊C₄	C₂²×D₁₂₇	1	8	yes	—
163	C₄²	—	127	8	no	—
164	C₄²	C₂²×D₁₂₇	1	8	yes	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	C₂×Dic₂₅₄	1	4	yes	—
168	C₂³	C₂³×D₁₂₇	1	2	yes	—
169	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
170	Q₈	—	508	16	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	127	4	no	—
174	C₂×C₄	—	127	4	no	—
175	C₂×C₄	—	127	8	no	—
176	C₂×C₄	—	508	8	no	—
177	C₂×C₄	—	254	8	no	—
178	C₂×C₄	—	254	4	no	—
179	C₂×C₄	—	254	4	no	—
180	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
181	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
182	C₂×C₄	C₂×Dic₂₅₄	1	16	yes	complemented
183	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
184	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
185	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
186	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	8	no	—
190	C₂²	C₂²×Dic₂₅₄	1	4	yes	—
191	C₂²	C₂²×Dic₂₅₄	1	2	yes	—
192	C₂²	#16861255	1	2	yes	—
193	C₂²	#16861256	1	2	yes	—
194	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	4	no	—
197	C₄	—	254	8	no	—
198	C₄	—	254	4	no	—
199	C₄	—	254	4	no	—
200	C₄	C₂²×Dic₂₅₄	1	4	yes	—
201	C₄	C₂²×Dic₂₅₄	1	4	yes	—
202	C₂	—	2	4	no	—
203	C₂	#22550504	1	4	yes	direct product factor
204	C₂	C₂³×Dic₂₅₄	1	1	yes	characteristic
205	C₂	C₂×#16861255	1	1	yes	characteristic
206	C₂	C₂×#16861256	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂×#117×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂⁴×C₁₂₇	C₂³
hypercenter	C₂×#117	D₁₂₇
hypocenter	C₁₂₇	C₂×#544
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22550504	C₂	direct (N × H)
2. C₂×#16861209	C₂	semidirect
3. C₂×#16861208	C₂	semidirect
4. C₂×#16861201	C₂	semidirect
5. C₂×#16861200	C₂	semidirect
12. #16861209	C₂²	semidirect
13. #16861208	C₂²	semidirect
14. #16861201	C₂²	semidirect
15. #16861200	C₂²	semidirect
51. C₂²×C₄×C₁₂₇	Q₈	semidirect
78. C₂×C₄×C₁₂₇	C₂×Q₈	semidirect
99. C₂×C₁₂₇	#544	semidirect
104. C₁₂₇	C₂×#544	semidirect
146. C₂²×C₄	Dic₂₅₄	semidirect
182. C₂×C₄	C₂×Dic₂₅₄	semidirect
203. C₂	#22550504	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550504	2
C₂×#16861209	2
C₂×#16861208	2
C₂×#16861201	2
C₂×#16861200	2
C₂×#16861186	2
C₂×#16861168	2
C₂²×C₄⋊Dic₁₂₇	2
C₂×#117×C₁₂₇	2
C₂×#544	127

Subgroups — C2×#22550504 #30652128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550504 #30652128