Subgroups — C₂×#22550587 #30652265

Order 16256.

1200 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550587	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16861200	C₂	1	2	yes	maximal, complemented
3	C₂×#16861189	C₂	1	2	yes	maximal, complemented
4	C₂×#16861185	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic
7	C₂×#16861171	C₂	1	2	yes	maximal
8	C₂×C₄⋊C₄×D₁₂₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₂²×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₂₇	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₂₇	C₂²	1	4	yes	—
14	#16861200	C₂²	1	16	yes	complemented
15	#16861189	C₂²	1	16	yes	complemented
16	#16861185	C₂²	1	32	yes	complemented
17	C₄×Dic₂₅₄	C₂²	1	8	yes	complemented
18	C₄×D₅₀₈	C₂²	1	8	yes	—
19	#16861171	C₂²	1	16	yes	—
20	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
23	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
24	C₂×Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
25	C₄⋊C₄×D₁₂₇	C₂²	1	16	yes	complemented
26	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
27	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
30	C₄⋊Q₈×C₁₂₇	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂³×D₁₂₇	C₂³	1	2	yes	—
35	C₂×Dic₂₅₄	—	2	4	no	—
36	C₂×Dic₂₅₄	C₂³	1	4	yes	—
37	C₂×D₅₀₈	Q₈	1	8	yes	complemented
38	C₂×D₅₀₈	C₂³	1	4	yes	—
39	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
41	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
42	C₂×C₄×D₁₂₇	—	2	8	no	—
43	C₂×C₄×D₁₂₇	—	4	8	no	—
44	C₂×C₄×D₁₂₇	C₂³	1	8	yes	—
45	C₂×C₄×D₁₂₇	C₂³	1	16	yes	—
46	C₄⋊Dic₁₂₇	C₂³	1	16	yes	—
47	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
48	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
49	Dic₁₂₇⋊C₄	C₂³	1	8	yes	—
50	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
51	(C₂×D₁₂₇)⋊C₄	C₂³	1	8	yes	—
52	(C₂×D₁₂₇)⋊C₄	C₂³	1	16	yes	—
53	C₄×Dic₁₂₇	C₂³	1	8	yes	—
54	C₂×Q₈×C₁₂₇	—	4	4	no	—
55	C₂×Q₈×C₁₂₇	C₂³	1	8	yes	—
56	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
61	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
62	C₄²×C₁₂₇	C₂³	1	4	yes	—
63	C₂²×D₁₂₇	—	2	4	no	—
64	C₂²×D₁₂₇	—	2	4	no	—
65	C₂²×D₁₂₇	C₂×Q₈	1	8	yes	—
66	C₂²×D₁₂₇	C₂⁴	1	4	yes	—
67	Dic₂₅₄	—	2	8	no	—
68	D₅₀₈	C₂×Q₈	1	16	yes	complemented
69	C₂×Dic₁₂₇	—	2	4	no	—
70	C₂×Dic₁₂₇	—	2	4	no	—
71	C₂×Dic₁₂₇	—	2	4	no	—
72	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
73	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
74	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
75	C₄×D₁₂₇	—	2	16	no	—
76	C₄×D₁₂₇	—	4	16	no	—
77	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
78	Q₈×C₁₂₇	—	4	8	no	—
79	C₂×C₄×C₁₂₇	—	2	2	no	—
80	C₂×C₄×C₁₂₇	—	2	8	no	—
81	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
82	C₂×C₄×C₁₂₇	#41	1	4	yes	—
83	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
85	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
87	C₂²×C₁₂₇	C₂²×Q₈	1	2	yes	—
88	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
89	C₂²×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
90	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
91	C₂×D₁₂₇	—	2	16	no	—
92	C₂×D₁₂₇	—	2	4	no	—
93	C₂×D₁₂₇	—	2	4	no	—
94	C₂×D₁₂₇	C₂²×Q₈	1	8	yes	—
95	C₄×C₁₂₇	—	2	2	no	—
96	C₄×C₁₂₇	—	2	8	no	—
97	C₄×C₁₂₇	C₂²×Q₈	1	4	yes	—
98	C₄×C₁₂₇	C₂×#41	1	4	yes	—
99	Dic₁₂₇	—	2	4	no	—
100	Dic₁₂₇	—	2	4	no	—
101	Dic₁₂₇	—	2	4	no	—
102	C₂×C₁₂₇	#554	1	4	yes	complemented
103	C₂×C₁₂₇	C₂³×Q₈	1	1	yes	characteristic
104	C₂×C₁₂₇	C₂²×#41	1	1	yes	characteristic
105	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
106	D₁₂₇	—	2	8	no	—
107	C₂×#554	—	127	1	no	maximal, Sylow
108	C₁₂₇	C₂×#554	1	1	yes	characteristic, complemented, Sylow
109	#554	—	127	16	no	—
110	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
111	C₂×#125	—	127	2	no	—
112	C₂×C₂²⋊Q₈	—	127	4	no	—
113	C₂×C₂²⋊Q₈	—	127	2	no	—
114	C₂×C₄×Q₈	—	127	1	no	—
115	C₂×C₄×D₄	—	127	2	no	—
116	C₂×C₄×D₄	—	127	1	no	—
117	C₂²×C₄⋊C₄	—	127	2	no	—
118	C₂²×Q₈	—	127	1	no	—
119	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
120	C₂²×D₄	—	127	1	no	—
121	C₂³×C₄	—	127	4	no	—
122	C₂³×C₄	—	127	2	no	—
123	C₄⋊Q₈	C₂×D₁₂₇	1	8	yes	complemented
124	#125	—	127	16	no	—
125	C₂²⋊Q₈	—	127	32	no	—
126	C₂²⋊Q₈	—	127	16	no	—
127	C₄×Q₈	—	127	8	no	—
128	C₄×D₄	—	127	16	no	—
129	C₄×D₄	—	127	8	no	—
130	C₂×C₄⋊C₄	—	127	1	no	—
131	C₂×C₄⋊C₄	—	127	4	no	—
132	C₂×C₄⋊C₄	—	127	1	no	—
133	C₂×C₄⋊C₄	—	127	2	no	—
134	C₂×C₄⋊C₄	—	127	4	no	—
135	C₂×C₄⋊C₄	—	127	16	no	—
136	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
137	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
138	C₂×C₂²⋊C₄	—	127	4	no	—
139	C₂×C₂²⋊C₄	—	127	2	no	—
140	C₂×C₄²	—	127	2	no	—
141	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
142	C₂⁴	—	127	2	no	—
143	C₂×Q₈	—	4	4	no	—
144	C₂×Q₈	—	127	4	no	—
145	C₂×Q₈	—	254	4	no	—
146	C₂×Q₈	C₂²×D₁₂₇	1	8	yes	—
147	C₂×D₄	—	127	8	no	—
148	C₂×D₄	—	127	4	no	—
149	C₂²×C₄	—	127	2	no	—
150	C₂²×C₄	—	127	2	no	—
151	C₂²×C₄	—	127	2	no	—
152	C₂²×C₄	—	127	16	no	—
153	C₂²×C₄	—	127	8	no	—
154	C₂²×C₄	—	508	8	no	—
155	C₂²×C₄	—	254	8	no	—
156	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
158	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
160	C₄⋊C₄	—	127	16	no	—
161	C₄⋊C₄	—	127	8	no	—
162	C₄⋊C₄	—	127	4	no	—
163	C₄⋊C₄	—	127	4	no	—
164	C₄⋊C₄	—	127	16	no	—
165	C₄⋊C₄	C₂²×D₁₂₇	1	8	yes	—
166	C₄⋊C₄	C₂²×D₁₂₇	1	8	yes	—
167	C₂²⋊C₄	—	127	16	no	—
168	C₂²⋊C₄	—	127	8	no	—
169	C₄²	—	127	8	no	—
170	C₄²	C₂²×D₁₂₇	1	4	yes	—
171	C₂³	—	127	8	no	—
172	C₂³	—	127	4	no	—
173	C₂³	—	254	4	no	—
174	C₂³	—	254	4	no	—
175	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
176	Q₈	—	4	8	no	—
177	Q₈	—	254	8	no	—
178	D₄	—	127	16	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	127	4	no	—
182	C₂×C₄	—	127	4	no	—
183	C₂×C₄	—	127	4	no	—
184	C₂×C₄	—	508	16	no	—
185	C₂×C₄	—	254	4	no	—
186	C₂×C₄	—	254	4	no	—
187	C₂×C₄	—	254	4	no	—
188	C₂×C₄	—	254	16	no	—
189	C₂×C₄	Q₈×D₁₂₇	1	4	yes	—
190	C₂×C₄	#14523249	1	4	yes	—
191	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
192	C₂×C₄	C₂³×D₁₂₇	1	8	yes	—
193	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
194	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
195	C₂²	—	127	8	no	—
196	C₂²	—	254	16	no	—
197	C₂²	—	254	4	no	—
198	C₂²	—	254	4	no	—
199	C₂²	C₂×Q₈×D₁₂₇	1	2	yes	—
200	C₂²	C₂×#14523249	1	2	yes	—
201	C₂²	#16861254	1	2	yes	—
202	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
203	C₄	—	2	8	no	—
204	C₄	—	2	2	no	—
205	C₄	—	254	4	no	—
206	C₄	—	254	4	no	—
207	C₄	—	254	4	no	—
208	C₄	C₂×Q₈×D₁₂₇	1	4	yes	—
209	C₄	C₂×#14523249	1	4	yes	—
210	C₂	—	254	8	no	—
211	C₂	#22550587	1	4	yes	direct product factor
212	C₂	C₂²×Q₈×D₁₂₇	1	1	yes	characteristic
213	C₂	C₂²×#14523249	1	1	yes	characteristic
214	C₂	C₂×#16861254	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₂₇
commutator, cosocle	C₂²×C₁₂₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂⁴
prime radical	C₂²×D₅₀₈	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#554
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #22550587	C₂	direct (N × H)
2. C₂×#16861200	C₂	semidirect
3. C₂×#16861189	C₂	semidirect
4. C₂×#16861185	C₂	semidirect
5. C₂×C₄×Dic₂₅₄	C₂	semidirect
8. C₂×C₄⋊C₄×D₁₂₇	C₂	semidirect
9. C₂×C₄⋊Q₈×C₁₂₇	C₂	semidirect
14. #16861200	C₂²	semidirect
15. #16861189	C₂²	semidirect
16. #16861185	C₂²	semidirect
17. C₄×Dic₂₅₄	C₂²	semidirect
25. C₄⋊C₄×D₁₂₇	C₂²	semidirect
30. C₄⋊Q₈×C₁₂₇	C₂²	semidirect
37. C₂×D₅₀₈	Q₈	semidirect
68. D₅₀₈	C₂×Q₈	semidirect
102. C₂×C₁₂₇	#554	semidirect
108. C₁₂₇	C₂×#554	semidirect
110. C₂×C₄⋊Q₈	D₁₂₇	semidirect
123. C₄⋊Q₈	C₂×D₁₂₇	semidirect
211. C₂	#22550587	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550587	2
C₂×#16861200	2
C₂×#16861189	2
C₂×#16861185	2
C₂×C₄×Dic₂₅₄	2
C₂×C₄×D₅₀₈	2
C₂×#16861171	2
C₂×C₄⋊C₄×D₁₂₇	2
C₂×C₄⋊Q₈×C₁₂₇	2
C₂×#554	127

Subgroups — C2×#22550587 #30652265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550587 #30652265