Subgroups — C₂×#22550604 #30652302

Order 16256.

1048 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550604	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16861209	C₂	1	1	yes	maximal, characteristic
3	C₂×#16861207	C₂	1	2	yes	maximal
4	C₂×#16861201	C₂	1	2	yes	maximal
5	C₂×#16861199	C₂	1	4	yes	maximal
6	C₂×C₄×Dic₂₅₄	C₂	1	2	yes	maximal
7	C₂×C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
8	C₂×#16861174	C₂	1	2	yes	maximal
9	C₂×C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
11	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
12	#16861209	C₂²	1	8	yes	—
13	#16861207	C₂²	1	16	yes	—
14	#16861201	C₂²	1	16	yes	—
15	#16861199	C₂²	1	32	yes	—
16	C₄×Dic₂₅₄	C₂²	1	16	yes	—
17	C₄×Dic₂₅₄	C₂²	1	8	yes	—
18	#16861174	C₂²	1	16	yes	—
19	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
22	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
23	C₂×Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
25	C₂×Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
26	C₂×C₄×Dic₁₂₇	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₁₂₇	C₂²	1	8	yes	—
30	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂×Dic₂₅₄	—	2	8	no	—
35	C₂×Dic₂₅₄	Q₈	1	8	yes	complemented
36	C₂×Dic₂₅₄	C₂³	1	8	yes	—
37	C₂×Dic₂₅₄	C₂³	1	4	yes	—
38	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
39	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
40	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
41	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
42	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
43	C₄⋊Dic₁₂₇	C₂³	1	8	yes	—
44	Dic₁₂₇⋊C₄	C₂³	1	8	yes	—
45	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
46	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
47	Dic₁₂₇⋊C₄	C₂³	1	16	yes	—
48	C₄×Dic₁₂₇	C₂³	1	16	yes	—
49	C₄×Dic₁₂₇	C₂³	1	8	yes	—
50	C₂×Q₈×C₁₂₇	—	2	4	no	—
51	C₂×Q₈×C₁₂₇	C₂³	1	4	yes	—
52	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₁₂₇	C₂³	1	8	yes	—
58	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
59	C₄²×C₁₂₇	C₂³	1	8	yes	—
60	C₄²×C₁₂₇	C₂³	1	4	yes	—
61	Dic₂₅₄	—	2	16	no	—
62	Dic₂₅₄	C₂×Q₈	1	16	yes	complemented
63	C₂×Dic₁₂₇	—	2	8	no	—
64	C₂×Dic₁₂₇	—	2	4	no	—
65	C₂×Dic₁₂₇	C₂×Q₈	1	8	yes	—
66	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
67	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
68	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
69	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
70	Q₈×C₁₂₇	—	2	8	no	—
71	C₂×C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₄×C₁₂₇	—	2	4	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
75	C₂×C₄×C₁₂₇	#41	1	4	yes	—
76	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
77	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
78	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
79	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
80	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
81	C₂²×C₁₂₇	C₂²×Q₈	1	2	yes	—
82	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
83	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
84	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
85	C₄×C₁₂₇	—	2	4	no	—
86	C₄×C₁₂₇	—	2	4	no	—
87	C₄×C₁₂₇	—	2	2	no	—
88	C₄×C₁₂₇	C₂²×Q₈	1	4	yes	—
89	C₄×C₁₂₇	C₂×#41	1	4	yes	—
90	Dic₁₂₇	—	2	8	no	—
91	Dic₁₂₇	—	2	4	no	—
92	Dic₁₂₇	C₂²×Q₈	1	8	yes	—
93	C₂×C₁₂₇	#557	1	4	yes	complemented
94	C₂×C₁₂₇	C₂³×Q₈	1	1	yes	characteristic
95	C₂×C₁₂₇	C₂²×#41	1	1	yes	characteristic
96	C₂×C₁₂₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
97	C₂×#557	—	127	1	no	maximal, Sylow
98	C₁₂₇	C₂×#557	1	1	yes	characteristic, complemented, Sylow
99	#557	—	127	16	no	—
100	C₂×C₄⋊Q₈	—	127	2	no	—
101	C₂×C₄⋊Q₈	—	127	1	no	—
102	C₂×#125	—	127	4	no	—
103	C₂×#125	—	127	2	no	—
104	C₂×C₄×Q₈	—	127	2	no	—
105	C₂×C₄×Q₈	—	127	2	no	—
106	C₂×C₄×Q₈	—	127	1	no	—
107	C₂×C₄×Q₈	D₁₂₇	1	1	yes	characteristic
108	C₂²×Q₈	—	127	1	no	—
109	C₂²×Q₈	—	127	2	no	—
110	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
111	C₄⋊Q₈	—	127	16	no	—
112	C₄⋊Q₈	—	127	8	no	—
113	#125	—	127	32	no	—
114	#125	—	127	16	no	—
115	C₄×Q₈	—	127	16	no	—
116	C₄×Q₈	—	127	16	no	—
117	C₄×Q₈	—	127	8	no	—
118	C₄×Q₈	C₂×D₁₂₇	1	8	yes	—
119	C₂×C₄⋊C₄	—	127	2	no	—
120	C₂×C₄⋊C₄	—	127	4	no	—
121	C₂×C₄⋊C₄	—	127	1	no	—
122	C₂×C₄⋊C₄	—	127	4	no	—
123	C₂×C₄⋊C₄	—	127	2	no	—
124	C₂×C₄⋊C₄	—	127	4	no	—
125	C₂×C₄⋊C₄	—	127	2	no	—
126	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
128	C₂×C₄²	—	127	4	no	—
129	C₂×C₄²	—	127	2	no	—
130	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
131	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
132	C₂×Q₈	—	2	4	no	—
133	C₂×Q₈	—	127	8	no	—
134	C₂×Q₈	—	127	4	no	—
135	C₂×Q₈	—	127	8	no	—
136	C₂×Q₈	—	254	8	no	—
137	C₂×Q₈	C₂²×D₁₂₇	1	4	yes	—
138	C₂²×C₄	—	127	2	no	—
139	C₂²×C₄	—	127	2	no	—
140	C₂²×C₄	—	127	4	no	—
141	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
142	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
144	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
146	C₄⋊C₄	—	127	8	no	—
147	C₄⋊C₄	—	127	16	no	—
148	C₄⋊C₄	—	127	16	no	—
149	C₄⋊C₄	—	127	16	no	—
150	C₄⋊C₄	—	127	4	no	—
151	C₄⋊C₄	—	127	8	no	—
152	C₄⋊C₄	—	127	8	no	—
153	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
154	C₄⋊C₄	C₂²×D₁₂₇	1	8	yes	—
155	C₄²	—	127	16	no	—
156	C₄²	—	127	8	no	—
157	C₄²	C₂²×D₁₂₇	1	8	yes	—
158	C₄²	C₂²×D₁₂₇	1	4	yes	—
159	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
160	Q₈	—	2	8	no	—
161	Q₈	—	127	16	no	—
162	Q₈	—	254	16	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	127	8	no	—
167	C₂×C₄	—	127	4	no	—
168	C₂×C₄	—	127	4	no	—
169	C₂×C₄	—	127	8	no	—
170	C₂×C₄	—	254	8	no	—
171	C₂×C₄	—	254	4	no	—
172	C₂×C₄	Q₈×D₁₂₇	1	4	yes	—
173	C₂×C₄	#14523248	1	4	yes	—
174	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
175	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
176	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
177	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
178	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
179	C₂²	C₂×Q₈×D₁₂₇	1	2	yes	—
180	C₂²	C₂×#14523248	1	2	yes	—
181	C₂²	#16861256	1	2	yes	—
182	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	2	2	no	—
186	C₄	—	127	8	no	—
187	C₄	—	254	8	no	—
188	C₄	—	254	4	no	—
189	C₄	C₂×Q₈×D₁₂₇	1	4	yes	—
190	C₄	C₂×#14523248	1	4	yes	—
191	C₂	#22550604	1	4	yes	direct product factor
192	C₂	C₂²×Q₈×D₁₂₇	1	1	yes	characteristic
193	C₂	C₂²×#14523248	1	1	yes	characteristic
194	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×C₄×Q₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#557
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. #22550604	C₂	direct (N × H)
35. C₂×Dic₂₅₄	Q₈	semidirect
62. Dic₂₅₄	C₂×Q₈	semidirect
93. C₂×C₁₂₇	#557	semidirect
98. C₁₂₇	C₂×#557	semidirect
191. C₂	#22550604	direct (N × H)

Maximal subgroups

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

subgroup	index
#22550604	2
C₂×#16861209	2
C₂×#16861207	2
C₂×#16861201	2
C₂×#16861199	2
C₂×C₄×Dic₂₅₄	2
C₂×#16861174	2
C₂×C₄×Q₈×C₁₂₇	2
C₂×#557	127

Subgroups — C2×#22550604 #30652302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550604 #30652302