Subgroups — C₂×C₄⋊Q₈×D₁₂₇ #30652003

Order 16256.

1624 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	C₂²×Q₈×D₁₂₇	C₂	1	2	yes	maximal
2	C₂×#16861209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16861208	C₂	1	4	yes	maximal, complemented
4	C₂×#16861207	C₂	1	2	yes	maximal, complemented
5	C₄⋊Q₈×D₁₂₇	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₄⋊C₄×D₁₂₇	C₂	1	4	yes	maximal
7	C₂×C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
10	C₂²×Dic₂₅₄	C₂²	1	4	yes	—
11	C₂×Q₈×D₁₂₇	—	2	16	no	—
12	C₂×Q₈×D₁₂₇	C₂²	1	16	yes	—
13	C₂²×C₄×D₁₂₇	C₂²	1	4	yes	—
14	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₂₇	C₂²	1	2	yes	—
16	#16861209	C₂²	1	8	yes	complemented
17	#16861208	C₂²	1	32	yes	complemented
18	#16861207	C₂²	1	16	yes	complemented
19	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
20	C₂×Dic₁₂₇⋊C₄	C₂²	1	8	yes	—
21	C₄⋊C₄×D₁₂₇	C₂²	1	32	yes	—
22	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
24	C₄²×D₁₂₇	C₂²	1	8	yes	—
25	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₁₂₇	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
28	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂³×D₁₂₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₅₄	—	2	16	no	—
31	C₂×Dic₂₅₄	—	2	8	no	—
32	C₂×Dic₂₅₄	C₂³	1	8	yes	—
33	C₂×Dic₂₅₄	C₂³	1	16	yes	—
34	Q₈×D₁₂₇	—	2	64	no	—
35	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
36	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
38	C₂×C₄×D₁₂₇	—	2	16	no	—
39	C₂×C₄×D₁₂₇	D₄	1	8	yes	—
40	C₂×C₄×D₁₂₇	Q₈	1	16	yes	complemented
41	C₂×C₄×D₁₂₇	C₂³	1	16	yes	—
42	C₂×C₄×D₁₂₇	C₂³	1	4	yes	—
43	C₂×C₄×D₁₂₇	C₂³	1	8	yes	—
44	C₄⋊Dic₁₂₇	C₂³	1	16	yes	—
45	Dic₁₂₇⋊C₄	C₂³	1	32	yes	—
46	C₄×Dic₁₂₇	C₂³	1	4	yes	—
47	C₄×Dic₁₂₇	C₂³	1	8	yes	—
48	C₂×Q₈×C₁₂₇	—	2	8	no	—
49	C₂×Q₈×C₁₂₇	C₂³	1	8	yes	—
50	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₂₇	C₂³	1	16	yes	—
54	C₄²×C₁₂₇	C₂³	1	4	yes	—
55	C₂²×D₁₂₇	C₂×D₄	1	4	yes	—
56	C₂²×D₁₂₇	C₂×Q₈	1	8	yes	—
57	C₂²×D₁₂₇	C₂⁴	1	2	yes	—
58	Dic₂₅₄	—	2	16	no	—
59	Dic₂₅₄	—	2	32	no	—
60	C₂×Dic₁₂₇	—	2	8	no	—
61	C₂×Dic₁₂₇	C₂×D₄	1	4	yes	—
62	C₂×Dic₁₂₇	C₂×Q₈	1	8	yes	complemented
63	C₂×Dic₁₂₇	C₂⁴	1	8	yes	—
64	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
65	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
66	C₄×D₁₂₇	—	2	32	no	—
67	C₄×D₁₂₇	C₂×D₄	1	16	yes	—
68	C₄×D₁₂₇	C₂×Q₈	1	32	yes	complemented
69	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
70	Q₈×C₁₂₇	—	2	16	no	—
71	C₂×C₄×C₁₂₇	—	2	8	no	—
72	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
73	C₂×C₄×C₁₂₇	C₂×Q₈	1	8	yes	complemented
74	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
75	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
76	C₂×C₄×C₁₂₇	C₂⁴	1	4	yes	—
77	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
78	C₂²×C₁₂₇	C₂²×Q₈	1	4	yes	—
79	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
80	C₂×D₁₂₇	C₄⋊Q₈	1	16	yes	direct product factor
81	C₂×D₁₂₇	C₂²×D₄	1	4	yes	—
82	C₂×D₁₂₇	C₂²×Q₈	1	8	yes	—
83	C₄×C₁₂₇	—	2	8	no	—
84	C₄×C₁₂₇	C₂²×D₄	1	4	yes	—
85	C₄×C₁₂₇	C₂²×Q₈	1	8	yes	complemented
86	Dic₁₂₇	—	2	8	no	—
87	Dic₁₂₇	C₂²×D₄	1	4	yes	—
88	Dic₁₂₇	C₂²×Q₈	1	8	yes	complemented
89	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	4	yes	complemented
90	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
91	C₂×C₁₂₇	C₂³×Q₈	1	2	yes	—
92	D₁₂₇	C₂×C₄⋊Q₈	1	8	yes	direct product factor
93	C₂²×C₄⋊Q₈	—	127	1	no	maximal, Sylow
94	C₁₂₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
95	C₂³×Q₈	—	127	2	no	—
96	C₂×C₄⋊Q₈	—	127	2	no	—
97	C₂×C₄⋊Q₈	—	127	16	no	—
98	C₂×C₄⋊Q₈	—	127	4	no	—
99	C₂×C₄⋊Q₈	—	127	1	no	—
100	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, direct product factor
101	C₂²×C₄⋊C₄	—	127	4	no	—
102	C₂²×C₄²	—	127	1	no	—
103	C₂²×Q₈	—	127	16	no	—
104	C₂²×Q₈	—	127	4	no	—
105	C₂²×Q₈	—	127	2	no	—
106	C₂²×Q₈	—	254	16	no	—
107	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
108	C₂³×C₄	—	127	2	no	—
109	C₂³×C₄	—	127	1	no	—
110	C₂³×C₄	—	127	4	no	—
111	C₄⋊Q₈	—	127	16	no	—
112	C₄⋊Q₈	—	127	32	no	—
113	C₄⋊Q₈	—	127	8	no	—
114	C₄⋊Q₈	C₂×D₁₂₇	1	8	yes	direct product factor
115	C₂×C₄⋊C₄	—	127	8	no	—
116	C₂×C₄⋊C₄	—	127	4	no	—
117	C₂×C₄⋊C₄	—	127	32	no	—
118	C₂×C₄⋊C₄	C₂×D₁₂₇	1	4	yes	—
119	C₂×C₄²	—	127	1	no	—
120	C₂×C₄²	—	127	2	no	—
121	C₂×C₄²	—	127	8	no	—
122	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
123	C₂⁴	—	127	1	no	—
124	C₂×Q₈	—	2	8	no	—
125	C₂×Q₈	—	127	16	no	—
126	C₂×Q₈	—	127	8	no	—
127	C₂×Q₈	—	254	16	no	—
128	C₂×Q₈	—	254	64	no	—
129	C₂×Q₈	—	254	8	no	—
130	C₂×Q₈	C₂²×D₁₂₇	1	8	yes	—
131	C₂²×C₄	—	127	2	no	—
132	C₂²×C₄	—	127	1	no	—
133	C₂²×C₄	—	127	8	no	—
134	C₂²×C₄	—	127	16	no	—
135	C₂²×C₄	—	127	8	no	—
136	C₂²×C₄	—	127	4	no	—
137	C₂²×C₄	—	127	4	no	—
138	C₂²×C₄	—	127	16	no	—
139	C₂²×C₄	—	254	16	no	—
140	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
141	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
143	C₄⋊C₄	—	127	32	no	—
144	C₄⋊C₄	—	127	16	no	—
145	C₄⋊C₄	C₂²×D₁₂₇	1	16	yes	—
146	C₄²	—	127	4	no	—
147	C₄²	—	127	8	no	—
148	C₄²	C₂²×D₁₂₇	1	4	yes	—
149	C₂³	—	127	4	no	—
150	C₂³	—	127	8	no	—
151	C₂³	—	127	2	no	—
152	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
153	Q₈	—	2	16	no	—
154	Q₈	—	254	32	no	—
155	Q₈	—	254	16	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	127	4	no	—
158	C₂×C₄	—	127	8	no	—
159	C₂×C₄	—	127	4	no	—
160	C₂×C₄	—	127	2	no	—
161	C₂×C₄	—	127	16	no	—
162	C₂×C₄	—	127	32	no	—
163	C₂×C₄	—	127	8	no	—
164	C₂×C₄	—	254	32	no	—
165	C₂×C₄	—	254	8	no	—
166	C₂×C₄	D₄×D₁₂₇	1	4	yes	—
167	C₂×C₄	Q₈×D₁₂₇	1	8	yes	complemented
168	C₂×C₄	C₂³×D₁₂₇	1	8	yes	—
169	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
170	C₂×C₄	C₂³×D₁₂₇	1	4	yes	—
171	C₂²	—	127	16	no	—
172	C₂²	—	127	4	no	—
173	C₂²	—	127	8	no	—
174	C₂²	C₂×D₄×D₁₂₇	1	2	yes	—
175	C₂²	C₂×Q₈×D₁₂₇	1	4	yes	—
176	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
177	C₄	—	2	8	no	—
178	C₄	—	127	4	no	—
179	C₄	—	127	8	no	—
180	C₄	—	254	8	no	—
181	C₄	C₂×D₄×D₁₂₇	1	4	yes	—
182	C₄	C₂×Q₈×D₁₂₇	1	8	yes	complemented
183	C₂	—	127	8	no	—
184	C₂	C₄⋊Q₈×D₁₂₇	1	4	yes	direct product factor
185	C₂	C₂²×D₄×D₁₂₇	1	1	yes	characteristic
186	C₂	C₂²×Q₈×D₁₂₇	1	2	yes	—

	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	C₂³×D₁₂₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂²×C₄⋊Q₈
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
2. C₂×#16861209	C₂	semidirect
3. C₂×#16861208	C₂	semidirect
4. C₂×#16861207	C₂	semidirect
5. C₄⋊Q₈×D₁₂₇	C₂	direct (N × H)
8. C₂×C₄⋊Q₈×C₁₂₇	C₂	semidirect
16. #16861209	C₂²	semidirect
17. #16861208	C₂²	semidirect
18. #16861207	C₂²	semidirect
26. C₄⋊Q₈×C₁₂₇	C₂²	semidirect
40. C₂×C₄×D₁₂₇	Q₈	semidirect
62. C₂×Dic₁₂₇	C₂×Q₈	semidirect
68. C₄×D₁₂₇	C₂×Q₈	semidirect
73. C₂×C₄×C₁₂₇	C₂×Q₈	semidirect
80. C₂×D₁₂₇	C₄⋊Q₈	direct (N × H)
85. C₄×C₁₂₇	C₂²×Q₈	semidirect
88. Dic₁₂₇	C₂²×Q₈	semidirect
89. C₂×C₁₂₇	C₂×C₄⋊Q₈	semidirect
92. D₁₂₇	C₂×C₄⋊Q₈	direct (N × H)
94. C₁₂₇	C₂²×C₄⋊Q₈	semidirect
100. C₂×C₄⋊Q₈	D₁₂₇	direct (N × H)
114. C₄⋊Q₈	C₂×D₁₂₇	direct (N × H)
167. C₂×C₄	Q₈×D₁₂₇	semidirect
182. C₄	C₂×Q₈×D₁₂₇	semidirect
184. C₂	C₄⋊Q₈×D₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₁₂₇	2
C₂×#16861209	2
C₂×#16861208	2
C₂×#16861207	2
C₄⋊Q₈×D₁₂₇	2
C₂×C₄⋊C₄×D₁₂₇	2
C₂×C₄²×D₁₂₇	2
C₂×C₄⋊Q₈×C₁₂₇	2
C₂²×C₄⋊Q₈	127

Subgroups — C2×C4⋊Q8×D127 #30652003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊Q₈×D₁₂₇ #30652003