Subgroups — D₄×Q₁₆×C₂₂₇ #44397242

Order 29056.

472 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	C₂²×Q₁₆×C₂₂₇	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈×C₂₂₇	C₂	1	2	yes	maximal
3	#494×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#467×C₂₂₇	C₂	1	2	yes	maximal, complemented
5	#462×C₂₂₇	C₂	1	2	yes	maximal, complemented
6	#451×C₂₂₇	C₂	1	4	yes	maximal, complemented
7	C₄×Q₁₆×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₂₂₇	C₂²	1	4	yes	—
10	C₂×Q₁₆×C₂₂₇	—	2	8	no	—
11	C₂×Q₁₆×C₂₂₇	C₂²	1	4	yes	—
12	C₂×Q₁₆×C₂₂₇	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₂₂₇	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₂₂₇	—	2	2	no	—
17	C₄⋊Q₈×C₂₂₇	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₂₂₇	—	2	4	no	—
19	C₂²⋊Q₈×C₂₂₇	C₂²	1	4	yes	—
20	C₄×Q₈×C₂₂₇	C₂²	1	2	yes	—
21	C₄×D₄×C₂₂₇	—	2	2	no	—
22	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
23	#107×C₂₂₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
25	#103×C₂₂₇	C₂²	1	2	yes	—
26	#103×C₂₂₇	C₂²	1	4	yes	—
27	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	—
28	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
29	C₂×Q₈×C₂₂₇	—	2	4	no	—
30	C₂×Q₈×C₂₂₇	—	2	8	no	—
31	C₂×Q₈×C₂₂₇	—	4	4	no	—
32	C₂×Q₈×C₂₂₇	C₂³	1	4	yes	—
33	C₂×Q₈×C₂₂₇	C₂³	1	2	yes	—
34	C₂×D₄×C₂₂₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₂₇	—	2	4	no	—
36	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
37	Q₁₆×C₂₂₇	—	2	8	no	—
38	Q₁₆×C₂₂₇	—	2	4	no	—
39	Q₁₆×C₂₂₇	—	4	2	no	—
40	Q₁₆×C₂₂₇	D₄	1	4	yes	direct product factor
41	C₂×C₈×C₂₂₇	—	2	4	no	—
42	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
44	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₂₇	—	2	2	no	—
46	C₄⋊C₄×C₂₂₇	—	2	2	no	—
47	C₄⋊C₄×C₂₂₇	—	2	2	no	—
48	C₄⋊C₄×C₂₂₇	—	2	4	no	—
49	C₄⋊C₄×C₂₂₇	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₂₂₇	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₂₂₇	—	2	4	no	—
52	C₂²⋊C₄×C₂₂₇	D₄	1	2	yes	—
53	C₄²×C₂₂₇	—	2	2	no	—
54	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
55	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
56	Q₈×C₂₂₇	—	2	2	no	—
57	Q₈×C₂₂₇	—	2	4	no	—
58	Q₈×C₂₂₇	—	2	4	no	—
59	Q₈×C₂₂₇	—	4	8	no	—
60	Q₈×C₂₂₇	C₂×D₄	1	4	yes	—
61	D₄×C₂₂₇	Q₁₆	1	4	yes	direct product factor
62	C₂×C₄×C₂₂₇	—	2	2	no	—
63	C₂×C₄×C₂₂₇	—	2	2	no	—
64	C₂×C₄×C₂₂₇	—	2	2	no	—
65	C₂×C₄×C₂₂₇	—	2	2	no	—
66	C₂×C₄×C₂₂₇	—	2	4	no	—
67	C₂×C₄×C₂₂₇	—	4	8	no	—
68	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
72	C₈×C₂₂₇	—	2	2	no	—
73	C₈×C₂₂₇	—	2	1	no	—
74	C₈×C₂₂₇	C₂×D₄	1	2	yes	—
75	C₂²×C₂₂₇	—	2	2	no	—
76	C₂²×C₂₂₇	—	2	2	no	—
77	C₂²×C₂₂₇	C₂×Q₁₆	1	4	yes	complemented
78	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₂₂₇	—	2	2	no	—
80	C₄×C₂₂₇	—	2	4	no	—
81	C₄×C₂₂₇	—	2	1	no	—
82	C₄×C₂₂₇	—	4	4	no	—
83	C₄×C₂₂₇	—	4	2	no	—
84	C₄×C₂₂₇	C₂×Q₁₆	1	2	yes	complemented
85	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂×C₂₂₇	—	2	4	no	—
88	C₂×C₂₂₇	D₄²	1	1	yes	characteristic
89	C₂×C₂₂₇	C₂²×Q₁₆	1	1	yes	characteristic
90	C₂×C₂₂₇	#578	1	1	yes	characteristic
91	C₂₂₇	D₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
92	D₄×Q₁₆	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×Q₁₆	C₂×C₂₂₇	1	2	yes	complemented
94	D₄×Q₈	C₂×C₂₂₇	1	2	yes	—
95	#494	C₂×C₂₂₇	1	1	yes	characteristic, complemented
96	#467	C₂×C₂₂₇	1	2	yes	complemented
97	#462	C₂×C₂₂₇	1	2	yes	complemented
98	#451	C₂×C₂₂₇	1	4	yes	complemented
99	C₄×Q₁₆	C₂×C₂₂₇	1	1	yes	characteristic, complemented
100	C₈×D₄	C₂×C₂₂₇	1	1	yes	characteristic
101	C₂²×Q₈	C₂²×C₂₂₇	1	4	yes	—
102	C₂×Q₁₆	—	2	8	no	—
103	C₂×Q₁₆	C₂²×C₂₂₇	1	4	yes	—
104	C₂×Q₁₆	C₂²×C₂₂₇	1	2	yes	—
105	C₂×Q₁₆	C₂²×C₂₂₇	1	2	yes	—
106	C₂×Q₁₆	C₂²×C₂₂₇	1	1	yes	characteristic
107	C₂²×C₈	C₂²×C₂₂₇	1	2	yes	—
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	C₂²×C₂₂₇	1	2	yes	—
110	C₂²⋊Q₈	—	2	4	no	—
111	C₂²⋊Q₈	C₂²×C₂₂₇	1	4	yes	—
112	C₄×Q₈	C₂²×C₂₂₇	1	2	yes	—
113	C₄×D₄	—	2	2	no	—
114	C₄×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
115	#107	C₂²×C₂₂₇	1	1	yes	characteristic
116	C₄⋊C₈	C₂²×C₂₂₇	1	1	yes	characteristic
117	#103	C₂²×C₂₂₇	1	2	yes	—
118	#103	C₂²×C₂₂₇	1	4	yes	—
119	C₂²⋊C₈	C₂²×C₂₂₇	1	2	yes	—
120	C₄×C₈	C₂²×C₂₂₇	1	1	yes	characteristic
121	C₂×Q₈	—	2	4	no	—
122	C₂×Q₈	—	2	8	no	—
123	C₂×Q₈	—	4	4	no	—
124	C₂×Q₈	C₂³×C₂₂₇	1	4	yes	—
125	C₂×Q₈	C₂³×C₂₂₇	1	2	yes	—
126	C₂×D₄	D₄×C₂₂₇	1	1	yes	characteristic
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
129	Q₁₆	—	2	8	no	—
130	Q₁₆	—	2	4	no	—
131	Q₁₆	—	4	2	no	—
132	Q₁₆	D₄×C₂₂₇	1	4	yes	direct product factor
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
135	C₂×C₈	C₂³×C₂₂₇	1	2	yes	—
136	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	D₄×C₂₂₇	1	1	yes	characteristic
142	C₄⋊C₄	C₂³×C₂₂₇	1	2	yes	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	D₄×C₂₂₇	1	2	yes	—
145	C₄²	—	2	2	no	—
146	C₄²	C₂³×C₂₂₇	1	1	yes	characteristic
147	C₂³	C₂×D₄×C₂₂₇	1	2	yes	—
148	Q₈	—	2	2	no	—
149	Q₈	—	2	4	no	—
150	Q₈	—	2	4	no	—
151	Q₈	—	4	8	no	—
152	Q₈	C₂×D₄×C₂₂₇	1	4	yes	—
153	D₄	Q₁₆×C₂₂₇	1	4	yes	direct product factor
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
162	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
163	C₂×C₄	C₂⁴×C₂₂₇	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	C₂×D₄×C₂₂₇	1	2	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	C₂×Q₁₆×C₂₂₇	1	4	yes	complemented
170	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	2	4	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	4	no	—
175	C₄	—	4	2	no	—
176	C₄	C₂×Q₁₆×C₂₂₇	1	2	yes	complemented
177	C₄	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
178	C₄	H(5,𝔽₂)×C₂₂₇	1	1	yes	characteristic
179	C₂	—	2	4	no	—
180	C₂	D₄²×C₂₂₇	1	1	yes	characteristic
181	C₂	C₂²×Q₁₆×C₂₂₇	1	1	yes	characteristic
182	C₂	#578×C₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₇
prime radical	C₂×D₄×C₂₂₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₇ ◁ C₄×D₄×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₁₆×C₂₂₇	C₂	semidirect
3. #494×C₂₂₇	C₂	semidirect
4. #467×C₂₂₇	C₂	semidirect
5. #462×C₂₂₇	C₂	semidirect
6. #451×C₂₂₇	C₂	semidirect
7. C₄×Q₁₆×C₂₂₇	C₂	semidirect
40. Q₁₆×C₂₂₇	D₄	direct (N × H)
61. D₄×C₂₂₇	Q₁₆	direct (N × H)
77. C₂²×C₂₂₇	C₂×Q₁₆	semidirect
84. C₄×C₂₂₇	C₂×Q₁₆	semidirect
91. C₂₂₇	D₄×Q₁₆	direct (N × H)
92. D₄×Q₁₆	C₂₂₇	direct (N × H)
93. C₂²×Q₁₆	C₂×C₂₂₇	semidirect
95. #494	C₂×C₂₂₇	semidirect
96. #467	C₂×C₂₂₇	semidirect
97. #462	C₂×C₂₂₇	semidirect
98. #451	C₂×C₂₂₇	semidirect
99. C₄×Q₁₆	C₂×C₂₂₇	semidirect
132. Q₁₆	D₄×C₂₂₇	direct (N × H)
153. D₄	Q₁₆×C₂₂₇	direct (N × H)
169. C₂²	C₂×Q₁₆×C₂₂₇	semidirect
176. C₄	C₂×Q₁₆×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₁₆×C₂₂₇	2
D₄×Q₈×C₂₂₇	2
#494×C₂₂₇	2
#467×C₂₂₇	2
#462×C₂₂₇	2
#451×C₂₂₇	2
C₄×Q₁₆×C₂₂₇	2
C₈×D₄×C₂₂₇	2
D₄×Q₁₆	227

Subgroups — D4×Q16×C227 #44397242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Q₁₆×C₂₂₇ #44397242