Subgroups — D₄×D₈×C₁₀₉ #28360533

Order 13952.

620 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈×C₁₀₉	C₂	1	2	yes	maximal, complemented
2	D₄²×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#466×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	#459×C₁₀₉	C₂	1	2	yes	maximal, complemented
6	#447×C₁₀₉	C₂	1	4	yes	maximal, complemented
7	C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₀₉	—	2	2	no	—
10	C₂²×D₄×C₁₀₉	C₂²	1	4	yes	complemented
11	C₂×D₈×C₁₀₉	—	2	8	no	—
12	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×D₈×C₁₀₉	C₂²	1	4	yes	complemented
14	C₂×D₈×C₁₀₉	C₂²	1	2	yes	complemented
15	C₂×D₈×C₁₀₉	C₂²	1	2	yes	complemented
16	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	complemented
17	#127×C₁₀₉	C₂²	1	2	yes	complemented
18	#121×C₁₀₉	—	2	2	no	—
19	#121×C₁₀₉	C₂²	1	4	yes	complemented
20	#120×C₁₀₉	—	2	4	no	—
21	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₁₀₉	C₂²	1	2	yes	complemented
23	#107×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
24	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
25	#102×C₁₀₉	C₂²	1	2	yes	complemented
26	#102×C₁₀₉	C₂²	1	4	yes	complemented
27	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	complemented
28	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
29	C₂⁴×C₁₀₉	—	2	4	no	—
30	C₂×D₄×C₁₀₉	—	2	8	no	—
31	C₂×D₄×C₁₀₉	—	2	4	no	—
32	C₂×D₄×C₁₀₉	—	2	4	no	—
33	C₂×D₄×C₁₀₉	—	2	2	no	—
34	C₂×D₄×C₁₀₉	—	2	2	no	—
35	C₂×D₄×C₁₀₉	—	4	4	no	—
36	C₂×D₄×C₁₀₉	—	4	8	no	—
37	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
38	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
39	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₀₉	—	2	2	no	—
41	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
42	D₈×C₁₀₉	—	2	8	no	—
43	D₈×C₁₀₉	—	2	4	no	—
44	D₈×C₁₀₉	—	4	2	no	—
45	D₈×C₁₀₉	D₄	1	4	yes	direct product factor
46	C₂×C₈×C₁₀₉	—	2	4	no	—
47	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
49	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
51	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
52	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
53	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
54	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
55	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₂³×C₁₀₉	—	2	2	no	—
57	C₂³×C₁₀₉	—	2	4	no	—
58	C₂³×C₁₀₉	—	4	8	no	—
59	C₂³×C₁₀₉	—	4	8	no	—
60	C₂³×C₁₀₉	—	8	4	no	—
61	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
62	D₄×C₁₀₉	—	2	4	no	—
63	D₄×C₁₀₉	—	2	2	no	—
64	D₄×C₁₀₉	—	2	4	no	—
65	D₄×C₁₀₉	—	4	8	no	—
66	D₄×C₁₀₉	—	4	8	no	—
67	D₄×C₁₀₉	—	4	4	no	—
68	D₄×C₁₀₉	—	8	2	no	—
69	D₄×C₁₀₉	D₈	1	4	yes	direct product factor
70	D₄×C₁₀₉	C₂×D₄	1	4	yes	complemented
71	C₂×C₄×C₁₀₉	—	2	2	no	—
72	C₂×C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄×C₁₀₉	—	2	2	no	—
74	C₂×C₄×C₁₀₉	—	4	4	no	—
75	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
79	C₈×C₁₀₉	—	2	2	no	—
80	C₈×C₁₀₉	—	2	1	no	—
81	C₈×C₁₀₉	C₂×D₄	1	2	yes	complemented
82	C₂²×C₁₀₉	—	2	2	no	—
83	C₂²×C₁₀₉	—	2	2	no	—
84	C₂²×C₁₀₉	—	2	4	no	—
85	C₂²×C₁₀₉	—	4	2	no	—
86	C₂²×C₁₀₉	—	4	2	no	—
87	C₂²×C₁₀₉	—	4	4	no	—
88	C₂²×C₁₀₉	—	4	4	no	—
89	C₂²×C₁₀₉	—	4	4	no	—
90	C₂²×C₁₀₉	—	8	16	no	—
91	C₂²×C₁₀₉	C₂×D₈	1	4	yes	complemented
92	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₁₀₉	—	2	2	no	—
94	C₄×C₁₀₉	—	2	1	no	—
95	C₄×C₁₀₉	—	4	2	no	—
96	C₄×C₁₀₉	C₂×D₈	1	2	yes	complemented
97	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂×C₁₀₉	—	2	4	no	—
100	C₂×C₁₀₉	—	4	4	no	—
101	C₂×C₁₀₉	—	8	4	no	—
102	C₂×C₁₀₉	D₄²	1	1	yes	characteristic
103	C₂×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
104	C₂×C₁₀₉	#576	1	1	yes	characteristic
105	D₄×D₈	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₁₀₉	D₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×D₈	C₂×C₁₀₉	1	2	yes	complemented
108	D₄²	C₂×C₁₀₉	1	2	yes	complemented
109	#493	C₂×C₁₀₉	1	1	yes	characteristic, complemented
110	#466	C₂×C₁₀₉	1	2	yes	complemented
111	#459	C₂×C₁₀₉	1	2	yes	complemented
112	#447	C₂×C₁₀₉	1	4	yes	complemented
113	C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
114	C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
115	C₂²×D₄	—	2	2	no	—
116	C₂²×D₄	C₂²×C₁₀₉	1	4	yes	complemented
117	C₂×D₈	—	2	8	no	—
118	C₂×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
119	C₂×D₈	C₂²×C₁₀₉	1	4	yes	complemented
120	C₂×D₈	C₂²×C₁₀₉	1	2	yes	complemented
121	C₂×D₈	C₂²×C₁₀₉	1	2	yes	complemented
122	C₂²×C₈	C₂²×C₁₀₉	1	2	yes	complemented
123	#127	C₂²×C₁₀₉	1	2	yes	complemented
124	#121	—	2	2	no	—
125	#121	C₂²×C₁₀₉	1	4	yes	complemented
126	#120	—	2	4	no	—
127	C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
128	C₄×D₄	C₂²×C₁₀₉	1	2	yes	complemented
129	#107	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
130	C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
131	#102	C₂²×C₁₀₉	1	2	yes	complemented
132	#102	C₂²×C₁₀₉	1	4	yes	complemented
133	C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	complemented
134	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
135	C₂⁴	—	2	4	no	—
136	C₂×D₄	—	2	8	no	—
137	C₂×D₄	—	2	4	no	—
138	C₂×D₄	—	2	4	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	4	4	no	—
142	C₂×D₄	—	4	8	no	—
143	C₂×D₄	D₄×C₁₀₉	1	1	yes	characteristic
144	C₂×D₄	C₂³×C₁₀₉	1	4	yes	—
145	C₂×D₄	C₂³×C₁₀₉	1	2	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	C₂³×C₁₀₉	1	2	yes	—
148	D₈	—	2	8	no	—
149	D₈	—	2	4	no	—
150	D₈	—	4	2	no	—
151	D₈	D₄×C₁₀₉	1	4	yes	direct product factor
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
154	C₂×C₈	C₂³×C₁₀₉	1	2	yes	—
155	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
156	C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
157	C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
161	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
162	C₂³	—	2	2	no	—
163	C₂³	—	2	4	no	—
164	C₂³	—	4	8	no	—
165	C₂³	—	4	8	no	—
166	C₂³	—	8	4	no	—
167	C₂³	C₂×D₄×C₁₀₉	1	2	yes	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	4	8	no	—
172	D₄	—	4	8	no	—
173	D₄	—	4	4	no	—
174	D₄	—	8	2	no	—
175	D₄	D₈×C₁₀₉	1	4	yes	direct product factor
176	D₄	C₂×D₄×C₁₀₉	1	4	yes	complemented
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
184	C₂×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	1	no	—
187	C₈	C₂×D₄×C₁₀₉	1	2	yes	complemented
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	4	no	—
195	C₂²	—	4	4	no	—
196	C₂²	—	8	16	no	—
197	C₂²	C₂×D₈×C₁₀₉	1	4	yes	complemented
198	C₂²	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	4	2	no	—
202	C₄	C₂×D₈×C₁₀₉	1	2	yes	complemented
203	C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
204	C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
205	C₂	—	2	4	no	—
206	C₂	—	4	4	no	—
207	C₂	—	8	4	no	—
208	C₂	D₄²×C₁₀₉	1	1	yes	characteristic
209	C₂	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
210	C₂	#576×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×D₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₈×C₁₀₉	C₂	semidirect
2. D₄²×C₁₀₉	C₂	semidirect
3. #493×C₁₀₉	C₂	semidirect
4. #466×C₁₀₉	C₂	semidirect
5. #459×C₁₀₉	C₂	semidirect
6. #447×C₁₀₉	C₂	semidirect
7. C₄×D₈×C₁₀₉	C₂	semidirect
8. C₈×D₄×C₁₀₉	C₂	semidirect
10. C₂²×D₄×C₁₀₉	C₂²	semidirect
13. C₂×D₈×C₁₀₉	C₂²	semidirect
14. C₂×D₈×C₁₀₉	C₂²	semidirect
15. C₂×D₈×C₁₀₉	C₂²	semidirect
16. C₂²×C₈×C₁₀₉	C₂²	semidirect
17. #127×C₁₀₉	C₂²	semidirect
19. #121×C₁₀₉	C₂²	semidirect
22. C₄×D₄×C₁₀₉	C₂²	semidirect
23. #107×C₁₀₉	C₂²	semidirect
24. C₄⋊C₈×C₁₀₉	C₂²	semidirect
25. #102×C₁₀₉	C₂²	semidirect
26. #102×C₁₀₉	C₂²	semidirect
27. C₂²⋊C₈×C₁₀₉	C₂²	semidirect
28. C₄×C₈×C₁₀₉	C₂²	semidirect
45. D₈×C₁₀₉	D₄	direct (N × H)
69. D₄×C₁₀₉	D₈	direct (N × H)
70. D₄×C₁₀₉	C₂×D₄	semidirect
81. C₈×C₁₀₉	C₂×D₄	semidirect
91. C₂²×C₁₀₉	C₂×D₈	semidirect
96. C₄×C₁₀₉	C₂×D₈	semidirect
105. D₄×D₈	C₁₀₉	direct (N × H)
106. C₁₀₉	D₄×D₈	direct (N × H)
107. C₂²×D₈	C₂×C₁₀₉	semidirect
108. D₄²	C₂×C₁₀₉	semidirect
109. #493	C₂×C₁₀₉	semidirect
110. #466	C₂×C₁₀₉	semidirect
111. #459	C₂×C₁₀₉	semidirect
112. #447	C₂×C₁₀₉	semidirect
113. C₄×D₈	C₂×C₁₀₉	semidirect
114. C₈×D₄	C₂×C₁₀₉	semidirect
116. C₂²×D₄	C₂²×C₁₀₉	semidirect
119. C₂×D₈	C₂²×C₁₀₉	semidirect
120. C₂×D₈	C₂²×C₁₀₉	semidirect
121. C₂×D₈	C₂²×C₁₀₉	semidirect
122. C₂²×C₈	C₂²×C₁₀₉	semidirect
123. #127	C₂²×C₁₀₉	semidirect
125. #121	C₂²×C₁₀₉	semidirect
128. C₄×D₄	C₂²×C₁₀₉	semidirect
129. #107	C₂²×C₁₀₉	semidirect
130. C₄⋊C₈	C₂²×C₁₀₉	semidirect
131. #102	C₂²×C₁₀₉	semidirect
132. #102	C₂²×C₁₀₉	semidirect
133. C₂²⋊C₈	C₂²×C₁₀₉	semidirect
134. C₄×C₈	C₂²×C₁₀₉	semidirect
151. D₈	D₄×C₁₀₉	direct (N × H)
175. D₄	D₈×C₁₀₉	direct (N × H)
176. D₄	C₂×D₄×C₁₀₉	semidirect
187. C₈	C₂×D₄×C₁₀₉	semidirect
197. C₂²	C₂×D₈×C₁₀₉	semidirect
202. C₄	C₂×D₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈×C₁₀₉	2
D₄²×C₁₀₉	2
#493×C₁₀₉	2
#466×C₁₀₉	2
#459×C₁₀₉	2
#447×C₁₀₉	2
C₄×D₈×C₁₀₉	2
C₈×D₄×C₁₀₉	2
D₄×D₈	109

Subgroups — D4×D8×C109 #28360533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×D₈×C₁₀₉ #28360533