Subgroups — C₂×C₄⋊C₄×Dic₁₀₉ #28365369

Order 13952.

928 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₀₉	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₀₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16812606	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₉	C₂²	1	2	yes	—
8	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₀₉	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₀₉	—	2	8	no	—
13	C₂×C₄×Dic₁₀₉	C₄	1	8	yes	complemented
14	C₂×C₄×Dic₁₀₉	C₂²	1	8	yes	—
15	C₂×C₄×Dic₁₀₉	C₂²	1	4	yes	—
16	#16812606	C₂²	1	8	yes	—
17	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₄	1	8	yes	complemented
20	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
21	C₂²×Dic₁₀₉	—	2	4	no	—
22	C₂²×Dic₁₀₉	—	2	2	no	—
23	C₂²×Dic₁₀₉	—	2	2	no	—
24	C₂²×Dic₁₀₉	—	2	4	no	—
25	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
26	C₂²×Dic₁₀₉	C₂×C₄	1	8	yes	—
27	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
28	C₂²×Dic₁₀₉	D₄	1	4	yes	—
29	C₂²×Dic₁₀₉	Q₈	1	4	yes	—
30	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
31	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
32	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
33	C₄⋊Dic₁₀₉	C₂×C₄	1	16	yes	complemented
34	C₄×Dic₁₀₉	—	2	16	no	—
35	C₄×Dic₁₀₉	C₂×C₄	1	16	yes	complemented
36	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	—	2	4	no	—
38	C₂²×C₄×C₁₀₉	—	2	4	no	—
39	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₁₀₉	C₂×C₄	1	8	yes	—
42	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
45	C₄⋊C₄×C₁₀₉	C₂×C₄	1	16	yes	complemented
46	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
47	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
48	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
49	C₂³×C₁₀₉	C₂×D₄	1	2	yes	—
50	C₂³×C₁₀₉	C₂×Q₈	1	2	yes	—
51	C₂³×C₁₀₉	#41	1	2	yes	—
52	C₂³×C₁₀₉	#41	1	2	yes	—
53	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	—	2	16	no	—
55	C₂×C₄×C₁₀₉	—	2	4	no	—
56	C₂×C₄×C₁₀₉	—	2	4	no	—
57	C₂×C₄×C₁₀₉	C₄²	1	16	yes	complemented
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
61	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
62	C₂×Dic₁₀₉	—	2	8	no	—
63	C₂×Dic₁₀₉	—	2	16	no	—
64	C₂×Dic₁₀₉	—	2	4	no	—
65	C₂×Dic₁₀₉	—	2	2	no	—
66	C₂×Dic₁₀₉	—	2	2	no	—
67	C₂×Dic₁₀₉	—	2	4	no	—
68	C₂×Dic₁₀₉	C₄⋊C₄	1	16	yes	direct product factor
69	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
70	C₂×Dic₁₀₉	C₂²×C₄	1	8	yes	—
71	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
72	C₂×Dic₁₀₉	C₂×D₄	1	4	yes	—
73	C₂×Dic₁₀₉	C₂×Q₈	1	4	yes	—
74	C₂²×C₁₀₉	C₂×C₄²	1	4	yes	—
75	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
76	C₂²×C₁₀₉	#117	1	4	yes	—
77	C₂²×C₁₀₉	C₄×D₄	1	4	yes	—
78	C₂²×C₁₀₉	C₄×D₄	1	4	yes	—
79	C₂²×C₁₀₉	C₄×Q₈	1	4	yes	—
80	C₂²×C₁₀₉	C₄×Q₈	1	4	yes	—
81	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
83	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
88	C₄×C₁₀₉	—	2	8	no	—
89	C₄×C₁₀₉	C₂×C₄²	1	8	yes	complemented
90	Dic₁₀₉	—	2	4	no	—
91	Dic₁₀₉	—	2	8	no	—
92	Dic₁₀₉	C₂×C₄⋊C₄	1	8	yes	direct product factor
93	C₂×C₁₀₉	C₄×C₄⋊C₄	1	8	yes	complemented
94	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
95	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₁₀₉	C₂×#117	1	1	yes	characteristic
97	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
99	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
100	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄⋊C₄	—	109	1	no	maximal, Sylow
102	C₁₀₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₄⋊C₄	—	109	1	no	—
104	C₂²×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
105	C₂²×C₄²	—	109	1	no	—
106	C₂²×C₄²	—	109	2	no	—
107	C₄×C₄⋊C₄	—	109	8	no	—
108	C₂×#95	—	109	2	no	—
109	C₂³×C₄	—	109	1	no	—
110	C₂³×C₄	—	109	2	no	—
111	C₂³×C₄	—	109	1	no	—
112	C₂³×C₄	C₂×D₁₀₉	1	2	yes	—
113	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	109	8	no	—
115	C₂×C₄⋊C₄	—	109	4	no	—
116	C₂×C₄⋊C₄	Dic₁₀₉	1	8	yes	direct product factor
117	C₂×C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
118	C₂×C₄²	—	109	8	no	—
119	C₂×C₄²	—	109	4	no	—
120	C₂×C₄²	—	109	8	no	—
121	C₂×C₄²	—	218	8	no	—
122	#95	—	109	8	no	—
123	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	109	4	no	—
127	C₂²×C₄	—	109	4	no	—
128	C₂²×C₄	—	109	4	no	—
129	C₂²×C₄	—	109	2	no	—
130	C₂²×C₄	—	109	4	no	—
131	C₂²×C₄	—	109	8	no	—
132	C₂²×C₄	—	109	2	no	—
133	C₂²×C₄	—	109	4	no	—
134	C₂²×C₄	—	218	4	no	—
135	C₂²×C₄	—	218	2	no	—
136	C₂²×C₄	—	218	2	no	—
137	C₂²×C₄	—	218	4	no	—
138	C₂²×C₄	C₄×D₁₀₉	1	4	yes	—
139	C₂²×C₄	C₄×D₁₀₉	1	4	yes	—
140	C₂²×C₄	C₂×Dic₁₀₉	1	8	yes	—
141	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
142	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
143	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
144	C₄⋊C₄	—	109	16	no	—
145	C₄⋊C₄	C₂×Dic₁₀₉	1	16	yes	direct product factor
146	C₄²	—	109	16	no	—
147	C₄²	—	218	16	no	—
148	C₂³	C₂×C₄×D₁₀₉	1	2	yes	—
149	C₂³	C₂×C₄×D₁₀₉	1	2	yes	—
150	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
151	C₂³	D₄×D₁₀₉	1	2	yes	—
152	C₂³	Q₈×D₁₀₉	1	2	yes	—
153	C₂³	#13182028	1	2	yes	—
154	C₂³	#13182030	1	2	yes	—
155	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
156	C₂×C₄	—	2	16	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	109	16	no	—
160	C₂×C₄	—	109	4	no	—
161	C₂×C₄	—	109	4	no	—
162	C₂×C₄	—	109	4	no	—
163	C₂×C₄	—	109	4	no	—
164	C₂×C₄	—	109	8	no	—
165	C₂×C₄	—	218	8	no	—
166	C₂×C₄	—	218	16	no	—
167	C₂×C₄	—	218	4	no	—
168	C₂×C₄	—	218	2	no	—
169	C₂×C₄	—	218	2	no	—
170	C₂×C₄	—	218	4	no	—
171	C₂×C₄	C₄×Dic₁₀₉	1	16	yes	complemented
172	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
173	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
174	C₂×C₄	C₂²×Dic₁₀₉	1	8	yes	—
175	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
176	C₂²	C₂×C₄×Dic₁₀₉	1	4	yes	—
177	C₂²	C₄⋊C₄×D₁₀₉	1	4	yes	—
178	C₂²	#16812657	1	4	yes	—
179	C₂²	#16812661	1	4	yes	—
180	C₂²	D₄×Dic₁₀₉	1	4	yes	—
181	C₂²	#16812664	1	4	yes	—
182	C₂²	Q₈×Dic₁₀₉	1	4	yes	—
183	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
184	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
185	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
186	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
187	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
188	C₂²	C₂×#13182028	1	1	yes	characteristic
189	C₂²	C₂×#13182030	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	—	109	8	no	—
192	C₄	—	218	4	no	—
193	C₄	—	218	8	no	—
194	C₄	C₂×C₄×Dic₁₀₉	1	8	yes	complemented
195	C₂	C₄⋊C₄×Dic₁₀₉	1	8	yes	direct product factor
196	C₂	C₂²×C₄×Dic₁₀₉	1	1	yes	characteristic
197	C₂	C₂×C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
198	C₂	C₂×#16812657	1	1	yes	characteristic
199	C₂	C₂×#16812661	1	1	yes	characteristic
200	C₂	C₂×D₄×Dic₁₀₉	1	1	yes	characteristic
201	C₂	C₂×#16812664	1	1	yes	characteristic
202	C₂	C₂×Q₈×Dic₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₀₉
Fitting	C₂²×C₄⋊C₄×C₁₀₉	C₂
socle, prime radical	C₂⁴×C₁₀₉	C₂³
cosocle	C₂³×C₁₀₉	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×Dic₁₀₉	C₂	direct (N × H)
10. C₂×C₄⋊Dic₁₀₉	C₄	semidirect
13. C₂×C₄×Dic₁₀₉	C₄	semidirect
19. C₂×C₄⋊C₄×C₁₀₉	C₄	semidirect
33. C₄⋊Dic₁₀₉	C₂×C₄	semidirect
35. C₄×Dic₁₀₉	C₂×C₄	semidirect
45. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
57. C₂×C₄×C₁₀₉	C₄²	semidirect
68. C₂×Dic₁₀₉	C₄⋊C₄	direct (N × H)
89. C₄×C₁₀₉	C₂×C₄²	semidirect
92. Dic₁₀₉	C₂×C₄⋊C₄	direct (N × H)
93. C₂×C₁₀₉	C₄×C₄⋊C₄	semidirect
102. C₁₀₉	C₂×C₄×C₄⋊C₄	semidirect
116. C₂×C₄⋊C₄	Dic₁₀₉	direct (N × H)
145. C₄⋊C₄	C₂×Dic₁₀₉	direct (N × H)
171. C₂×C₄	C₄×Dic₁₀₉	semidirect
194. C₄	C₂×C₄×Dic₁₀₉	semidirect
195. C₂	C₄⋊C₄×Dic₁₀₉	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×C₄⋊Dic₁₀₉	2
C₂²×C₄×Dic₁₀₉	2
C₄⋊C₄×Dic₁₀₉	2
C₂×#16812606	2
C₂²×C₄⋊C₄×C₁₀₉	2
C₂×C₄×C₄⋊C₄	109

Subgroups — C2×C4⋊C4×Dic109 #28365369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₄×Dic₁₀₉ #28365369