Subgroups — C₂×Q₈×C₂²⋊D₁₀₉ #28376504

Order 13952.

1568 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×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₂²⋊D₁₀₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16812697	C₂	1	3	yes	maximal, complemented
4	C₂×#16812678	C₂	1	3	yes	maximal, complemented
5	C₂×#16812675	C₂	1	3	yes	maximal, complemented
6	C₂×Q₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₂²⋊D₁₀₉	C₂	1	3	yes	maximal
8	C₂³×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₂₁₈	C₂²	1	3	yes	—
10	C₂×Q₈×D₁₀₉	—	2	8	no	—
11	C₂×Q₈×D₁₀₉	C₂²	1	8	yes	complemented
12	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₉	C₂²	1	3	yes	—
14	#16812697	C₂²	1	24	yes	complemented
15	#16812678	C₂²	1	24	yes	complemented
16	#16812675	C₂²	1	24	yes	complemented
17	Q₈×Dic₁₀₉	C₂²	1	8	yes	complemented
18	C₄×C₂²⋊D₁₀₉	C₂²	1	24	yes	—
19	C₂×C₄⋊Dic₁₀₉	C₂²	1	3	yes	—
20	C₂×#13182008	C₂²	1	3	yes	—
21	C₂×#13182008	C₂²	1	6	yes	—
22	C₂×C₂²⋊Dic₁₀₉	C₂²	1	3	yes	—
23	C₂×#13182006	C₂²	1	3	yes	—
24	C₂×C₄×Dic₁₀₉	C₂²	1	3	yes	—
25	C₂²×Q₈×C₁₀₉	—	2	8	no	—
26	C₂²×Q₈×C₁₀₉	C₂²	1	3	yes	—
27	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²×Q₈×C₁₀₉	C₂²	1	8	yes	complemented
29	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
30	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₂₁₈	—	2	12	no	—
32	C₂×Dic₂₁₈	C₂³	1	12	yes	—
33	Q₈×D₁₀₉	—	2	32	no	—
34	C₂×C₂²⋊D₁₀₉	Q₈	1	8	yes	direct product factor
35	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
36	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
37	C₂²×Dic₁₀₉	C₂³	1	3	yes	—
38	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×D₁₀₉	—	2	12	no	—
40	C₂×C₄×D₁₀₉	C₂³	1	12	yes	—
41	C₄⋊Dic₁₀₉	C₂³	1	12	yes	—
42	#13182008	C₂³	1	12	yes	—
43	#13182008	C₂³	1	24	yes	—
44	C₂²⋊Dic₁₀₉	C₂³	1	12	yes	—
45	#13182006	C₂³	1	12	yes	—
46	C₄×Dic₁₀₉	C₂³	1	12	yes	—
47	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₀₉	—	2	32	no	—
49	C₂×Q₈×C₁₀₉	—	2	12	no	—
50	C₂×Q₈×C₁₀₉	D₄	1	8	yes	complemented
51	C₂×Q₈×C₁₀₉	C₂³	1	12	yes	—
52	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
53	C₂²×C₄×C₁₀₉	—	2	12	no	—
54	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
55	C₂²×C₄×C₁₀₉	C₂³	1	12	yes	—
56	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
57	C₂³×C₁₀₉	—	2	2	no	—
58	C₂³×C₁₀₉	—	2	2	no	—
59	C₂³×C₁₀₉	C₂×Q₈	1	4	yes	complemented
60	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
61	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
62	C₂²×D₁₀₉	—	2	2	no	—
63	C₂²×D₁₀₉	—	2	2	no	—
64	C₂²×D₁₀₉	C₂×Q₈	1	4	yes	complemented
65	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
66	Q₈×C₁₀₉	—	2	24	no	—
67	Q₈×C₁₀₉	C₂×D₄	1	16	yes	complemented
68	C₂×C₄×C₁₀₉	—	2	6	no	—
69	C₂×C₄×C₁₀₉	—	2	24	no	—
70	C₂×C₄×C₁₀₉	C₂×D₄	1	12	yes	—
71	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
72	C₂×C₄×C₁₀₉	C₂⁴	1	6	yes	—
73	C₂²⋊D₁₀₉	C₂×Q₈	1	16	yes	direct product factor
74	C₂×Dic₁₀₉	—	2	6	no	—
75	C₂×Dic₁₀₉	—	2	6	no	—
76	C₂×Dic₁₀₉	C₂×Q₈	1	4	yes	complemented
77	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
78	C₂×Dic₁₀₉	C₂⁴	1	6	yes	—
79	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
80	C₄×D₁₀₉	—	2	24	no	—
81	Dic₂₁₈	—	2	24	no	—
82	C₂²×C₁₀₉	—	2	8	no	—
83	C₂²×C₁₀₉	—	2	2	no	—
84	C₂²×C₁₀₉	—	2	2	no	—
85	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
86	C₂²×C₁₀₉	C₂²×Q₈	1	2	yes	—
87	C₂²×C₁₀₉	C₂²×Q₈	1	4	yes	complemented
88	C₂²×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
89	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
90	C₂×D₁₀₉	—	2	8	no	—
91	C₂×D₁₀₉	—	2	2	no	—
92	C₂×D₁₀₉	—	2	2	no	—
93	C₂×D₁₀₉	C₂²×Q₈	1	4	yes	complemented
94	C₄×C₁₀₉	—	2	6	no	—
95	C₄×C₁₀₉	C₂²×D₄	1	12	yes	—
96	Dic₁₀₉	—	2	6	no	—
97	Dic₁₀₉	—	2	6	no	—
98	Dic₁₀₉	C₂²×Q₈	1	4	yes	complemented
99	C₂×C₁₀₉	—	2	4	no	—
100	C₂×C₁₀₉	D₄×Q₈	1	4	yes	complemented
101	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
102	C₂×C₁₀₉	C₂³×Q₈	1	1	yes	characteristic
103	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
104	D₁₀₉	—	2	4	no	—
105	C₂×D₄×Q₈	—	109	1	no	maximal, Sylow
106	C₁₀₉	C₂×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
107	C₂³×Q₈	—	109	1	no	—
108	C₂³×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
109	D₄×Q₈	—	109	16	no	—
110	C₂×C₄⋊Q₈	—	109	3	no	—
111	C₂×C₂²⋊Q₈	—	109	3	no	—
112	C₂×C₂²⋊Q₈	—	109	3	no	—
113	C₂×C₄×Q₈	—	109	1	no	—
114	C₂×C₄×D₄	—	109	3	no	—
115	C₂²×Q₈	—	2	8	no	—
116	C₂²×Q₈	—	109	8	no	—
117	C₂²×Q₈	—	109	3	no	—
118	C₂²×Q₈	—	218	8	no	—
119	C₂²×Q₈	C₂×D₁₀₉	1	3	yes	—
120	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
121	C₂²×Q₈	C₂×D₁₀₉	1	8	yes	complemented
122	C₂²×D₄	—	109	1	no	—
123	C₂³×C₄	—	109	3	no	—
124	C₂³×C₄	C₂×D₁₀₉	1	3	yes	—
125	C₄⋊Q₈	—	109	24	no	—
126	C₂²⋊Q₈	—	109	24	no	—
127	C₂²⋊Q₈	—	109	24	no	—
128	C₄×Q₈	—	109	8	no	—
129	C₄×D₄	—	109	24	no	—
130	C₂×C₄⋊C₄	—	109	3	no	—
131	C₂×C₄⋊C₄	—	109	3	no	—
132	C₂×C₄⋊C₄	—	109	6	no	—
133	C₂×C₂²⋊C₄	—	109	3	no	—
134	C₂×C₂²⋊C₄	—	109	3	no	—
135	C₂×C₄²	—	109	3	no	—
136	C₂⁴	—	109	1	no	—
137	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
138	C₂×Q₈	—	2	32	no	—
139	C₂×Q₈	—	2	12	no	—
140	C₂×Q₈	—	109	12	no	—
141	C₂×Q₈	—	218	32	no	—
142	C₂×Q₈	—	218	12	no	—
143	C₂×Q₈	C₂²⋊D₁₀₉	1	8	yes	direct product factor
144	C₂×Q₈	C₂²×D₁₀₉	1	12	yes	—
145	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
146	C₂×D₄	—	109	8	no	—
147	C₂×D₄	—	109	4	no	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	109	1	no	—
150	C₂²×C₄	—	109	3	no	—
151	C₂²×C₄	—	109	3	no	—
152	C₂²×C₄	—	109	12	no	—
153	C₂²×C₄	—	218	12	no	—
154	C₂²×C₄	C₂²×D₁₀₉	1	12	yes	—
155	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
156	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
157	C₄⋊C₄	—	109	12	no	—
158	C₄⋊C₄	—	109	24	no	—
159	C₄⋊C₄	—	109	12	no	—
160	C₂²⋊C₄	—	109	12	no	—
161	C₂²⋊C₄	—	109	12	no	—
162	C₄²	—	109	12	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	109	4	no	—
166	C₂³	—	109	2	no	—
167	C₂³	—	218	2	no	—
168	C₂³	—	218	2	no	—
169	C₂³	Q₈×D₁₀₉	1	4	yes	complemented
170	C₂³	C₂³×D₁₀₉	1	2	yes	—
171	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
172	Q₈	—	2	24	no	—
173	Q₈	—	218	24	no	—
174	Q₈	C₂×C₂²⋊D₁₀₉	1	16	yes	direct product factor
175	D₄	—	109	16	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	24	no	—
178	C₂×C₄	—	109	4	no	—
179	C₂×C₄	—	109	2	no	—
180	C₂×C₄	—	109	6	no	—
181	C₂×C₄	—	109	6	no	—
182	C₂×C₄	—	218	6	no	—
183	C₂×C₄	—	218	6	no	—
184	C₂×C₄	—	218	24	no	—
185	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	12	yes	—
186	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
187	C₂×C₄	C₂³×D₁₀₉	1	6	yes	—
188	C₂²	—	2	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	109	4	no	—
192	C₂²	—	218	8	no	—
193	C₂²	—	218	2	no	—
194	C₂²	—	218	2	no	—
195	C₂²	C₂²×C₂²⋊D₁₀₉	1	2	yes	—
196	C₂²	C₂×Q₈×D₁₀₉	1	2	yes	—
197	C₂²	C₂×Q₈×D₁₀₉	1	4	yes	complemented
198	C₂²	#16812747	1	2	yes	—
199	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
200	C₄	—	2	6	no	—
201	C₄	—	109	4	no	—
202	C₄	—	218	6	no	—
203	C₄	—	218	6	no	—
204	C₄	C₂²×C₂²⋊D₁₀₉	1	12	yes	—
205	C₂	—	2	4	no	—
206	C₂	—	218	4	no	—
207	C₂	Q₈×C₂²⋊D₁₀₉	1	4	yes	direct product factor
208	C₂	C₂³×C₂²⋊D₁₀₉	1	1	yes	characteristic
209	C₂	C₂²×Q₈×D₁₀₉	1	1	yes	characteristic
210	C₂	C₂×#16812747	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂³×Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂²×C₂²⋊D₁₀₉	C₂²
hypercenter	C₂³×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×D₄×Q₈
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×D₁₀₉	C₂	semidirect
2. Q₈×C₂²⋊D₁₀₉	C₂	direct (N × H)
3. C₂×#16812697	C₂	semidirect
4. C₂×#16812678	C₂	semidirect
5. C₂×#16812675	C₂	semidirect
6. C₂×Q₈×Dic₁₀₉	C₂	semidirect
8. C₂³×Q₈×C₁₀₉	C₂	semidirect
11. C₂×Q₈×D₁₀₉	C₂²	semidirect
14. #16812697	C₂²	semidirect
15. #16812678	C₂²	semidirect
16. #16812675	C₂²	semidirect
17. Q₈×Dic₁₀₉	C₂²	semidirect
28. C₂²×Q₈×C₁₀₉	C₂²	semidirect
34. C₂×C₂²⋊D₁₀₉	Q₈	direct (N × H)
50. C₂×Q₈×C₁₀₉	D₄	semidirect
59. C₂³×C₁₀₉	C₂×Q₈	semidirect
64. C₂²×D₁₀₉	C₂×Q₈	semidirect
67. Q₈×C₁₀₉	C₂×D₄	semidirect
73. C₂²⋊D₁₀₉	C₂×Q₈	direct (N × H)
76. C₂×Dic₁₀₉	C₂×Q₈	semidirect
87. C₂²×C₁₀₉	C₂²×Q₈	semidirect
93. C₂×D₁₀₉	C₂²×Q₈	semidirect
98. Dic₁₀₉	C₂²×Q₈	semidirect
100. C₂×C₁₀₉	D₄×Q₈	semidirect
106. C₁₀₉	C₂×D₄×Q₈	semidirect
108. C₂³×Q₈	D₁₀₉	semidirect
121. C₂²×Q₈	C₂×D₁₀₉	semidirect
143. C₂×Q₈	C₂²⋊D₁₀₉	direct (N × H)
169. C₂³	Q₈×D₁₀₉	semidirect
174. Q₈	C₂×C₂²⋊D₁₀₉	direct (N × H)
197. C₂²	C₂×Q₈×D₁₀₉	semidirect
207. C₂	Q₈×C₂²⋊D₁₀₉	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×D₁₀₉	2
Q₈×C₂²⋊D₁₀₉	2
C₂×#16812697	2
C₂×#16812678	2
C₂×#16812675	2
C₂×Q₈×Dic₁₀₉	2
C₂×C₄×C₂²⋊D₁₀₉	2
C₂³×Q₈×C₁₀₉	2
C₂×D₄×Q₈	109

Subgroups — C2×Q8×C22⋊D109 #28376504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₂²⋊D₁₀₉ #28376504