Subgroups — C₂×C₄⋊Q₈×D₁₀₉ #28376305

Order 13952.

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×D₁₀₉	C₂	1	2	yes	maximal
2	C₂×#16812699	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812698	C₂	1	4	yes	maximal, complemented
4	C₂×#16812697	C₂	1	2	yes	maximal, complemented
5	C₄⋊Q₈×D₁₀₉	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₄⋊C₄×D₁₀₉	C₂	1	4	yes	maximal
7	C₂×C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₂₁₈	C₂²	1	2	yes	—
10	C₂²×Dic₂₁₈	C₂²	1	4	yes	—
11	C₂×Q₈×D₁₀₉	—	2	16	no	—
12	C₂×Q₈×D₁₀₉	C₂²	1	16	yes	—
13	C₂²×C₄×D₁₀₉	C₂²	1	4	yes	—
14	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₉	C₂²	1	2	yes	—
16	#16812699	C₂²	1	8	yes	complemented
17	#16812698	C₂²	1	32	yes	complemented
18	#16812697	C₂²	1	16	yes	complemented
19	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
20	C₂×#13182008	C₂²	1	8	yes	—
21	C₄⋊C₄×D₁₀₉	C₂²	1	32	yes	—
22	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
24	C₄²×D₁₀₉	C₂²	1	8	yes	—
25	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
26	C₄⋊Q₈×C₁₀₉	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
28	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₁₈	—	2	8	no	—
31	C₂×Dic₂₁₈	—	2	16	no	—
32	C₂×Dic₂₁₈	C₂³	1	8	yes	—
33	C₂×Dic₂₁₈	C₂³	1	16	yes	—
34	Q₈×D₁₀₉	—	2	64	no	—
35	C₂²×Dic₁₀₉	C₂³	1	4	yes	—
36	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
38	C₂×C₄×D₁₀₉	—	2	16	no	—
39	C₂×C₄×D₁₀₉	D₄	1	8	yes	—
40	C₂×C₄×D₁₀₉	Q₈	1	16	yes	complemented
41	C₂×C₄×D₁₀₉	C₂³	1	16	yes	—
42	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
43	C₂×C₄×D₁₀₉	C₂³	1	8	yes	—
44	C₄⋊Dic₁₀₉	C₂³	1	16	yes	—
45	#13182008	C₂³	1	32	yes	—
46	C₄×Dic₁₀₉	C₂³	1	4	yes	—
47	C₄×Dic₁₀₉	C₂³	1	8	yes	—
48	C₂×Q₈×C₁₀₉	—	2	8	no	—
49	C₂×Q₈×C₁₀₉	C₂³	1	8	yes	—
50	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₁₀₉	C₂³	1	16	yes	—
54	C₄²×C₁₀₉	C₂³	1	4	yes	—
55	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	C₂²×D₁₀₉	C₂×D₄	1	4	yes	—
57	C₂²×D₁₀₉	C₂×Q₈	1	8	yes	—
58	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
59	Q₈×C₁₀₉	—	2	16	no	—
60	C₂×C₄×C₁₀₉	—	2	8	no	—
61	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₁₀₉	C₂×Q₈	1	8	yes	complemented
63	C₂×C₄×C₁₀₉	C₂⁴	1	8	yes	—
64	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
65	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
66	C₂×Dic₁₀₉	—	2	8	no	—
67	C₂×Dic₁₀₉	C₂×D₄	1	4	yes	—
68	C₂×Dic₁₀₉	C₂×Q₈	1	8	yes	complemented
69	C₂×Dic₁₀₉	C₂⁴	1	8	yes	—
70	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
71	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
72	C₄×D₁₀₉	—	2	32	no	—
73	C₄×D₁₀₉	C₂×D₄	1	16	yes	—
74	C₄×D₁₀₉	C₂×Q₈	1	32	yes	complemented
75	Dic₂₁₈	—	2	16	no	—
76	Dic₂₁₈	—	2	32	no	—
77	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
78	C₂²×C₁₀₉	C₂²×Q₈	1	4	yes	—
79	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
80	C₂×D₁₀₉	C₄⋊Q₈	1	16	yes	direct product factor
81	C₂×D₁₀₉	C₂²×D₄	1	4	yes	—
82	C₂×D₁₀₉	C₂²×Q₈	1	8	yes	—
83	C₄×C₁₀₉	—	2	8	no	—
84	C₄×C₁₀₉	C₂²×D₄	1	4	yes	—
85	C₄×C₁₀₉	C₂²×Q₈	1	8	yes	complemented
86	Dic₁₀₉	—	2	8	no	—
87	Dic₁₀₉	C₂²×D₄	1	4	yes	—
88	Dic₁₀₉	C₂²×Q₈	1	8	yes	complemented
89	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	4	yes	complemented
90	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
91	C₂×C₁₀₉	C₂³×Q₈	1	2	yes	—
92	D₁₀₉	C₂×C₄⋊Q₈	1	8	yes	direct product factor
93	C₂²×C₄⋊Q₈	—	109	1	no	maximal, Sylow
94	C₁₀₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
95	C₂³×Q₈	—	109	2	no	—
96	C₂×C₄⋊Q₈	—	109	2	no	—
97	C₂×C₄⋊Q₈	—	109	16	no	—
98	C₂×C₄⋊Q₈	—	109	4	no	—
99	C₂×C₄⋊Q₈	—	109	1	no	—
100	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, direct product factor
101	C₂²×C₄⋊C₄	—	109	4	no	—
102	C₂²×C₄²	—	109	1	no	—
103	C₂²×Q₈	—	109	16	no	—
104	C₂²×Q₈	—	109	4	no	—
105	C₂²×Q₈	—	109	2	no	—
106	C₂²×Q₈	—	218	16	no	—
107	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
108	C₂³×C₄	—	109	2	no	—
109	C₂³×C₄	—	109	1	no	—
110	C₂³×C₄	—	109	4	no	—
111	C₄⋊Q₈	—	109	16	no	—
112	C₄⋊Q₈	—	109	32	no	—
113	C₄⋊Q₈	—	109	8	no	—
114	C₄⋊Q₈	C₂×D₁₀₉	1	8	yes	direct product factor
115	C₂×C₄⋊C₄	—	109	8	no	—
116	C₂×C₄⋊C₄	—	109	4	no	—
117	C₂×C₄⋊C₄	—	109	32	no	—
118	C₂×C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
119	C₂×C₄²	—	109	1	no	—
120	C₂×C₄²	—	109	2	no	—
121	C₂×C₄²	—	109	8	no	—
122	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
123	C₂⁴	—	109	1	no	—
124	C₂×Q₈	—	2	8	no	—
125	C₂×Q₈	—	109	16	no	—
126	C₂×Q₈	—	109	8	no	—
127	C₂×Q₈	—	218	16	no	—
128	C₂×Q₈	—	218	64	no	—
129	C₂×Q₈	—	218	8	no	—
130	C₂×Q₈	C₂²×D₁₀₉	1	8	yes	—
131	C₂²×C₄	—	109	2	no	—
132	C₂²×C₄	—	109	1	no	—
133	C₂²×C₄	—	109	8	no	—
134	C₂²×C₄	—	109	16	no	—
135	C₂²×C₄	—	109	8	no	—
136	C₂²×C₄	—	109	4	no	—
137	C₂²×C₄	—	109	4	no	—
138	C₂²×C₄	—	109	16	no	—
139	C₂²×C₄	—	218	16	no	—
140	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
141	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
143	C₄⋊C₄	—	109	32	no	—
144	C₄⋊C₄	—	109	16	no	—
145	C₄⋊C₄	C₂²×D₁₀₉	1	16	yes	—
146	C₄²	—	109	4	no	—
147	C₄²	—	109	8	no	—
148	C₄²	C₂²×D₁₀₉	1	4	yes	—
149	C₂³	—	109	4	no	—
150	C₂³	—	109	8	no	—
151	C₂³	—	109	2	no	—
152	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
153	Q₈	—	2	16	no	—
154	Q₈	—	218	32	no	—
155	Q₈	—	218	16	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	109	4	no	—
158	C₂×C₄	—	109	8	no	—
159	C₂×C₄	—	109	4	no	—
160	C₂×C₄	—	109	2	no	—
161	C₂×C₄	—	109	16	no	—
162	C₂×C₄	—	109	32	no	—
163	C₂×C₄	—	109	8	no	—
164	C₂×C₄	—	218	32	no	—
165	C₂×C₄	—	218	8	no	—
166	C₂×C₄	D₄×D₁₀₉	1	4	yes	—
167	C₂×C₄	Q₈×D₁₀₉	1	8	yes	complemented
168	C₂×C₄	C₂³×D₁₀₉	1	8	yes	—
169	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
170	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
171	C₂²	—	109	16	no	—
172	C₂²	—	109	4	no	—
173	C₂²	—	109	8	no	—
174	C₂²	C₂×D₄×D₁₀₉	1	2	yes	—
175	C₂²	C₂×Q₈×D₁₀₉	1	4	yes	—
176	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
177	C₄	—	2	8	no	—
178	C₄	—	109	4	no	—
179	C₄	—	109	8	no	—
180	C₄	—	218	8	no	—
181	C₄	C₂×D₄×D₁₀₉	1	4	yes	—
182	C₄	C₂×Q₈×D₁₀₉	1	8	yes	complemented
183	C₂	—	109	8	no	—
184	C₂	C₄⋊Q₈×D₁₀₉	1	4	yes	direct product factor
185	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic
186	C₂	C₂²×Q₈×D₁₀₉	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
prime radical	C₂³×D₁₀₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16812699	C₂	semidirect
3. C₂×#16812698	C₂	semidirect
4. C₂×#16812697	C₂	semidirect
5. C₄⋊Q₈×D₁₀₉	C₂	direct (N × H)
8. C₂×C₄⋊Q₈×C₁₀₉	C₂	semidirect
16. #16812699	C₂²	semidirect
17. #16812698	C₂²	semidirect
18. #16812697	C₂²	semidirect
26. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
40. C₂×C₄×D₁₀₉	Q₈	semidirect
62. C₂×C₄×C₁₀₉	C₂×Q₈	semidirect
68. C₂×Dic₁₀₉	C₂×Q₈	semidirect
74. C₄×D₁₀₉	C₂×Q₈	semidirect
80. C₂×D₁₀₉	C₄⋊Q₈	direct (N × H)
85. C₄×C₁₀₉	C₂²×Q₈	semidirect
88. Dic₁₀₉	C₂²×Q₈	semidirect
89. C₂×C₁₀₉	C₂×C₄⋊Q₈	semidirect
92. D₁₀₉	C₂×C₄⋊Q₈	direct (N × H)
94. C₁₀₉	C₂²×C₄⋊Q₈	semidirect
100. C₂×C₄⋊Q₈	D₁₀₉	direct (N × H)
114. C₄⋊Q₈	C₂×D₁₀₉	direct (N × H)
167. C₂×C₄	Q₈×D₁₀₉	semidirect
182. C₄	C₂×Q₈×D₁₀₉	semidirect
184. C₂	C₄⋊Q₈×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
C₂×#16812699	2
C₂×#16812698	2
C₂×#16812697	2
C₄⋊Q₈×D₁₀₉	2
C₂×C₄⋊C₄×D₁₀₉	2
C₂×C₄²×D₁₀₉	2
C₂×C₄⋊Q₈×C₁₀₉	2
C₂²×C₄⋊Q₈	109

Subgroups — C2×C4⋊Q8×D109 #28376305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊Q₈×D₁₀₉ #28376305