Subgroups — C₂×D₁₀₉⋊#108 #28379689

Order 13952.

480 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	C₂×D₁₀₉⋊M₁₆	C₂	1	2	yes	maximal
2	C₂×#16812770	C₂	1	2	yes	maximal, complemented
3	D₁₀₉⋊#108	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16812769	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	4	yes	—
7	D₁₀₉⋊M₁₆	—	2	4	no	—
8	D₁₀₉⋊M₁₆	C₂²	1	8	yes	—
9	C₂×#13182038	—	2	2	no	—
10	C₂×D₁₀₉⋊C₈	—	2	2	no	—
11	#16812770	C₂²	1	8	yes	complemented
12	#16812769	C₂²	1	8	yes	complemented
13	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₂²×#11724784	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
16	C₂×C₈×D₁₀₉	C₄	1	4	yes	—
17	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
22	#13182039	—	2	4	no	—
23	#13182039	C₂³	1	8	yes	—
24	#13182038	—	2	8	no	—
25	D₁₀₉⋊C₈	—	2	8	no	—
26	C₂×#11724786	—	2	4	no	—
27	C₂³×D₁₀₉	Q₈	1	1	yes	characteristic
28	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
29	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
32	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
33	C₂×C₄×D₁₀₉	Q₈	1	2	yes	—
34	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
35	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
36	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
38	C₂×#11724784	C₂×C₄	1	4	yes	—
39	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
40	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
41	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
42	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
43	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
44	C₈×D₁₀₉	C₂³	1	2	yes	—
45	C₈×D₁₀₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
49	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×D₁₀₉	—	2	4	no	—
52	C₂²×D₁₀₉	C₄⋊C₄	1	2	yes	—
53	C₂²×D₁₀₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×D₁₀₉	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
61	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
64	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
65	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
66	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
67	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
69	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
70	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
72	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
73	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
74	#11724786	—	2	8	no	—
75	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
76	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
77	#11724784	C₂²×C₄	1	2	yes	—
78	#11724784	C₂²×C₄	1	2	yes	—
79	C₂²×C₁₀₉	—	2	2	no	—
80	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
81	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×D₁₀₉	—	2	2	no	—
83	C₂×D₁₀₉	—	2	4	no	—
84	C₂×D₁₀₉	—	2	4	no	—
85	C₂×D₁₀₉	#108	1	4	yes	complemented
86	C₂×D₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×D₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×D₁₀₉	C₂²×Q₈	1	1	yes	characteristic
89	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
92	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	Dic₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
94	Dic₁₀₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₁₀₉	—	2	2	no	—
96	C₂×C₁₀₉	C₂×#108	1	2	yes	complemented
97	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	D₁₀₉	—	2	1	no	—
99	D₁₀₉	—	2	1	no	—
100	D₁₀₉	C₂×#108	1	4	yes	complemented
101	C₂²×#108	—	109	1	no	maximal, Sylow
102	C₁₀₉	C₂²×#108	1	1	yes	characteristic, complemented, Sylow
103	C₂²×M₁₆	—	109	2	no	—
104	C₂³×C₈	—	109	1	no	—
105	C₂×#108	—	109	2	no	—
106	C₂×#108	—	109	8	no	—
107	C₂×#108	—	109	2	no	—
108	C₂³×C₄	—	109	1	no	—
109	C₂×M₁₆	—	109	8	no	—
110	C₂×M₁₆	—	109	4	no	—
111	C₂×M₁₆	—	218	2	no	—
112	C₂×M₁₆	—	218	4	no	—
113	C₂²×C₈	—	109	1	no	—
114	C₂²×C₈	—	109	1	no	—
115	C₂²×C₈	—	109	1	no	—
116	C₂²×C₈	—	109	4	no	—
117	C₂²×C₈	—	109	1	no	—
118	C₂²×C₈	—	109	4	no	—
119	C₂²×C₈	—	109	1	no	—
120	C₂²×C₈	—	218	2	no	—
121	C₂²×C₈	#89137	1	1	yes	characteristic
122	#108	—	109	8	no	—
123	#108	—	109	8	no	—
124	C₂⁴	—	109	1	no	—
125	C₂²×C₄	—	109	1	no	—
126	C₂²×C₄	—	109	2	no	—
127	C₂²×C₄	—	109	2	no	—
128	C₂²×C₄	—	109	1	no	—
129	C₂²×C₄	—	109	1	no	—
130	C₂²×C₄	—	109	1	no	—
131	C₂²×C₄	—	109	2	no	—
132	C₂²×C₄	—	109	1	no	—
133	C₂²×C₄	—	109	2	no	—
134	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
135	M₁₆	—	109	8	no	—
136	M₁₆	—	218	8	no	—
137	M₁₆	—	218	4	no	—
138	C₂×C₈	—	109	1	no	—
139	C₂×C₈	—	109	1	no	—
140	C₂×C₈	—	109	4	no	—
141	C₂×C₈	—	109	4	no	—
142	C₂×C₈	—	109	4	no	—
143	C₂×C₈	—	109	2	no	—
144	C₂×C₈	—	109	2	no	—
145	C₂×C₈	—	109	2	no	—
146	C₂×C₈	—	109	2	no	—
147	C₂×C₈	—	218	4	no	—
148	C₂×C₈	—	218	8	no	—
149	C₂×C₈	C₂×#89137	1	1	yes	characteristic
150	C₂×C₈	C₂×#89137	1	1	yes	characteristic
151	C₂×C₈	C₂×#89137	1	4	yes	—
152	C₂³	—	109	1	no	—
153	C₂³	—	109	2	no	—
154	C₂³	—	109	2	no	—
155	C₂³	—	109	1	no	—
156	C₂³	—	218	4	no	—
157	C₂³	C₄⋊#89137	1	1	yes	characteristic
158	C₂×C₄	—	109	1	no	—
159	C₂×C₄	—	109	2	no	—
160	C₂×C₄	—	109	2	no	—
161	C₂×C₄	—	109	1	no	—
162	C₂×C₄	—	109	2	no	—
163	C₂×C₄	—	109	1	no	—
164	C₂×C₄	—	109	2	no	—
165	C₂×C₄	—	109	2	no	—
166	C₂×C₄	—	109	2	no	—
167	C₂×C₄	—	109	1	no	—
168	C₂×C₄	—	109	2	no	—
169	C₂×C₄	—	109	1	no	—
170	C₂×C₄	—	109	2	no	—
171	C₂×C₄	—	109	1	no	—
172	C₂×C₄	C₄⋊#89137	1	2	yes	—
173	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
174	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
175	C₂×C₄	C₂²×#89137	1	2	yes	—
176	C₈	—	109	2	no	—
177	C₈	—	109	2	no	—
178	C₈	—	218	8	no	—
179	C₈	C₂²×#89137	1	2	yes	—
180	C₈	C₂²×#89137	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	—	109	4	no	—
183	C₂²	—	109	2	no	—
184	C₂²	—	109	1	no	—
185	C₂²	—	109	1	no	—
186	C₂²	—	218	4	no	—
187	C₂²	—	218	4	no	—
188	C₂²	—	218	2	no	—
189	C₂²	C₂×C₄⋊#89137	1	2	yes	—
190	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
191	C₄	—	109	1	no	—
192	C₄	—	109	1	no	—
193	C₄	—	109	2	no	—
194	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
195	C₄	C₂×C₄⋊#89137	1	2	yes	—
196	C₄	C₂³×#89137	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	109	4	no	—
199	C₂	—	218	1	no	—
200	C₂	—	218	1	no	—
201	C₂	D₁₀₉⋊#108	1	2	yes	direct product factor
202	C₂	C₂²×C₄⋊#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#89137
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₄	C₂³×#89137
Fitting	C₂²×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₀₉	Q₈
cosocle	C₄×D₁₀₉	C₂⁴
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	C₂²×#108
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16812770	C₂	semidirect
3. D₁₀₉⋊#108	C₂	direct (N × H)
4. C₂×#16812769	C₂	semidirect
11. #16812770	C₂²	semidirect
12. #16812769	C₂²	semidirect
85. C₂×D₁₀₉	#108	semidirect
96. C₂×C₁₀₉	C₂×#108	semidirect
100. D₁₀₉	C₂×#108	semidirect
102. C₁₀₉	C₂²×#108	semidirect
201. C₂	D₁₀₉⋊#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₁₀₉⋊M₁₆	2
C₂×#16812770	2
D₁₀₉⋊#108	2
C₂×#16812769	2
C₂²×C₈×D₁₀₉	2
C₂²×#108	109

Subgroups — C2×D109⋊#108 #28379689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₀₉⋊#108 #28379689