Subgroups — C₄×#16812786 #28379600

Order 13952.

572 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812786	C₂	1	2	yes	maximal
2	#20295991	C₂	1	2	yes	maximal, complemented
3	#20295990	C₂	1	4	yes	maximal, complemented
4	C₄×#13182039	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#13182038	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#11724786	C₂	1	2	yes	maximal, complemented
8	C₄×#13182028	C₂	1	1	yes	maximal, characteristic
9	#16812786	C₄	1	16	yes	direct product factor
10	C₂×#13182039	C₂²	1	2	yes	—
11	C₂×#13182038	C₂²	1	4	yes	—
12	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
13	C₂²×#11724786	C₂²	1	4	yes	—
14	#16812754	C₂²	1	4	yes	—
15	#16812754	C₂²	1	2	yes	—
16	C₄×#11724786	C₂²	1	1	yes	characteristic
17	C₄×#11724786	C₂²	1	2	yes	—
18	C₄×#11724786	C₂²	1	4	yes	—
19	C₄×#11724786	C₂²	1	1	yes	characteristic
20	C₄×#11724786	C₂²	1	2	yes	—
21	C₂×#13182028	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
23	#16812664	C₄	1	1	yes	characteristic
24	D₄×Dic₁₀₉	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₀₉	C₄	1	2	yes	—
26	#16812659	C₄	1	2	yes	—
27	#16812657	C₂²	1	1	yes	characteristic
28	#16812655	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
30	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
32	#13182039	C₂×C₄	1	8	yes	complemented
33	#13182038	C₂×C₄	1	16	yes	complemented
34	D₁₀₉⋊C₈	C₂×C₄	1	8	yes	complemented
35	C₂×#11724786	C₂×C₄	1	16	yes	complemented
36	C₂×#11724786	C₂³	1	2	yes	—
37	C₂×#11724786	C₂³	1	4	yes	—
38	C₂×#11724786	C₂³	1	2	yes	—
39	#13182028	C₂×C₄	1	8	yes	—
40	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
42	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
43	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
44	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
48	#13182008	C₂×C₄	1	2	yes	—
49	#13182008	C₂×C₄	1	2	yes	—
50	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
51	#13182006	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
56	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
57	C₄×Dic₁₀₉	C₂³	1	2	yes	—
58	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
60	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
62	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
64	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	D₄×C₁₀₉	C₄²	1	4	yes	—
66	C₂×C₄×C₁₀₉	—	2	4	no	—
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²⋊D₁₀₉	C₄²	1	8	yes	—
72	C₂×Dic₁₀₉	—	2	4	no	—
73	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₀₉	C₂²×C₄	1	8	yes	—
77	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
79	C₄×D₁₀₉	—	2	2	no	—
80	C₄×D₁₀₉	—	2	2	no	—
81	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
82	Dic₂₁₈	C₄²	1	4	yes	—
83	#11724786	C₂²×C₄	1	4	yes	—
84	#11724786	C₂²×C₄	1	8	yes	—
85	#11724786	C₂²×C₄	1	4	yes	—
86	C₂²×C₁₀₉	—	2	2	no	—
87	C₂²×C₁₀₉	—	2	2	no	—
88	C₂²×C₁₀₉	C₂×C₄²	1	4	yes	—
89	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₁₀₉	—	2	1	no	—
91	C₂×D₁₀₉	—	2	1	no	—
92	C₂×D₁₀₉	C₂×C₄²	1	2	yes	—
93	C₄×C₁₀₉	—	2	1	no	—
94	C₄×C₁₀₉	—	2	2	no	—
95	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
96	C₄×C₁₀₉	#131	1	2	yes	complemented
97	Dic₁₀₉	—	2	1	no	—
98	Dic₁₀₉	—	2	2	no	—
99	Dic₁₀₉	C₂×C₄²	1	4	yes	—
100	Dic₁₀₉	#131	1	2	yes	complemented
101	Dic₁₀₉	C₂³×C₄	1	2	yes	—
102	C₂×C₁₀₉	—	2	4	no	—
103	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
104	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
105	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
106	D₁₀₉	—	2	2	no	—
107	C₄×#131	—	109	1	no	maximal, Sylow
108	C₁₀₉	C₄×#131	1	1	yes	characteristic, complemented, Sylow
109	C₂×#131	—	109	2	no	—
110	C₄×#41	—	109	1	no	—
111	#405	—	109	4	no	—
112	#405	—	109	2	no	—
113	C₄×M₁₆	—	109	2	no	—
114	C₄×M₁₆	—	109	1	no	—
115	C₂×C₄×C₈	—	109	2	no	—
116	C₂×C₄×C₈	—	109	1	no	—
117	C₂×#41	—	109	1	no	—
118	#131	—	109	16	no	—
119	C₂×M₁₆	—	109	4	no	—
120	C₂×M₁₆	—	109	2	no	—
121	C₂²×C₈	—	109	4	no	—
122	C₂²×C₈	—	109	2	no	—
123	C₄×Q₈	—	109	1	no	—
124	C₄×Q₈	—	109	1	no	—
125	C₄×D₄	—	109	1	no	—
126	C₄×D₄	—	109	2	no	—
127	C₄×D₄	—	109	2	no	—
128	C₄×D₄	#89137	1	1	yes	characteristic
129	#117	—	109	1	no	—
130	#117	—	109	2	no	—
131	C₂×C₄²	—	109	2	no	—
132	C₂×C₄²	—	109	1	no	—
133	#97	—	109	4	no	—
134	#97	—	109	2	no	—
135	C₄×C₈	—	109	1	no	—
136	C₄×C₈	—	109	2	no	—
137	C₄×C₈	—	109	1	no	—
138	C₄×C₈	—	109	4	no	—
139	C₄×C₈	—	109	2	no	—
140	#41	—	109	8	no	—
141	C₂×Q₈	—	109	1	no	—
142	C₂×D₄	—	109	2	no	—
143	C₂×D₄	C₂×#89137	1	1	yes	characteristic
144	C₂²×C₄	—	109	2	no	—
145	C₂²×C₄	—	109	1	no	—
146	C₂²×C₄	—	109	2	no	—
147	C₂²×C₄	—	109	1	no	—
148	C₂²×C₄	—	109	1	no	—
149	C₂²×C₄	C₂×#89137	1	2	yes	—
150	M₁₆	—	109	16	no	—
151	M₁₆	—	109	8	no	—
152	C₂×C₈	—	109	16	no	—
153	C₂×C₈	—	109	2	no	—
154	C₂×C₈	—	109	4	no	—
155	C₂×C₈	—	109	2	no	—
156	C₂×C₈	—	109	8	no	—
157	C₄⋊C₄	—	109	2	no	—
158	C₄⋊C₄	—	109	2	no	—
159	C₄⋊C₄	—	109	1	no	—
160	C₄⋊C₄	C₂×#89137	1	1	yes	characteristic
161	C₂²⋊C₄	—	109	2	no	—
162	C₂²⋊C₄	—	109	2	no	—
163	C₂²⋊C₄	C₂×#89137	1	2	yes	—
164	C₄²	—	109	2	no	—
165	C₄²	—	109	1	no	—
166	C₄²	—	109	2	no	—
167	C₄²	—	109	1	no	—
168	C₄²	—	109	2	no	—
169	C₄²	—	109	1	no	—
170	C₄²	C₂×#89137	1	1	yes	characteristic
171	C₂³	—	109	1	no	—
172	C₂³	C₂²×#89137	1	2	yes	—
173	Q₈	—	109	4	no	—
174	D₄	—	109	8	no	—
175	D₄	C₄×#89137	1	4	yes	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	109	8	no	—
178	C₂×C₄	—	109	1	no	—
179	C₂×C₄	—	109	1	no	—
180	C₂×C₄	—	109	2	no	—
181	C₂×C₄	—	109	4	no	—
182	C₂×C₄	—	109	1	no	—
183	C₂×C₄	—	109	2	no	—
184	C₂×C₄	—	218	2	no	—
185	C₂×C₄	—	218	4	no	—
186	C₂×C₄	—	218	2	no	—
187	C₂×C₄	C₂²×#89137	1	2	yes	—
188	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
189	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
190	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
191	C₈	—	109	4	no	—
192	C₈	—	109	8	no	—
193	C₈	—	109	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	109	2	no	—
197	C₂²	—	218	1	no	—
198	C₂²	—	218	1	no	—
199	C₂²	C₂×C₄×#89137	1	4	yes	—
200	C₂²	C₂³×#89137	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	109	2	no	—
204	C₄	—	109	4	no	—
205	C₄	—	109	2	no	—
206	C₄	—	218	1	no	—
207	C₄	—	218	2	no	—
208	C₄	C₂×C₄×#89137	1	2	yes	—
209	C₄	#16812786	1	2	yes	direct product factor
210	C₂	—	2	4	no	—
211	C₂	—	218	2	no	—
212	C₂	C₂²×C₄×#89137	1	1	yes	characteristic
213	C₂	C₂×#16812786	1	1	yes	characteristic
214	C₂	D₁₀₉⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#89137
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂²	C₂³×#89137
Fitting	C₄×D₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂×#13182028	C₂²
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₄×D₄	#89137
hypocenter	C₁₀₉	C₄×#131
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #20295991	C₂	semidirect
3. #20295990	C₂	semidirect
4. C₄×#13182039	C₂	semidirect
5. C₄×#13182038	C₂	semidirect
6. C₄×D₁₀₉⋊C₈	C₂	semidirect
7. C₂×C₄×#11724786	C₂	semidirect
9. #16812786	C₄	direct (N × H)
32. #13182039	C₂×C₄	semidirect
33. #13182038	C₂×C₄	semidirect
34. D₁₀₉⋊C₈	C₂×C₄	semidirect
35. C₂×#11724786	C₂×C₄	semidirect
96. C₄×C₁₀₉	#131	semidirect
100. Dic₁₀₉	#131	semidirect
108. C₁₀₉	C₄×#131	semidirect
209. C₄	#16812786	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×#16812786	2
#20295991	2
#20295990	2
C₄×#13182039	2
C₄×#13182038	2
C₄×D₁₀₉⋊C₈	2
C₂×C₄×#11724786	2
C₄×#13182028	2
C₄×#131	109

Subgroups — C4×#16812786 #28379600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16812786 #28379600