Subgroups — C₂×#20294788 #28362745

Order 13952.

512 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11724784	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294788	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₉	—	2	1	no	—
6	C₂²×#11724784	—	2	2	no	—
7	C₂²×#11724784	—	2	2	no	—
8	C₂²×#11724784	C₄	1	4	yes	complemented
9	C₂²×#11724784	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₀₉	—	2	2	no	—
11	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
12	C₂×C₄⋊Dic₁₀₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₀₉	—	2	1	no	—
15	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
17	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₀₉	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
20	C₂²×Dic₁₀₉	—	2	4	no	—
21	C₂²×Dic₁₀₉	—	2	2	no	—
22	C₂²×Dic₁₀₉	—	4	2	no	—
23	C₂²×Dic₁₀₉	—	4	2	no	—
24	C₂×#11724784	—	2	8	no	—
25	C₂×#11724784	—	2	2	no	—
26	C₂×#11724784	—	2	2	no	—
27	C₂×#11724784	C₂×C₄	1	4	yes	complemented
28	C₄⋊Dic₁₀₉	—	2	4	no	—
29	C₄⋊Dic₁₀₉	—	2	2	no	—
30	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	complemented
31	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	—	2	4	no	—
33	C₂²×C₄×C₁₀₉	—	2	2	no	—
34	C₂²×C₄×C₁₀₉	—	4	2	no	—
35	C₂²×C₄×C₁₀₉	—	4	2	no	—
36	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
40	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
42	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₀₉	—	2	4	no	—
45	C₄⋊C₄×C₁₀₉	—	2	2	no	—
46	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
47	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₁₀₉	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₀₉	D₈	1	2	yes	—
51	C₂³×C₁₀₉	SD₁₆	1	2	yes	—
52	C₂³×C₁₀₉	SD₁₆	1	2	yes	—
53	C₂³×C₁₀₉	Q₁₆	1	2	yes	—
54	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	—	2	4	no	—
56	C₂×C₄×C₁₀₉	—	4	8	no	—
57	C₂×C₄×C₁₀₉	—	4	2	no	—
58	C₂×C₄×C₁₀₉	—	4	2	no	—
59	C₂×C₄×C₁₀₉	C₄²	1	4	yes	—
60	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
62	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
66	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×Dic₁₀₉	—	2	4	no	—
73	C₂×Dic₁₀₉	—	4	8	no	—
74	C₂×Dic₁₀₉	—	4	2	no	—
75	C₂×Dic₁₀₉	—	4	2	no	—
76	#11724784	—	2	4	no	—
77	C₂²×C₁₀₉	#95	1	4	yes	—
78	C₂²×C₁₀₉	#102	1	4	yes	—
79	C₂²×C₁₀₉	#102	1	4	yes	—
80	C₂²×C₁₀₉	#103	1	4	yes	—
81	C₂²×C₁₀₉	#103	1	4	yes	—
82	C₂²×C₁₀₉	#106	1	4	yes	—
83	C₂²×C₁₀₉	#107	1	4	yes	—
84	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
88	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
89	C₂²×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
91	C₄×C₁₀₉	—	4	4	no	—
92	C₄×C₁₀₉	#95	1	4	yes	—
93	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
94	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	Dic₁₀₉	—	4	4	no	—
98	C₂×C₁₀₉	#340	1	8	yes	complemented
99	C₂×C₁₀₉	C₂×#95	1	1	yes	characteristic
100	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
101	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
102	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
103	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
104	C₂×C₁₀₉	C₂×#106	1	1	yes	characteristic
105	C₂×C₁₀₉	C₂×#107	1	1	yes	characteristic
106	C₂×#340	—	109	1	no	maximal, Sylow
107	C₁₀₉	C₂×#340	1	1	yes	characteristic, complemented, Sylow
108	C₂³×C₈	—	109	1	no	—
109	C₂²×C₄⋊C₄	—	109	1	no	—
110	C₂²×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
111	#340	—	109	4	no	—
112	C₂³×C₄	—	2	1	no	—
113	C₂³×C₄	—	218	1	no	—
114	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
115	C₂²×C₈	—	109	4	no	—
116	C₂²×C₈	—	109	2	no	—
117	C₂²×C₈	—	218	2	no	—
118	C₂²×C₈	—	218	2	no	—
119	C₂×C₄⋊C₄	—	2	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	109	4	no	—
122	C₂×C₄⋊C₄	—	109	2	no	—
123	C₂×C₄⋊C₄	—	218	2	no	—
124	C₂×C₄⋊C₄	—	218	1	no	—
125	C₂×C₄⋊C₄	Dic₁₀₉	1	4	yes	complemented
126	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
127	C₂⁴	C₂²⋊D₁₀₉	1	1	yes	characteristic
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	436	2	no	—
133	C₂²×C₄	—	436	2	no	—
134	C₂²×C₄	—	218	4	no	—
135	C₂²×C₄	—	218	2	no	—
136	C₂²×C₄	Dic₂₁₈	1	2	yes	—
137	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
138	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
139	C₂²×C₄	D₄₃₆	1	2	yes	—
140	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
141	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
143	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
144	C₂×C₈	—	109	4	no	—
145	C₂×C₈	—	218	8	no	—
146	C₂×C₈	—	218	2	no	—
147	C₂×C₈	—	218	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	109	4	no	—
151	C₄⋊C₄	—	218	4	no	—
152	C₄⋊C₄	—	218	2	no	—
153	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	complemented
154	C₂³	#13182006	1	2	yes	—
155	C₂³	C₂²⋊Dic₁₀₉	1	2	yes	—
156	C₂³	#13182008	1	2	yes	—
157	C₂³	#13182014	1	2	yes	—
158	C₂³	#13182016	1	2	yes	—
159	C₂³	#13182017	1	2	yes	—
160	C₂³	#13182019	1	2	yes	—
161	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	436	8	no	—
167	C₂×C₄	—	436	2	no	—
168	C₂×C₄	—	436	2	no	—
169	C₂×C₄	—	218	4	no	—
170	C₂×C₄	C₄×Dic₁₀₉	1	4	yes	—
171	C₂×C₄	#13182006	1	2	yes	—
172	C₂×C₄	#13182006	1	4	yes	—
173	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
174	C₂×C₄	#13182008	1	2	yes	—
175	C₂×C₄	#13182008	1	4	yes	—
176	C₂×C₄	C₄⋊Dic₁₀₉	1	4	yes	—
177	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
180	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
183	C₈	—	218	4	no	—
184	C₂²	#16812606	1	4	yes	—
185	C₂²	#16812619	1	4	yes	—
186	C₂²	#16812620	1	4	yes	—
187	C₂²	#16812622	1	4	yes	—
188	C₂²	#16812623	1	4	yes	—
189	C₂²	#16812630	1	4	yes	—
190	C₂²	#16812632	1	4	yes	—
191	C₂²	C₂×#13182006	1	1	yes	characteristic
192	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
193	C₂²	C₂×#13182008	1	1	yes	characteristic
194	C₂²	C₂×#13182014	1	1	yes	characteristic
195	C₂²	C₂×#13182016	1	1	yes	characteristic
196	C₂²	C₂×#13182017	1	1	yes	characteristic
197	C₂²	C₂×#13182019	1	1	yes	characteristic
198	C₄	—	4	4	no	—
199	C₄	—	436	4	no	—
200	C₄	#16812606	1	4	yes	—
201	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
202	C₄	C₂×#13182006	1	1	yes	characteristic
203	C₄	C₂×#13182008	1	1	yes	characteristic
204	C₄	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
205	C₂	#20294788	1	8	yes	direct product factor
206	C₂	C₂×#16812606	1	1	yes	characteristic
207	C₂	C₂×#16812619	1	1	yes	characteristic
208	C₂	C₂×#16812620	1	1	yes	characteristic
209	C₂	C₂×#16812622	1	1	yes	characteristic
210	C₂	C₂×#16812623	1	1	yes	characteristic
211	C₂	C₂×#16812630	1	1	yes	characteristic
212	C₂	C₂×#16812632	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂²×C₄⋊C₄×C₁₀₉	C₂
socle, prime radical	C₂⁴×C₁₀₉	D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	C₂×#340
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #20294788	C₂	direct (N × H)
8. C₂²×#11724784	C₄	semidirect
12. C₂×C₄⋊Dic₁₀₉	C₄	semidirect
18. C₂×C₄⋊C₄×C₁₀₉	C₄	semidirect
27. C₂×#11724784	C₂×C₄	semidirect
30. C₄⋊Dic₁₀₉	C₂×C₄	semidirect
46. C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
98. C₂×C₁₀₉	#340	semidirect
107. C₁₀₉	C₂×#340	semidirect
125. C₂×C₄⋊C₄	Dic₁₀₉	semidirect
153. C₄⋊C₄	C₂×Dic₁₀₉	semidirect
205. C₂	#20294788	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11724784	2
C₂²×C₄⋊Dic₁₀₉	2
#20294788	2
C₂²×C₄⋊C₄×C₁₀₉	2
C₂×#340	109

Subgroups — C2×#20294788 #28362745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20294788 #28362745