Subgroups — C₂×#20294980 #28371443

Order 13952.

592 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182012	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812656	C₂	1	1	yes	maximal, characteristic
4	#20294980	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16812609	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×#13182012	C₄	1	8	yes	complemented
11	C₂×#13182012	C₂²	1	4	yes	—
12	C₂²×#11724784	C₂²	1	1	yes	characteristic
13	C₂²×#11724784	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₀₉	C₄	1	8	yes	complemented
15	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
16	#16812656	C₄	1	4	yes	—
17	#16812656	C₂²	1	4	yes	—
18	C₂×#13182008	C₄	1	2	yes	—
19	C₂×#13182006	C₄	1	2	yes	—
20	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
21	#16812610	C₂²	1	4	yes	complemented
22	#16812609	C₂²	1	4	yes	complemented
23	C₈×Dic₁₀₉	C₂²	1	4	yes	complemented
24	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₁₀₉	C₂²	1	4	yes	complemented
28	C₂³×D₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
30	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
35	#13182012	C₂×C₄	1	16	yes	complemented
36	C₂×#11724784	C₂×C₄	1	4	yes	—
37	C₂×#11724784	C₂×C₄	1	4	yes	—
38	C₂×#11724784	C₂³	1	2	yes	—
39	C₂×#11724784	C₂³	1	2	yes	—
40	C₈×D₁₀₉	C₂×C₄	1	16	yes	complemented
41	#13182008	C₂×C₄	1	8	yes	—
42	#13182006	C₂×C₄	1	8	yes	—
43	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₀₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
48	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
49	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
50	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
51	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₄²×C₁₀₉	C₂³	1	2	yes	—
53	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₁₀₉	—	2	4	no	—
55	C₂²×D₁₀₉	C₄²	1	4	yes	—
56	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	—	2	4	no	—
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₁₀₉	—	2	4	no	—
64	C₂×Dic₁₀₉	C₄²	1	4	yes	—
65	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
66	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
67	C₄×D₁₀₉	C₄²	1	8	yes	—
68	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
69	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
70	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
71	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
72	#11724784	C₂²×C₄	1	4	yes	—
73	#11724784	C₂²×C₄	1	4	yes	—
74	C₂²×C₁₀₉	C₂×C₄²	1	2	yes	—
75	C₂²×C₁₀₉	#131	1	4	yes	—
76	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₁₀₉	—	2	4	no	—
78	C₂×D₁₀₉	—	2	8	no	—
79	C₂×D₁₀₉	C₂×C₄²	1	4	yes	—
80	C₄×C₁₀₉	—	2	4	no	—
81	C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
82	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
84	Dic₁₀₉	—	2	4	no	—
85	Dic₁₀₉	C₂×C₄²	1	4	yes	—
86	C₂×C₁₀₉	#405	1	4	yes	complemented
87	C₂×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
88	C₂×C₁₀₉	C₂×#131	1	2	yes	—
89	D₁₀₉	—	2	4	no	—
90	C₂×#405	—	109	1	no	maximal, Sylow
91	C₁₀₉	C₂×#405	1	1	yes	characteristic, complemented, Sylow
92	C₂²×M₁₆	—	109	1	no	—
93	C₂³×C₈	—	109	1	no	—
94	C₂×#117	—	109	1	no	—
95	#405	—	109	8	no	—
96	C₂×#97	—	109	1	no	—
97	C₂×#97	—	109	1	no	—
98	C₂×C₄×C₈	—	109	1	no	—
99	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic, complemented
100	C₂³×C₄	—	109	1	no	—
101	C₂×M₁₆	—	109	8	no	—
102	C₂×M₁₆	—	109	4	no	—
103	C₂²×C₈	—	109	1	no	—
104	C₂²×C₈	—	109	1	no	—
105	C₂²×C₈	—	109	8	no	—
106	C₂²×C₈	—	109	4	no	—
107	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
109	#117	—	109	4	no	—
110	#117	—	109	4	no	—
111	C₂×C₄⋊C₄	—	109	2	no	—
112	C₂×C₂²⋊C₄	—	109	2	no	—
113	C₂×C₄²	—	109	1	no	—
114	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
115	#97	—	109	4	no	—
116	#97	—	109	4	no	—
117	C₄×C₈	—	109	4	no	—
118	C₄×C₈	C₂×D₁₀₉	1	4	yes	complemented
119	C₂⁴	—	109	1	no	—
120	C₂²×C₄	—	109	1	no	—
121	C₂²×C₄	—	109	2	no	—
122	C₂²×C₄	—	109	4	no	—
123	C₂²×C₄	—	109	4	no	—
124	C₂²×C₄	—	109	2	no	—
125	C₂²×C₄	—	109	2	no	—
126	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
127	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
128	M₁₆	—	109	16	no	—
129	C₂×C₈	—	109	4	no	—
130	C₂×C₈	—	109	4	no	—
131	C₂×C₈	—	109	2	no	—
132	C₂×C₈	—	109	2	no	—
133	C₂×C₈	—	109	16	no	—
134	C₂×C₈	C₄×D₁₀₉	1	4	yes	—
135	C₂×C₈	C₄×D₁₀₉	1	4	yes	—
136	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
137	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
138	C₄⋊C₄	—	109	8	no	—
139	C₂²⋊C₄	—	109	8	no	—
140	C₄²	—	109	2	no	—
141	C₄²	—	109	2	no	—
142	C₄²	C₄×D₁₀₉	1	2	yes	—
143	C₄²	C₂²×D₁₀₉	1	2	yes	—
144	C₂³	—	109	4	no	—
145	C₂³	—	109	2	no	—
146	C₂³	—	218	4	no	—
147	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	109	4	no	—
150	C₂×C₄	—	109	2	no	—
151	C₂×C₄	—	109	8	no	—
152	C₂×C₄	—	109	4	no	—
153	C₂×C₄	—	109	4	no	—
154	C₂×C₄	—	109	4	no	—
155	C₂×C₄	—	218	4	no	—
156	C₂×C₄	C₂×C₄×D₁₀₉	1	4	yes	—
157	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
160	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
161	C₈	—	109	4	no	—
162	C₈	—	109	4	no	—
163	C₈	C₂×C₄×D₁₀₉	1	4	yes	—
164	C₈	C₂×C₄×D₁₀₉	1	4	yes	—
165	C₂²	—	109	4	no	—
166	C₂²	—	218	8	no	—
167	C₂²	—	218	4	no	—
168	C₂²	C₄²×D₁₀₉	1	2	yes	—
169	C₂²	#16812705	1	4	yes	—
170	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	109	4	no	—
173	C₄	—	218	4	no	—
174	C₄	C₄²×D₁₀₉	1	2	yes	—
175	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
176	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
177	C₂	—	218	4	no	—
178	C₂	#20294980	1	4	yes	direct product factor
179	C₂	C₂×C₄²×D₁₀₉	1	1	yes	characteristic
180	C₂	C₂×#16812705	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂²×C₄
prime radical	C₂³×D₁₀₉	C₂×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#405
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20294980	C₂	direct (N × H)
5. C₂×#16812610	C₂	semidirect
6. C₂×#16812609	C₂	semidirect
7. C₂×C₈×Dic₁₀₉	C₂	semidirect
8. C₂×C₄×C₈×C₁₀₉	C₂	semidirect
10. C₂×#13182012	C₄	semidirect
14. C₂×C₈×D₁₀₉	C₄	semidirect
21. #16812610	C₂²	semidirect
22. #16812609	C₂²	semidirect
23. C₈×Dic₁₀₉	C₂²	semidirect
27. C₄×C₈×C₁₀₉	C₂²	semidirect
35. #13182012	C₂×C₄	semidirect
40. C₈×D₁₀₉	C₂×C₄	semidirect
86. C₂×C₁₀₉	#405	semidirect
91. C₁₀₉	C₂×#405	semidirect
99. C₂×C₄×C₈	D₁₀₉	semidirect
118. C₄×C₈	C₂×D₁₀₉	semidirect
178. C₂	#20294980	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×#13182012	2
C₂²×C₈×D₁₀₉	2
C₂×#16812656	2
#20294980	2
C₂×#16812610	2
C₂×#16812609	2
C₂×C₈×Dic₁₀₉	2
C₂×C₄×C₈×C₁₀₉	2
C₂×#405	109

Subgroups — C2×#20294980 #28371443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20294980 #28371443