Finite Group Databasebeta

← C₂×#20295883 #28378480

Subgroups — C₂×#20295883 #28378480

Order 13952.

232 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812754	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812754	C₂	1	1	yes	maximal, characteristic
3	#20295883	C₂	1	4	yes	maximal, direct product factor
4	C₂×#16812690	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724786	—	2	1	no	—
6	C₂²×#11724786	—	2	1	no	—
7	#16812754	—	2	1	no	—
8	#16812754	—	2	1	no	—
9	#16812754	C₂²	1	2	yes	—
10	#16812754	C₂²	1	2	yes	—
11	#16812690	—	2	1	no	—
12	#16812690	—	2	1	no	—
13	#16812690	C₄	1	2	yes	—
14	#16812690	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
16	C₂×C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
17	C₂×#13182008	—	2	1	no	—
18	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
20	C₂×#11724786	—	2	4	no	—
21	C₂×#11724786	—	2	4	no	—
22	C₂×#11724786	—	2	2	no	—
23	C₂×#11724786	—	2	2	no	—
24	C₂²×Dic₁₀₉	—	2	1	no	—
25	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
26	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	C₄⋊Dic₁₀₉	—	2	2	no	—
29	C₄⋊Dic₁₀₉	—	2	2	no	—
30	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
31	#13182008	—	2	2	no	—
32	#13182008	—	2	2	no	—
33	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
35	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
36	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	—	2	1	no	—
38	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	—	2	1	no	—
40	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
41	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄×C₁₀₉	—	4	2	no	—
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₁₀₉	—	2	2	no	—
48	C₂×Dic₁₀₉	—	2	2	no	—
49	C₂×Dic₁₀₉	—	2	2	no	—
50	C₂×Dic₁₀₉	—	4	2	no	—
51	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
55	#11724786	—	2	4	no	—
56	#11724786	—	2	4	no	—
57	C₂²×C₁₀₉	#101	1	2	yes	—
58	C₂²×C₁₀₉	#104	1	2	yes	—
59	C₂²×C₁₀₉	#104	1	2	yes	—
60	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₁₀₉	—	2	1	no	—
62	C₄×C₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	—	4	2	no	—
64	Dic₁₀₉	—	2	1	no	—
65	Dic₁₀₉	—	2	1	no	—
66	Dic₁₀₉	—	2	1	no	—
67	Dic₁₀₉	—	2	1	no	—
68	Dic₁₀₉	—	4	2	no	—
69	C₂×C₁₀₉	#330	1	4	yes	complemented
70	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
72	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
73	C₂×#330	—	109	1	no	maximal, Sylow
74	C₁₀₉	C₂×#330	1	1	yes	characteristic, complemented, Sylow
75	C₂×#125	—	109	1	no	—
76	C₂×#97	—	109	1	no	—
77	C₂×#97	—	109	1	no	—
78	#330	—	109	4	no	—
79	C₂²×C₈	—	218	1	no	—
80	C₂²×C₈	—	218	1	no	—
81	#125	—	109	2	no	—
82	#125	—	109	2	no	—
83	#125	—	218	1	no	—
84	#125	—	218	1	no	—
85	C₂×C₄⋊C₄	—	109	1	no	—
86	C₂×C₄⋊C₄	—	218	1	no	—
87	C₂×C₄⋊C₄	—	218	1	no	—
88	C₂×C₄⋊C₄	#89137	1	1	yes	characteristic
89	C₂×C₄²	—	109	1	no	—
90	#97	—	109	2	no	—
91	#97	—	109	2	no	—
92	#97	—	218	1	no	—
93	#97	—	218	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	109	1	no	—
96	C₂²×C₄	—	109	1	no	—
97	C₂²×C₄	—	218	1	no	—
98	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
99	C₂×C₈	—	218	2	no	—
100	C₂×C₈	—	218	2	no	—
101	C₂×C₈	—	218	4	no	—
102	C₂×C₈	—	218	4	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	109	2	no	—
105	C₄⋊C₄	—	218	2	no	—
106	C₄⋊C₄	—	218	2	no	—
107	C₄⋊C₄	—	218	2	no	—
108	C₄⋊C₄	—	218	2	no	—
109	C₄⋊C₄	—	218	1	no	—
110	C₄⋊C₄	C₂×#89137	1	2	yes	—
111	C₄²	—	109	1	no	—
112	C₄²	—	109	1	no	—
113	C₄²	—	109	1	no	—
114	C₄²	—	109	1	no	—
115	C₂³	C₂²⋊#89137	1	1	yes	characteristic
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	109	1	no	—
120	C₂×C₄	—	109	1	no	—
121	C₂×C₄	—	109	1	no	—
122	C₂×C₄	—	109	1	no	—
123	C₂×C₄	—	436	2	no	—
124	C₂×C₄	—	218	2	no	—
125	C₂×C₄	—	218	2	no	—
126	C₂×C₄	—	218	2	no	—
127	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
128	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
129	C₈	—	218	4	no	—
130	C₈	—	218	4	no	—
131	C₂²	#16812760	1	2	yes	—
132	C₂²	#16812763	1	2	yes	—
133	C₂²	#16812764	1	2	yes	—
134	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	2	no	—
138	C₄	—	436	2	no	—
139	C₄	—	218	1	no	—
140	C₄	—	218	1	no	—
141	C₄	—	218	1	no	—
142	C₄	—	218	1	no	—
143	C₂	#20295883	1	4	yes	direct product factor
144	C₂	C₂×#16812760	1	1	yes	characteristic
145	C₂	C₂×#16812763	1	1	yes	characteristic
146	C₂	C₂×#16812764	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₄
socle, prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₄	#89137
hypocenter	C₁₀₉	C₂×#330
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #20295883	C₂	direct (N × H)
69. C₂×C₁₀₉	#330	semidirect
74. C₁₀₉	C₂×#330	semidirect
143. C₂	#20295883	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×#16812754	2
#20295883	2
C₂×#16812690	2
C₂×#330	109