Subgroups — C₂×D₄×#11724784 #28371705

Order 13952.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 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	2	yes	maximal, complemented
2	D₄×#11724784	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊#11724784	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11724784	—	2	8	no	—
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	C₂²×#11724784	C₂²	1	2	yes	—
10	C₂²×#11724784	C₂²	1	1	yes	characteristic
11	C₂²×#11724784	C₂²	1	8	yes	complemented
12	C₄⋊#11724784	C₂²	1	4	yes	complemented
13	C₂²⋊#11724784	C₂²	1	8	yes	complemented
14	C₄×#11724784	C₂²	1	4	yes	complemented
15	C₂²×D₄×C₁₀₉	C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₀₉	C₂²	1	2	yes	—
17	C₄×D₄×C₁₀₉	C₄	1	4	yes	—
18	C₄×D₄×C₁₀₉	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₁₀₉	C₄	1	2	yes	—
21	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×#11724784	—	2	4	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₂×#11724784	—	2	16	no	—
25	C₂×#11724784	D₄	1	4	yes	direct product factor
26	C₂×#11724784	C₂³	1	2	yes	—
27	C₂×#11724784	C₂³	1	4	yes	—
28	C₂×#11724784	C₂³	1	2	yes	—
29	C₂⁴×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₁₀₉	C₈	1	8	yes	complemented
31	C₂×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₁₀₉	—	2	4	no	—
33	C₂²×C₄×C₁₀₉	—	2	4	no	—
34	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
42	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄²×C₁₀₉	C₂³	1	2	yes	—
44	C₂³×C₁₀₉	—	2	4	no	—
45	C₂³×C₁₀₉	—	2	4	no	—
46	C₂³×C₁₀₉	C₂×C₈	1	8	yes	complemented
47	C₂³×C₁₀₉	C₂²×C₄	1	4	yes	—
48	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	D₄×C₁₀₉	C₂×C₈	1	16	yes	complemented
50	C₂×C₄×C₁₀₉	—	2	4	no	—
51	C₂×C₄×C₁₀₉	—	2	2	no	—
52	C₂×C₄×C₁₀₉	—	2	8	no	—
53	C₂×C₄×C₁₀₉	—	2	4	no	—
54	C₂×C₄×C₁₀₉	—	2	4	no	—
55	C₂×C₄×C₁₀₉	C₂×C₈	1	4	yes	complemented
56	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	#41	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
63	#11724784	—	2	4	no	—
64	#11724784	—	2	2	no	—
65	#11724784	C₂×D₄	1	4	yes	direct product factor
66	C₂²×C₁₀₉	—	2	4	no	—
67	C₂²×C₁₀₉	—	2	4	no	—
68	C₂²×C₁₀₉	—	2	16	no	—
69	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
70	C₂²×C₁₀₉	C₂²×C₈	1	2	yes	—
71	C₂²×C₁₀₉	C₂²×C₈	1	8	yes	complemented
72	C₂²×C₁₀₉	#131	1	2	yes	—
73	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₁₀₉	—	2	4	no	—
75	C₄×C₁₀₉	—	2	2	no	—
76	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
77	C₄×C₁₀₉	C₂²×C₈	1	4	yes	complemented
78	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
80	C₂×C₁₀₉	—	2	8	no	—
81	C₂×C₁₀₉	C₈×D₄	1	4	yes	complemented
82	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
83	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
84	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
85	C₂×C₈×D₄	—	109	1	no	maximal, Sylow
86	C₁₀₉	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₈	—	109	2	no	—
88	C₂×C₄×D₄	D₁₀₉	1	1	yes	characteristic
89	C₈×D₄	—	109	8	no	—
90	C₂×C₄⋊C₈	—	109	1	no	—
91	C₂×C₂²⋊C₈	—	109	2	no	—
92	C₂×C₄×C₈	—	109	1	no	—
93	C₂²×D₄	Dic₁₀₉	1	1	yes	characteristic
94	C₂³×C₄	C₂×D₁₀₉	1	2	yes	—
95	C₂²×C₈	—	109	1	no	—
96	C₂²×C₈	—	109	8	no	—
97	C₂²×C₈	—	109	2	no	—
98	C₂²×C₈	—	109	1	no	—
99	C₂²×C₈	—	218	8	no	—
100	C₄×D₄	Dic₁₀₉	1	4	yes	—
101	C₄×D₄	C₂×D₁₀₉	1	4	yes	—
102	C₂×C₄⋊C₄	Dic₁₀₉	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	Dic₁₀₉	1	2	yes	—
104	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
105	C₄⋊C₈	—	109	4	no	—
106	C₂²⋊C₈	—	109	8	no	—
107	C₄×C₈	—	109	4	no	—
108	C₂⁴	C₂×Dic₁₀₉	1	2	yes	—
109	C₂×D₄	#11724784	1	8	yes	direct product factor
110	C₂×D₄	C₂×Dic₁₀₉	1	4	yes	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
115	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
116	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	—
118	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
119	C₂×C₈	—	109	4	no	—
120	C₂×C₈	—	109	2	no	—
121	C₂×C₈	—	109	2	no	—
122	C₂×C₈	—	109	4	no	—
123	C₂×C₈	—	218	16	no	—
124	C₂×C₈	—	218	4	no	—
125	C₂×C₈	—	218	2	no	—
126	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
127	C₂²⋊C₄	C₂×Dic₁₀₉	1	8	yes	—
128	C₄²	C₂×Dic₁₀₉	1	2	yes	—
129	C₄²	C₂²×D₁₀₉	1	2	yes	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	4	no	—
132	C₂³	C₂×#11724784	1	8	yes	complemented
133	C₂³	C₂²×Dic₁₀₉	1	4	yes	—
134	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
135	D₄	C₂×#11724784	1	16	yes	direct product factor
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₂×#11724784	1	4	yes	complemented
142	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
143	C₂×C₄	C₂²×Dic₁₀₉	1	4	yes	—
144	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
145	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
146	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
147	C₂×C₄	#13182028	1	2	yes	—
148	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
149	C₈	—	109	4	no	—
150	C₈	—	218	4	no	—
151	C₈	—	218	2	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	16	no	—
155	C₂²	D₄×Dic₁₀₉	1	2	yes	—
156	C₂²	C₂²×#11724784	1	2	yes	—
157	C₂²	C₂²×#11724784	1	8	yes	complemented
158	C₂²	#16812706	1	2	yes	—
159	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	2	2	no	—
162	C₄	D₄×Dic₁₀₉	1	2	yes	—
163	C₄	C₂²×#11724784	1	4	yes	complemented
164	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
165	C₄	C₂×#13182028	1	1	yes	characteristic
166	C₂	—	2	8	no	—
167	C₂	D₄×#11724784	1	4	yes	direct product factor
168	C₂	C₂×D₄×Dic₁₀₉	1	1	yes	characteristic
169	C₂	C₂³×#11724784	1	1	yes	characteristic
170	C₂	C₂×#16812706	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂×C₁₀₉	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄×D₄×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂²×C₄
prime radical	C₂²×D₄×C₁₀₉	C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₀₉
hypocenter	C₁₀₉	C₂×C₈×D₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11724784	C₂	semidirect
2. D₄×#11724784	C₂	direct (N × H)
3. C₂×C₄⋊#11724784	C₂	semidirect
4. C₂×C₂²⋊#11724784	C₂	semidirect
5. C₂×C₄×#11724784	C₂	semidirect
11. C₂²×#11724784	C₂²	semidirect
12. C₄⋊#11724784	C₂²	semidirect
13. C₂²⋊#11724784	C₂²	semidirect
14. C₄×#11724784	C₂²	semidirect
25. C₂×#11724784	D₄	direct (N × H)
30. C₂×D₄×C₁₀₉	C₈	semidirect
46. C₂³×C₁₀₉	C₂×C₈	semidirect
49. D₄×C₁₀₉	C₂×C₈	semidirect
55. C₂×C₄×C₁₀₉	C₂×C₈	semidirect
65. #11724784	C₂×D₄	direct (N × H)
71. C₂²×C₁₀₉	C₂²×C₈	semidirect
77. C₄×C₁₀₉	C₂²×C₈	semidirect
81. C₂×C₁₀₉	C₈×D₄	semidirect
86. C₁₀₉	C₂×C₈×D₄	semidirect
109. C₂×D₄	#11724784	direct (N × H)
132. C₂³	C₂×#11724784	semidirect
135. D₄	C₂×#11724784	direct (N × H)
141. C₂×C₄	C₂×#11724784	semidirect
157. C₂²	C₂²×#11724784	semidirect
163. C₄	C₂²×#11724784	semidirect
167. C₂	D₄×#11724784	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
D₄×#11724784	2
C₂×C₄⋊#11724784	2
C₂×C₂²⋊#11724784	2
C₂×C₄×#11724784	2
C₂×C₄×D₄×C₁₀₉	2
C₂×C₈×D₄	109

Subgroups — C2×D4×#11724784 #28371705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×#11724784 #28371705