Subgroups — C₂×C₄×D₁₀₈ #13136211

Order 1728.

852 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₄²×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₂₇)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊Dic₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₁₀₈	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄×D₃₆	—	3	1	no	maximal
8	C₂³×D₂₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₇	C₂²	1	2	yes	—
13	C₂×D₁₀₈	C₄	1	8	yes	direct product factor
14	C₂×D₁₀₈	C₂²	1	4	yes	—
15	C₂×C₄×D₂₇	—	2	8	no	—
16	C₂×C₄×D₂₇	C₂²	1	8	yes	complemented
17	C₄²×C₂₇	C₂²	1	4	yes	complemented
18	(C₂×D₂₇)⋊C₄	C₂²	1	8	yes	complemented
19	C₄⋊Dic₂₇	C₂²	1	4	yes	complemented
20	C₂²×D₃₆	—	3	1	no	—
21	C₂²×C₄×D₉	—	3	2	no	—
22	C₂×C₄²×C₉	D₃	1	1	yes	characteristic
23	C₂×(C₂×D₉)⋊C₄	—	3	2	no	—
24	C₂×C₄⋊Dic₉	—	3	1	no	—
25	C₄×D₃₆	—	3	8	no	—
26	C₂³×C₂₇	C₂³	1	1	yes	characteristic
27	C₂²×D₂₇	—	2	4	no	—
28	C₂²×D₂₇	—	2	4	no	—
29	C₂²×D₂₇	C₂×C₄	1	8	yes	complemented
30	C₂²×D₂₇	C₂³	1	4	yes	—
31	C₂×C₄×C₂₇	—	2	2	no	—
32	C₂×C₄×C₂₇	C₂×C₄	1	4	yes	complemented
33	C₂×C₄×C₂₇	D₄	1	4	yes	complemented
34	C₂×C₄×C₂₇	C₂³	1	2	yes	—
35	C₂×C₄×C₂₇	C₂³	1	2	yes	—
36	C₂×C₄×C₂₇	C₂³	1	2	yes	—
37	C₂×Dic₂₇	—	2	4	no	—
38	C₂×Dic₂₇	C₂³	1	4	yes	—
39	D₁₀₈	C₂×C₄	1	16	yes	direct product factor
40	C₄×D₂₇	—	2	16	no	—
41	C₂×C₄×D₁₂	—	9	1	no	—
42	C₂³×D₉	—	3	2	no	—
43	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
46	C₂²×Dic₉	—	3	2	no	—
47	C₂×D₃₆	—	3	8	no	—
48	C₂×D₃₆	—	3	4	no	—
49	C₂×C₄×D₉	—	3	8	no	—
50	C₂×C₄×D₉	—	6	8	no	—
51	C₄²×C₉	C₂×D₃	1	4	yes	—
52	(C₂×D₉)⋊C₄	—	3	8	no	—
53	C₄⋊Dic₉	—	3	4	no	—
54	C₂²×C₂₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₂₇	C₂×D₄	1	2	yes	—
56	C₂²×C₂₇	#41	1	2	yes	—
57	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
58	C₂×D₂₇	—	2	16	no	—
59	C₂×D₂₇	—	2	4	no	—
60	C₂×D₂₇	—	2	4	no	—
61	C₂×D₂₇	C₂²×C₄	1	8	yes	complemented
62	C₄×C₂₇	—	2	2	no	—
63	C₄×C₂₇	C₂²×C₄	1	4	yes	complemented
64	C₄×C₂₇	C₂×D₄	1	4	yes	complemented
65	Dic₂₇	—	2	4	no	—
66	C₂²×D₁₂	—	9	1	no	—
67	C₂²×C₄×D₃	—	9	2	no	—
68	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
69	C₂×(C₂×D₃)⋊C₄	—	9	2	no	—
70	C₂×C₄⋊Dic₃	—	9	1	no	—
71	C₄×D₁₂	—	9	8	no	—
72	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
73	C₂²×D₉	—	3	8	no	—
74	C₂²×D₉	—	3	4	no	—
75	C₂²×D₉	—	6	4	no	—
76	C₂²×D₉	—	6	4	no	—
77	C₂×C₄×C₉	—	2	2	no	—
78	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
79	C₂×C₄×C₉	D₁₂	1	4	yes	—
80	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
81	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
82	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
83	C₂×Dic₉	—	3	4	no	—
84	C₂×Dic₉	—	6	4	no	—
85	D₃₆	—	3	16	no	—
86	C₄×D₉	—	6	16	no	—
87	C₂×C₄×D₄	—	27	1	no	Sylow
88	C₂×C₂₇	C₄×D₄	1	4	yes	complemented
89	C₂×C₂₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₂₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₂₇	C₂×#41	1	1	yes	characteristic
92	D₂₇	—	2	8	no	—
93	C₂³×D₃	—	9	2	no	—
94	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
97	C₂²×Dic₃	—	9	2	no	—
98	C₂×D₁₂	—	9	8	no	—
99	C₂×D₁₂	—	9	4	no	—
100	C₂×C₄×D₃	—	9	8	no	—
101	C₂×C₄×D₃	—	18	8	no	—
102	C₃×C₄²	C₂×D₉	1	4	yes	—
103	(C₂×D₃)⋊C₄	—	9	8	no	—
104	C₄⋊Dic₃	—	9	4	no	—
105	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
106	C₂²×C₉	C₂×D₁₂	1	2	yes	—
107	C₂²×C₉	#235	1	2	yes	—
108	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
109	C₂×D₉	—	3	8	no	—
110	C₂×D₉	—	6	16	no	—
111	C₂×D₉	—	6	4	no	—
112	C₂×D₉	—	6	4	no	—
113	C₄×C₉	—	2	2	no	—
114	C₄×C₉	C₂×C₄×D₃	1	4	yes	—
115	C₄×C₉	C₂×D₁₂	1	4	yes	—
116	Dic₉	—	6	4	no	—
117	C₂²×D₄	—	27	1	no	—
118	C₂³×C₄	—	27	2	no	—
119	C₄×D₄	—	27	8	no	—
120	C₂×C₄⋊C₄	—	27	1	no	—
121	C₂×C₂²⋊C₄	—	27	2	no	—
122	C₂×C₄²	D₂₇	1	1	yes	characteristic, complemented
123	C₂₇	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
125	C₂²×D₃	—	9	8	no	—
126	C₂²×D₃	—	9	4	no	—
127	C₂²×D₃	—	18	4	no	—
128	C₂²×D₃	—	18	4	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	C₄×D₉	1	4	yes	—
131	C₂×C₃×C₄	D₃₆	1	4	yes	—
132	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
133	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
134	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
135	C₂×Dic₃	—	9	4	no	—
136	C₂×Dic₃	—	18	4	no	—
137	D₁₂	—	9	16	no	—
138	C₄×D₃	—	18	16	no	—
139	C₂×C₉	C₄×D₁₂	1	4	yes	—
140	C₂×C₉	C₂²×C₄×D₃	1	1	yes	characteristic
141	C₂×C₉	C₂²×D₁₂	1	1	yes	characteristic
142	C₂×C₉	C₂×#235	1	1	yes	characteristic
143	D₉	—	6	8	no	—
144	C₂⁴	—	27	2	no	—
145	C₂×D₄	—	27	8	no	—
146	C₂×D₄	—	27	4	no	—
147	C₂²×C₄	—	27	2	no	—
148	C₂²×C₄	—	27	8	no	—
149	C₂²×C₄	—	54	8	no	—
150	C₂²×C₄	C₂×D₂₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₂₇	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₂₇	1	1	yes	characteristic
153	C₄⋊C₄	—	27	4	no	—
154	C₂²⋊C₄	—	27	8	no	—
155	C₄²	C₂×D₂₇	1	4	yes	complemented
156	C₂²×C₃	C₂×C₄×D₉	1	2	yes	—
157	C₂²×C₃	C₂×D₃₆	1	2	yes	—
158	C₂²×C₃	#3697	1	2	yes	—
159	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
160	C₂×D₃	—	9	8	no	—
161	C₂×D₃	—	18	16	no	—
162	C₂×D₃	—	18	4	no	—
163	C₂×D₃	—	18	4	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	C₂×C₄×D₉	1	4	yes	—
166	C₃×C₄	C₂×D₃₆	1	4	yes	—
167	Dic₃	—	18	4	no	—
168	C₉	C₂×C₄×D₁₂	1	1	yes	characteristic
169	C₂³	—	27	4	no	—
170	C₂³	—	27	8	no	—
171	C₂³	—	54	4	no	—
172	C₂³	—	54	4	no	—
173	C₂³	C₂²×D₂₇	1	1	yes	characteristic
174	D₄	—	27	16	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	27	4	no	—
177	C₂×C₄	—	54	4	no	—
178	C₂×C₄	—	54	16	no	—
179	C₂×C₄	C₄×D₂₇	1	4	yes	complemented
180	C₂×C₄	D₁₀₈	1	4	yes	direct product factor
181	C₂×C₄	C₂²×D₂₇	1	2	yes	—
182	C₂×C₄	C₂²×D₂₇	1	2	yes	—
183	C₂×C₄	C₂²×D₂₇	1	2	yes	—
184	C₂×C₃	C₄×D₃₆	1	4	yes	—
185	C₂×C₃	C₂²×C₄×D₉	1	1	yes	characteristic
186	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
187	C₂×C₃	C₂×#3697	1	1	yes	characteristic
188	D₃	—	18	8	no	—
189	C₂²	—	27	8	no	—
190	C₂²	—	54	16	no	—
191	C₂²	—	54	4	no	—
192	C₂²	—	54	4	no	—
193	C₂²	C₂×C₄×D₂₇	1	2	yes	—
194	C₂²	C₂×D₁₀₈	1	2	yes	—
195	C₂²	#88399	1	2	yes	—
196	C₂²	C₂³×D₂₇	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	54	4	no	—
199	C₄	C₂×C₄×D₂₇	1	4	yes	complemented
200	C₄	C₂×D₁₀₈	1	4	yes	direct product factor
201	C₃	C₂×C₄×D₃₆	1	1	yes	characteristic
202	C₂	—	54	8	no	—
203	C₂	C₄×D₁₀₈	1	4	yes	direct product factor
204	C₂	C₂²×C₄×D₂₇	1	1	yes	characteristic
205	C₂	C₂²×D₁₀₈	1	1	yes	characteristic
206	C₂	C₂×#88399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₇
commutator	C₂×C₂₇	C₂³×C₄
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₂×C₄²×C₂₇	C₂
socle	C₂³×C₃	C₂²×D₉
prime radical	C₂²×D₁₀₈	C₂
cosocle	C₂²×C₂₇	C₂⁴
hypercenter	C₂×C₄²	D₂₇
hypocenter	C₂₇	C₂×C₄×D₄
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₂₇	C₂	semidirect
3. C₂×C₄²×C₂₇	C₂	semidirect
4. C₂×(C₂×D₂₇)⋊C₄	C₂	semidirect
5. C₂×C₄⋊Dic₂₇	C₂	semidirect
6. C₄×D₁₀₈	C₂	direct (N × H)
13. C₂×D₁₀₈	C₄	direct (N × H)
16. C₂×C₄×D₂₇	C₂²	semidirect
17. C₄²×C₂₇	C₂²	semidirect
18. (C₂×D₂₇)⋊C₄	C₂²	semidirect
19. C₄⋊Dic₂₇	C₂²	semidirect
29. C₂²×D₂₇	C₂×C₄	semidirect
32. C₂×C₄×C₂₇	C₂×C₄	semidirect
33. C₂×C₄×C₂₇	D₄	semidirect
39. D₁₀₈	C₂×C₄	direct (N × H)
61. C₂×D₂₇	C₂²×C₄	semidirect
63. C₄×C₂₇	C₂²×C₄	semidirect
64. C₄×C₂₇	C₂×D₄	semidirect
88. C₂×C₂₇	C₄×D₄	semidirect
122. C₂×C₄²	D₂₇	semidirect
123. C₂₇	C₂×C₄×D₄	semidirect
155. C₄²	C₂×D₂₇	semidirect
179. C₂×C₄	C₄×D₂₇	semidirect
180. C₂×C₄	D₁₀₈	direct (N × H)
199. C₄	C₂×C₄×D₂₇	semidirect
200. C₄	C₂×D₁₀₈	direct (N × H)
203. C₂	C₄×D₁₀₈	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×D₁₀₈	2
C₂²×C₄×D₂₇	2
C₂×C₄²×C₂₇	2
C₂×(C₂×D₂₇)⋊C₄	2
C₂×C₄⋊Dic₂₇	2
C₄×D₁₀₈	2
C₂×C₄×D₃₆	3

Subgroups — C2×C4×D108 #13136211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×D₁₀₈ #13136211