Subgroups — C₂×C₄×Dic₁₀₄ #16016195

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₅₂	C₂	1	2	yes	maximal
3	C₄×Dic₁₀₄	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718096	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718095	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₂	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₆	—	2	2	no	—
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88058	C₂²	1	2	yes	—
13	C₂×#88056	—	2	2	no	—
14	C₂×Dic₁₀₄	C₄	1	8	yes	direct product factor
15	C₂×Dic₁₀₄	C₂²	1	4	yes	—
16	C₄×Dic₅₂	—	2	4	no	—
17	C₄×Dic₅₂	C₂²	1	8	yes	—
18	C₄×C₁₃×C₁₆	C₂²	1	4	yes	—
19	#11718096	C₂²	1	4	yes	—
20	#11718095	C₂²	1	8	yes	—
21	C₂²×Dic₂₆	—	2	2	no	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×#6101	—	4	2	no	—
27	C₂×C₄×Dic₁₃	—	4	2	no	—
28	C₂×Dic₅₂	—	2	2	no	—
29	C₂×Dic₅₂	—	2	4	no	—
30	C₂×Dic₅₂	C₂×C₄	1	8	yes	—
31	C₂×Dic₅₂	C₂³	1	4	yes	—
32	C₄×Dic₂₆	—	2	8	no	—
33	C₄×Dic₂₆	—	4	4	no	—
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
36	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
37	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	C₄×C₈×C₁₃	C₂³	1	2	yes	—
40	#88058	—	2	2	no	—
41	#88058	C₂³	1	4	yes	—
42	#88056	—	2	8	no	—
43	Dic₁₀₄	C₂×C₄	1	16	yes	direct product factor
44	C₂×C₄×Q₃₂	—	13	1	no	maximal, Sylow
45	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	—	4	2	no	—
49	C₂²×Dic₁₃	—	4	2	no	—
50	C₂×Dic₂₆	—	2	8	no	—
51	C₂×Dic₂₆	—	2	4	no	—
52	C₂×Dic₂₆	—	4	2	no	—
53	C₂×Dic₂₆	—	4	4	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	#41	1	2	yes	—
59	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₄²×C₁₃	D₈	1	2	yes	—
61	C₄²×C₁₃	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₁₃	—	2	4	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	#6101	—	4	8	no	—
65	C₄×Dic₁₃	—	4	8	no	—
66	Dic₅₂	—	2	8	no	—
67	Dic₅₂	—	2	4	no	—
68	Dic₅₂	C₂²×C₄	1	8	yes	—
69	C₁₃×C₁₆	—	2	2	no	—
70	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
71	C₂²×Q₃₂	—	13	1	no	—
72	C₂×C₄×Q₁₆	—	13	2	no	—
73	C₄×Q₃₂	—	13	8	no	—
74	C₂×#366	—	13	1	no	—
75	C₂×#358	—	13	2	no	—
76	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
77	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	Q₃₂	1	4	yes	complemented
80	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₁₃	#135	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₁₃	—	4	4	no	—
87	C₂×Dic₁₃	—	4	8	no	—
88	C₂×Dic₁₃	—	4	4	no	—
89	C₂×Dic₁₃	—	8	4	no	—
90	Dic₂₆	—	2	8	no	—
91	Dic₂₆	—	4	4	no	—
92	Dic₂₆	—	4	8	no	—
93	C₈×C₁₃	—	2	2	no	—
94	C₈×C₁₃	C₄×D₄	1	2	yes	—
95	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
97	C₂²×Q₁₆	—	13	2	no	—
98	C₂×C₄×Q₈	—	26	2	no	—
99	C₂×Q₃₂	—	13	8	no	—
100	C₂×Q₃₂	—	13	4	no	—
101	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
103	C₄×Q₁₆	—	13	8	no	—
104	C₄×Q₁₆	—	26	4	no	—
105	C₂×#107	—	13	2	no	—
106	C₂×#103	—	26	2	no	—
107	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
108	#366	—	13	4	no	—
109	#358	—	13	8	no	—
110	C₄×C₁₆	C₂×D₁₃	1	4	yes	—
111	C₂²×C₁₃	C₄×D₈	1	2	yes	—
112	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
113	C₂²×C₁₃	#508	1	2	yes	—
114	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	C₄×D₈	1	2	yes	—
117	C₄×C₁₃	C₂×Q₃₂	1	4	yes	complemented
118	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
120	Dic₁₃	—	4	8	no	—
121	Dic₁₃	—	8	4	no	—
122	C₂²×Q₈	—	26	2	no	—
123	C₂×Q₁₆	—	13	8	no	—
124	C₂×Q₁₆	—	13	4	no	—
125	C₂×Q₁₆	—	26	2	no	—
126	C₂×Q₁₆	—	26	4	no	—
127	C₂²×C₈	D₅₂	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
129	C₄×Q₈	—	26	8	no	—
130	C₄×Q₈	—	52	4	no	—
131	C₂×C₄⋊C₄	—	26	2	no	—
132	C₂×C₄⋊C₄	—	52	2	no	—
133	C₂×C₄²	—	52	2	no	—
134	C₂×C₄²	D₅₂	1	1	yes	characteristic
135	Q₃₂	—	13	16	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
138	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
139	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
140	#107	—	13	4	no	—
141	#107	—	26	2	no	—
142	#103	—	26	8	no	—
143	C₄×C₈	D₅₂	1	2	yes	—
144	C₄×C₈	C₂²×D₁₃	1	2	yes	—
145	C₂×C₁₃	C₄×Q₃₂	1	4	yes	complemented
146	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
147	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
148	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
149	C₂×Q₈	—	26	8	no	—
150	C₂×Q₈	—	26	4	no	—
151	C₂×Q₈	—	52	2	no	—
152	C₂×Q₈	—	52	4	no	—
153	C₂²×C₄	—	52	2	no	—
154	C₂²×C₄	—	52	2	no	—
155	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
156	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
158	Q₁₆	—	13	8	no	—
159	Q₁₆	—	26	8	no	—
160	Q₁₆	—	26	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
163	C₂×C₈	C₂×D₅₂	1	2	yes	—
164	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
165	C₂×C₈	#6127	1	2	yes	—
166	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
167	C₄⋊C₄	—	26	4	no	—
168	C₄⋊C₄	—	52	2	no	—
169	C₄⋊C₄	—	52	8	no	—
170	C₄²	—	52	8	no	—
171	C₄²	D₁₀₄	1	2	yes	—
172	C₄²	C₂×D₅₂	1	2	yes	—
173	C₁₆	—	2	2	no	—
174	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
175	C₁₃	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
176	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
177	Q₈	—	26	8	no	—
178	Q₈	—	52	4	no	—
179	Q₈	—	52	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	52	4	no	—
182	C₂×C₄	—	52	8	no	—
183	C₂×C₄	—	52	4	no	—
184	C₂×C₄	—	104	4	no	—
185	C₂×C₄	Dic₁₀₄	1	4	yes	direct product factor
186	C₂×C₄	C₄×D₅₂	1	2	yes	—
187	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
188	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
189	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
190	C₂×C₄	#88158	1	2	yes	—
191	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₄×D₅₂	1	2	yes	—
194	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
195	C₈	C₂×#6127	1	1	yes	characteristic
196	C₂²	C₄×D₁₀₄	1	2	yes	—
197	C₂²	#11718565	1	2	yes	—
198	C₂²	C₂×Dic₁₀₄	1	2	yes	—
199	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	52	8	no	—
202	C₄	—	104	4	no	—
203	C₄	C₄×D₁₀₄	1	2	yes	—
204	C₄	C₂×Dic₁₀₄	1	4	yes	direct product factor
205	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
206	C₄	C₂×#88158	1	1	yes	characteristic
207	C₂	C₄×Dic₁₀₄	1	4	yes	direct product factor
208	C₂	C₂×C₄×D₁₀₄	1	1	yes	characteristic
209	C₂	C₂²×Dic₁₀₄	1	1	yes	characteristic
210	C₂	C₂×#11718565	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₀₄
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₁₀₄	C₂	direct (N × H)
14. C₂×Dic₁₀₄	C₄	direct (N × H)
43. Dic₁₀₄	C₂×C₄	direct (N × H)
79. C₂×C₄×C₁₃	Q₃₂	semidirect
117. C₄×C₁₃	C₂×Q₃₂	semidirect
145. C₂×C₁₃	C₄×Q₃₂	semidirect
175. C₁₃	C₂×C₄×Q₃₂	semidirect
185. C₂×C₄	Dic₁₀₄	direct (N × H)
204. C₄	C₂×Dic₁₀₄	direct (N × H)
207. C₂	C₄×Dic₁₀₄	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×Dic₁₀₄	2
C₂×C₄×Dic₅₂	2
C₄×Dic₁₀₄	2
C₂×#11718096	2
C₂×#11718095	2
C₂×C₄×C₁₃×C₁₆	2
C₂×C₄×Q₃₂	13

Subgroups — C2×C4×Dic104 #16016195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₁₀₄ #16016195