Subgroups — C₂×C₄×#13110105 #27208324

Order 13184.

1728 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13110105	C₂	1	1	yes	maximal, characteristic
2	C₄×#13110105	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₀₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₄₁₂	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₂²⋊D₁₀₃	C₂	1	4	yes	maximal, complemented
6	C₂×#15692980	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15692978	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₁₀₃	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13110105	C₄	1	16	yes	direct product factor
11	C₂×#13110105	C₂²	1	8	yes	—
12	C₂²×Dic₂₀₆	C₂²	1	1	yes	characteristic
13	C₂²×D₄₁₂	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₀₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₃	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₀₃	C₂²	1	2	yes	—
17	C₄×Dic₂₀₆	C₂²	1	16	yes	complemented
18	C₄×D₄₁₂	C₂²	1	16	yes	complemented
19	C₄×C₂²⋊D₁₀₃	C₂²	1	32	yes	complemented
20	#15692980	C₂²	1	8	yes	complemented
21	#15692978	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
23	C₂×Dic₁₀₃⋊C₄	C₂²	1	4	yes	—
24	C₂×C₂²⋊Dic₁₀₃	C₂²	1	2	yes	—
25	C₂×(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
26	C₂×C₄×Dic₁₀₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₀₃	C₂²	1	4	yes	—
28	C₄²×D₁₀₃	C₂²	1	16	yes	complemented
29	C₂³×C₄×C₁₀₃	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₀₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₀₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₃	C₂²	1	8	yes	complemented
35	C₂³×D₁₀₃	C₂³	1	2	yes	—
36	#13110105	C₂×C₄	1	64	yes	direct product factor
37	C₂×Dic₂₀₆	C₂×C₄	1	8	yes	complemented
38	C₂×Dic₂₀₆	C₂³	1	4	yes	—
39	C₂×D₄₁₂	C₂×C₄	1	8	yes	complemented
40	C₂×D₄₁₂	C₂³	1	4	yes	—
41	C₂×C₂²⋊D₁₀₃	C₂×C₄	1	16	yes	complemented
42	C₂×C₂²⋊D₁₀₃	C₂³	1	8	yes	—
43	C₂²×Dic₁₀₃	C₂³	1	4	yes	—
44	C₂²×Dic₁₀₃	C₂³	1	2	yes	—
45	C₂×C₄×D₁₀₃	—	2	16	no	—
46	C₂×C₄×D₁₀₃	C₂×C₄	1	16	yes	complemented
47	C₂×C₄×D₁₀₃	C₂³	1	16	yes	—
48	C₂×C₄×D₁₀₃	C₂³	1	8	yes	—
49	C₄⋊Dic₁₀₃	C₂³	1	8	yes	—
50	Dic₁₀₃⋊C₄	C₂³	1	16	yes	—
51	C₂²⋊Dic₁₀₃	C₂³	1	8	yes	—
52	(C₂×D₁₀₃)⋊C₄	C₂³	1	16	yes	—
53	C₄×Dic₁₀₃	C₂³	1	8	yes	—
54	C₄×Dic₁₀₃	C₂³	1	16	yes	—
55	C₂⁴×C₁₀₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₃	—	2	8	no	—
57	C₂²×C₄×C₁₀₃	C₂×C₄	1	8	yes	complemented
58	C₂²×C₄×C₁₀₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₀₃	C₂³	1	8	yes	—
60	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₁₀₃	C₂³	1	2	yes	—
64	C₄²×C₁₀₃	C₂³	1	4	yes	—
65	C₄²×C₁₀₃	C₂³	1	4	yes	—
66	C₄²×C₁₀₃	C₂³	1	8	yes	—
67	C₂³×C₁₀₃	—	2	4	no	—
68	C₂³×C₁₀₃	C₂²×C₄	1	4	yes	—
69	C₂³×C₁₀₃	C₂⁴	1	2	yes	—
70	C₂³×C₁₀₃	C₂⁴	1	1	yes	characteristic
71	C₂²×D₁₀₃	—	2	8	no	—
72	C₂²×D₁₀₃	C₂²×C₄	1	8	yes	—
73	C₂²×D₁₀₃	C₂⁴	1	4	yes	—
74	C₂×C₄×C₁₀₃	—	2	4	no	—
75	C₂×C₄×C₁₀₃	—	2	16	no	—
76	C₂×C₄×C₁₀₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₁₀₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₀₃	C₂²×C₄	1	16	yes	complemented
79	C₂×C₄×C₁₀₃	#41	1	8	yes	complemented
80	C₂×C₄×C₁₀₃	C₂⁴	1	4	yes	—
81	C₂×C₄×C₁₀₃	C₂⁴	1	2	yes	—
82	C₂×C₄×C₁₀₃	C₂⁴	1	2	yes	—
83	C₂×C₄×C₁₀₃	C₂⁴	1	4	yes	—
84	C₂²⋊D₁₀₃	C₂²×C₄	1	32	yes	complemented
85	C₂×Dic₁₀₃	—	2	8	no	—
86	C₂×Dic₁₀₃	C₂²×C₄	1	8	yes	—
87	C₂×Dic₁₀₃	C₂⁴	1	8	yes	—
88	C₂×Dic₁₀₃	C₂⁴	1	4	yes	—
89	D₄₁₂	C₂²×C₄	1	16	yes	complemented
90	C₄×D₁₀₃	—	2	32	no	—
91	C₄×D₁₀₃	C₂²×C₄	1	32	yes	complemented
92	Dic₂₀₆	C₂²×C₄	1	16	yes	complemented
93	C₂²×C₁₀₃	—	2	8	no	—
94	C₂²×C₁₀₃	—	2	4	no	—
95	C₂²×C₁₀₃	C₂³×C₄	1	4	yes	—
96	C₂²×C₁₀₃	C₂³×C₄	1	2	yes	—
97	C₂²×C₁₀₃	C₂×#41	1	4	yes	—
98	C₂²×C₁₀₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₁₀₃	—	2	16	no	—
100	C₂×D₁₀₃	—	2	8	no	—
101	C₂×D₁₀₃	C₂³×C₄	1	8	yes	—
102	C₄×C₁₀₃	—	2	4	no	—
103	C₄×C₁₀₃	C₂³×C₄	1	4	yes	—
104	C₄×C₁₀₃	C₂³×C₄	1	4	yes	—
105	C₄×C₁₀₃	C₂×#41	1	8	yes	complemented
106	Dic₁₀₃	—	2	8	no	—
107	Dic₁₀₃	C₂³×C₄	1	8	yes	—
108	C₂×C₁₀₃	—	2	4	no	—
109	C₂×C₁₀₃	C₄×#41	1	4	yes	complemented
110	C₂×C₁₀₃	C₂⁴×C₄	1	1	yes	characteristic
111	C₂×C₁₀₃	C₂²×#41	1	2	yes	—
112	D₁₀₃	—	2	8	no	—
113	C₂×C₄×#41	—	103	1	no	maximal, Sylow
114	C₁₀₃	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
115	C₂²×#41	—	103	1	no	—
116	C₄×#41	—	103	16	no	—
117	C₂×C₄×Q₈	—	103	2	no	—
118	C₂×C₄×D₄	—	103	4	no	—
119	C₂×C₄×D₄	—	103	2	no	—
120	C₂×#117	—	103	1	no	—
121	C₂×#117	—	103	2	no	—
122	C₂²×C₄²	—	103	2	no	—
123	C₂²×C₄²	D₁₀₃	1	1	yes	characteristic, complemented
124	C₂×#41	—	103	16	no	—
125	C₂×#41	—	103	8	no	—
126	C₂²×Q₈	—	103	1	no	—
127	C₂²×D₄	—	103	2	no	—
128	C₂²×D₄	—	103	1	no	—
129	C₂³×C₄	—	103	2	no	—
130	C₂³×C₄	—	103	4	no	—
131	C₂³×C₄	C₂×D₁₀₃	1	2	yes	—
132	C₂³×C₄	C₂×D₁₀₃	1	1	yes	characteristic
133	C₄×Q₈	—	103	16	no	—
134	C₄×D₄	—	103	32	no	—
135	C₄×D₄	—	103	16	no	—
136	#117	—	103	8	no	—
137	#117	—	103	16	no	—
138	C₂×C₄⋊C₄	—	103	4	no	—
139	C₂×C₄⋊C₄	—	103	2	no	—
140	C₂×C₂²⋊C₄	—	103	2	no	—
141	C₂×C₂²⋊C₄	—	103	4	no	—
142	C₂×C₄²	—	103	2	no	—
143	C₂×C₄²	—	103	4	no	—
144	C₂×C₄²	—	103	16	no	—
145	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
146	C₂×C₄²	C₂×D₁₀₃	1	2	yes	—
147	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×D₁₀₃	1	8	yes	complemented
149	C₂⁴	—	103	2	no	—
150	C₂⁴	C₂²×D₁₀₃	1	1	yes	characteristic
151	#41	—	103	64	no	—
152	C₂×Q₈	—	103	8	no	—
153	C₂×Q₈	—	103	4	no	—
154	C₂×D₄	—	103	16	no	—
155	C₂×D₄	—	103	8	no	—
156	C₂×D₄	—	103	8	no	—
157	C₂×D₄	—	103	4	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	103	2	no	—
160	C₂²×C₄	—	103	16	no	—
161	C₂²×C₄	—	103	8	no	—
162	C₂²×C₄	—	103	4	no	—
163	C₂²×C₄	—	103	16	no	—
164	C₂²×C₄	—	206	16	no	—
165	C₂²×C₄	C₄×D₁₀₃	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₁₀₃	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₀₃	1	8	yes	—
168	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₀₃	1	4	yes	—
171	C₂²×C₄	C₂²×D₁₀₃	1	2	yes	—
172	C₄⋊C₄	—	103	16	no	—
173	C₄⋊C₄	—	103	8	no	—
174	C₂²⋊C₄	—	103	8	no	—
175	C₂²⋊C₄	—	103	16	no	—
176	C₄²	—	103	8	no	—
177	C₄²	—	103	16	no	—
178	C₄²	C₂²×D₁₀₃	1	4	yes	—
179	C₄²	C₂²×D₁₀₃	1	4	yes	—
180	C₄²	C₂²×D₁₀₃	1	8	yes	—
181	C₂³	—	2	4	no	—
182	C₂³	—	103	4	no	—
183	C₂³	—	103	8	no	—
184	C₂³	—	206	8	no	—
185	C₂³	C₂×C₄×D₁₀₃	1	4	yes	—
186	C₂³	C₂³×D₁₀₃	1	2	yes	—
187	C₂³	C₂³×D₁₀₃	1	1	yes	characteristic
188	Q₈	—	103	16	no	—
189	D₄	—	103	32	no	—
190	D₄	—	103	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	103	4	no	—
194	C₂×C₄	—	103	8	no	—
195	C₂×C₄	—	103	32	no	—
196	C₂×C₄	—	103	8	no	—
197	C₂×C₄	—	206	8	no	—
198	C₂×C₄	—	206	32	no	—
199	C₂×C₄	C₂×C₄×D₁₀₃	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₁₀₃	1	4	yes	—
201	C₂×C₄	C₂×C₄×D₁₀₃	1	16	yes	complemented
202	C₂×C₄	#13110105	1	8	yes	direct product factor
203	C₂×C₄	C₂³×D₁₀₃	1	4	yes	—
204	C₂×C₄	C₂³×D₁₀₃	1	4	yes	—
205	C₂×C₄	C₂³×D₁₀₃	1	2	yes	—
206	C₂×C₄	C₂³×D₁₀₃	1	2	yes	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	103	8	no	—
210	C₂²	—	206	8	no	—
211	C₂²	—	206	16	no	—
212	C₂²	C₂²×C₄×D₁₀₃	1	4	yes	—
213	C₂²	C₂²×C₄×D₁₀₃	1	2	yes	—
214	C₂²	C₂×#13110105	1	4	yes	—
215	C₂²	C₂⁴×D₁₀₃	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	103	8	no	—
218	C₄	—	206	8	no	—
219	C₄	C₂²×C₄×D₁₀₃	1	4	yes	—
220	C₄	C₂²×C₄×D₁₀₃	1	4	yes	—
221	C₄	C₂×#13110105	1	8	yes	direct product factor
222	C₂	—	2	4	no	—
223	C₂	—	206	8	no	—
224	C₂	C₄×#13110105	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₁₀₃	1	1	yes	characteristic
226	C₂	C₂²×#13110105	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₀₃
commutator	C₂×C₁₀₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₀₃
Fitting	C₂²×C₄²×C₁₀₃	C₂
socle	C₂³×C₁₀₃	C₂⁴
prime radical	C₂²×#13110105	C₂
cosocle	C₂²×C₁₀₃	C₂⁵
hypercenter	C₂²×C₄²	D₁₀₃
hypocenter	C₁₀₃	C₂×C₄×#41
LCS	G ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#13110105	C₂	direct (N × H)
3. C₂×C₄×Dic₂₀₆	C₂	semidirect
4. C₂×C₄×D₄₁₂	C₂	semidirect
5. C₂×C₄×C₂²⋊D₁₀₃	C₂	semidirect
6. C₂×#15692980	C₂	semidirect
7. C₂×#15692978	C₂	semidirect
8. C₂×C₄²×D₁₀₃	C₂	semidirect
9. C₂²×C₄²×C₁₀₃	C₂	semidirect
10. C₂×#13110105	C₄	direct (N × H)
17. C₄×Dic₂₀₆	C₂²	semidirect
18. C₄×D₄₁₂	C₂²	semidirect
19. C₄×C₂²⋊D₁₀₃	C₂²	semidirect
20. #15692980	C₂²	semidirect
21. #15692978	C₂²	semidirect
28. C₄²×D₁₀₃	C₂²	semidirect
34. C₂×C₄²×C₁₀₃	C₂²	semidirect
36. #13110105	C₂×C₄	direct (N × H)
37. C₂×Dic₂₀₆	C₂×C₄	semidirect
39. C₂×D₄₁₂	C₂×C₄	semidirect
41. C₂×C₂²⋊D₁₀₃	C₂×C₄	semidirect
46. C₂×C₄×D₁₀₃	C₂×C₄	semidirect
57. C₂²×C₄×C₁₀₃	C₂×C₄	semidirect
78. C₂×C₄×C₁₀₃	C₂²×C₄	semidirect
79. C₂×C₄×C₁₀₃	#41	semidirect
84. C₂²⋊D₁₀₃	C₂²×C₄	semidirect
89. D₄₁₂	C₂²×C₄	semidirect
91. C₄×D₁₀₃	C₂²×C₄	semidirect
92. Dic₂₀₆	C₂²×C₄	semidirect
105. C₄×C₁₀₃	C₂×#41	semidirect
109. C₂×C₁₀₃	C₄×#41	semidirect
114. C₁₀₃	C₂×C₄×#41	semidirect
123. C₂²×C₄²	D₁₀₃	semidirect
148. C₂×C₄²	C₂×D₁₀₃	semidirect
165. C₂²×C₄	C₄×D₁₀₃	semidirect
201. C₂×C₄	C₂×C₄×D₁₀₃	semidirect
202. C₂×C₄	#13110105	direct (N × H)
221. C₄	C₂×#13110105	direct (N × H)
224. C₂	C₄×#13110105	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×#13110105	2
C₄×#13110105	2
C₂×C₄×Dic₂₀₆	2
C₂×C₄×D₄₁₂	2
C₂×C₄×C₂²⋊D₁₀₃	2
C₂×#15692980	2
C₂×#15692978	2
C₂×C₄²×D₁₀₃	2
C₂²×C₄²×C₁₀₃	2
C₂×C₄×#41	103

Subgroups — C2×C4×#13110105 #27208324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13110105 #27208324