Subgroups — C₂×C₄×#14523514 #30679142

Order 16768.

1728 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523514	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523514	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₅₂₄	C₂	1	3	yes	maximal, complemented
5	C₂×#16862208	C₂	1	3	yes	maximal, complemented
6	C₂×#16862204	C₂	1	3	yes	maximal, complemented
7	C₂×C₄²×D₁₃₁	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523514	C₄	1	16	yes	direct product factor
10	C₂×#14523514	C₂²	1	8	yes	—
11	C₂²×D₅₂₄	C₂²	1	3	yes	—
12	C₂²×C₄×D₁₃₁	C₂²	1	3	yes	—
13	C₂²×C₄×D₁₃₁	C₂²	1	3	yes	—
14	C₂²×C₄×D₁₃₁	C₂²	1	3	yes	—
15	Q₈×Dic₁₃₁	C₂²	1	8	yes	complemented
16	C₄×D₅₂₄	C₂²	1	24	yes	complemented
17	#16862208	C₂²	1	24	yes	complemented
18	#16862204	C₂²	1	24	yes	complemented
19	C₂×C₄⋊Dic₁₃₁	C₂²	1	3	yes	—
20	C₂×(C₂×D₁₃₁)⋊C₄	C₂²	1	6	yes	—
21	C₂×C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃₁	C₂²	1	3	yes	—
23	C₂×C₄×Dic₁₃₁	C₂²	1	3	yes	—
24	C₄²×D₁₃₁	C₂²	1	24	yes	complemented
25	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₃₁	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	3	yes	—
28	C₂×C₄²×C₁₃₁	C₂²	1	3	yes	—
29	C₂³×D₁₃₁	C₂³	1	3	yes	—
30	#14523514	C₂×C₄	1	64	yes	direct product factor
31	C₂×D₅₂₄	C₂×C₄	1	24	yes	complemented
32	C₂×D₅₂₄	C₂³	1	12	yes	—
33	C₂²×Dic₁₃₁	C₂³	1	3	yes	—
34	C₂²×Dic₁₃₁	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₁₃₁	—	2	12	no	—
37	C₂×C₄×D₁₃₁	—	2	12	no	—
38	C₂×C₄×D₁₃₁	C₂×C₄	1	24	yes	complemented
39	C₂×C₄×D₁₃₁	C₂³	1	12	yes	—
40	C₂×C₄×D₁₃₁	C₂³	1	12	yes	—
41	C₂×C₄×D₁₃₁	C₂³	1	12	yes	—
42	C₄⋊Dic₁₃₁	C₂³	1	12	yes	—
43	(C₂×D₁₃₁)⋊C₄	C₂³	1	24	yes	—
44	C₄×Dic₁₃₁	C₂³	1	12	yes	—
45	C₄×Dic₁₃₁	C₂³	1	4	yes	—
46	C₄×Dic₁₃₁	C₂³	1	12	yes	—
47	C₂×Q₈×C₁₃₁	C₂×C₄	1	8	yes	complemented
48	C₂×Q₈×C₁₃₁	C₂³	1	4	yes	—
49	C₂²×C₄×C₁₃₁	C₂³	1	3	yes	—
50	C₂²×C₄×C₁₃₁	C₂³	1	3	yes	—
51	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₁₃₁	C₂³	1	12	yes	—
53	C₄²×C₁₃₁	C₂³	1	12	yes	—
54	C₂²×D₁₃₁	—	2	6	no	—
55	C₂²×D₁₃₁	—	2	6	no	—
56	C₂²×D₁₃₁	C₂²×C₄	1	12	yes	—
57	C₂²×D₁₃₁	C₂⁴	1	6	yes	—
58	D₅₂₄	C₂²×C₄	1	48	yes	complemented
59	C₂×Dic₁₃₁	—	2	6	no	—
60	C₂×Dic₁₃₁	C₂²×C₄	1	4	yes	—
61	C₂×Dic₁₃₁	#41	1	4	yes	complemented
62	C₂×Dic₁₃₁	C₂⁴	1	6	yes	—
63	C₂×Dic₁₃₁	C₂⁴	1	2	yes	—
64	C₂×Dic₁₃₁	C₂⁴	1	2	yes	—
65	C₄×D₁₃₁	—	2	24	no	—
66	C₄×D₁₃₁	—	2	24	no	—
67	C₄×D₁₃₁	C₂²×C₄	1	48	yes	complemented
68	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
69	Q₈×C₁₃₁	C₂²×C₄	1	16	yes	complemented
70	C₂×C₄×C₁₃₁	—	2	6	no	—
71	C₂×C₄×C₁₃₁	C₂²×C₄	1	12	yes	—
72	C₂×C₄×C₁₃₁	#41	1	4	yes	complemented
73	C₂×C₄×C₁₃₁	C₂⁴	1	6	yes	—
74	C₂×C₄×C₁₃₁	C₂⁴	1	6	yes	—
75	C₂×C₄×C₁₃₁	C₂⁴	1	2	yes	—
76	C₂²×C₁₃₁	C₂³×C₄	1	2	yes	—
77	C₂²×C₁₃₁	C₂×#41	1	2	yes	—
78	C₂²×C₁₃₁	C₂×#41	1	2	yes	—
79	C₂²×C₁₃₁	C₂⁵	1	1	yes	characteristic
80	C₂×D₁₃₁	—	2	24	no	—
81	C₂×D₁₃₁	—	2	6	no	—
82	C₂×D₁₃₁	—	2	6	no	—
83	C₂×D₁₃₁	C₂³×C₄	1	12	yes	—
84	C₄×C₁₃₁	—	2	6	no	—
85	C₄×C₁₃₁	C₂³×C₄	1	12	yes	—
86	C₄×C₁₃₁	C₂×#41	1	4	yes	complemented
87	Dic₁₃₁	—	2	6	no	—
88	Dic₁₃₁	C₂³×C₄	1	4	yes	—
89	Dic₁₃₁	C₂×#41	1	4	yes	complemented
90	C₂×C₁₃₁	C₄×#41	1	4	yes	complemented
91	C₂×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
92	C₂×C₁₃₁	C₂²×#41	1	1	yes	characteristic
93	C₂×C₁₃₁	C₂²×#41	1	1	yes	characteristic
94	D₁₃₁	—	2	12	no	—
95	C₁₃₁	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×#41	—	131	1	no	maximal, Sylow
97	C₂²×#41	—	131	1	no	—
98	C₄×#41	—	131	16	no	—
99	C₂×C₄×Q₈	—	131	1	no	—
100	C₂×C₄×Q₈	D₁₃₁	1	1	yes	characteristic, complemented
101	C₂×C₄×D₄	—	131	3	no	—
102	C₂×C₄×D₄	—	131	3	no	—
103	C₂×#117	—	131	3	no	—
104	C₂²×C₄²	—	131	3	no	—
105	C₂×#41	—	131	16	no	—
106	C₂×#41	—	131	8	no	—
107	C₂²×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
108	C₂²×D₄	—	131	3	no	—
109	C₂³×C₄	—	131	3	no	—
110	C₂³×C₄	—	131	3	no	—
111	C₂³×C₄	—	131	3	no	—
112	C₄×Q₈	—	131	8	no	—
113	C₄×Q₈	C₂×D₁₃₁	1	8	yes	complemented
114	C₄×D₄	—	131	24	no	—
115	C₄×D₄	—	131	24	no	—
116	#117	—	131	24	no	—
117	C₂×C₄⋊C₄	—	131	3	no	—
118	C₂×C₄⋊C₄	C₂×D₁₃₁	1	3	yes	—
119	C₂×C₂²⋊C₄	—	131	6	no	—
120	C₂×C₄²	—	131	3	no	—
121	C₂×C₄²	—	131	3	no	—
122	C₂×C₄²	—	131	1	no	—
123	C₂×C₄²	—	131	24	no	—
124	C₂×C₄²	C₂×D₁₃₁	1	3	yes	—
125	C₂⁴	—	131	3	no	—
126	#41	—	131	64	no	—
127	C₂×Q₈	C₄×D₁₃₁	1	8	yes	complemented
128	C₂×Q₈	C₂²×D₁₃₁	1	4	yes	—
129	C₂×D₄	—	131	24	no	—
130	C₂×D₄	—	131	12	no	—
131	C₂²×C₄	—	262	12	no	—
132	C₂²×C₄	—	262	12	no	—
133	C₂²×C₄	—	131	3	no	—
134	C₂²×C₄	—	131	12	no	—
135	C₂²×C₄	—	131	12	no	—
136	C₂²×C₄	—	131	1	no	—
137	C₂²×C₄	—	131	1	no	—
138	C₂²×C₄	—	131	24	no	—
139	C₂²×C₄	—	131	12	no	—
140	C₂²×C₄	C₂²×D₁₃₁	1	3	yes	—
141	C₂²×C₄	C₂²×D₁₃₁	1	3	yes	—
142	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
143	C₄⋊C₄	—	131	12	no	—
144	C₄⋊C₄	C₂²×D₁₃₁	1	12	yes	—
145	C₂²⋊C₄	—	131	24	no	—
146	C₄²	—	131	4	no	—
147	C₄²	—	131	12	no	—
148	C₄²	—	131	12	no	—
149	C₄²	C₂²×D₁₃₁	1	12	yes	—
150	C₂³	—	262	6	no	—
151	C₂³	—	262	6	no	—
152	C₂³	—	131	12	no	—
153	C₂³	—	131	6	no	—
154	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
155	Q₈	C₂×C₄×D₁₃₁	1	16	yes	complemented
156	D₄	—	131	48	no	—
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	262	24	no	—
159	C₂×C₄	—	262	6	no	—
160	C₂×C₄	—	262	24	no	—
161	C₂×C₄	—	131	6	no	—
162	C₂×C₄	—	131	4	no	—
163	C₂×C₄	—	131	4	no	—
164	C₂×C₄	—	131	2	no	—
165	C₂×C₄	—	131	2	no	—
166	C₂×C₄	—	131	48	no	—
167	C₂×C₄	C₂×C₄×D₁₃₁	1	12	yes	—
168	C₂×C₄	#14523514	1	4	yes	direct product factor
169	C₂×C₄	C₂³×D₁₃₁	1	6	yes	—
170	C₂×C₄	C₂³×D₁₃₁	1	6	yes	—
171	C₂×C₄	C₂³×D₁₃₁	1	2	yes	—
172	C₂²	—	262	6	no	—
173	C₂²	—	262	6	no	—
174	C₂²	—	262	24	no	—
175	C₂²	—	131	12	no	—
176	C₂²	C₂²×C₄×D₁₃₁	1	2	yes	—
177	C₂²	#41×D₁₃₁	1	2	yes	—
178	C₂²	C₂×#14523514	1	2	yes	—
179	C₂²	C₂⁴×D₁₃₁	1	1	yes	characteristic
180	C₄	—	2	6	no	—
181	C₄	—	262	6	no	—
182	C₄	—	131	4	no	—
183	C₄	—	131	4	no	—
184	C₄	C₂²×C₄×D₁₃₁	1	12	yes	—
185	C₄	C₂×#14523514	1	4	yes	direct product factor
186	C₂	—	262	12	no	—
187	C₂	C₄×#14523514	1	4	yes	direct product factor
188	C₂	C₂³×C₄×D₁₃₁	1	1	yes	characteristic
189	C₂	C₂×#41×D₁₃₁	1	1	yes	characteristic
190	C₂	C₂²×#14523514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₁
commutator	C₂×C₁₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃₁
Fitting	C₂×C₄×Q₈×C₁₃₁	C₂
socle	C₂³×C₁₃₁	C₂⁴
prime radical	C₂²×#14523514	C₂
cosocle	C₂²×C₁₃₁	C₂⁵
hypercenter	C₂×C₄×Q₈	D₁₃₁
hypocenter	C₁₃₁	C₂×C₄×#41
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×#14523514	C₂	direct (N × H)
3. C₂×Q₈×Dic₁₃₁	C₂	semidirect
4. C₂×C₄×D₅₂₄	C₂	semidirect
5. C₂×#16862208	C₂	semidirect
6. C₂×#16862204	C₂	semidirect
7. C₂×C₄²×D₁₃₁	C₂	semidirect
8. C₂×C₄×Q₈×C₁₃₁	C₂	semidirect
9. C₂×#14523514	C₄	direct (N × H)
15. Q₈×Dic₁₃₁	C₂²	semidirect
16. C₄×D₅₂₄	C₂²	semidirect
17. #16862208	C₂²	semidirect
18. #16862204	C₂²	semidirect
24. C₄²×D₁₃₁	C₂²	semidirect
26. C₄×Q₈×C₁₃₁	C₂²	semidirect
30. #14523514	C₂×C₄	direct (N × H)
31. C₂×D₅₂₄	C₂×C₄	semidirect
38. C₂×C₄×D₁₃₁	C₂×C₄	semidirect
47. C₂×Q₈×C₁₃₁	C₂×C₄	semidirect
58. D₅₂₄	C₂²×C₄	semidirect
61. C₂×Dic₁₃₁	#41	semidirect
67. C₄×D₁₃₁	C₂²×C₄	semidirect
69. Q₈×C₁₃₁	C₂²×C₄	semidirect
72. C₂×C₄×C₁₃₁	#41	semidirect
86. C₄×C₁₃₁	C₂×#41	semidirect
89. Dic₁₃₁	C₂×#41	semidirect
90. C₂×C₁₃₁	C₄×#41	semidirect
95. C₁₃₁	C₂×C₄×#41	semidirect
100. C₂×C₄×Q₈	D₁₃₁	semidirect
113. C₄×Q₈	C₂×D₁₃₁	semidirect
127. C₂×Q₈	C₄×D₁₃₁	semidirect
155. Q₈	C₂×C₄×D₁₃₁	semidirect
168. C₂×C₄	#14523514	direct (N × H)
185. C₄	C₂×#14523514	direct (N × H)
187. C₂	C₄×#14523514	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×#14523514	2
C₄×#14523514	2
C₂×Q₈×Dic₁₃₁	2
C₂×C₄×D₅₂₄	2
C₂×#16862208	2
C₂×#16862204	2
C₂×C₄²×D₁₃₁	2
C₂×C₄×Q₈×C₁₃₁	2
C₂×C₄×#41	131

Subgroups — C2×C4×#14523514 #30679142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14523514 #30679142