Subgroups — C₂×C₄×C₄⋊#10587739 #31797219

Order 17536.

928 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×C₄⋊#10587739	C₂	1	2	yes	maximal
2	C₂²×C₄×#10587739	C₂	1	2	yes	maximal
3	C₄×C₄⋊#10587739	C₂	1	8	yes	maximal, direct product factor
4	C₂×(C₂×#10587739)⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10587739	C₂²	1	4	yes	—
7	C₂×C₄⋊#10587739	C₄	1	16	yes	direct product factor
8	C₂×C₄⋊#10587739	C₂²	1	8	yes	—
9	C₂×C₄×#10587739	—	2	8	no	—
10	C₂×C₄×#10587739	C₂²	1	8	yes	—
11	(C₂×#10587739)⋊C₄	C₂²	1	8	yes	—
12	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
17	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
18	C₄²×D₁₃₇	C₄	1	4	yes	complemented
19	C₄²×D₁₃₇	C₂²	1	4	yes	—
20	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic, complemented
21	C₂²×#10587739	—	2	8	no	—
22	C₂²×#10587739	—	2	8	no	—
23	C₂²×#10587739	C₂×C₄	1	16	yes	—
24	C₂²×#10587739	C₂³	1	8	yes	—
25	C₄⋊#10587739	C₂×C₄	1	32	yes	direct product factor
26	C₄×#10587739	—	2	16	no	—
27	C₂³×D₁₃₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₃₇	—	2	4	no	—
32	C₂×C₄×D₁₃₇	C₂×C₄	1	8	yes	—
33	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₃₇	D₄	1	4	yes	—
37	C₂×C₄×D₁₃₇	Q₈	1	4	yes	—
38	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
39	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
40	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
41	C₄×Dic₁₃₇	C₂×C₄	1	4	yes	complemented
42	C₄×Dic₁₃₇	C₂×C₄	1	4	yes	complemented
43	C₄×Dic₁₃₇	C₂×C₄	1	4	yes	complemented
44	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₁₃₇	C₂×C₄	1	4	yes	complemented
48	C₂×#10587739	—	2	32	no	—
49	C₂×#10587739	—	2	8	no	—
50	C₂×#10587739	—	2	8	no	—
51	C₂×#10587739	C₂²×C₄	1	16	yes	—
52	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×D₁₃₇	C₂²×C₄	1	4	yes	—
54	C₂²×D₁₃₇	C₂×D₄	1	2	yes	—
55	C₂²×D₁₃₇	C₂×Q₈	1	2	yes	—
56	C₂²×D₁₃₇	#41	1	4	yes	—
57	C₂²×D₁₃₇	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₁₃₇	—	2	2	no	—
59	C₂×Dic₁₃₇	C₄²	1	4	yes	complemented
60	C₂×Dic₁₃₇	C₄⋊C₄	1	4	yes	complemented
61	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
62	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
63	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
64	C₄×D₁₃₇	—	2	4	no	—
65	C₄×D₁₃₇	—	2	4	no	—
66	C₄×D₁₃₇	C₄²	1	8	yes	complemented
67	C₄×D₁₃₇	C₄⋊C₄	1	8	yes	complemented
68	C₄×D₁₃₇	C₂²×C₄	1	8	yes	—
69	C₄×D₁₃₇	C₂×D₄	1	4	yes	—
70	C₄×D₁₃₇	C₂×Q₈	1	4	yes	—
71	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄×C₁₃₇	C₄²	1	4	yes	complemented
74	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	complemented
75	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₁₃₇	C₂×C₄²	1	2	yes	—
79	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂²×C₁₃₇	#117	1	2	yes	—
81	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
82	C₂×D₁₃₇	C₂×C₄²	1	2	yes	—
83	C₂×D₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×D₁₃₇	#117	1	2	yes	—
85	C₂×D₁₃₇	C₄×D₄	1	8	yes	—
86	C₂×D₁₃₇	C₄×Q₈	1	8	yes	—
87	C₂×D₁₃₇	C₂³×C₄	1	2	yes	—
88	C₂×D₁₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₁₃₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×D₁₃₇	C₂×#41	1	2	yes	—
91	#10587739	—	2	16	no	—
92	C₄×C₁₃₇	—	2	2	no	—
93	C₄×C₁₃₇	C₂×C₄²	1	4	yes	complemented
94	C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	complemented
95	Dic₁₃₇	—	2	2	no	—
96	Dic₁₃₇	C₂×C₄²	1	4	yes	complemented
97	Dic₁₃₇	C₂×C₄⋊C₄	1	4	yes	complemented
98	C₂×C₁₃₇	C₄×C₄⋊C₄	1	4	yes	complemented
99	C₂×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
100	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
102	D₁₃₇	C₄×C₄⋊C₄	1	4	yes	complemented
103	D₁₃₇	C₂×C₄×D₄	1	2	yes	—
104	D₁₃₇	C₂×C₄×Q₈	1	2	yes	—
105	C₁₃₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
106	C₂×C₄×C₄⋊C₄	—	137	1	no	maximal, Sylow
107	C₂²×C₄⋊C₄	—	137	2	no	—
108	C₂²×C₄²	—	137	2	no	—
109	C₂²×C₄²	—	137	1	no	—
110	C₄×C₄⋊C₄	—	137	8	no	—
111	C₂×#95	—	137	2	no	—
112	C₂³×C₄	—	137	4	no	—
113	C₂³×C₄	—	137	1	no	—
114	C₂³×C₄	—	137	1	no	—
115	C₂³×C₄	—	137	1	no	—
116	C₂×C₄⋊C₄	—	137	8	no	—
117	C₂×C₄⋊C₄	—	137	16	no	—
118	C₂×C₄²	—	274	8	no	—
119	C₂×C₄²	—	137	8	no	—
120	C₂×C₄²	—	137	1	no	—
121	C₂×C₄²	—	137	1	no	—
122	C₂×C₄²	—	137	1	no	—
123	C₂×C₄²	—	137	4	no	—
124	C₂×C₄²	—	137	4	no	—
125	C₂×C₄²	#10587739	1	1	yes	characteristic, complemented
126	#95	—	137	8	no	—
127	C₂⁴	—	137	1	no	—
128	C₂²×C₄	—	274	4	no	—
129	C₂²×C₄	—	274	8	no	—
130	C₂²×C₄	—	274	8	no	—
131	C₂²×C₄	—	137	8	no	—
132	C₂²×C₄	—	137	16	no	—
133	C₂²×C₄	—	137	1	no	—
134	C₂²×C₄	—	137	2	no	—
135	C₂²×C₄	—	137	2	no	—
136	C₂²×C₄	—	137	1	no	—
137	C₂²×C₄	—	137	1	no	—
138	C₂²×C₄	—	137	8	no	—
139	C₂²×C₄	—	137	4	no	—
140	C₂²×C₄	—	137	4	no	—
141	C₂²×C₄	—	137	2	no	—
142	C₂²×C₄	—	137	2	no	—
143	C₂²×C₄	—	137	2	no	—
144	C₂²×C₄	—	137	2	no	—
145	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
146	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
147	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
148	C₄⋊C₄	—	137	32	no	—
149	C₄²	—	274	16	no	—
150	C₄²	—	137	4	no	—
151	C₄²	—	137	4	no	—
152	C₄²	—	137	4	no	—
153	C₄²	C₂×#10587739	1	4	yes	complemented
154	C₂³	—	137	4	no	—
155	C₂³	—	137	2	no	—
156	C₂³	—	137	2	no	—
157	C₂³	—	137	4	no	—
158	C₂³	—	137	1	no	—
159	C₂³	—	137	1	no	—
160	C₂³	C₂²×#10587739	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	274	4	no	—
163	C₂×C₄	—	274	4	no	—
164	C₂×C₄	—	274	8	no	—
165	C₂×C₄	—	274	8	no	—
166	C₂×C₄	—	274	32	no	—
167	C₂×C₄	—	274	2	no	—
168	C₂×C₄	—	137	16	no	—
169	C₂×C₄	—	137	2	no	—
170	C₂×C₄	—	137	4	no	—
171	C₂×C₄	—	137	4	no	—
172	C₂×C₄	—	137	2	no	—
173	C₂×C₄	—	137	2	no	—
174	C₂×C₄	—	137	8	no	—
175	C₂×C₄	—	137	8	no	—
176	C₂×C₄	—	137	4	no	—
177	C₂×C₄	—	137	8	no	—
178	C₂×C₄	—	137	4	no	—
179	C₂×C₄	C₄×#10587739	1	4	yes	complemented
180	C₂×C₄	C₄⋊#10587739	1	4	yes	direct product factor
181	C₂×C₄	C₂²×#10587739	1	2	yes	—
182	C₂×C₄	C₂²×#10587739	1	2	yes	—
183	C₂×C₄	C₂²×#10587739	1	2	yes	—
184	C₂²	—	137	2	no	—
185	C₂²	—	137	2	no	—
186	C₂²	—	137	1	no	—
187	C₂²	—	137	2	no	—
188	C₂²	—	137	1	no	—
189	C₂²	—	137	2	no	—
190	C₂²	—	137	2	no	—
191	C₂²	—	137	8	no	—
192	C₂²	—	137	8	no	—
193	C₂²	C₂×C₄×#10587739	1	2	yes	—
194	C₂²	C₂×C₄⋊#10587739	1	2	yes	—
195	C₂²	#17984167	1	2	yes	—
196	C₂²	C₂³×#10587739	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	274	16	no	—
199	C₄	—	274	2	no	—
200	C₄	—	137	4	no	—
201	C₄	—	137	4	no	—
202	C₄	C₂×C₄×#10587739	1	4	yes	complemented
203	C₄	C₂×C₄⋊#10587739	1	4	yes	direct product factor
204	C₂	—	137	2	no	—
205	C₂	—	137	2	no	—
206	C₂	—	137	4	no	—
207	C₂	C₄×C₄⋊#10587739	1	4	yes	direct product factor
208	C₂	C₂²×C₄×#10587739	1	1	yes	characteristic
209	C₂	C₂²×C₄⋊#10587739	1	1	yes	characteristic
210	C₂	C₂×#17984167	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#10587739
commutator	C₂×C₁₃₇	C₂²×C₄²
Frattini	C₂²	C₂³×#10587739
Fitting	C₂×C₄²×C₁₃₇	C₄
socle	C₂³×C₁₃₇	C₂²×C₄
prime radical	C₂³×D₁₃₇	C₂³
cosocle	C₂²×D₁₃₇	C₂⁴
hypercenter	C₂×C₄²	#10587739
hypocenter	C₁₃₇	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄×C₄⋊#10587739	C₂	direct (N × H)
7. C₂×C₄⋊#10587739	C₄	direct (N × H)
15. C₂×C₄×Dic₁₃₇	C₄	semidirect
16. C₂×C₄×Dic₁₃₇	C₄	semidirect
17. C₂×C₄×Dic₁₃₇	C₄	semidirect
18. C₄²×D₁₃₇	C₄	semidirect
20. C₂×C₄²×C₁₃₇	C₄	semidirect
25. C₄⋊#10587739	C₂×C₄	direct (N × H)
41. C₄×Dic₁₃₇	C₂×C₄	semidirect
42. C₄×Dic₁₃₇	C₂×C₄	semidirect
43. C₄×Dic₁₃₇	C₂×C₄	semidirect
47. C₄²×C₁₃₇	C₂×C₄	semidirect
59. C₂×Dic₁₃₇	C₄²	semidirect
60. C₂×Dic₁₃₇	C₄⋊C₄	semidirect
66. C₄×D₁₃₇	C₄²	semidirect
67. C₄×D₁₃₇	C₄⋊C₄	semidirect
73. C₂×C₄×C₁₃₇	C₄²	semidirect
74. C₂×C₄×C₁₃₇	C₄⋊C₄	semidirect
93. C₄×C₁₃₇	C₂×C₄²	semidirect
94. C₄×C₁₃₇	C₂×C₄⋊C₄	semidirect
96. Dic₁₃₇	C₂×C₄²	semidirect
97. Dic₁₃₇	C₂×C₄⋊C₄	semidirect
98. C₂×C₁₃₇	C₄×C₄⋊C₄	semidirect
102. D₁₃₇	C₄×C₄⋊C₄	semidirect
105. C₁₃₇	C₂×C₄×C₄⋊C₄	semidirect
125. C₂×C₄²	#10587739	semidirect
153. C₄²	C₂×#10587739	semidirect
179. C₂×C₄	C₄×#10587739	semidirect
180. C₂×C₄	C₄⋊#10587739	direct (N × H)
202. C₄	C₂×C₄×#10587739	semidirect
203. C₄	C₂×C₄⋊#10587739	direct (N × H)
207. C₂	C₄×C₄⋊#10587739	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×C₄⋊#10587739	2
C₂²×C₄×#10587739	2
C₄×C₄⋊#10587739	2
C₂×(C₂×#10587739)⋊C₄	2
C₂×C₄²×D₁₃₇	2
C₂×C₄×C₄⋊C₄	137

Subgroups — C2×C4×C4⋊#10587739 #31797219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₄⋊#10587739 #31797219