Subgroups — C₂×C₄×Dic₁₃₆ #17187600

Order 4352.

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×Dic₁₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₆₈	C₂	1	2	yes	maximal
3	C₄×Dic₁₃₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762635	C₂	1	1	yes	maximal, characteristic
5	C₂×#11762614	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₆₈	C₂²	1	2	yes	—
8	C₂×C₄×Dic₃₄	—	2	2	no	—
9	C₂×Dic₁₃₆	C₄	1	8	yes	direct product factor
10	C₂×Dic₁₃₆	C₂²	1	4	yes	—
11	C₄×Dic₆₈	—	2	4	no	—
12	C₄×Dic₆₈	C₂²	1	8	yes	—
13	C₂×#10587489	C₂²	1	2	yes	—
14	C₂×#10587487	—	2	2	no	—
15	#11762635	C₂²	1	4	yes	—
16	#11762614	C₂²	1	8	yes	—
17	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₁₇	C₂²	1	4	yes	—
21	C₂²×Dic₃₄	—	2	2	no	—
22	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₇	—	2	2	no	—
26	C₂×#63225	—	4	2	no	—
27	C₂×C₄×Dic₁₇	—	4	2	no	—
28	C₂×Dic₆₈	—	2	2	no	—
29	C₂×Dic₆₈	—	2	4	no	—
30	C₂×Dic₆₈	C₂×C₄	1	8	yes	—
31	C₂×Dic₆₈	C₂³	1	4	yes	—
32	C₄×Dic₃₄	—	2	8	no	—
33	C₄×Dic₃₄	—	4	4	no	—
34	C₂×C₁₆×C₁₇	—	2	2	no	—
35	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
36	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
37	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
38	C₄×C₈×C₁₇	D₄	1	2	yes	—
39	C₄×C₈×C₁₇	C₂³	1	2	yes	—
40	#10587489	—	2	2	no	—
41	#10587489	C₂³	1	4	yes	—
42	#10587487	—	2	8	no	—
43	Dic₁₃₆	C₂×C₄	1	16	yes	direct product factor
44	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	—	4	2	no	—
48	C₂²×Dic₁₇	—	4	2	no	—
49	C₂×Dic₃₄	—	2	8	no	—
50	C₂×Dic₃₄	—	2	4	no	—
51	C₂×Dic₃₄	—	4	2	no	—
52	C₂×Dic₃₄	—	4	4	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	#41	1	2	yes	—
58	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
59	C₄²×C₁₇	D₈	1	2	yes	—
60	C₄²×C₁₇	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₁₇	—	2	4	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	#63225	—	4	8	no	—
64	C₄×Dic₁₇	—	4	8	no	—
65	Dic₆₈	—	2	8	no	—
66	Dic₆₈	—	2	4	no	—
67	Dic₆₈	C₂²×C₄	1	8	yes	—
68	C₁₆×C₁₇	—	2	2	no	—
69	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
70	C₂×C₄×Q₃₂	—	17	1	no	maximal, Sylow
71	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	Q₃₂	1	4	yes	complemented
74	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₁₇	#135	1	2	yes	—
79	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₁₇	—	4	4	no	—
81	C₂×Dic₁₇	—	4	8	no	—
82	C₂×Dic₁₇	—	4	4	no	—
83	C₂×Dic₁₇	—	8	4	no	—
84	Dic₃₄	—	2	8	no	—
85	Dic₃₄	—	4	4	no	—
86	Dic₃₄	—	4	8	no	—
87	C₈×C₁₇	—	2	2	no	—
88	C₈×C₁₇	C₄×D₄	1	2	yes	—
89	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
91	C₂²×Q₃₂	—	17	1	no	—
92	C₂×C₄×Q₁₆	—	17	2	no	—
93	C₄×Q₃₂	—	17	8	no	—
94	C₂×#366	—	17	1	no	—
95	C₂×#358	—	17	2	no	—
96	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
97	C₂²×C₁₇	C₄×D₈	1	2	yes	—
98	C₂²×C₁₇	C₂×Q₃₂	1	2	yes	—
99	C₂²×C₁₇	#508	1	2	yes	—
100	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
101	C₄×C₁₇	—	2	2	no	—
102	C₄×C₁₇	C₄×D₈	1	2	yes	—
103	C₄×C₁₇	C₂×Q₃₂	1	4	yes	complemented
104	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
105	C₄×C₁₇	C₂×#135	1	1	yes	characteristic
106	Dic₁₇	—	4	8	no	—
107	Dic₁₇	—	8	4	no	—
108	C₂²×Q₁₆	—	17	2	no	—
109	C₂×C₄×Q₈	—	34	2	no	—
110	C₂×Q₃₂	—	17	8	no	—
111	C₂×Q₃₂	—	17	4	no	—
112	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
113	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
114	C₄×Q₁₆	—	17	8	no	—
115	C₄×Q₁₆	—	34	4	no	—
116	C₂×#107	—	17	2	no	—
117	C₂×#103	—	34	2	no	—
118	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
119	#366	—	17	4	no	—
120	#358	—	17	8	no	—
121	C₄×C₁₆	C₂×D₁₇	1	4	yes	—
122	C₂×C₁₇	C₄×Q₃₂	1	4	yes	complemented
123	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
124	C₂×C₁₇	C₂²×Q₃₂	1	1	yes	characteristic
125	C₂×C₁₇	C₂×#508	1	1	yes	characteristic
126	C₂²×Q₈	—	34	2	no	—
127	C₂×Q₁₆	—	17	8	no	—
128	C₂×Q₁₆	—	17	4	no	—
129	C₂×Q₁₆	—	34	2	no	—
130	C₂×Q₁₆	—	34	4	no	—
131	C₂²×C₈	D₆₈	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
133	C₄×Q₈	—	34	8	no	—
134	C₄×Q₈	—	68	4	no	—
135	C₂×C₄⋊C₄	—	34	2	no	—
136	C₂×C₄⋊C₄	—	68	2	no	—
137	C₂×C₄²	—	68	2	no	—
138	C₂×C₄²	D₆₈	1	1	yes	characteristic
139	Q₃₂	—	17	16	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	C₄×D₁₇	1	4	yes	—
142	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
143	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
144	#107	—	17	4	no	—
145	#107	—	34	2	no	—
146	#103	—	34	8	no	—
147	C₄×C₈	D₆₈	1	2	yes	—
148	C₄×C₈	C₂²×D₁₇	1	2	yes	—
149	C₁₇	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
150	C₂×Q₈	—	34	4	no	—
151	C₂×Q₈	—	34	8	no	—
152	C₂×Q₈	—	68	2	no	—
153	C₂×Q₈	—	68	4	no	—
154	C₂²×C₄	—	68	2	no	—
155	C₂²×C₄	—	68	2	no	—
156	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
157	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
159	Q₁₆	—	17	8	no	—
160	Q₁₆	—	34	4	no	—
161	Q₁₆	—	34	8	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
164	C₂×C₈	C₂×D₆₈	1	2	yes	—
165	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
166	C₂×C₈	#63252	1	2	yes	—
167	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
168	C₄⋊C₄	—	34	4	no	—
169	C₄⋊C₄	—	68	2	no	—
170	C₄⋊C₄	—	68	8	no	—
171	C₄²	—	68	8	no	—
172	C₄²	D₁₃₆	1	2	yes	—
173	C₄²	C₂×D₆₈	1	2	yes	—
174	C₁₆	—	2	2	no	—
175	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
176	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
177	Q₈	—	34	8	no	—
178	Q₈	—	68	4	no	—
179	Q₈	—	68	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	68	4	no	—
182	C₂×C₄	—	68	8	no	—
183	C₂×C₄	—	68	4	no	—
184	C₂×C₄	—	136	4	no	—
185	C₂×C₄	Dic₁₃₆	1	4	yes	direct product factor
186	C₂×C₄	C₄×D₆₈	1	2	yes	—
187	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
188	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
189	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
190	C₂×C₄	#10587590	1	2	yes	—
191	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₄×D₆₈	1	2	yes	—
194	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
195	C₈	C₂×#63252	1	1	yes	characteristic
196	C₂²	C₄×D₁₃₆	1	2	yes	—
197	C₂²	C₂×Dic₁₃₆	1	2	yes	—
198	C₂²	#11763241	1	2	yes	—
199	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	68	8	no	—
202	C₄	—	136	4	no	—
203	C₄	C₄×D₁₃₆	1	2	yes	—
204	C₄	C₂×Dic₁₃₆	1	4	yes	direct product factor
205	C₄	C₂×C₄×D₆₈	1	1	yes	characteristic
206	C₄	C₂×#10587590	1	1	yes	characteristic
207	C₂	C₄×Dic₁₃₆	1	4	yes	direct product factor
208	C₂	C₂×C₄×D₁₃₆	1	1	yes	characteristic
209	C₂	C₂²×Dic₁₃₆	1	1	yes	characteristic
210	C₂	C₂×#11763241	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₃₆
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
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×Dic₁₃₆	C₂	direct (N × H)
9. C₂×Dic₁₃₆	C₄	direct (N × H)
43. Dic₁₃₆	C₂×C₄	direct (N × H)
73. C₂×C₄×C₁₇	Q₃₂	semidirect
103. C₄×C₁₇	C₂×Q₃₂	semidirect
122. C₂×C₁₇	C₄×Q₃₂	semidirect
149. C₁₇	C₂×C₄×Q₃₂	semidirect
185. C₂×C₄	Dic₁₃₆	direct (N × H)
204. C₄	C₂×Dic₁₃₆	direct (N × H)
207. C₂	C₄×Dic₁₃₆	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×Dic₁₃₆	2
C₂×C₄×Dic₆₈	2
C₄×Dic₁₃₆	2
C₂×#11762635	2
C₂×#11762614	2
C₂×C₄×C₁₆×C₁₇	2
C₂×C₄×Q₃₂	17

Subgroups — C2×C4×Dic136 #17187600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₁₃₆ #17187600