Subgroups — C₂²×C₄×D₁₃₆ #17898327

Order 4352.

3694 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₆₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₁₃₆	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#10587489	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#10587492	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₆₈	C₂²	1	2	yes	—
8	C₂³×C₄×D₁₇	—	2	2	no	—
9	C₂²×D₁₃₆	C₄	1	16	yes	direct product factor
10	C₂²×D₁₃₆	C₂²	1	12	yes	—
11	C₂×C₄×D₆₈	—	2	12	no	—
12	C₂×C₄×D₆₈	C₂²	1	24	yes	complemented
13	C₂²×C₄⋊Dic₁₇	C₂²	1	2	yes	—
14	C₂²×#63227	—	2	2	no	—
15	C₄×D₁₃₆	C₂²	1	64	yes	direct product factor
16	C₂×#10587489	C₂²	1	12	yes	complemented
17	C₂×#10587492	C₂²	1	24	yes	complemented
18	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₇	C₂²	1	12	yes	complemented
22	C₂⁴×D₁₇	—	2	2	no	—
23	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂³×Dic₁₇	—	2	2	no	—
27	C₂²×D₆₈	—	2	6	no	—
28	C₂²×D₆₈	—	2	8	no	—
29	C₂²×D₆₈	C₂×C₄	1	16	yes	complemented
30	C₂²×D₆₈	C₂³	1	12	yes	—
31	C₂²×C₄×D₁₇	—	2	24	no	—
32	C₂²×C₄×D₁₇	—	4	16	no	—
33	C₂²×C₈×C₁₇	—	2	4	no	—
34	C₂²×C₈×C₁₇	C₂×C₄	1	8	yes	complemented
35	C₂²×C₈×C₁₇	C₂³	1	6	yes	—
36	C₂²×C₈×C₁₇	C₂³	1	6	yes	—
37	C₂×C₄²×C₁₇	D₄	1	6	yes	—
38	C₂×C₄²×C₁₇	C₂³	1	6	yes	—
39	C₂×#63227	—	2	24	no	—
40	C₂×C₄⋊Dic₁₇	—	2	6	no	—
41	C₂×C₄⋊Dic₁₇	C₂³	1	12	yes	—
42	C₂×D₁₃₆	C₂×C₄	1	96	yes	direct product factor
43	C₂×D₁₃₆	C₂³	1	16	yes	—
44	C₄×D₆₈	—	2	48	no	—
45	C₄×D₆₈	C₂³	1	32	yes	complemented
46	C₄×C₈×C₁₇	C₂³	1	16	yes	complemented
47	#10587492	C₂³	1	32	yes	complemented
48	#10587489	C₂³	1	16	yes	complemented
49	C₂⁴×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂³×D₁₇	—	2	16	no	—
51	C₂³×D₁₇	—	2	12	no	—
52	C₂³×D₁₇	—	4	8	no	—
53	C₂³×D₁₇	—	4	8	no	—
54	C₂²×C₄×C₁₇	—	2	4	no	—
55	C₂²×C₄×C₁₇	D₈	1	8	yes	complemented
56	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₁₇	C₂×D₄	1	6	yes	—
58	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₁₇	C₂×D₄	1	6	yes	—
60	C₂²×C₄×C₁₇	#41	1	4	yes	—
61	C₂²×C₄×C₁₇	C₂⁴	1	3	yes	—
62	C₂²×Dic₁₇	—	2	12	no	—
63	C₂²×Dic₁₇	—	4	8	no	—
64	C₂×D₆₈	—	2	48	no	—
65	C₂×D₆₈	—	2	12	no	—
66	C₂×D₆₈	—	2	24	no	—
67	C₂×D₆₈	C₂²×C₄	1	48	yes	complemented
68	C₂×D₆₈	C₂⁴	1	8	yes	—
69	C₂×C₄×D₁₇	—	2	32	no	—
70	C₂×C₄×D₁₇	—	4	96	no	—
71	C₂×C₈×C₁₇	—	2	12	no	—
72	C₂×C₈×C₁₇	C₂²×C₄	1	24	yes	complemented
73	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
74	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
75	C₄²×C₁₇	C₂×D₄	1	12	yes	—
76	C₄²×C₁₇	C₂⁴	1	4	yes	—
77	#63227	—	2	32	no	—
78	C₄⋊Dic₁₇	—	2	12	no	—
79	C₄⋊Dic₁₇	C₂⁴	1	8	yes	—
80	D₁₃₆	C₂²×C₄	1	64	yes	direct product factor
81	C₂²×C₄×D₈	—	17	1	no	maximal, Sylow
82	C₂³×C₁₇	C₄×D₄	1	4	yes	—
83	C₂³×C₁₇	C₂×D₈	1	4	yes	—
84	C₂³×C₁₇	#135	1	4	yes	—
85	C₂³×C₁₇	C₂²×D₄	1	3	yes	—
86	C₂²×D₁₇	—	2	48	no	—
87	C₂²×D₁₇	—	2	8	no	—
88	C₂²×D₁₇	—	4	24	no	—
89	C₂²×D₁₇	—	4	24	no	—
90	C₂²×D₁₇	—	4	64	no	—
91	C₂×C₄×C₁₇	—	2	12	no	—
92	C₂×C₄×C₁₇	C₄×D₄	1	12	yes	—
93	C₂×C₄×C₁₇	C₂×D₈	1	24	yes	complemented
94	C₂×C₄×C₁₇	C₂³×C₄	1	6	yes	—
95	C₂×C₄×C₁₇	C₂²×D₄	1	4	yes	—
96	C₂×C₄×C₁₇	C₂²×D₄	1	3	yes	—
97	C₂×C₄×C₁₇	C₂²×D₄	1	4	yes	—
98	C₂×C₄×C₁₇	C₂×#41	1	6	yes	—
99	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
100	C₂×Dic₁₇	—	2	8	no	—
101	C₂×Dic₁₇	—	4	24	no	—
102	D₆₈	—	2	48	no	—
103	D₆₈	—	2	8	no	—
104	D₆₈	C₂³×C₄	1	16	yes	complemented
105	C₄×D₁₇	—	4	64	no	—
106	C₈×C₁₇	—	2	4	no	—
107	C₈×C₁₇	C₂³×C₄	1	8	yes	complemented
108	C₂³×D₈	—	17	1	no	—
109	C₂²×C₄×D₄	—	17	2	no	—
110	C₂×C₄×D₈	—	17	24	no	—
111	C₂²×#107	—	17	1	no	—
112	C₂²×#102	—	17	2	no	—
113	C₂²×C₄×C₈	D₁₇	1	1	yes	characteristic, complemented
114	C₂²×C₁₇	C₄×D₈	1	16	yes	complemented
115	C₂²×C₁₇	C₂×C₄×D₄	1	6	yes	—
116	C₂²×C₁₇	C₂²×D₈	1	6	yes	—
117	C₂²×C₁₇	C₂×#135	1	6	yes	—
118	C₂²×C₁₇	C₂³×D₄	1	1	yes	characteristic
119	C₂×D₁₇	—	2	16	no	—
120	C₂×D₁₇	—	4	96	no	—
121	C₂×D₁₇	—	4	8	no	—
122	C₂×D₁₇	—	4	8	no	—
123	C₄×C₁₇	—	2	4	no	—
124	C₄×C₁₇	C₂×C₄×D₄	1	6	yes	—
125	C₄×C₁₇	C₂²×D₈	1	8	yes	complemented
126	C₄×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
127	C₄×C₁₇	C₂²×#41	1	1	yes	characteristic
128	Dic₁₇	—	4	8	no	—
129	C₂³×D₄	—	17	2	no	—
130	C₂⁴×C₄	—	34	2	no	—
131	C₂²×D₈	—	17	16	no	—
132	C₂²×D₈	—	17	12	no	—
133	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
134	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
135	C₂×C₄×D₄	—	17	24	no	—
136	C₂×C₄×D₄	—	34	12	no	—
137	C₂²×C₄⋊C₄	—	17	2	no	—
138	C₂²×C₂²⋊C₄	—	34	2	no	—
139	C₂²×C₄²	C₂×D₁₇	1	1	yes	characteristic
140	C₄×D₈	—	17	64	no	—
141	C₂×#107	—	17	12	no	—
142	C₂×#102	—	17	24	no	—
143	C₂×C₄×C₈	C₂×D₁₇	1	12	yes	complemented
144	C₂×C₁₇	C₂×C₄×D₈	1	12	yes	complemented
145	C₂×C₁₇	C₂²×C₄×D₄	1	1	yes	characteristic
146	C₂×C₁₇	C₂³×D₈	1	1	yes	characteristic
147	C₂×C₁₇	C₂²×#135	1	1	yes	characteristic
148	D₁₇	—	4	16	no	—
149	C₂⁵	—	34	2	no	—
150	C₂²×D₄	—	17	16	no	—
151	C₂²×D₄	—	17	12	no	—
152	C₂²×D₄	—	34	6	no	—
153	C₂²×D₄	—	34	8	no	—
154	C₂³×C₄	—	34	2	no	—
155	C₂³×C₄	—	34	24	no	—
156	C₂³×C₄	—	68	16	no	—
157	C₂³×C₄	D₆₈	1	1	yes	characteristic
158	C₂³×C₄	D₆₈	1	1	yes	characteristic
159	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
160	C₂×D₈	—	17	96	no	—
161	C₂×D₈	—	17	16	no	—
162	C₂²×C₈	—	2	4	no	—
163	C₂²×C₈	C₄×D₁₇	1	8	yes	complemented
164	C₂²×C₈	C₂²×D₁₇	1	6	yes	—
165	C₂²×C₈	C₂²×D₁₇	1	6	yes	—
166	C₄×D₄	—	17	32	no	—
167	C₄×D₄	—	34	48	no	—
168	C₂×C₄⋊C₄	—	17	12	no	—
169	C₂×C₄⋊C₄	—	34	6	no	—
170	C₂×C₂²⋊C₄	—	34	24	no	—
171	C₂×C₄²	D₆₈	1	6	yes	—
172	C₂×C₄²	C₂²×D₁₇	1	6	yes	—
173	#107	—	17	16	no	—
174	#102	—	17	32	no	—
175	C₄×C₈	C₂²×D₁₇	1	16	yes	complemented
176	C₁₇	C₂²×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
177	C₂⁴	—	34	12	no	—
178	C₂⁴	—	34	16	no	—
179	C₂⁴	—	68	8	no	—
180	C₂⁴	—	68	8	no	—
181	C₂⁴	C₂×D₆₈	1	1	yes	characteristic
182	C₂×D₄	—	17	48	no	—
183	C₂×D₄	—	17	8	no	—
184	C₂×D₄	—	34	12	no	—
185	C₂×D₄	—	34	24	no	—
186	C₂×D₄	—	34	48	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	34	12	no	—
189	C₂²×C₄	—	34	32	no	—
190	C₂²×C₄	—	68	8	no	—
191	C₂²×C₄	—	68	96	no	—
192	C₂²×C₄	D₁₃₆	1	8	yes	direct product factor
193	C₂²×C₄	C₂×C₄×D₁₇	1	4	yes	—
194	C₂²×C₄	C₂×D₆₈	1	6	yes	—
195	C₂²×C₄	C₂×D₆₈	1	3	yes	—
196	C₂²×C₄	C₂×D₆₈	1	6	yes	—
197	C₂²×C₄	#63252	1	4	yes	—
198	C₂²×C₄	C₂³×D₁₇	1	3	yes	—
199	D₈	—	17	64	no	—
200	C₂×C₈	—	2	12	no	—
201	C₂×C₈	C₂×C₄×D₁₇	1	24	yes	complemented
202	C₂×C₈	C₂³×D₁₇	1	4	yes	—
203	C₂×C₈	C₂³×D₁₇	1	4	yes	—
204	C₄⋊C₄	—	17	8	no	—
205	C₄⋊C₄	—	34	12	no	—
206	C₂²⋊C₄	—	34	32	no	—
207	C₄²	C₂×D₆₈	1	12	yes	—
208	C₄²	C₂³×D₁₇	1	4	yes	—
209	C₂³	—	34	48	no	—
210	C₂³	—	34	8	no	—
211	C₂³	—	68	24	no	—
212	C₂³	—	68	24	no	—
213	C₂³	—	68	64	no	—
214	C₂³	C₄×D₆₈	1	4	yes	—
215	C₂³	C₂×D₁₃₆	1	4	yes	—
216	C₂³	#10587590	1	4	yes	—
217	C₂³	C₂²×D₆₈	1	3	yes	—
218	D₄	—	17	16	no	—
219	D₄	—	34	48	no	—
220	D₄	—	34	8	no	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	34	8	no	—
223	C₂×C₄	—	68	24	no	—
224	C₂×C₄	—	68	64	no	—
225	C₂×C₄	C₄×D₆₈	1	12	yes	—
226	C₂×C₄	C₂×D₁₃₆	1	24	yes	direct product factor
227	C₂×C₄	C₂²×C₄×D₁₇	1	6	yes	—
228	C₂×C₄	C₂²×D₆₈	1	4	yes	—
229	C₂×C₄	C₂²×D₆₈	1	3	yes	—
230	C₂×C₄	C₂²×D₆₈	1	4	yes	—
231	C₂×C₄	C₂×#63252	1	6	yes	—
232	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	C₂²×C₄×D₁₇	1	8	yes	complemented
235	C₂²	—	34	16	no	—
236	C₂²	—	68	96	no	—
237	C₂²	—	68	8	no	—
238	C₂²	—	68	8	no	—
239	C₂²	C₄×D₁₃₆	1	16	yes	direct product factor
240	C₂²	C₂×C₄×D₆₈	1	6	yes	—
241	C₂²	C₂²×D₁₃₆	1	6	yes	—
242	C₂²	C₂×#10587590	1	6	yes	—
243	C₂²	C₂³×D₆₈	1	1	yes	characteristic
244	C₄	—	2	4	no	—
245	C₄	—	68	8	no	—
246	C₄	C₂×C₄×D₆₈	1	6	yes	—
247	C₄	C₂²×D₁₃₆	1	8	yes	direct product factor
248	C₄	C₂³×C₄×D₁₇	1	1	yes	characteristic
249	C₄	C₂²×#63252	1	1	yes	characteristic
250	C₂	—	68	16	no	—
251	C₂	C₂×C₄×D₁₃₆	1	12	yes	direct product factor
252	C₂	C₂²×C₄×D₆₈	1	1	yes	characteristic
253	C₂	C₂³×D₁₃₆	1	1	yes	characteristic
254	C₂	C₂²×#10587590	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₆₈
commutator	C₄×C₁₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₇
Fitting	C₂²×C₄×C₈×C₁₇	C₂
socle	C₂⁴×C₁₇	C₂×D₄
prime radical	C₂³×D₁₃₆	C₂
cosocle	C₂×C₄×C₁₇	C₂⁵
hypercenter	C₂²×C₄×C₈	D₁₇
hypocenter	C₁₇	C₂²×C₄×D₈
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₆₈	C₂	semidirect
3. C₂×C₄×D₁₃₆	C₂	direct (N × H)
4. C₂²×#10587489	C₂	semidirect
5. C₂²×#10587492	C₂	semidirect
6. C₂²×C₄×C₈×C₁₇	C₂	semidirect
9. C₂²×D₁₃₆	C₄	direct (N × H)
12. C₂×C₄×D₆₈	C₂²	semidirect
15. C₄×D₁₃₆	C₂²	direct (N × H)
16. C₂×#10587489	C₂²	semidirect
17. C₂×#10587492	C₂²	semidirect
21. C₂×C₄×C₈×C₁₇	C₂²	semidirect
29. C₂²×D₆₈	C₂×C₄	semidirect
34. C₂²×C₈×C₁₇	C₂×C₄	semidirect
42. C₂×D₁₃₆	C₂×C₄	direct (N × H)
45. C₄×D₆₈	C₂³	semidirect
46. C₄×C₈×C₁₇	C₂³	semidirect
47. #10587492	C₂³	semidirect
48. #10587489	C₂³	semidirect
55. C₂²×C₄×C₁₇	D₈	semidirect
67. C₂×D₆₈	C₂²×C₄	semidirect
72. C₂×C₈×C₁₇	C₂²×C₄	semidirect
80. D₁₃₆	C₂²×C₄	direct (N × H)
93. C₂×C₄×C₁₇	C₂×D₈	semidirect
104. D₆₈	C₂³×C₄	semidirect
107. C₈×C₁₇	C₂³×C₄	semidirect
113. C₂²×C₄×C₈	D₁₇	semidirect
114. C₂²×C₁₇	C₄×D₈	semidirect
125. C₄×C₁₇	C₂²×D₈	semidirect
143. C₂×C₄×C₈	C₂×D₁₇	semidirect
144. C₂×C₁₇	C₂×C₄×D₈	semidirect
163. C₂²×C₈	C₄×D₁₇	semidirect
175. C₄×C₈	C₂²×D₁₇	semidirect
176. C₁₇	C₂²×C₄×D₈	semidirect
192. C₂²×C₄	D₁₃₆	direct (N × H)
201. C₂×C₈	C₂×C₄×D₁₇	semidirect
226. C₂×C₄	C₂×D₁₃₆	direct (N × H)
234. C₈	C₂²×C₄×D₁₇	semidirect
239. C₂²	C₄×D₁₃₆	direct (N × H)
247. C₄	C₂²×D₁₃₆	direct (N × H)
251. C₂	C₂×C₄×D₁₃₆	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×D₁₃₆	2
C₂²×C₄×D₆₈	2
C₂×C₄×D₁₃₆	2
C₂²×#10587489	2
C₂²×#10587492	2
C₂²×C₄×C₈×C₁₇	2
C₂²×C₄×D₈	17

Subgroups — C22×C4×D136 #17898327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×D₁₃₆ #17898327