Subgroups — C₂×Q₈×D₁₃₆ #17914739

Order 4352.

2286 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₆₈	C₂	1	2	yes	maximal, complemented
2	Q₈×D₁₃₆	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11763214	C₂	1	3	yes	maximal, complemented
4	C₂×#11763090	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₁₃₆	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×D₁₇	—	2	2	no	—
8	C₂²×D₁₃₆	C₂²	1	1	yes	characteristic
9	Q₈×D₆₈	—	2	8	no	—
10	Q₈×D₆₈	C₂²	1	16	yes	complemented
11	C₂×#10587555	C₂²	1	6	yes	—
12	C₂×#10587583	—	2	6	no	—
13	C₂×C₄×D₆₈	C₂²	1	6	yes	—
14	#11763214	C₂²	1	24	yes	complemented
15	#11763090	C₂²	1	48	yes	complemented
16	C₄×D₁₃₆	C₂²	1	24	yes	—
17	C₂×#10587489	C₂²	1	3	yes	—
18	C₂×#10587489	C₂²	1	6	yes	—
19	C₂×#10587492	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₁₇	C₂²	1	8	yes	complemented
22	C₂×C₁₇×C₄⋊C₈	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₁₇	C₂²	1	3	yes	—
24	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂×Q₈×D₁₇	—	2	16	no	—
26	C₂×Q₈×D₁₇	—	4	16	no	—
27	C₂²×D₆₈	C₂³	1	2	yes	—
28	C₂²×C₄×D₁₇	—	2	6	no	—
29	C₂²×Dic₃₄	—	2	6	no	—
30	C₂²×C₈×C₁₇	C₂³	1	3	yes	—
31	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
32	C₄×Q₈×C₁₇	D₄	1	4	yes	—
33	C₄×Q₈×C₁₇	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₇	D₄	1	3	yes	—
35	C₂×C₄²×C₁₇	C₂³	1	3	yes	—
36	C₂×#63227	—	2	6	no	—
37	C₂×C₄⋊Dic₁₇	—	2	6	no	—
38	C₂×C₄⋊Dic₁₇	C₂³	1	6	yes	—
39	C₂×D₁₃₆	Q₈	1	8	yes	direct product factor
40	C₂×D₁₃₆	C₂³	1	4	yes	—
41	#10587583	—	2	48	no	—
42	C₄×D₆₈	—	2	12	no	—
43	C₄×D₆₈	C₂³	1	24	yes	—
44	#10587555	—	2	12	no	—
45	#10587555	C₂³	1	24	yes	—
46	C₁₇×C₄⋊C₈	C₂³	1	12	yes	—
47	C₄×C₈×C₁₇	C₂³	1	12	yes	—
48	#10587492	C₂³	1	24	yes	—
49	#10587489	C₂³	1	12	yes	—
50	#10587489	C₂³	1	24	yes	—
51	C₂³×D₁₇	—	2	2	no	—
52	C₂×Q₈×C₁₇	D₈	1	8	yes	complemented
53	C₂×Q₈×C₁₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₁₇	—	2	6	no	—
58	Q₈×D₁₇	—	4	64	no	—
59	C₂×D₆₈	—	2	2	no	—
60	C₂×D₆₈	—	2	4	no	—
61	C₂×D₆₈	C₂×Q₈	1	8	yes	complemented
62	C₂×D₆₈	C₂⁴	1	4	yes	—
63	C₂×C₄×D₁₇	—	2	24	no	—
64	C₂×C₄×D₁₇	—	4	24	no	—
65	C₂×Dic₃₄	—	2	24	no	—
66	C₂×Dic₃₄	—	4	24	no	—
67	C₂×C₈×C₁₇	—	2	6	no	—
68	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	complemented
69	C₂×C₈×C₁₇	C₂⁴	1	6	yes	—
70	C₂×C₈×C₁₇	C₂⁴	1	2	yes	—
71	C₄⋊C₄×C₁₇	C₂×D₄	1	12	yes	—
72	C₄²×C₁₇	C₂×D₄	1	6	yes	—
73	C₄²×C₁₇	C₂⁴	1	6	yes	—
74	#63227	—	2	24	no	—
75	C₄⋊Dic₁₇	—	2	6	no	—
76	C₄⋊Dic₁₇	—	2	24	no	—
77	C₄⋊Dic₁₇	C₂⁴	1	12	yes	—
78	D₁₃₆	C₂×Q₈	1	16	yes	direct product factor
79	C₂×Q₈×D₈	—	17	1	no	maximal, Sylow
80	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
81	C₂²×D₁₇	—	2	8	no	—
82	C₂²×D₁₇	—	2	4	no	—
83	C₂²×D₁₇	—	4	4	no	—
84	C₂²×D₁₇	—	4	4	no	—
85	Q₈×C₁₇	C₂×D₈	1	16	yes	complemented
86	C₂×C₄×C₁₇	—	2	6	no	—
87	C₂×C₄×C₁₇	C₂×D₈	1	12	yes	—
88	C₂×C₄×C₁₇	C₂²×D₄	1	6	yes	—
89	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂²×D₄	1	6	yes	—
91	C₂×C₄×C₁₇	C₂²×Q₈	1	2	yes	—
92	C₂×C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
93	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
94	C₂×Dic₁₇	—	2	12	no	—
95	C₂×Dic₁₇	—	4	12	no	—
96	D₆₈	—	2	8	no	—
97	D₆₈	—	2	4	no	—
98	D₆₈	C₂²×Q₈	1	8	yes	complemented
99	C₄×D₁₇	—	4	48	no	—
100	Dic₃₄	—	4	48	no	—
101	C₈×C₁₇	—	2	6	no	—
102	C₈×C₁₇	C₂²×Q₈	1	4	yes	complemented
103	C₂×D₄×Q₈	—	17	2	no	—
104	Q₈×D₈	—	17	16	no	—
105	C₂×#500	—	17	3	no	—
106	C₂×#474	—	17	6	no	—
107	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic, complemented
108	C₂×C₄×D₈	—	17	3	no	—
109	C₂²×C₁₇	D₄×Q₈	1	2	yes	—
110	C₂²×C₁₇	C₂²×D₈	1	2	yes	—
111	C₂²×C₁₇	#578	1	2	yes	—
112	C₂²×C₁₇	C₂³×D₄	1	1	yes	characteristic
113	C₂×D₁₇	—	2	8	no	—
114	C₂×D₁₇	—	4	16	no	—
115	C₂×D₁₇	—	4	4	no	—
116	C₂×D₁₇	—	4	4	no	—
117	C₄×C₁₇	—	2	6	no	—
118	C₄×C₁₇	D₄×Q₈	1	2	yes	—
119	C₄×C₁₇	C₂²×D₈	1	12	yes	—
120	C₄×C₁₇	C₂³×Q₈	1	1	yes	characteristic
121	C₄×C₁₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
122	Dic₁₇	—	4	12	no	—
123	C₂³×Q₈	—	34	2	no	—
124	C₂²×D₈	—	17	1	no	—
125	D₄×Q₈	—	17	16	no	—
126	D₄×Q₈	—	34	8	no	—
127	C₂×C₄⋊Q₈	—	17	6	no	—
128	C₂×C₂²⋊Q₈	—	34	6	no	—
129	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
130	C₂×C₄×D₄	—	17	6	no	—
131	#500	—	17	24	no	—
132	#474	—	17	48	no	—
133	C₈×Q₈	C₂×D₁₇	1	8	yes	complemented
134	C₄×D₈	—	17	24	no	—
135	C₂×#107	—	17	3	no	—
136	C₂×#107	—	17	6	no	—
137	C₂×C₄⋊C₈	C₂×D₁₇	1	3	yes	—
138	C₂×#102	—	17	6	no	—
139	C₂×C₄×C₈	C₂×D₁₇	1	3	yes	—
140	C₂×C₁₇	Q₈×D₈	1	4	yes	complemented
141	C₂×C₁₇	C₂×D₄×Q₈	1	1	yes	characteristic
142	C₂×C₁₇	C₂³×D₈	1	1	yes	characteristic
143	C₂×C₁₇	C₂×#578	1	1	yes	characteristic
144	D₁₇	—	4	8	no	—
145	C₂²×Q₈	—	34	16	no	—
146	C₂²×Q₈	—	34	6	no	—
147	C₂²×Q₈	—	68	16	no	—
148	C₂²×Q₈	D₆₈	1	1	yes	characteristic
149	C₂²×D₄	—	17	2	no	—
150	C₂³×C₄	—	34	6	no	—
151	C₂×D₈	—	17	8	no	—
152	C₂×D₈	—	17	4	no	—
153	C₂²×C₈	C₂²×D₁₇	1	3	yes	—
154	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
155	C₄⋊Q₈	—	17	24	no	—
156	C₄⋊Q₈	—	34	12	no	—
157	C₂²⋊Q₈	—	34	48	no	—
158	C₄×Q₈	D₆₈	1	4	yes	—
159	C₄×Q₈	C₂²×D₁₇	1	4	yes	—
160	C₄×D₄	—	17	24	no	—
161	C₄×D₄	—	34	12	no	—
162	C₂×C₄⋊C₄	—	17	6	no	—
163	C₂×C₄⋊C₄	—	34	6	no	—
164	C₂×C₄⋊C₄	D₆₈	1	3	yes	—
165	C₂×C₂²⋊C₄	—	34	6	no	—
166	C₂×C₄²	C₂²×D₁₇	1	3	yes	—
167	#107	—	17	12	no	—
168	#107	—	17	24	no	—
169	C₄⋊C₈	C₂²×D₁₇	1	12	yes	—
170	#102	—	17	24	no	—
171	C₄×C₈	C₂²×D₁₇	1	12	yes	—
172	C₁₇	C₂×Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
173	C₂⁴	—	34	2	no	—
174	C₂×Q₈	—	34	24	no	—
175	C₂×Q₈	—	68	64	no	—
176	C₂×Q₈	—	68	24	no	—
177	C₂×Q₈	D₁₃₆	1	8	yes	direct product factor
178	C₂×Q₈	C₂×D₆₈	1	4	yes	—
179	C₂×D₄	—	17	8	no	—
180	C₂×D₄	—	17	4	no	—
181	C₂×D₄	—	34	2	no	—
182	C₂×D₄	—	34	4	no	—
183	C₂²×C₄	—	34	6	no	—
184	C₂²×C₄	—	34	24	no	—
185	C₂²×C₄	—	68	24	no	—
186	C₂²×C₄	C₂×D₆₈	1	3	yes	—
187	C₂²×C₄	C₂×D₆₈	1	3	yes	—
188	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
189	D₈	—	17	16	no	—
190	C₂×C₈	—	2	6	no	—
191	C₂×C₈	Q₈×D₁₇	1	4	yes	complemented
192	C₂×C₈	C₂³×D₁₇	1	6	yes	—
193	C₂×C₈	C₂³×D₁₇	1	2	yes	—
194	C₄⋊C₄	—	17	12	no	—
195	C₄⋊C₄	—	34	6	no	—
196	C₄⋊C₄	—	34	24	no	—
197	C₄⋊C₄	C₂×D₆₈	1	12	yes	—
198	C₂²⋊C₄	—	34	24	no	—
199	C₄²	C₂×D₆₈	1	6	yes	—
200	C₄²	C₂³×D₁₇	1	6	yes	—
201	C₂³	—	34	8	no	—
202	C₂³	—	34	4	no	—
203	C₂³	—	68	4	no	—
204	C₂³	—	68	4	no	—
205	C₂³	C₂²×D₆₈	1	1	yes	characteristic
206	Q₈	—	68	48	no	—
207	Q₈	C₂×D₁₃₆	1	16	yes	direct product factor
208	D₄	—	17	8	no	—
209	D₄	—	34	8	no	—
210	D₄	—	34	4	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	34	12	no	—
213	C₂×C₄	—	68	12	no	—
214	C₂×C₄	—	68	48	no	—
215	C₂×C₄	C₂×D₁₃₆	1	12	yes	—
216	C₂×C₄	C₂²×D₆₈	1	6	yes	—
217	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₆₈	1	6	yes	—
219	C₂×C₄	C₂×Q₈×D₁₇	1	2	yes	—
220	C₂×C₄	#10587697	1	2	yes	—
221	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
222	C₈	—	2	6	no	—
223	C₈	C₂×Q₈×D₁₇	1	4	yes	complemented
224	C₂²	—	34	8	no	—
225	C₂²	—	68	16	no	—
226	C₂²	—	68	4	no	—
227	C₂²	—	68	4	no	—
228	C₂²	Q₈×D₆₈	1	2	yes	—
229	C₂²	C₂²×D₁₃₆	1	2	yes	—
230	C₂²	#11763626	1	2	yes	—
231	C₂²	C₂³×D₆₈	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	68	12	no	—
234	C₄	Q₈×D₆₈	1	2	yes	—
235	C₄	C₂²×D₁₃₆	1	12	yes	—
236	C₄	C₂²×Q₈×D₁₇	1	1	yes	characteristic
237	C₄	C₂×#10587697	1	1	yes	characteristic
238	C₂	—	68	8	no	—
239	C₂	Q₈×D₁₃₆	1	4	yes	direct product factor
240	C₂	C₂×Q₈×D₆₈	1	1	yes	characteristic
241	C₂	C₂³×D₁₃₆	1	1	yes	characteristic
242	C₂	C₂×#11763626	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₇
Fitting	C₂×C₈×Q₈×C₁₇	C₂
socle	C₂³×C₁₇	C₂²×D₄
prime radical	C₂²×D₁₃₆	C₂²
hypercenter	C₂×C₈×Q₈	D₁₇
hypocenter	C₁₇	C₂×Q₈×D₈
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₆₈	C₂	semidirect
2. Q₈×D₁₃₆	C₂	direct (N × H)
3. C₂×#11763214	C₂	semidirect
4. C₂×#11763090	C₂	semidirect
6. C₂×C₈×Q₈×C₁₇	C₂	semidirect
10. Q₈×D₆₈	C₂²	semidirect
14. #11763214	C₂²	semidirect
15. #11763090	C₂²	semidirect
21. C₈×Q₈×C₁₇	C₂²	semidirect
39. C₂×D₁₃₆	Q₈	direct (N × H)
52. C₂×Q₈×C₁₇	D₈	semidirect
61. C₂×D₆₈	C₂×Q₈	semidirect
68. C₂×C₈×C₁₇	C₂×Q₈	semidirect
78. D₁₃₆	C₂×Q₈	direct (N × H)
85. Q₈×C₁₇	C₂×D₈	semidirect
98. D₆₈	C₂²×Q₈	semidirect
102. C₈×C₁₇	C₂²×Q₈	semidirect
107. C₂×C₈×Q₈	D₁₇	semidirect
133. C₈×Q₈	C₂×D₁₇	semidirect
140. C₂×C₁₇	Q₈×D₈	semidirect
172. C₁₇	C₂×Q₈×D₈	semidirect
177. C₂×Q₈	D₁₃₆	direct (N × H)
191. C₂×C₈	Q₈×D₁₇	semidirect
207. Q₈	C₂×D₁₃₆	direct (N × H)
223. C₈	C₂×Q₈×D₁₇	semidirect
239. C₂	Q₈×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
C₂×Q₈×D₆₈	2
Q₈×D₁₃₆	2
C₂×#11763214	2
C₂×#11763090	2
C₂×C₄×D₁₃₆	2
C₂×C₈×Q₈×C₁₇	2
C₂×Q₈×D₈	17

Subgroups — C2×Q8×D136 #17914739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₁₃₆ #17914739