Subgroups — C₂×Q₈×D₅₃₆ #31736246

Order 17152.

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×D₂₆₈	C₂	1	2	yes	maximal, complemented
2	Q₈×D₅₃₆	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863801	C₂	1	3	yes	maximal, complemented
4	C₂×#16863677	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₅₃₆	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×D₆₇	—	2	2	no	—
8	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
9	Q₈×D₂₆₈	—	2	8	no	—
10	Q₈×D₂₆₈	C₂²	1	16	yes	complemented
11	C₂×#14524043	C₂²	1	6	yes	—
12	C₂×#14524023	—	2	6	no	—
13	C₂×C₄×D₂₆₈	C₂²	1	6	yes	—
14	#16863801	C₂²	1	24	yes	complemented
15	#16863677	C₂²	1	48	yes	complemented
16	C₄×D₅₃₆	C₂²	1	24	yes	—
17	C₂×#14523977	C₂²	1	3	yes	—
18	C₂×#14523977	C₂²	1	6	yes	—
19	C₂×#14523965	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₆₇	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
24	C₂²×Dic₁₃₄	—	2	6	no	—
25	C₂×Q₈×D₆₇	—	2	16	no	—
26	C₂×Q₈×D₆₇	—	4	16	no	—
27	C₂²×D₂₆₈	C₂³	1	2	yes	—
28	C₂²×C₄×D₆₇	—	2	6	no	—
29	C₂×D₅₃₆	Q₈	1	8	yes	direct product factor
30	C₂×D₅₃₆	C₂³	1	4	yes	—
31	#14524043	—	2	12	no	—
32	#14524043	C₂³	1	24	yes	—
33	#14524023	—	2	48	no	—
34	C₄×D₂₆₈	—	2	12	no	—
35	C₄×D₂₆₈	C₂³	1	24	yes	—
36	C₂×C₄⋊Dic₆₇	—	2	6	no	—
37	C₂×C₄⋊Dic₆₇	C₂³	1	6	yes	—
38	C₂×(C₂×D₆₇)⋊C₄	—	2	6	no	—
39	#14523977	C₂³	1	12	yes	—
40	#14523977	C₂³	1	24	yes	—
41	#14523965	C₂³	1	24	yes	—
42	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₆₇	D₄	1	4	yes	—
46	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
49	C₄⋊C₈×C₆₇	C₂³	1	12	yes	—
50	C₄×C₈×C₆₇	C₂³	1	12	yes	—
51	C₂³×D₆₇	—	2	2	no	—
52	C₂×Dic₁₃₄	—	2	24	no	—
53	C₂×Dic₁₃₄	—	4	24	no	—
54	Q₈×D₆₇	—	4	64	no	—
55	C₂×D₂₆₈	—	2	2	no	—
56	C₂×D₂₆₈	—	2	4	no	—
57	C₂×D₂₆₈	C₂×Q₈	1	8	yes	complemented
58	C₂×D₂₆₈	C₂⁴	1	4	yes	—
59	C₂²×Dic₆₇	—	2	6	no	—
60	C₂×C₄×D₆₇	—	2	24	no	—
61	C₂×C₄×D₆₇	—	4	24	no	—
62	D₅₃₆	C₂×Q₈	1	16	yes	direct product factor
63	C₄⋊Dic₆₇	—	2	24	no	—
64	C₄⋊Dic₆₇	—	2	6	no	—
65	C₄⋊Dic₆₇	C₂⁴	1	12	yes	—
66	(C₂×D₆₇)⋊C₄	—	2	24	no	—
67	C₂×Q₈×C₆₇	D₈	1	8	yes	complemented
68	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	6	no	—
73	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
74	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
75	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
76	C₄⋊C₄×C₆₇	C₂×D₄	1	12	yes	—
77	C₄²×C₆₇	C₂×D₄	1	6	yes	—
78	C₄²×C₆₇	C₂⁴	1	6	yes	—
79	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
80	C₂²×D₆₇	—	2	8	no	—
81	C₂²×D₆₇	—	2	4	no	—
82	C₂²×D₆₇	—	4	4	no	—
83	C₂²×D₆₇	—	4	4	no	—
84	Q₈×C₆₇	C₂×D₈	1	16	yes	complemented
85	C₂×C₄×C₆₇	—	2	6	no	—
86	C₂×C₄×C₆₇	C₂×D₈	1	12	yes	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
88	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
90	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
92	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₆₇	—	2	12	no	—
94	C₂×Dic₆₇	—	4	12	no	—
95	D₂₆₈	—	2	8	no	—
96	D₂₆₈	—	2	4	no	—
97	D₂₆₈	C₂²×Q₈	1	8	yes	complemented
98	C₄×D₆₇	—	4	48	no	—
99	Dic₁₃₄	—	4	48	no	—
100	C₈×C₆₇	—	2	6	no	—
101	C₈×C₆₇	C₂²×Q₈	1	4	yes	complemented
102	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
103	C₂²×C₆₇	C₂²×D₈	1	2	yes	—
104	C₂²×C₆₇	#578	1	2	yes	—
105	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
106	C₂×D₆₇	—	2	8	no	—
107	C₂×D₆₇	—	4	16	no	—
108	C₂×D₆₇	—	4	4	no	—
109	C₂×D₆₇	—	4	4	no	—
110	C₄×C₆₇	—	2	6	no	—
111	C₄×C₆₇	D₄×Q₈	1	2	yes	—
112	C₄×C₆₇	C₂²×D₈	1	12	yes	—
113	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
114	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
115	Dic₆₇	—	4	12	no	—
116	C₂×Q₈×D₈	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	Q₈×D₈	1	4	yes	complemented
118	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
119	C₂×C₆₇	C₂³×D₈	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
121	D₆₇	—	4	8	no	—
122	C₂×D₄×Q₈	—	67	2	no	—
123	Q₈×D₈	—	67	16	no	—
124	C₂×#500	—	67	3	no	—
125	C₂×#474	—	67	6	no	—
126	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic, complemented
127	C₂×C₄×D₈	—	67	3	no	—
128	C₆₇	C₂×Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
129	C₂³×Q₈	—	134	2	no	—
130	C₂²×D₈	—	67	1	no	—
131	D₄×Q₈	—	67	16	no	—
132	D₄×Q₈	—	134	8	no	—
133	C₂×C₄⋊Q₈	—	67	6	no	—
134	C₂×C₂²⋊Q₈	—	134	6	no	—
135	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
136	C₂×C₄×D₄	—	67	6	no	—
137	#500	—	67	24	no	—
138	#474	—	67	48	no	—
139	C₈×Q₈	C₂×D₆₇	1	8	yes	complemented
140	C₄×D₈	—	67	24	no	—
141	C₂×#107	—	67	3	no	—
142	C₂×#107	—	67	6	no	—
143	C₂×C₄⋊C₈	C₂×D₆₇	1	3	yes	—
144	C₂×#102	—	67	6	no	—
145	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
146	C₂²×Q₈	—	268	16	no	—
147	C₂²×Q₈	—	134	16	no	—
148	C₂²×Q₈	—	134	6	no	—
149	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
150	C₂²×D₄	—	67	2	no	—
151	C₂³×C₄	—	134	6	no	—
152	C₂×D₈	—	67	8	no	—
153	C₂×D₈	—	67	4	no	—
154	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
155	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
156	C₄⋊Q₈	—	67	24	no	—
157	C₄⋊Q₈	—	134	12	no	—
158	C₂²⋊Q₈	—	134	48	no	—
159	C₄×Q₈	D₂₆₈	1	4	yes	—
160	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
161	C₄×D₄	—	67	24	no	—
162	C₄×D₄	—	134	12	no	—
163	C₂×C₄⋊C₄	—	67	6	no	—
164	C₂×C₄⋊C₄	—	134	6	no	—
165	C₂×C₄⋊C₄	D₂₆₈	1	3	yes	—
166	C₂×C₂²⋊C₄	—	134	6	no	—
167	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
168	#107	—	67	12	no	—
169	#107	—	67	24	no	—
170	C₄⋊C₈	C₂²×D₆₇	1	12	yes	—
171	#102	—	67	24	no	—
172	C₄×C₈	C₂²×D₆₇	1	12	yes	—
173	C₂⁴	—	134	2	no	—
174	C₂×Q₈	—	268	64	no	—
175	C₂×Q₈	—	268	24	no	—
176	C₂×Q₈	—	134	24	no	—
177	C₂×Q₈	D₅₃₆	1	8	yes	direct product factor
178	C₂×Q₈	C₂×D₂₆₈	1	4	yes	—
179	C₂×D₄	—	67	8	no	—
180	C₂×D₄	—	67	4	no	—
181	C₂×D₄	—	134	2	no	—
182	C₂×D₄	—	134	4	no	—
183	C₂²×C₄	—	268	24	no	—
184	C₂²×C₄	—	134	6	no	—
185	C₂²×C₄	—	134	24	no	—
186	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
187	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
188	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
189	D₈	—	67	16	no	—
190	C₂×C₈	—	2	6	no	—
191	C₂×C₈	Q₈×D₆₇	1	4	yes	complemented
192	C₂×C₈	C₂³×D₆₇	1	6	yes	—
193	C₂×C₈	C₂³×D₆₇	1	2	yes	—
194	C₄⋊C₄	—	67	12	no	—
195	C₄⋊C₄	—	134	6	no	—
196	C₄⋊C₄	—	134	24	no	—
197	C₄⋊C₄	C₂×D₂₆₈	1	12	yes	—
198	C₂²⋊C₄	—	134	24	no	—
199	C₄²	C₂×D₂₆₈	1	6	yes	—
200	C₄²	C₂³×D₆₇	1	6	yes	—
201	C₂³	—	268	4	no	—
202	C₂³	—	268	4	no	—
203	C₂³	—	134	8	no	—
204	C₂³	—	134	4	no	—
205	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
206	Q₈	—	268	48	no	—
207	Q₈	C₂×D₅₃₆	1	16	yes	direct product factor
208	D₄	—	67	8	no	—
209	D₄	—	134	8	no	—
210	D₄	—	134	4	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	268	12	no	—
213	C₂×C₄	—	268	48	no	—
214	C₂×C₄	—	134	12	no	—
215	C₂×C₄	C₂×D₅₃₆	1	12	yes	—
216	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
217	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
219	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
220	C₂×C₄	#14524090	1	2	yes	—
221	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
222	C₈	—	2	6	no	—
223	C₈	C₂×Q₈×D₆₇	1	4	yes	complemented
224	C₂²	—	268	16	no	—
225	C₂²	—	268	4	no	—
226	C₂²	—	268	4	no	—
227	C₂²	—	134	8	no	—
228	C₂²	Q₈×D₂₆₈	1	2	yes	—
229	C₂²	C₂²×D₅₃₆	1	2	yes	—
230	C₂²	#16864213	1	2	yes	—
231	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	268	12	no	—
234	C₄	Q₈×D₂₆₈	1	2	yes	—
235	C₄	C₂²×D₅₃₆	1	12	yes	—
236	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
237	C₄	C₂×#14524090	1	1	yes	characteristic
238	C₂	—	268	8	no	—
239	C₂	Q₈×D₅₃₆	1	4	yes	direct product factor
240	C₂	C₂×Q₈×D₂₆₈	1	1	yes	characteristic
241	C₂	C₂³×D₅₃₆	1	1	yes	characteristic
242	C₂	C₂×#16864213	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂²×D₅₃₆	C₂²
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	C₂×Q₈×D₈
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
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×D₂₆₈	C₂	semidirect
2. Q₈×D₅₃₆	C₂	direct (N × H)
3. C₂×#16863801	C₂	semidirect
4. C₂×#16863677	C₂	semidirect
6. C₂×C₈×Q₈×C₆₇	C₂	semidirect
10. Q₈×D₂₆₈	C₂²	semidirect
14. #16863801	C₂²	semidirect
15. #16863677	C₂²	semidirect
21. C₈×Q₈×C₆₇	C₂²	semidirect
29. C₂×D₅₃₆	Q₈	direct (N × H)
57. C₂×D₂₆₈	C₂×Q₈	semidirect
62. D₅₃₆	C₂×Q₈	direct (N × H)
67. C₂×Q₈×C₆₇	D₈	semidirect
73. C₂×C₈×C₆₇	C₂×Q₈	semidirect
84. Q₈×C₆₇	C₂×D₈	semidirect
97. D₂₆₈	C₂²×Q₈	semidirect
101. C₈×C₆₇	C₂²×Q₈	semidirect
117. C₂×C₆₇	Q₈×D₈	semidirect
126. C₂×C₈×Q₈	D₆₇	semidirect
128. C₆₇	C₂×Q₈×D₈	semidirect
139. C₈×Q₈	C₂×D₆₇	semidirect
177. C₂×Q₈	D₅₃₆	direct (N × H)
191. C₂×C₈	Q₈×D₆₇	semidirect
207. Q₈	C₂×D₅₃₆	direct (N × H)
223. C₈	C₂×Q₈×D₆₇	semidirect
239. C₂	Q₈×D₅₃₆	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×D₂₆₈	2
Q₈×D₅₃₆	2
C₂×#16863801	2
C₂×#16863677	2
C₂×C₄×D₅₃₆	2
C₂×C₈×Q₈×C₆₇	2
C₂×Q₈×D₈	67

Subgroups — C2×Q8×D536 #31736246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₅₃₆ #31736246