Subgroups — C₂×#22562089 #31009230

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863828	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863512	C₂	1	2	yes	maximal, complemented
3	#22562089	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863202	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#368×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14524065	C₂²	1	2	yes	—
8	C₂×#14524051	—	2	2	no	—
9	#16863828	C₄	1	4	yes	complemented
10	#16863828	C₂²	1	4	yes	—
11	C₂×Dic₅₃₆	C₄	1	1	yes	characteristic, complemented
12	C₂×#14523988	C₄	1	2	yes	complemented
13	C₂×D₁₀₇₂	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	#16863512	—	2	4	no	—
16	#16863512	C₂²	1	8	yes	complemented
17	C₂×#14523969	—	2	2	no	—
18	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
19	#16863202	C₂²	1	8	yes	complemented
20	C₁₆×Dic₆₇	C₂²	1	4	yes	complemented
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#108×C₆₇	C₂²	1	2	yes	—
23	#368×C₆₇	C₂²	1	4	yes	complemented
24	C₂×#11761628	—	2	2	no	—
25	#14524065	—	2	2	no	—
26	#14524065	—	2	2	no	—
27	#14524065	C₂×C₄	1	4	yes	—
28	#14524065	C₂³	1	4	yes	—
29	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
30	C₂×#11761617	—	2	2	no	—
31	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
32	#14524051	—	2	8	no	—
33	#14524051	—	4	4	no	—
34	C₂×#11761611	—	4	2	no	—
35	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
36	C₂×C₈×D₆₇	—	4	2	no	—
37	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
38	Dic₅₃₆	C₂×C₄	1	4	yes	complemented
39	#14523988	C₂×C₄	1	8	yes	complemented
40	D₁₀₇₂	C₂×C₄	1	4	yes	complemented
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₂×C₆₇⋊C₁₆	—	2	1	no	—
43	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
44	#14523969	—	2	8	no	—
45	C₈×Dic₆₇	D₄	1	2	yes	—
46	C₈×Dic₆₇	C₂³	1	2	yes	—
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
53	#108×C₆₇	—	2	2	no	—
54	#108×C₆₇	C₂³	1	4	yes	—
55	#11761628	—	2	4	no	—
56	#11761628	—	2	4	no	—
57	#11761628	—	4	2	no	—
58	#11761628	—	4	2	no	—
59	C₂×Dic₁₃₄	—	2	2	no	—
60	C₂×D₂₆₈	—	2	2	no	—
61	C₂×C₂²⋊D₆₇	—	4	2	no	—
62	C₂²×Dic₆₇	—	2	1	no	—
63	C₂×C₄×D₆₇	—	4	2	no	—
64	Dic₂₆₈	—	2	2	no	—
65	Dic₂₆₈	C₂²×C₄	1	4	yes	—
66	#11761617	—	2	4	no	—
67	#11761617	—	2	4	no	—
68	D₅₃₆	—	2	2	no	—
69	D₅₃₆	C₂²×C₄	1	4	yes	—
70	#11761611	—	4	8	no	—
71	C₂×C₆₇⋊C₈	D₈	1	1	yes	characteristic
72	C₂×C₆₇⋊C₈	D₈	1	1	yes	characteristic
73	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
74	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
75	C₈×D₆₇	—	4	8	no	—
76	C₄×Dic₆₇	—	2	1	no	—
77	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
78	C₆₇⋊C₁₆	—	2	2	no	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	M₁₆×C₆₇	—	2	4	no	—
81	M₁₆×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	—	4	2	no	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
87	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
90	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂³×C₆₇	#135	1	1	yes	characteristic
92	C₂²×D₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
99	C₂²⋊D₆₇	—	4	4	no	—
100	C₂²⋊D₆₇	—	4	4	no	—
101	C₂×Dic₆₇	—	2	4	no	—
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	4	2	no	—
104	D₂₆₈	—	2	4	no	—
105	D₂₆₈	—	4	2	no	—
106	C₄×D₆₇	—	4	4	no	—
107	C₄×D₆₇	—	4	4	no	—
108	Dic₁₃₄	—	2	4	no	—
109	Dic₁₃₄	—	4	2	no	—
110	C₈×C₆₇	—	4	4	no	—
111	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
114	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₆₇⋊C₈	C₂×D₈	1	2	yes	—
116	C₆₇⋊C₈	C₂×D₈	1	2	yes	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
119	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
121	C₂×D₆₇	—	4	4	no	—
122	C₂×D₆₇	—	8	4	no	—
123	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
124	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
125	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
127	Dic₆₇	—	2	4	no	—
128	Dic₆₇	—	4	4	no	—
129	C₂×#2178	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	—	2	1	no	—
131	C₂×C₆₇	—	2	1	no	—
132	C₂×C₆₇	#2178	1	2	yes	complemented
133	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
134	D₆₇	—	8	4	no	—
135	C₂×#508	—	67	1	no	—
136	C₂×#443	—	67	2	no	—
137	#2178	—	67	8	no	—
138	C₂×#368	D₆₇	1	1	yes	characteristic, complemented
139	C₂×#359	—	67	2	no	—
140	C₂×C₄×C₁₆	—	67	1	no	—
141	C₆₇	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	—	67	2	no	—
143	C₂×#131	—	134	2	no	—
144	#508	—	67	4	no	—
145	#508	—	67	4	no	—
146	C₂×Q₃₂	—	67	1	no	—
147	C₂×SD₃₂	—	67	2	no	—
148	C₂×D₁₆	—	67	1	no	—
149	C₂²×C₁₆	—	67	1	no	—
150	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
151	#443	—	67	8	no	—
152	#443	—	134	4	no	—
153	C₂×#108	C₂×D₆₇	1	2	yes	—
154	C₂×#104	—	134	2	no	—
155	C₂×C₄×C₈	—	67	1	no	—
156	#368	C₂×D₆₇	1	4	yes	complemented
157	#359	—	67	8	no	—
158	C₄×C₁₆	—	67	4	no	—
159	C₂×#41	—	134	2	no	—
160	#135	—	67	4	no	—
161	#135	—	67	4	no	—
162	#135	—	134	2	no	—
163	#135	—	134	2	no	—
164	C₂×Q₁₆	—	67	2	no	—
165	C₂×SD₁₆	—	134	2	no	—
166	C₂×D₈	—	67	2	no	—
167	#131	—	268	4	no	—
168	#131	—	134	8	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	268	2	no	—
171	C₂²×C₈	—	268	2	no	—
172	C₂²×C₈	—	67	1	no	—
173	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
174	C₂×C₄²	—	67	1	no	—
175	Q₃₂	—	67	4	no	—
176	SD₃₂	—	67	8	no	—
177	D₁₆	—	67	4	no	—
178	C₂×C₁₆	—	67	2	no	—
179	C₂×C₁₆	—	134	1	no	—
180	C₂×C₁₆	—	134	1	no	—
181	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
182	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
183	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
184	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
185	#108	—	2	2	no	—
186	#108	C₂²×D₆₇	1	4	yes	—
187	#104	—	134	8	no	—
188	C₄×C₈	—	67	2	no	—
189	C₄×C₈	—	67	2	no	—
190	#41	—	268	2	no	—
191	#41	—	268	2	no	—
192	#41	—	134	4	no	—
193	#41	—	134	4	no	—
194	C₂×Q₈	—	134	2	no	—
195	C₂×D₄	—	268	2	no	—
196	C₂×D₄	—	134	2	no	—
197	C₂²×C₄	—	268	2	no	—
198	C₂²×C₄	—	134	1	no	—
199	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
200	Q₁₆	—	67	4	no	—
201	Q₁₆	—	134	2	no	—
202	SD₁₆	—	134	4	no	—
203	SD₁₆	—	134	4	no	—
204	D₈	—	67	4	no	—
205	D₈	—	134	2	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	268	8	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	268	8	no	—
211	C₂×C₈	—	67	2	no	—
212	C₂×C₈	—	67	1	no	—
213	C₂×C₈	—	67	1	no	—
214	C₂×C₈	—	67	2	no	—
215	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
217	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
218	C₂×C₈	#11761629	1	1	yes	characteristic
219	C₂×C₈	#11761629	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
221	C₄²	—	67	2	no	—
222	C₄²	—	134	1	no	—
223	C₁₆	—	134	2	no	—
224	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
225	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
226	C₂³	—	268	2	no	—
227	C₂³	#14524066	1	1	yes	characteristic
228	Q₈	—	268	2	no	—
229	Q₈	—	134	4	no	—
230	D₄	—	268	4	no	—
231	D₄	—	268	4	no	—
232	D₄	—	268	2	no	—
233	D₄	—	134	4	no	—
234	C₂×C₄	—	268	2	no	—
235	C₂×C₄	—	268	4	no	—
236	C₂×C₄	—	268	4	no	—
237	C₂×C₄	—	134	4	no	—
238	C₂×C₄	—	134	2	no	—
239	C₂×C₄	#14524005	1	1	yes	characteristic
240	C₂×C₄	#14524005	1	1	yes	characteristic
241	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
242	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
243	C₂×C₄	#14524066	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
245	C₈	—	4	4	no	—
246	C₈	—	67	2	no	—
247	C₈	—	67	2	no	—
248	C₈	#14524005	1	1	yes	characteristic
249	C₈	#14524005	1	1	yes	characteristic
250	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
251	C₈	C₂×#11761629	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	268	4	no	—
254	C₂²	—	536	4	no	—
255	C₂²	#16863478	1	1	yes	characteristic
256	C₂²	#16863478	1	1	yes	characteristic
257	C₂²	C₂×#14524066	1	1	yes	characteristic
258	C₄	—	268	4	no	—
259	C₄	—	134	4	no	—
260	C₄	#16863478	1	1	yes	characteristic
261	C₄	#16863478	1	1	yes	characteristic
262	C₄	C₂×#14524005	1	1	yes	characteristic
263	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	536	4	no	—
267	C₂	#22562089	1	2	yes	direct product factor
268	C₂	C₂×#16863478	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#368×C₆₇	C₂
socle	C₂²×C₆₇	C₄×D₈
prime radical	C₂×#16863828	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#368	D₆₇
hypocenter	C₆₇	C₂×#2178
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
2. C₂×#16863512	C₂	semidirect
3. #22562089	C₂	direct (N × H)
4. C₂×#16863202	C₂	semidirect
5. C₂×C₁₆×Dic₆₇	C₂	semidirect
6. C₂×#368×C₆₇	C₂	semidirect
9. #16863828	C₄	semidirect
11. C₂×Dic₅₃₆	C₄	semidirect
12. C₂×#14523988	C₄	semidirect
13. C₂×D₁₀₇₂	C₄	semidirect
16. #16863512	C₂²	semidirect
19. #16863202	C₂²	semidirect
20. C₁₆×Dic₆₇	C₂²	semidirect
23. #368×C₆₇	C₂²	semidirect
38. Dic₅₃₆	C₂×C₄	semidirect
39. #14523988	C₂×C₄	semidirect
40. D₁₀₇₂	C₂×C₄	semidirect
132. C₂×C₆₇	#2178	semidirect
138. C₂×#368	D₆₇	semidirect
141. C₆₇	C₂×#2178	semidirect
156. #368	C₂×D₆₇	semidirect
267. C₂	#22562089	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×#16863828	2
C₂×#16863512	2
#22562089	2
C₂×#16863202	2
C₂×C₁₆×Dic₆₇	2
C₂×#368×C₆₇	2
C₂×#2178	67

Subgroups — C2×#22562089 #31009230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562089 #31009230