Subgroups — C₂×#22572008 #31732056

Order 17152.

2766 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864197	C₂	1	4	yes	maximal, complemented
2	#22572008	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16863663	C₂	1	4	yes	maximal, complemented
4	C₂×#16863563	C₂	1	4	yes	maximal, complemented
5	C₂×#16863514	C₂	1	2	yes	maximal, complemented
6	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14524088	C₂²	1	4	yes	complemented
8	#16864197	—	2	16	no	—
9	#16864197	C₂²	1	32	yes	complemented
10	C₂×#14524069	C₂²	1	8	yes	complemented
11	C₂×#14524067	—	2	8	no	—
12	C₂×#14524065	C₂²	1	2	yes	—
13	C₂×#14524062	—	2	4	no	—
14	C₂×D₈×D₆₇	—	2	4	no	—
15	C₂×#14524049	C₂²	1	4	yes	complemented
16	#16863663	C₂²	1	32	yes	complemented
17	#16863563	C₂²	1	32	yes	complemented
18	#16863514	C₂²	1	16	yes	complemented
19	C₂×#14523979	C₂²	1	2	yes	complemented
20	C₂×#14523971	C₂²	1	4	yes	complemented
21	C₂×#14523960	C₂²	1	4	yes	complemented
22	C₂²×D₈×C₆₇	C₂²	1	4	yes	complemented
23	C₂×#127×C₆₇	C₂²	1	2	yes	complemented
24	#493×C₆₇	C₂²	1	8	yes	complemented
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	#14524088	—	2	16	no	—
27	#14524088	—	4	4	no	—
28	#14524088	C₂³	1	16	yes	complemented
29	C₂×#11761628	C₂³	1	4	yes	—
30	C₂×#11761627	—	2	8	no	—
31	C₂²×C₂²⋊D₆₇	—	4	4	no	—
32	C₂×D₄×D₆₇	—	2	8	no	—
33	#14524069	—	2	16	no	—
34	#14524069	C₂³	1	32	yes	complemented
35	#14524067	—	2	64	no	—
36	#14524065	D₄	1	8	yes	complemented
37	#14524065	C₂³	1	8	yes	—
38	#14524062	—	2	32	no	—
39	C₂×Dic₂₆₈	D₄	1	2	yes	complemented
40	C₂×#11761617	D₄	1	4	yes	complemented
41	C₂×#11761616	—	2	8	no	—
42	C₂×D₅₃₆	D₄	1	2	yes	complemented
43	C₂×#11761613	—	2	8	no	—
44	D₈×D₆₇	—	2	32	no	—
45	#14524049	—	2	8	no	—
46	#14524049	C₂³	1	16	yes	complemented
47	C₂×#11761612	C₂³	1	4	yes	—
48	C₂×#11761611	—	2	4	no	—
49	C₂×C₈×D₆₇	—	2	4	no	—
50	#14523979	C₂³	1	8	yes	complemented
51	#14523971	C₂³	1	16	yes	complemented
52	#14523960	C₂³	1	16	yes	complemented
53	C₂²×D₄×C₆₇	—	4	2	no	—
54	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
55	C₂×D₈×C₆₇	—	2	4	no	—
56	C₂×D₈×C₆₇	—	2	4	no	—
57	C₂×D₈×C₆₇	—	2	8	no	—
58	C₂×D₈×C₆₇	C₂³	1	16	yes	complemented
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	#127×C₆₇	—	4	2	no	—
61	#127×C₆₇	C₂³	1	8	yes	complemented
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂³	1	4	yes	complemented
64	C₂³×D₆₇	—	4	4	no	—
65	#11761628	—	2	4	no	—
66	#11761628	—	2	4	no	—
67	#11761628	C₂×D₄	1	8	yes	—
68	#11761628	C₂⁴	1	8	yes	—
69	#11761627	—	2	32	no	—
70	#11761627	—	4	16	no	—
71	C₂×Dic₁₃₄	C₂×D₄	1	4	yes	—
72	C₂×D₂₆₈	C₂×D₄	1	4	yes	—
73	C₂×C₂²⋊D₆₇	—	2	4	no	—
74	C₂×C₂²⋊D₆₇	—	4	8	no	—
75	C₂×C₂²⋊D₆₇	—	4	16	no	—
76	C₂×C₂²⋊D₆₇	—	4	16	no	—
77	D₄×D₆₇	—	2	32	no	—
78	D₄×D₆₇	—	4	16	no	—
79	C₂²×Dic₆₇	—	4	4	no	—
80	C₂×C₄×D₆₇	—	2	4	no	—
81	Dic₂₆₈	C₂×D₄	1	8	yes	complemented
82	#11761617	C₂×D₄	1	16	yes	complemented
83	#11761616	—	2	32	no	—
84	D₅₃₆	C₂×D₄	1	8	yes	complemented
85	#11761613	—	2	32	no	—
86	#11761612	—	2	4	no	—
87	#11761612	C₂⁴	1	8	yes	—
88	#11761611	—	2	16	no	—
89	C₂×C₆₇⋊C₈	—	2	4	no	—
90	C₈×D₆₇	—	2	16	no	—
91	C₂⁴×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	2	8	no	—
93	C₂×D₄×C₆₇	—	2	8	no	—
94	C₂×D₄×C₆₇	—	2	4	no	—
95	C₂×D₄×C₆₇	—	4	8	no	—
96	C₂×D₄×C₆₇	—	8	8	no	—
97	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	D₈×C₆₇	—	2	16	no	—
101	D₈×C₆₇	—	2	16	no	—
102	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
105	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
106	C₄²×C₆₇	—	2	1	no	—
107	C₄²×C₆₇	C₂⁴	1	2	yes	—
108	C₂³×C₆₇	—	2	8	no	—
109	C₂³×C₆₇	—	4	4	no	—
110	C₂³×C₆₇	—	4	4	no	—
111	C₂³×C₆₇	—	8	8	no	—
112	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×D₆₇	—	2	4	no	—
114	C₂²×D₆₇	—	4	16	no	—
115	C₂²×D₆₇	—	16	8	no	—
116	D₄×C₆₇	—	2	16	no	—
117	D₄×C₆₇	—	4	8	no	—
118	D₄×C₆₇	—	8	16	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
123	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
125	C₂²⋊D₆₇	—	2	8	no	—
126	C₂²⋊D₆₇	—	2	8	no	—
127	C₂²⋊D₆₇	—	4	32	no	—
128	C₂×Dic₆₇	—	2	4	no	—
129	C₂×Dic₆₇	—	4	16	no	—
130	D₂₆₈	—	2	4	no	—
131	D₂₆₈	C₂²×D₄	1	8	yes	—
132	C₄×D₆₇	—	2	8	no	—
133	C₄×D₆₇	—	2	8	no	—
134	Dic₁₃₄	—	2	4	no	—
135	Dic₁₃₄	C₂²×D₄	1	8	yes	—
136	C₈×C₆₇	C₂²×D₄	1	4	yes	—
137	C₈×C₆₇	C₂²×D₄	1	4	yes	—
138	C₆₇⋊C₈	—	2	8	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	4	8	no	—
141	C₂²×C₆₇	—	8	8	no	—
142	C₂²×C₆₇	—	8	8	no	—
143	C₂²×C₆₇	—	16	4	no	—
144	C₂²×C₆₇	D₄²	1	1	yes	characteristic
145	C₂²×C₆₇	D₄²	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×D₆₇	—	2	8	no	—
148	C₂×D₆₇	—	4	8	no	—
149	C₂×D₆₇	—	16	16	no	—
150	C₄×C₆₇	—	2	4	no	—
151	C₄×C₆₇	D₄²	1	2	yes	—
152	C₄×C₆₇	C₂³×D₄	1	2	yes	—
153	Dic₆₇	—	2	8	no	—
154	C₂×#3292	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	—	8	8	no	—
158	C₂×C₆₇	#3292	1	2	yes	complemented
159	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
160	D₆₇	—	4	8	no	—
161	C₂×#576	—	67	4	no	—
162	#3292	—	67	16	no	—
163	C₂×#493	D₆₇	1	1	yes	characteristic, complemented
164	C₂×#472	—	67	4	no	—
165	C₂×#453	—	67	4	no	—
166	C₂×#443	—	67	2	no	—
167	C₆₇	C₂×#3292	1	1	yes	characteristic, complemented, Sylow
168	C₂×H(5,𝔽₂)	—	67	4	no	—
169	#576	—	67	32	no	—
170	#576	—	134	16	no	—
171	C₂×AGL(1,ℤ/8)	—	67	8	no	—
172	C₂×AGL(1,ℤ/8)	—	134	8	no	—
173	C₂×#135	—	67	2	no	—
174	C₂×#135	—	134	4	no	—
175	C₂²×D₈	—	134	4	no	—
176	C₂²×D₈	C₂×D₆₇	1	4	yes	complemented
177	C₂×#131	—	67	4	no	—
178	C₂×#127	C₂×D₆₇	1	2	yes	complemented
179	#493	C₂×D₆₇	1	8	yes	complemented
180	#472	—	67	32	no	—
181	#453	—	67	32	no	—
182	#443	—	67	16	no	—
183	C₂×#108	—	67	2	no	—
184	C₂×#104	—	67	4	no	—
185	C₂×#100	—	67	4	no	—
186	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
187	H(5,𝔽₂)	—	268	4	no	—
188	H(5,𝔽₂)	—	67	16	no	—
189	H(5,𝔽₂)	—	134	16	no	—
190	C₂×#41	—	67	4	no	—
191	C₂×#41	—	134	8	no	—
192	C₂²×D₄	—	4	2	no	—
193	C₂²×D₄	—	268	4	no	—
194	C₂²×D₄	—	134	8	no	—
195	C₂²×D₄	C₂²×D₆₇	1	4	yes	—
196	AGL(1,ℤ/8)	—	67	32	no	—
197	AGL(1,ℤ/8)	—	134	64	no	—
198	AGL(1,ℤ/8)	—	134	16	no	—
199	#135	—	67	8	no	—
200	#135	—	67	8	no	—
201	#135	—	134	32	no	—
202	C₂×Q₁₆	—	67	2	no	—
203	C₂×SD₁₆	—	67	4	no	—
204	C₂×SD₁₆	—	134	8	no	—
205	C₂×D₈	—	2	4	no	—
206	C₂×D₈	—	2	4	no	—
207	C₂×D₈	—	2	8	no	—
208	C₂×D₈	—	67	2	no	—
209	C₂×D₈	—	134	32	no	—
210	C₂×D₈	—	134	8	no	—
211	C₂×D₈	C₂²×D₆₇	1	16	yes	complemented
212	#131	—	67	16	no	—
213	#131	—	134	8	no	—
214	C₂×M₁₆	—	67	4	no	—
215	C₂×M₁₆	—	134	4	no	—
216	C₂²×C₈	—	134	4	no	—
217	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
218	#127	—	4	2	no	—
219	#127	C₂²×D₆₇	1	8	yes	complemented
220	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
221	#108	—	67	8	no	—
222	#104	—	67	16	no	—
223	#100	—	67	16	no	—
224	C₄×C₈	C₂²×D₆₇	1	4	yes	complemented
225	C₂⁴	—	2	4	no	—
226	C₂⁴	—	268	4	no	—
227	#41	—	268	16	no	—
228	#41	—	67	8	no	—
229	#41	—	67	8	no	—
230	#41	—	134	32	no	—
231	#41	—	134	4	no	—
232	#41	—	134	4	no	—
233	C₂×Q₈	—	67	4	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	4	8	no	—
238	C₂×D₄	—	8	8	no	—
239	C₂×D₄	—	268	16	no	—
240	C₂×D₄	—	268	16	no	—
241	C₂×D₄	—	268	16	no	—
242	C₂×D₄	—	268	8	no	—
243	C₂×D₄	—	67	4	no	—
244	C₂×D₄	—	134	32	no	—
245	C₂×D₄	—	134	4	no	—
246	C₂×D₄	C₂³×D₆₇	1	8	yes	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	268	4	no	—
249	C₂²×C₄	—	134	4	no	—
250	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
251	Q₁₆	—	67	8	no	—
252	SD₁₆	—	67	16	no	—
253	SD₁₆	—	134	32	no	—
254	D₈	—	2	16	no	—
255	D₈	—	2	16	no	—
256	D₈	—	67	8	no	—
257	D₈	—	134	32	no	—
258	M₁₆	—	67	8	no	—
259	M₁₆	—	134	4	no	—
260	M₁₆	—	134	16	no	—
261	C₂×C₈	—	134	4	no	—
262	C₂×C₈	—	134	16	no	—
263	C₂×C₈	D₄×D₆₇	1	2	yes	—
264	C₂×C₈	D₄×D₆₇	1	2	yes	—
265	C₂×C₈	D₄×D₆₇	1	4	yes	—
266	C₂×C₈	C₂³×D₆₇	1	4	yes	—
267	C₄²	—	2	1	no	—
268	C₄²	C₂³×D₆₇	1	2	yes	—
269	C₂³	—	2	8	no	—
270	C₂³	—	4	4	no	—
271	C₂³	—	4	4	no	—
272	C₂³	—	8	8	no	—
273	C₂³	—	268	16	no	—
274	C₂³	—	1072	8	no	—
275	C₂³	—	134	4	no	—
276	C₂³	#14524088	1	1	yes	characteristic
277	Q₈	—	67	8	no	—
278	Q₈	—	134	4	no	—
279	D₄	—	2	16	no	—
280	D₄	—	4	8	no	—
281	D₄	—	8	16	no	—
282	D₄	—	268	32	no	—
283	D₄	—	67	8	no	—
284	D₄	—	134	8	no	—
285	D₄	—	134	8	no	—
286	D₄	—	134	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	268	16	no	—
290	C₂×C₄	—	134	4	no	—
291	C₂×C₄	—	134	8	no	—
292	C₂×C₄	—	134	8	no	—
293	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
294	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
295	C₂×C₄	#14524088	1	1	yes	characteristic
296	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
297	C₈	—	134	8	no	—
298	C₈	C₂×D₄×D₆₇	1	4	yes	—
299	C₈	C₂×D₄×D₆₇	1	4	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	8	8	no	—
303	C₂²	—	8	8	no	—
304	C₂²	—	268	8	no	—
305	C₂²	—	16	4	no	—
306	C₂²	—	1072	16	no	—
307	C₂²	—	134	8	no	—
308	C₂²	#16864031	1	1	yes	characteristic
309	C₂²	#16864031	1	1	yes	characteristic
310	C₂²	C₂×#14524088	1	1	yes	characteristic
311	C₄	—	2	4	no	—
312	C₄	—	134	8	no	—
313	C₄	#16864031	1	2	yes	—
314	C₄	C₂²×D₄×D₆₇	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	8	8	no	—
318	C₂	—	268	8	no	—
319	C₂	#22572008	1	2	yes	direct product factor
320	C₂	C₂×#16864031	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16864031
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#493×C₆₇	C₂
socle	C₂²×C₆₇	D₄²
hypercenter	C₂×#493	D₆₇
hypocenter	C₆₇	C₂×#3292
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16864197	C₂	semidirect
2. #22572008	C₂	direct (N × H)
3. C₂×#16863663	C₂	semidirect
4. C₂×#16863563	C₂	semidirect
5. C₂×#16863514	C₂	semidirect
6. C₂×#493×C₆₇	C₂	semidirect
7. C₂×#14524088	C₂²	semidirect
9. #16864197	C₂²	semidirect
10. C₂×#14524069	C₂²	semidirect
15. C₂×#14524049	C₂²	semidirect
16. #16863663	C₂²	semidirect
17. #16863563	C₂²	semidirect
18. #16863514	C₂²	semidirect
19. C₂×#14523979	C₂²	semidirect
20. C₂×#14523971	C₂²	semidirect
21. C₂×#14523960	C₂²	semidirect
22. C₂²×D₈×C₆₇	C₂²	semidirect
23. C₂×#127×C₆₇	C₂²	semidirect
24. #493×C₆₇	C₂²	semidirect
25. C₂×C₄×C₈×C₆₇	C₂²	semidirect
28. #14524088	C₂³	semidirect
34. #14524069	C₂³	semidirect
36. #14524065	D₄	semidirect
39. C₂×Dic₂₆₈	D₄	semidirect
40. C₂×#11761617	D₄	semidirect
42. C₂×D₅₃₆	D₄	semidirect
46. #14524049	C₂³	semidirect
50. #14523979	C₂³	semidirect
51. #14523971	C₂³	semidirect
52. #14523960	C₂³	semidirect
58. C₂×D₈×C₆₇	C₂³	semidirect
61. #127×C₆₇	C₂³	semidirect
63. C₄×C₈×C₆₇	C₂³	semidirect
81. Dic₂₆₈	C₂×D₄	semidirect
82. #11761617	C₂×D₄	semidirect
84. D₅₃₆	C₂×D₄	semidirect
158. C₂×C₆₇	#3292	semidirect
163. C₂×#493	D₆₇	semidirect
167. C₆₇	C₂×#3292	semidirect
176. C₂²×D₈	C₂×D₆₇	semidirect
178. C₂×#127	C₂×D₆₇	semidirect
179. #493	C₂×D₆₇	semidirect
186. C₂×C₄×C₈	C₂×D₆₇	semidirect
211. C₂×D₈	C₂²×D₆₇	semidirect
219. #127	C₂²×D₆₇	semidirect
224. C₄×C₈	C₂²×D₆₇	semidirect
319. C₂	#22572008	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×#16864197	2
#22572008	2
C₂×#16863663	2
C₂×#16863563	2
C₂×#16863514	2
C₂×#493×C₆₇	2
C₂×#3292	67

Subgroups — C2×#22572008 #31732056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22572008 #31732056