Subgroups — C₂×#22572009 #31732057

Order 17152.

2766 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864201	C₂	1	2	yes	maximal, complemented
2	C₂×#16864198	C₂	1	2	yes	maximal, complemented
3	#22572009	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16863773	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863667	C₂	1	2	yes	maximal, complemented
6	C₂×#16863666	C₂	1	2	yes	maximal, complemented
7	C₂×#16863565	C₂	1	4	yes	maximal, complemented
8	C₂×#16863514	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524089	C₂²	1	4	yes	complemented
11	#16864201	—	2	8	no	—
12	#16864201	C₂²	1	16	yes	complemented
13	#16864198	—	2	8	no	—
14	#16864198	C₂²	1	16	yes	complemented
15	C₂×#14524071	—	2	4	no	—
16	C₂×#14524071	C₂²	1	4	yes	complemented
17	C₂×#14524070	—	2	4	no	—
18	C₂×#14524068	C₂²	1	4	yes	complemented
19	C₂×#14524066	—	2	2	no	—
20	C₂×#14524065	—	2	2	no	—
21	C₂×#14524065	C₂²	1	1	yes	characteristic
22	C₂²×D₅₃₆	—	2	2	no	—
23	C₂²×D₅₃₆	C₂²	1	2	yes	complemented
24	C₂²×D₅₃₆	C₂²	1	2	yes	complemented
25	C₂×D₈×D₆₇	—	2	2	no	—
26	C₂×#14524049	C₂²	1	2	yes	complemented
27	C₂×#14524040	C₂²	1	2	yes	complemented
28	#16863773	C₂²	1	8	yes	complemented
29	#16863667	C₂²	1	16	yes	complemented
30	#16863666	C₂²	1	16	yes	complemented
31	#16863565	C₂²	1	32	yes	complemented
32	#16863514	C₂²	1	8	yes	complemented
33	C₂×#14523979	C₂²	1	1	yes	characteristic, complemented
34	C₂×#14523971	C₂²	1	2	yes	complemented
35	C₂×#14523959	C₂²	1	4	yes	complemented
36	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
38	#443×C₆₇	C₂²	1	8	yes	complemented
39	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	C₂×#104×C₆₇	C₂²	1	2	yes	complemented
41	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	#14524089	—	2	8	no	—
43	#14524089	—	2	8	no	—
44	#14524089	—	4	4	no	—
45	#14524089	C₂³	1	16	yes	complemented
46	C₂×#11761629	—	2	4	no	—
47	C₂×#11761628	—	2	4	no	—
48	C₂×#11761628	C₂³	1	2	yes	—
49	C₂²×D₂₆₈	—	2	4	no	—
50	C₂²×D₂₆₈	—	4	2	no	—
51	C₂²×D₂₆₈	C₂³	1	4	yes	—
52	C₂×D₄×D₆₇	—	2	4	no	—
53	C₂×D₄×D₆₇	—	4	4	no	—
54	#14524071	—	2	8	no	—
55	#14524071	—	2	32	no	—
56	#14524071	C₂³	1	16	yes	complemented
57	#14524070	—	2	32	no	—
58	#14524068	—	2	8	no	—
59	#14524068	C₂³	1	16	yes	complemented
60	#14524066	—	2	16	no	—
61	#14524065	—	2	16	no	—
62	#14524065	D₄	1	4	yes	complemented
63	#14524065	C₂³	1	4	yes	—
64	C₂×Dic₂₆₈	D₄	1	1	yes	characteristic, complemented
65	C₂×#11761617	—	2	4	no	—
66	C₂×#11761617	D₄	1	2	yes	complemented
67	C₂×#11761615	—	2	4	no	—
68	C₂×D₅₃₆	—	2	4	no	—
69	C₂×D₅₃₆	—	2	16	no	—
70	C₂×D₅₃₆	—	2	2	no	—
71	C₂×D₅₃₆	—	2	2	no	—
72	C₂×D₅₃₆	—	2	2	no	—
73	C₂×D₅₃₆	—	2	2	no	—
74	C₂×D₅₃₆	—	2	4	no	—
75	C₂×D₅₃₆	—	2	4	no	—
76	C₂×D₅₃₆	D₄	1	1	yes	characteristic, complemented
77	C₂×D₅₃₆	C₂³	1	8	yes	complemented
78	C₂×D₅₃₆	C₂³	1	8	yes	complemented
79	C₂×#11761613	—	2	4	no	—
80	D₈×D₆₇	—	2	16	no	—
81	#14524049	—	2	4	no	—
82	#14524049	C₂³	1	8	yes	complemented
83	C₂×#11761612	C₂³	1	2	yes	—
84	C₂×#11761611	—	2	2	no	—
85	C₂×C₈×D₆₇	—	2	2	no	—
86	#14524040	—	4	2	no	—
87	#14524040	C₂³	1	8	yes	complemented
88	#14523979	C₂³	1	4	yes	complemented
89	#14523971	C₂³	1	8	yes	complemented
90	#14523959	C₂³	1	16	yes	complemented
91	C₂×#41×C₆₇	C₂³	1	2	yes	—
92	#135×C₆₇	D₄	1	4	yes	complemented
93	#135×C₆₇	C₂³	1	4	yes	—
94	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic, complemented
95	C₂×SD₁₆×C₆₇	D₄	1	2	yes	complemented
96	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic, complemented
97	#131×C₆₇	—	2	4	no	—
98	#131×C₆₇	C₂³	1	8	yes	complemented
99	C₂×M₁₆×C₆₇	—	2	2	no	—
100	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
101	C₂²×C₈×C₆₇	—	2	2	no	—
102	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	#108×C₆₇	C₂³	1	4	yes	complemented
106	#104×C₆₇	C₂³	1	8	yes	complemented
107	C₄×C₈×C₆₇	C₂³	1	4	yes	complemented
108	C₂³×D₆₇	—	2	4	no	—
109	C₂³×D₆₇	—	4	4	no	—
110	#11761629	—	2	16	no	—
111	#11761629	—	4	8	no	—
112	#11761628	—	2	2	no	—
113	#11761628	—	2	2	no	—
114	#11761628	—	2	16	no	—
115	#11761628	—	4	8	no	—
116	#11761628	C₂×D₄	1	4	yes	—
117	#11761628	C₂⁴	1	4	yes	—
118	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
119	C₂×D₂₆₈	—	2	4	no	—
120	C₂×D₂₆₈	—	2	4	no	—
121	C₂×D₂₆₈	—	2	4	no	—
122	C₂×D₂₆₈	—	2	4	no	—
123	C₂×D₂₆₈	—	2	16	no	—
124	C₂×D₂₆₈	—	2	4	no	—
125	C₂×D₂₆₈	—	4	8	no	—
126	C₂×D₂₆₈	—	4	4	no	—
127	C₂×D₂₆₈	—	4	8	no	—
128	C₂×D₂₆₈	—	8	8	no	—
129	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
130	C₂×D₂₆₈	C₂⁴	1	8	yes	—
131	C₂×C₂²⋊D₆₇	—	2	2	no	—
132	C₂×C₂²⋊D₆₇	—	4	4	no	—
133	D₄×D₆₇	—	2	16	no	—
134	D₄×D₆₇	—	4	8	no	—
135	D₄×D₆₇	—	4	16	no	—
136	D₄×D₆₇	—	4	16	no	—
137	C₂×C₄×D₆₇	—	2	2	no	—
138	C₂×C₄×D₆₇	—	4	4	no	—
139	Dic₂₆₈	C₂×D₄	1	4	yes	complemented
140	#11761617	—	2	16	no	—
141	#11761617	C₂×D₄	1	8	yes	complemented
142	#11761615	—	2	16	no	—
143	D₅₃₆	—	2	16	no	—
144	D₅₃₆	—	2	8	no	—
145	D₅₃₆	—	2	8	no	—
146	D₅₃₆	—	2	8	no	—
147	D₅₃₆	—	2	8	no	—
148	D₅₃₆	C₂×D₄	1	4	yes	complemented
149	#11761613	—	2	16	no	—
150	#11761612	—	2	2	no	—
151	#11761612	C₂⁴	1	4	yes	—
152	#11761611	—	2	8	no	—
153	C₂×C₆₇⋊C₈	—	2	2	no	—
154	C₈×D₆₇	—	2	8	no	—
155	#41×C₆₇	—	2	2	no	—
156	#41×C₆₇	—	2	2	no	—
157	#41×C₆₇	C₂×D₄	1	4	yes	—
158	#41×C₆₇	C₂⁴	1	4	yes	—
159	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
160	C₂×D₄×C₆₇	—	2	2	no	—
161	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
165	Q₁₆×C₆₇	C₂×D₄	1	4	yes	complemented
166	SD₁₆×C₆₇	C₂×D₄	1	8	yes	complemented
167	D₈×C₆₇	C₂×D₄	1	4	yes	complemented
168	M₁₆×C₆₇	—	2	2	no	—
169	M₁₆×C₆₇	—	2	8	no	—
170	M₁₆×C₆₇	C₂⁴	1	4	yes	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	2	8	no	—
173	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
176	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
179	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
180	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
181	C₄²×C₆₇	—	2	1	no	—
182	C₄²×C₆₇	C₂⁴	1	2	yes	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂²×D₆₇	—	2	2	no	—
186	C₂²×D₆₇	—	2	8	no	—
187	C₂²×D₆₇	—	4	16	no	—
188	C₂²×D₆₇	—	4	4	no	—
189	C₂²×D₆₇	—	4	4	no	—
190	C₂²×D₆₇	—	8	8	no	—
191	C₂²×D₆₇	—	16	8	no	—
192	Q₈×C₆₇	—	2	2	no	—
193	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
194	D₄×C₆₇	—	2	4	no	—
195	D₄×C₆₇	—	2	4	no	—
196	D₄×C₆₇	—	2	2	no	—
197	D₄×C₆₇	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
209	C₂²⋊D₆₇	—	2	4	no	—
210	C₂²⋊D₆₇	—	2	4	no	—
211	C₂²⋊D₆₇	—	4	16	no	—
212	C₂×Dic₆₇	—	2	2	no	—
213	D₂₆₈	—	2	2	no	—
214	D₂₆₈	—	2	8	no	—
215	D₂₆₈	—	2	8	no	—
216	D₂₆₈	—	4	16	no	—
217	D₂₆₈	—	4	4	no	—
218	D₂₆₈	—	4	4	no	—
219	D₂₆₈	—	8	16	no	—
220	D₂₆₈	C₂²×D₄	1	4	yes	—
221	C₄×D₆₇	—	2	4	no	—
222	C₄×D₆₇	—	2	4	no	—
223	C₄×D₆₇	—	4	16	no	—
224	Dic₁₃₄	—	2	2	no	—
225	Dic₁₃₄	C₂²×D₄	1	4	yes	—
226	C₈×C₆₇	—	2	4	no	—
227	C₈×C₆₇	C₂²×D₄	1	2	yes	—
228	C₈×C₆₇	C₂²×D₄	1	2	yes	—
229	C₈×C₆₇	C₂²×D₄	1	2	yes	—
230	C₈×C₆₇	C₂²×D₄	1	2	yes	—
231	C₆₇⋊C₈	—	2	4	no	—
232	C₂²×C₆₇	—	2	2	no	—
233	C₂²×C₆₇	—	2	4	no	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	D₄²	1	1	yes	characteristic
236	C₂²×C₆₇	D₄²	1	1	yes	characteristic
237	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
238	C₂×D₆₇	—	2	4	no	—
239	C₂×D₆₇	—	4	4	no	—
240	C₂×D₆₇	—	4	8	no	—
241	C₂×D₆₇	—	8	8	no	—
242	C₂×D₆₇	—	8	8	no	—
243	C₂×D₆₇	—	16	16	no	—
244	C₂×D₆₇	—	16	4	no	—
245	C₄×C₆₇	—	2	4	no	—
246	C₄×C₆₇	—	2	4	no	—
247	C₄×C₆₇	D₄²	1	1	yes	characteristic
248	C₄×C₆₇	D₄²	1	1	yes	characteristic
249	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
250	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
251	Dic₆₇	—	2	4	no	—
252	C₂×#3292	—	67	1	no	maximal, Sylow
253	C₂×C₆₇	—	2	1	no	—
254	C₂×C₆₇	—	2	1	no	—
255	C₂×C₆₇	—	4	4	no	—
256	C₂×C₆₇	#3292	1	2	yes	complemented
257	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
258	D₆₇	—	4	4	no	—
259	D₆₇	—	8	8	no	—
260	C₂×#576	—	67	2	no	—
261	C₂×#576	—	67	2	no	—
262	#3292	—	67	16	no	—
263	C₂×#493	—	67	1	no	—
264	C₂×#472	—	67	2	no	—
265	C₂×#472	—	67	2	no	—
266	C₂×#453	—	67	4	no	—
267	C₂×#443	—	67	1	no	—
268	C₂×#443	D₆₇	1	1	yes	characteristic, complemented
269	C₆₇	C₂×#3292	1	1	yes	characteristic, complemented, Sylow
270	C₂×H(5,𝔽₂)	—	67	4	no	—
271	#576	—	67	16	no	—
272	#576	—	67	16	no	—
273	#576	—	134	8	no	—
274	#576	—	134	8	no	—
275	C₂×AGL(1,ℤ/8)	—	67	4	no	—
276	C₂×AGL(1,ℤ/8)	—	67	4	no	—
277	C₂×AGL(1,ℤ/8)	—	134	4	no	—
278	C₂×AGL(1,ℤ/8)	—	134	4	no	—
279	C₂×#135	—	67	1	no	—
280	C₂×#135	—	134	2	no	—
281	C₂×#135	—	134	2	no	—
282	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
283	C₂²×D₈	—	67	2	no	—
284	C₂²×D₈	—	67	2	no	—
285	C₂²×D₈	—	134	2	no	—
286	C₂²×D₈	—	134	2	no	—
287	C₂×#131	—	67	2	no	—
288	C₂×#131	C₂×D₆₇	1	2	yes	complemented
289	C₂×#127	—	67	2	no	—
290	#493	—	67	8	no	—
291	#472	—	67	16	no	—
292	#472	—	67	16	no	—
293	#453	—	67	32	no	—
294	#443	—	67	8	no	—
295	#443	C₂×D₆₇	1	8	yes	complemented
296	C₂×#108	—	67	1	no	—
297	C₂×#108	C₂×D₆₇	1	1	yes	characteristic, complemented
298	C₂×#104	—	67	2	no	—
299	C₂×#104	C₂×D₆₇	1	2	yes	complemented
300	C₂×#100	—	67	4	no	—
301	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
302	H(5,𝔽₂)	—	268	4	no	—
303	H(5,𝔽₂)	—	67	16	no	—
304	H(5,𝔽₂)	—	134	8	no	—
305	H(5,𝔽₂)	—	134	8	no	—
306	C₂×#41	—	67	2	no	—
307	C₂×#41	—	134	4	no	—
308	C₂×#41	—	134	4	no	—
309	C₂×#41	C₂²×D₆₇	1	2	yes	—
310	C₂²×D₄	—	268	2	no	—
311	C₂²×D₄	—	268	4	no	—
312	C₂²×D₄	—	67	4	no	—
313	C₂²×D₄	—	134	4	no	—
314	C₂²×D₄	—	134	4	no	—
315	AGL(1,ℤ/8)	—	67	16	no	—
316	AGL(1,ℤ/8)	—	67	16	no	—
317	AGL(1,ℤ/8)	—	134	32	no	—
318	AGL(1,ℤ/8)	—	134	8	no	—
319	AGL(1,ℤ/8)	—	134	8	no	—
320	AGL(1,ℤ/8)	—	134	32	no	—
321	#135	—	67	4	no	—
322	#135	—	67	4	no	—
323	#135	—	134	16	no	—
324	#135	—	134	16	no	—
325	#135	D₂₆₈	1	4	yes	complemented
326	#135	C₂²×D₆₇	1	4	yes	—
327	C₂×Q₁₆	—	67	1	no	—
328	C₂×Q₁₆	D₂₆₈	1	1	yes	characteristic, complemented
329	C₂×SD₁₆	—	67	2	no	—
330	C₂×SD₁₆	—	134	4	no	—
331	C₂×SD₁₆	—	134	4	no	—
332	C₂×SD₁₆	D₂₆₈	1	2	yes	complemented
333	C₂×D₈	—	67	1	no	—
334	C₂×D₈	—	67	8	no	—
335	C₂×D₈	—	67	8	no	—
336	C₂×D₈	—	134	16	no	—
337	C₂×D₈	—	134	4	no	—
338	C₂×D₈	—	134	4	no	—
339	C₂×D₈	—	134	16	no	—
340	C₂×D₈	—	134	2	no	—
341	C₂×D₈	—	134	2	no	—
342	C₂×D₈	—	134	4	no	—
343	C₂×D₈	—	134	2	no	—
344	C₂×D₈	—	134	2	no	—
345	C₂×D₈	—	134	4	no	—
346	C₂×D₈	D₂₆₈	1	1	yes	characteristic, complemented
347	#131	—	2	4	no	—
348	#131	—	67	8	no	—
349	#131	—	134	4	no	—
350	#131	C₂²×D₆₇	1	8	yes	complemented
351	C₂×M₁₆	—	2	2	no	—
352	C₂×M₁₆	—	67	2	no	—
353	C₂×M₁₆	—	134	2	no	—
354	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
355	C₂²×C₈	—	2	2	no	—
356	C₂²×C₈	—	134	2	no	—
357	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
358	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
359	#127	—	268	2	no	—
360	#127	—	67	8	no	—
361	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
362	#108	—	67	4	no	—
363	#108	C₂²×D₆₇	1	4	yes	complemented
364	#104	—	67	8	no	—
365	#104	C₂²×D₆₇	1	8	yes	complemented
366	#100	—	67	16	no	—
367	C₄×C₈	C₂²×D₆₇	1	4	yes	complemented
368	C₂⁴	—	268	4	no	—
369	C₂⁴	—	134	4	no	—
370	#41	—	2	2	no	—
371	#41	—	2	2	no	—
372	#41	—	268	8	no	—
373	#41	—	268	8	no	—
374	#41	—	67	4	no	—
375	#41	—	67	4	no	—
376	#41	—	134	16	no	—
377	#41	—	134	16	no	—
378	#41	—	134	2	no	—
379	#41	—	134	2	no	—
380	#41	C₂×D₂₆₈	1	4	yes	—
381	#41	C₂³×D₆₇	1	4	yes	—
382	C₂×Q₈	—	67	2	no	—
383	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	268	4	no	—
386	C₂×D₄	—	268	8	no	—
387	C₂×D₄	—	268	8	no	—
388	C₂×D₄	—	268	16	no	—
389	C₂×D₄	—	268	16	no	—
390	C₂×D₄	—	268	8	no	—
391	C₂×D₄	—	268	4	no	—
392	C₂×D₄	—	536	8	no	—
393	C₂×D₄	—	67	2	no	—
394	C₂×D₄	—	67	8	no	—
395	C₂×D₄	—	134	16	no	—
396	C₂×D₄	—	134	16	no	—
397	C₂×D₄	—	134	2	no	—
398	C₂×D₄	—	134	4	no	—
399	C₂×D₄	—	134	4	no	—
400	C₂×D₄	—	134	4	no	—
401	C₂×D₄	—	134	4	no	—
402	C₂×D₄	—	134	4	no	—
403	C₂×D₄	C₂×D₂₆₈	1	2	yes	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	268	4	no	—
407	C₂²×C₄	—	134	2	no	—
408	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
409	Q₁₆	—	67	4	no	—
410	Q₁₆	C₂×D₂₆₈	1	4	yes	complemented
411	SD₁₆	—	67	8	no	—
412	SD₁₆	—	134	16	no	—
413	SD₁₆	—	134	16	no	—
414	SD₁₆	C₂×D₂₆₈	1	8	yes	complemented
415	D₈	—	67	4	no	—
416	D₈	—	134	16	no	—
417	D₈	—	134	16	no	—
418	D₈	—	134	8	no	—
419	D₈	—	134	8	no	—
420	D₈	—	134	8	no	—
421	D₈	—	134	8	no	—
422	D₈	C₂×D₂₆₈	1	4	yes	complemented
423	M₁₆	—	2	2	no	—
424	M₁₆	—	2	8	no	—
425	M₁₆	—	67	4	no	—
426	M₁₆	—	134	2	no	—
427	M₁₆	—	134	8	no	—
428	M₁₆	C₂³×D₆₇	1	4	yes	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	8	no	—
431	C₂×C₈	—	134	2	no	—
432	C₂×C₈	—	134	8	no	—
433	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
434	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
435	C₂×C₈	D₄×D₆₇	1	2	yes	—
436	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
437	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
438	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
439	C₂×C₈	C₂³×D₆₇	1	2	yes	—
440	C₂×C₈	C₂³×D₆₇	1	2	yes	—
441	C₄²	—	2	1	no	—
442	C₄²	C₂³×D₆₇	1	2	yes	—
443	C₂³	—	2	2	no	—
444	C₂³	—	268	16	no	—
445	C₂³	—	268	4	no	—
446	C₂³	—	268	4	no	—
447	C₂³	—	536	8	no	—
448	C₂³	—	1072	8	no	—
449	C₂³	—	134	2	no	—
450	C₂³	—	134	8	no	—
451	C₂³	#14524089	1	1	yes	characteristic
452	Q₈	—	2	2	no	—
453	Q₈	—	67	4	no	—
454	Q₈	—	134	2	no	—
455	Q₈	C₂²×D₂₆₈	1	4	yes	—
456	D₄	—	2	4	no	—
457	D₄	—	2	4	no	—
458	D₄	—	2	2	no	—
459	D₄	—	268	16	no	—
460	D₄	—	268	16	no	—
461	D₄	—	268	4	no	—
462	D₄	—	268	4	no	—
463	D₄	—	536	16	no	—
464	D₄	—	67	4	no	—
465	D₄	—	134	4	no	—
466	D₄	—	134	4	no	—
467	D₄	—	134	2	no	—
468	D₄	—	134	8	no	—
469	D₄	—	134	8	no	—
470	D₄	C₂²×D₂₆₈	1	4	yes	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	268	16	no	—
477	C₂×C₄	—	134	2	no	—
478	C₂×C₄	—	134	4	no	—
479	C₂×C₄	—	134	4	no	—
480	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
481	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
484	C₂×C₄	#14524089	1	1	yes	characteristic
485	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
486	C₈	—	2	4	no	—
487	C₈	—	134	4	no	—
488	C₈	C₂×D₄×D₆₇	1	2	yes	—
489	C₈	C₂×D₄×D₆₇	1	2	yes	—
490	C₈	C₂²×D₂₆₈	1	2	yes	—
491	C₈	C₂²×D₂₆₈	1	2	yes	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	268	4	no	—
496	C₂²	—	268	8	no	—
497	C₂²	—	536	8	no	—
498	C₂²	—	536	8	no	—
499	C₂²	—	1072	4	no	—
500	C₂²	—	1072	16	no	—
501	C₂²	—	134	4	no	—
502	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
503	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
504	C₂²	C₂×#14524089	1	1	yes	characteristic
505	C₄	—	2	4	no	—
506	C₄	—	2	4	no	—
507	C₄	—	134	4	no	—
508	C₄	D₄×D₂₆₈	1	1	yes	characteristic
509	C₄	D₄×D₂₆₈	1	1	yes	characteristic
510	C₄	C₂²×D₄×D₆₇	1	1	yes	characteristic
511	C₄	C₂³×D₂₆₈	1	1	yes	characteristic
512	C₂	—	2	1	no	—
513	C₂	—	2	1	no	—
514	C₂	—	4	4	no	—
515	C₂	—	268	4	no	—
516	C₂	—	536	8	no	—
517	C₂	#22572009	1	2	yes	direct product factor
518	C₂	C₂×D₄×D₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#443×C₆₇	C₂
socle	C₂²×C₆₇	D₄²
hypercenter	C₂×#443	D₆₇
hypocenter	C₆₇	C₂×#3292
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#16864201	C₂	semidirect
2. C₂×#16864198	C₂	semidirect
3. #22572009	C₂	direct (N × H)
4. C₂×#16863773	C₂	semidirect
5. C₂×#16863667	C₂	semidirect
6. C₂×#16863666	C₂	semidirect
7. C₂×#16863565	C₂	semidirect
8. C₂×#16863514	C₂	semidirect
9. C₂×#443×C₆₇	C₂	semidirect
10. C₂×#14524089	C₂²	semidirect
12. #16864201	C₂²	semidirect
14. #16864198	C₂²	semidirect
16. C₂×#14524071	C₂²	semidirect
18. C₂×#14524068	C₂²	semidirect
23. C₂²×D₅₃₆	C₂²	semidirect
24. C₂²×D₅₃₆	C₂²	semidirect
26. C₂×#14524049	C₂²	semidirect
27. C₂×#14524040	C₂²	semidirect
28. #16863773	C₂²	semidirect
29. #16863667	C₂²	semidirect
30. #16863666	C₂²	semidirect
31. #16863565	C₂²	semidirect
32. #16863514	C₂²	semidirect
33. C₂×#14523979	C₂²	semidirect
34. C₂×#14523971	C₂²	semidirect
35. C₂×#14523959	C₂²	semidirect
37. C₂×#131×C₆₇	C₂²	semidirect
38. #443×C₆₇	C₂²	semidirect
39. C₂×#108×C₆₇	C₂²	semidirect
40. C₂×#104×C₆₇	C₂²	semidirect
41. C₂×C₄×C₈×C₆₇	C₂²	semidirect
45. #14524089	C₂³	semidirect
56. #14524071	C₂³	semidirect
59. #14524068	C₂³	semidirect
62. #14524065	D₄	semidirect
64. C₂×Dic₂₆₈	D₄	semidirect
66. C₂×#11761617	D₄	semidirect
76. C₂×D₅₃₆	D₄	semidirect
77. C₂×D₅₃₆	C₂³	semidirect
78. C₂×D₅₃₆	C₂³	semidirect
82. #14524049	C₂³	semidirect
87. #14524040	C₂³	semidirect
88. #14523979	C₂³	semidirect
89. #14523971	C₂³	semidirect
90. #14523959	C₂³	semidirect
92. #135×C₆₇	D₄	semidirect
94. C₂×Q₁₆×C₆₇	D₄	semidirect
95. C₂×SD₁₆×C₆₇	D₄	semidirect
96. C₂×D₈×C₆₇	D₄	semidirect
98. #131×C₆₇	C₂³	semidirect
105. #108×C₆₇	C₂³	semidirect
106. #104×C₆₇	C₂³	semidirect
107. C₄×C₈×C₆₇	C₂³	semidirect
139. Dic₂₆₈	C₂×D₄	semidirect
141. #11761617	C₂×D₄	semidirect
148. D₅₃₆	C₂×D₄	semidirect
165. Q₁₆×C₆₇	C₂×D₄	semidirect
166. SD₁₆×C₆₇	C₂×D₄	semidirect
167. D₈×C₆₇	C₂×D₄	semidirect
256. C₂×C₆₇	#3292	semidirect
268. C₂×#443	D₆₇	semidirect
269. C₆₇	C₂×#3292	semidirect
288. C₂×#131	C₂×D₆₇	semidirect
295. #443	C₂×D₆₇	semidirect
297. C₂×#108	C₂×D₆₇	semidirect
299. C₂×#104	C₂×D₆₇	semidirect
301. C₂×C₄×C₈	C₂×D₆₇	semidirect
325. #135	D₂₆₈	semidirect
328. C₂×Q₁₆	D₂₆₈	semidirect
332. C₂×SD₁₆	D₂₆₈	semidirect
346. C₂×D₈	D₂₆₈	semidirect
350. #131	C₂²×D₆₇	semidirect
363. #108	C₂²×D₆₇	semidirect
365. #104	C₂²×D₆₇	semidirect
367. C₄×C₈	C₂²×D₆₇	semidirect
410. Q₁₆	C₂×D₂₆₈	semidirect
414. SD₁₆	C₂×D₂₆₈	semidirect
422. D₈	C₂×D₂₆₈	semidirect
517. C₂	#22572009	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×#16864201	2
C₂×#16864198	2
#22572009	2
C₂×#16863773	2
C₂×#16863667	2
C₂×#16863666	2
C₂×#16863565	2
C₂×#16863514	2
C₂×#443×C₆₇	2
C₂×#3292	67

Subgroups — C2×#22572009 #31732057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22572009 #31732057