Subgroups — C₂×#22559832 #30843545

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863792	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863770	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863769	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863746	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863744	C₂	1	1	yes	maximal, characteristic
6	#22559832	C₂	1	8	yes	maximal, direct product factor
7	C₂×#16863117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11761617	—	2	2	no	—
10	C₂²×#11761615	—	2	2	no	—
11	C₂×#14524043	C₂²	1	1	yes	characteristic
12	C₂×#14524040	C₂²	1	1	yes	characteristic
13	#16863792	—	2	2	no	—
14	#16863792	C₂²	1	4	yes	complemented
15	#16863770	—	2	2	no	—
16	#16863770	C₂²	1	4	yes	—
17	#16863769	—	2	2	no	—
18	#16863769	C₂²	1	4	yes	—
19	#16863746	—	2	2	no	—
20	#16863746	C₂²	1	4	yes	complemented
21	#16863744	—	2	2	no	—
22	#16863744	C₂²	1	4	yes	—
23	C₂×#14523974	—	2	2	no	—
24	C₂×#14523967	—	2	2	no	—
25	C₂×#14523963	—	2	2	no	—
26	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
28	#16863117	C₂²	1	4	yes	complemented
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	#500×C₆₇	—	2	2	no	—
32	#500×C₆₇	C₂²	1	4	yes	complemented
33	C₂×#107×C₆₇	—	2	2	no	—
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×Dic₁₃₄	—	2	1	no	—
36	C₂²×D₂₆₈	—	2	1	no	—
37	C₂²×D₂₆₈	—	2	1	no	—
38	C₂²×D₂₆₈	—	2	1	no	—
39	C₂×#11761617	—	2	8	no	—
40	C₂×#11761617	—	4	8	no	—
41	C₂×#11761615	—	2	8	no	—
42	C₂×#11761615	—	4	8	no	—
43	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
44	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
45	#14524043	—	2	1	no	—
46	#14524043	—	2	1	no	—
47	#14524043	—	2	1	no	—
48	#14524043	C₂³	1	2	yes	—
49	#14524040	—	2	1	no	—
50	#14524040	—	2	1	no	—
51	#14524040	—	2	1	no	—
52	#14524040	C₂³	1	2	yes	—
53	C₂×C₄⋊Dic₆₇	—	2	1	no	—
54	C₂×C₄⋊Dic₆₇	—	2	1	no	—
55	#14523974	—	2	8	no	—
56	#14523967	—	2	8	no	—
57	#14523963	—	2	8	no	—
58	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
59	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
60	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
61	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
62	C₂²×Q₈×C₆₇	—	2	1	no	—
63	C₂²×Q₈×C₆₇	—	2	1	no	—
64	C₂²×C₈×C₆₇	D₄	1	2	yes	—
65	C₄⋊Q₈×C₆₇	—	2	1	no	—
66	C₄⋊Q₈×C₆₇	—	2	1	no	—
67	C₄⋊Q₈×C₆₇	—	2	1	no	—
68	C₄⋊Q₈×C₆₇	—	2	1	no	—
69	C₄⋊Q₈×C₆₇	—	2	1	no	—
70	C₄⋊Q₈×C₆₇	—	2	1	no	—
71	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	—	2	8	no	—
79	C₄×C₈×C₆₇	D₄	1	2	yes	—
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂³×D₆₇	—	4	1	no	—
82	C₂×Dic₁₃₄	—	2	2	no	—
83	C₂×Dic₁₃₄	—	4	4	no	—
84	C₂×Dic₁₃₄	—	4	1	no	—
85	C₂×D₂₆₈	—	2	2	no	—
86	C₂×D₂₆₈	—	2	2	no	—
87	C₂×D₂₆₈	—	2	2	no	—
88	C₂×D₂₆₈	—	4	4	no	—
89	C₂×D₂₆₈	—	4	4	no	—
90	C₂×D₂₆₈	—	4	4	no	—
91	C₂×D₂₆₈	—	4	1	no	—
92	C₂×D₂₆₈	—	4	1	no	—
93	C₂×D₂₆₈	—	4	1	no	—
94	C₂²×Dic₆₇	—	4	1	no	—
95	#11761617	—	4	16	no	—
96	#11761615	—	4	16	no	—
97	C₂×C₆₇⋊C₈	—	2	4	no	—
98	C₂×C₆₇⋊C₈	—	2	4	no	—
99	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
100	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
101	C₄⋊Dic₆₇	—	2	2	no	—
102	C₄⋊Dic₆₇	—	2	2	no	—
103	C₄⋊Dic₆₇	—	4	1	no	—
104	C₄⋊Dic₆₇	—	4	1	no	—
105	C₂×Q₈×C₆₇	—	2	2	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	4	1	no	—
108	C₂×Q₈×C₆₇	—	4	1	no	—
109	C₂×Q₈×C₆₇	—	4	4	no	—
110	C₂×Q₈×C₆₇	—	4	4	no	—
111	C₂²×C₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	—	4	1	no	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	SD₁₆	1	8	yes	complemented
117	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₄⋊C₄×C₆₇	—	4	1	no	—
124	C₄⋊C₄×C₆₇	—	4	1	no	—
125	C₄⋊C₄×C₆₇	—	4	1	no	—
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₂³×C₆₇	#127	1	1	yes	characteristic
131	C₂²×D₆₇	—	4	2	no	—
132	C₂²×D₆₇	—	8	2	no	—
133	C₂²×D₆₇	—	8	2	no	—
134	C₂²×D₆₇	—	8	2	no	—
135	Q₈×C₆₇	—	4	4	no	—
136	Q₈×C₆₇	—	4	4	no	—
137	Q₈×C₆₇	—	8	2	no	—
138	Q₈×C₆₇	—	8	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	8	2	no	—
142	C₂×C₄×C₆₇	—	8	2	no	—
143	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
145	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
147	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
148	C₂×C₄×C₆₇	#137	1	4	yes	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₆₇	—	4	2	no	—
153	C₂×Dic₆₇	—	8	2	no	—
154	D₂₆₈	—	4	4	no	—
155	D₂₆₈	—	4	4	no	—
156	D₂₆₈	—	4	4	no	—
157	D₂₆₈	—	8	2	no	—
158	D₂₆₈	—	8	2	no	—
159	D₂₆₈	—	8	2	no	—
160	Dic₁₃₄	—	4	4	no	—
161	Dic₁₃₄	—	8	2	no	—
162	C₈×C₆₇	C₂×SD₁₆	1	8	yes	complemented
163	C₆₇⋊C₈	—	2	4	no	—
164	C₆₇⋊C₈	—	2	4	no	—
165	C₂²×C₆₇	#492	1	2	yes	—
166	C₂²×C₆₇	#496	1	2	yes	—
167	C₂²×C₆₇	#497	1	2	yes	—
168	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
169	C₂×D₆₇	—	8	2	no	—
170	C₂×D₆₇	—	8	2	no	—
171	C₂×D₆₇	—	8	2	no	—
172	C₂×D₆₇	—	16	4	no	—
173	C₄×C₆₇	—	8	2	no	—
174	C₄×C₆₇	—	8	2	no	—
175	C₄×C₆₇	#492	1	2	yes	—
176	C₄×C₆₇	#496	1	2	yes	—
177	C₄×C₆₇	#497	1	2	yes	—
178	C₄×C₆₇	C₂²×SD₁₆	1	2	yes	—
179	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	C₄×C₆₇	C₂×#137	1	2	yes	—
181	Dic₆₇	—	8	2	no	—
182	C₂×#1714	—	67	1	no	maximal, Sylow
183	C₂×C₆₇	#1714	1	4	yes	complemented
184	C₂×C₆₇	C₂×#492	1	1	yes	characteristic
185	C₂×C₆₇	C₂×#496	1	1	yes	characteristic
186	C₂×C₆₇	C₂×#497	1	1	yes	characteristic
187	D₆₇	—	16	2	no	—
188	C₂×#500	D₆₇	1	1	yes	characteristic, complemented
189	C₂×#498	—	67	1	no	—
190	C₂×#492	—	67	1	no	—
191	C₂×#492	—	67	1	no	—
192	C₂×#487	—	67	1	no	—
193	C₂×#486	—	67	1	no	—
194	#1714	—	67	8	no	—
195	C₂×#322	—	67	1	no	—
196	C₆₇	C₂×#1714	1	1	yes	characteristic, complemented, Sylow
197	C₂²×SD₁₆	—	134	2	no	—
198	C₂²×SD₁₆	—	134	2	no	—
199	C₂×C₄⋊Q₈	—	67	1	no	—
200	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
201	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
202	C₂×#127	—	67	1	no	—
203	#500	—	2	2	no	—
204	#500	C₂×D₆₇	1	4	yes	complemented
205	#498	—	67	4	no	—
206	#498	—	134	2	no	—
207	#492	—	67	4	no	—
208	#492	—	67	4	no	—
209	#492	—	134	2	no	—
210	#492	—	134	2	no	—
211	#487	—	67	4	no	—
212	#487	—	134	2	no	—
213	#486	—	67	4	no	—
214	#486	—	134	2	no	—
215	C₂×#107	—	2	2	no	—
216	C₂×#106	—	134	2	no	—
217	C₂×#103	—	134	2	no	—
218	C₂×#102	—	134	2	no	—
219	C₂×C₄×C₈	—	67	1	no	—
220	C₂×C₄×C₈	—	67	1	no	—
221	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
222	#322	—	67	4	no	—
223	C₂²×Q₈	—	2	1	no	—
224	C₂²×Q₈	—	2	1	no	—
225	C₂²×Q₈	—	134	1	no	—
226	C₂²×D₄	—	134	1	no	—
227	C₂²×D₄	—	134	1	no	—
228	C₂²×D₄	—	134	1	no	—
229	C₂×SD₁₆	—	268	8	no	—
230	C₂×SD₁₆	—	268	8	no	—
231	C₂×SD₁₆	—	134	8	no	—
232	C₂×SD₁₆	—	134	8	no	—
233	C₂²×C₈	—	67	2	no	—
234	C₂²×C₈	—	67	2	no	—
235	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	—	67	2	no	—
243	C₄⋊Q₈	—	134	1	no	—
244	C₄⋊Q₈	—	134	1	no	—
245	C₄⋊Q₈	—	134	1	no	—
246	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
247	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
248	#127	—	67	2	no	—
249	#127	—	134	1	no	—
250	#127	—	134	1	no	—
251	#127	—	134	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	134	1	no	—
257	C₂×C₄⋊C₄	—	134	1	no	—
258	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
259	#107	—	2	8	no	—
260	#106	—	134	8	no	—
261	#103	—	134	8	no	—
262	#102	—	134	8	no	—
263	C₄×C₈	—	67	2	no	—
264	C₄×C₈	—	67	2	no	—
265	C₄×C₈	—	67	2	no	—
266	C₄×C₈	—	67	2	no	—
267	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
268	C₄×C₈	C₂²×D₆₇	1	2	yes	—
269	C₂⁴	—	268	1	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	4	4	no	—
275	C₂×Q₈	—	4	4	no	—
276	C₂×Q₈	—	268	1	no	—
277	C₂×Q₈	—	268	4	no	—
278	C₂×Q₈	—	134	2	no	—
279	C₂×D₄	—	268	1	no	—
280	C₂×D₄	—	268	1	no	—
281	C₂×D₄	—	268	1	no	—
282	C₂×D₄	—	268	4	no	—
283	C₂×D₄	—	268	4	no	—
284	C₂×D₄	—	268	4	no	—
285	C₂×D₄	—	134	2	no	—
286	C₂×D₄	—	134	2	no	—
287	C₂×D₄	—	134	2	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	268	1	no	—
291	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
294	SD₁₆	—	268	16	no	—
295	SD₁₆	—	268	16	no	—
296	C₂×C₈	—	67	4	no	—
297	C₂×C₈	—	67	4	no	—
298	C₂×C₈	—	134	4	no	—
299	C₂×C₈	—	134	4	no	—
300	C₂×C₈	#11761615	1	8	yes	complemented
301	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	268	1	no	—
311	C₄⋊C₄	—	268	1	no	—
312	C₄⋊C₄	—	134	2	no	—
313	C₄⋊C₄	—	134	2	no	—
314	C₄²	D₄×D₆₇	1	1	yes	characteristic
315	C₄²	D₄×D₆₇	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
317	C₄²	C₂³×D₆₇	1	1	yes	characteristic
318	C₂³	—	268	2	no	—
319	C₂³	—	536	2	no	—
320	C₂³	—	536	2	no	—
321	C₂³	—	536	2	no	—
322	C₂³	#14524039	1	1	yes	characteristic
323	Q₈	—	4	4	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	8	2	no	—
326	Q₈	—	8	2	no	—
327	Q₈	—	268	4	no	—
328	Q₈	—	536	2	no	—
329	D₄	—	268	4	no	—
330	D₄	—	268	4	no	—
331	D₄	—	268	4	no	—
332	D₄	—	536	2	no	—
333	D₄	—	536	2	no	—
334	D₄	—	536	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	268	2	no	—
340	C₂×C₄	—	536	2	no	—
341	C₂×C₄	#14524039	1	1	yes	characteristic
342	C₂×C₄	#14524039	1	1	yes	characteristic
343	C₂×C₄	#14524039	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761615	1	4	yes	—
345	C₂×C₄	#14524067	1	4	yes	—
346	C₂×C₄	#14524075	1	4	yes	—
347	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
350	C₈	—	134	4	no	—
351	C₈	—	134	4	no	—
352	C₈	C₂×#11761615	1	8	yes	complemented
353	C₂²	—	536	2	no	—
354	C₂²	—	536	2	no	—
355	C₂²	—	536	2	no	—
356	C₂²	—	1072	4	no	—
357	C₂²	#16863770	1	2	yes	—
358	C₂²	#16863784	1	2	yes	—
359	C₂²	#16863789	1	2	yes	—
360	C₂²	C₂×#14524039	1	1	yes	characteristic
361	C₄	—	8	2	no	—
362	C₄	—	8	2	no	—
363	C₄	—	536	2	no	—
364	C₄	#16863770	1	2	yes	—
365	C₄	#16863784	1	2	yes	—
366	C₄	#16863789	1	2	yes	—
367	C₄	C₂²×#11761615	1	2	yes	—
368	C₄	C₂×#14524067	1	2	yes	—
369	C₄	C₂×#14524075	1	2	yes	—
370	C₂	—	1072	2	no	—
371	C₂	#22559832	1	4	yes	direct product factor
372	C₂	C₂×#16863770	1	1	yes	characteristic
373	C₂	C₂×#16863784	1	1	yes	characteristic
374	C₂	C₂×#16863789	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#14524040	C₂²
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	C₂×#1714
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
1. C₂×#16863792	C₂	semidirect
4. C₂×#16863746	C₂	semidirect
6. #22559832	C₂	direct (N × H)
7. C₂×#16863117	C₂	semidirect
8. C₂×#500×C₆₇	C₂	semidirect
14. #16863792	C₂²	semidirect
20. #16863746	C₂²	semidirect
28. #16863117	C₂²	semidirect
32. #500×C₆₇	C₂²	semidirect
116. C₂×C₈×C₆₇	SD₁₆	semidirect
162. C₈×C₆₇	C₂×SD₁₆	semidirect
183. C₂×C₆₇	#1714	semidirect
188. C₂×#500	D₆₇	semidirect
196. C₆₇	C₂×#1714	semidirect
204. #500	C₂×D₆₇	semidirect
300. C₂×C₈	#11761615	semidirect
352. C₈	C₂×#11761615	semidirect
371. C₂	#22559832	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×#16863792	2
C₂×#16863770	2
C₂×#16863769	2
C₂×#16863746	2
C₂×#16863744	2
#22559832	2
C₂×#16863117	2
C₂×#500×C₆₇	2
C₂×#1714	67

Subgroups — C2×#22559832 #30843545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559832 #30843545