Subgroups — C₂×#22559861 #30843586

Order 17152.

850 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863768	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863748	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863746	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863742	C₂	1	1	yes	maximal, characteristic
6	#22559861	C₂	1	8	yes	maximal, direct product factor
7	C₂×#16863118	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11761616	—	2	2	no	—
10	C₂×#14524043	C₂²	1	1	yes	characteristic
11	C₂×#14524043	C₂²	1	1	yes	characteristic
12	#16863801	—	2	2	no	—
13	#16863801	C₂²	1	4	yes	complemented
14	#16863768	—	2	2	no	—
15	#16863768	C₂²	1	4	yes	—
16	#16863748	—	2	2	no	—
17	#16863748	C₂²	1	4	yes	complemented
18	#16863746	—	2	2	no	—
19	#16863746	C₂²	1	4	yes	complemented
20	#16863742	—	2	2	no	—
21	#16863742	C₂²	1	4	yes	—
22	C₂×#14523977	—	2	2	no	—
23	C₂×#14523967	—	2	2	no	—
24	C₂×#14523966	—	2	2	no	—
25	C₂×#14523964	—	2	2	no	—
26	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
28	#16863118	C₂²	1	4	yes	complemented
29	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
31	#486×C₆₇	—	2	2	no	—
32	#486×C₆₇	C₂²	1	4	yes	—
33	C₂×#102×C₆₇	—	2	2	no	—
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×Dic₁₃₄	—	2	1	no	—
36	C₂²×Dic₁₃₄	—	2	1	no	—
37	C₂×#11761616	—	2	8	no	—
38	C₂×#11761616	—	4	8	no	—
39	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
40	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
41	#14524043	—	2	1	no	—
42	#14524043	—	2	1	no	—
43	#14524043	—	2	1	no	—
44	#14524043	—	2	1	no	—
45	#14524043	—	2	1	no	—
46	#14524043	—	2	1	no	—
47	#14524043	C₂³	1	2	yes	—
48	#14524043	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₆₇	—	2	1	no	—
50	C₂×C₄⋊Dic₆₇	—	2	1	no	—
51	C₂×C₄⋊Dic₆₇	—	2	1	no	—
52	C₂×C₄⋊Dic₆₇	—	2	1	no	—
53	#14523977	—	2	8	no	—
54	#14523967	—	2	8	no	—
55	#14523966	—	2	8	no	—
56	#14523964	—	2	8	no	—
57	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
58	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
59	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
60	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
61	C₂²×Q₈×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	—	2	1	no	—
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	—	2	1	no	—
65	C₂²×C₈×C₆₇	D₄	1	2	yes	—
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	C₂³	1	2	yes	—
70	#127×C₆₇	—	2	1	no	—
71	#127×C₆₇	—	2	1	no	—
72	#127×C₆₇	—	2	1	no	—
73	#127×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#102×C₆₇	—	2	8	no	—
78	C₄×C₈×C₆₇	D₄	1	2	yes	—
79	C₄×C₈×C₆₇	C₂³	1	2	yes	—
80	C₂×Dic₁₃₄	—	2	2	no	—
81	C₂×Dic₁₃₄	—	2	2	no	—
82	C₂×Dic₁₃₄	—	4	4	no	—
83	C₂×Dic₁₃₄	—	4	4	no	—
84	C₂×Dic₁₃₄	—	4	1	no	—
85	C₂×Dic₁₃₄	—	4	1	no	—
86	C₂²×Dic₆₇	—	4	1	no	—
87	C₂²×Dic₆₇	—	4	1	no	—
88	#11761616	—	4	16	no	—
89	C₂×C₆₇⋊C₈	—	2	4	no	—
90	C₂×C₆₇⋊C₈	—	2	4	no	—
91	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
92	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
93	C₄⋊Dic₆₇	—	2	2	no	—
94	C₄⋊Dic₆₇	—	2	2	no	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	C₄⋊Dic₆₇	—	2	2	no	—
97	C₄⋊Dic₆₇	—	4	1	no	—
98	C₄⋊Dic₆₇	—	4	1	no	—
99	C₄⋊Dic₆₇	—	4	1	no	—
100	C₄⋊Dic₆₇	—	4	1	no	—
101	C₂⁴×C₆₇	—	4	1	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	4	1	no	—
104	C₂×Q₈×C₆₇	—	4	4	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	4	1	no	—
109	C₂×D₄×C₆₇	—	4	1	no	—
110	C₂×D₄×C₆₇	—	4	1	no	—
111	C₂×D₄×C₆₇	—	4	4	no	—
112	C₂×D₄×C₆₇	—	4	4	no	—
113	C₂×D₄×C₆₇	—	4	4	no	—
114	C₂²×C₄×C₆₇	—	4	1	no	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	—	2	4	no	—
119	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
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	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₂³×C₆₇	—	4	2	no	—
129	C₂³×C₆₇	—	8	2	no	—
130	C₂³×C₆₇	—	8	2	no	—
131	C₂³×C₆₇	—	8	2	no	—
132	C₂³×C₆₇	#127	1	1	yes	characteristic
133	Q₈×C₆₇	—	4	4	no	—
134	Q₈×C₆₇	—	8	2	no	—
135	D₄×C₆₇	—	4	4	no	—
136	D₄×C₆₇	—	4	4	no	—
137	D₄×C₆₇	—	4	4	no	—
138	D₄×C₆₇	—	8	2	no	—
139	D₄×C₆₇	—	8	2	no	—
140	D₄×C₆₇	—	8	2	no	—
141	C₂×C₄×C₆₇	—	4	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	#135	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	—	4	2	no	—
154	C₂×Dic₆₇	—	8	2	no	—
155	C₂×Dic₆₇	—	8	2	no	—
156	Dic₁₃₄	—	4	4	no	—
157	Dic₁₃₄	—	4	4	no	—
158	Dic₁₃₄	—	8	2	no	—
159	Dic₁₃₄	—	8	2	no	—
160	C₈×C₆₇	—	2	4	no	—
161	C₆₇⋊C₈	—	2	4	no	—
162	C₆₇⋊C₈	—	2	4	no	—
163	C₂²×C₆₇	—	8	2	no	—
164	C₂²×C₆₇	—	8	2	no	—
165	C₂²×C₆₇	—	8	2	no	—
166	C₂²×C₆₇	—	16	4	no	—
167	C₂²×C₆₇	#495	1	2	yes	—
168	C₂²×C₆₇	#496	1	2	yes	—
169	C₂²×C₆₇	#497	1	2	yes	—
170	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
171	C₄×C₆₇	—	8	2	no	—
172	C₄×C₆₇	#495	1	2	yes	—
173	C₄×C₆₇	#496	1	2	yes	—
174	C₄×C₆₇	#497	1	2	yes	—
175	C₄×C₆₇	C₂×#135	1	2	yes	—
176	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
177	C₄×C₆₇	C₂×#137	1	2	yes	—
178	Dic₆₇	—	8	2	no	—
179	Dic₆₇	—	8	2	no	—
180	C₂×#1719	—	67	1	no	maximal, Sylow
181	C₂×C₆₇	—	16	2	no	—
182	C₂×C₆₇	#1719	1	4	yes	complemented
183	C₂×C₆₇	C₂×#495	1	1	yes	characteristic
184	C₂×C₆₇	C₂×#496	1	1	yes	characteristic
185	C₂×C₆₇	C₂×#497	1	1	yes	characteristic
186	C₂×#500	—	67	1	no	—
187	C₂×#492	—	67	1	no	—
188	C₂×#487	—	67	1	no	—
189	C₂×#487	—	67	1	no	—
190	C₂×#486	—	67	1	no	—
191	C₂×#486	D₆₇	1	1	yes	characteristic
192	#1719	—	67	8	no	—
193	C₂×#322	—	67	1	no	—
194	C₆₇	C₂×#1719	1	1	yes	characteristic, complemented, Sylow
195	C₂²×SD₁₆	—	134	2	no	—
196	C₂×C₄⋊Q₈	—	67	1	no	—
197	C₂×C₄⋊Q₈	—	67	1	no	—
198	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
199	C₂×#127	C₂×D₆₇	1	1	yes	characteristic
200	#500	—	67	4	no	—
201	#500	—	134	2	no	—
202	#492	—	67	4	no	—
203	#492	—	134	2	no	—
204	#487	—	67	4	no	—
205	#487	—	67	4	no	—
206	#487	—	134	2	no	—
207	#487	—	134	2	no	—
208	#486	—	2	2	no	—
209	#486	—	67	4	no	—
210	#486	—	134	2	no	—
211	#486	C₂×D₆₇	1	4	yes	—
212	C₂×#107	—	134	2	no	—
213	C₂×#103	—	134	2	no	—
214	C₂×#103	—	134	2	no	—
215	C₂×#102	—	2	2	no	—
216	C₂×#102	—	134	2	no	—
217	C₂×C₄×C₈	—	67	1	no	—
218	C₂×C₄×C₈	—	67	1	no	—
219	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
220	#322	—	67	4	no	—
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×Q₈	—	134	1	no	—
223	C₂²×Q₈	—	134	1	no	—
224	C₂²×D₄	—	2	1	no	—
225	C₂²×D₄	—	2	1	no	—
226	C₂²×D₄	—	2	1	no	—
227	C₂×SD₁₆	—	268	8	no	—
228	C₂×SD₁₆	—	134	8	no	—
229	C₂²×C₈	—	67	2	no	—
230	C₂²×C₈	—	67	2	no	—
231	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	67	2	no	—
236	C₄⋊Q₈	—	67	2	no	—
237	C₄⋊Q₈	—	134	1	no	—
238	C₄⋊Q₈	—	134	1	no	—
239	C₄⋊Q₈	—	134	1	no	—
240	C₄⋊Q₈	—	134	1	no	—
241	C₄⋊Q₈	—	134	1	no	—
242	C₄⋊Q₈	—	134	1	no	—
243	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
244	#127	—	2	1	no	—
245	#127	—	2	1	no	—
246	#127	—	2	1	no	—
247	#127	C₂²×D₆₇	1	2	yes	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	134	1	no	—
251	C₂×C₄⋊C₄	—	134	1	no	—
252	C₂×C₄⋊C₄	—	134	1	no	—
253	C₂×C₄⋊C₄	—	134	1	no	—
254	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
255	#107	—	134	8	no	—
256	#103	—	134	8	no	—
257	#103	—	134	8	no	—
258	#102	—	2	8	no	—
259	#102	—	134	8	no	—
260	C₄×C₈	—	67	2	no	—
261	C₄×C₈	—	67	2	no	—
262	C₄×C₈	—	67	2	no	—
263	C₄×C₈	—	67	2	no	—
264	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
265	C₄×C₈	C₂²×D₆₇	1	2	yes	—
266	C₂⁴	—	4	1	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	4	4	no	—
270	C₂×Q₈	—	268	1	no	—
271	C₂×Q₈	—	268	1	no	—
272	C₂×Q₈	—	268	4	no	—
273	C₂×Q₈	—	268	4	no	—
274	C₂×Q₈	—	134	2	no	—
275	C₂×Q₈	—	134	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	—	4	1	no	—
282	C₂×D₄	—	4	4	no	—
283	C₂×D₄	—	4	4	no	—
284	C₂×D₄	—	4	4	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	268	1	no	—
287	C₂²×C₄	—	268	1	no	—
288	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
291	SD₁₆	—	268	16	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	67	4	no	—
294	C₂×C₈	—	67	4	no	—
295	C₂×C₈	—	134	4	no	—
296	C₂×C₈	—	134	4	no	—
297	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	268	1	no	—
303	C₄⋊C₄	—	268	1	no	—
304	C₄⋊C₄	—	268	1	no	—
305	C₄⋊C₄	—	268	1	no	—
306	C₄⋊C₄	—	134	2	no	—
307	C₄⋊C₄	—	134	2	no	—
308	C₄⋊C₄	—	134	2	no	—
309	C₄⋊C₄	—	134	2	no	—
310	C₄²	D₄×D₆₇	1	1	yes	characteristic
311	C₄²	D₄×D₆₇	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
313	C₄²	C₂³×D₆₇	1	1	yes	characteristic
314	C₂³	—	4	2	no	—
315	C₂³	—	8	2	no	—
316	C₂³	—	8	2	no	—
317	C₂³	—	8	2	no	—
318	C₂³	#14524039	1	1	yes	characteristic
319	Q₈	—	4	4	no	—
320	Q₈	—	8	2	no	—
321	Q₈	—	268	4	no	—
322	Q₈	—	268	4	no	—
323	Q₈	—	536	2	no	—
324	Q₈	—	536	2	no	—
325	D₄	—	4	4	no	—
326	D₄	—	4	4	no	—
327	D₄	—	4	4	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	2	no	—
330	D₄	—	8	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	8	2	no	—
333	C₂×C₄	—	268	2	no	—
334	C₂×C₄	—	268	2	no	—
335	C₂×C₄	—	536	2	no	—
336	C₂×C₄	—	536	2	no	—
337	C₂×C₄	#14524039	1	1	yes	characteristic
338	C₂×C₄	#14524039	1	1	yes	characteristic
339	C₂×C₄	#14524039	1	1	yes	characteristic
340	C₂×C₄	#14524062	1	4	yes	—
341	C₂×C₄	#14524070	1	4	yes	—
342	C₂×C₄	#14524073	1	4	yes	—
343	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
346	C₈	—	2	4	no	—
347	C₈	—	134	4	no	—
348	C₈	—	134	4	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	16	4	no	—
353	C₂²	#16863779	1	2	yes	—
354	C₂²	#16863782	1	2	yes	—
355	C₂²	#16863791	1	2	yes	—
356	C₂²	C₂×#14524039	1	1	yes	characteristic
357	C₄	—	8	2	no	—
358	C₄	—	536	2	no	—
359	C₄	—	536	2	no	—
360	C₄	#16863779	1	2	yes	—
361	C₄	#16863782	1	2	yes	—
362	C₄	#16863791	1	2	yes	—
363	C₄	C₂×#14524062	1	2	yes	—
364	C₄	C₂×#14524070	1	2	yes	—
365	C₄	C₂×#14524073	1	2	yes	—
366	C₂	—	16	2	no	—
367	C₂	#22559861	1	4	yes	direct product factor
368	C₂	C₂×#16863779	1	1	yes	characteristic
369	C₂	C₂×#16863782	1	1	yes	characteristic
370	C₂	C₂×#16863791	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524039
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#486×C₆₇	C₂
socle	C₂³×C₆₇	#127
prime radical	C₂×#127×C₆₇	C₂²
hypercenter	C₂×#486	D₆₇
hypocenter	C₆₇	C₂×#1719
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#486

Complements

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

N	H ≅ G/N	type
1. C₂×#16863801	C₂	semidirect
3. C₂×#16863748	C₂	semidirect
4. C₂×#16863746	C₂	semidirect
6. #22559861	C₂	direct (N × H)
7. C₂×#16863118	C₂	semidirect
13. #16863801	C₂²	semidirect
17. #16863748	C₂²	semidirect
19. #16863746	C₂²	semidirect
28. #16863118	C₂²	semidirect
182. C₂×C₆₇	#1719	semidirect
194. C₆₇	C₂×#1719	semidirect
367. C₂	#22559861	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×#16863801	2
C₂×#16863768	2
C₂×#16863748	2
C₂×#16863746	2
C₂×#16863742	2
#22559861	2
C₂×#16863118	2
C₂×#486×C₆₇	2
C₂×#1719	67

Subgroups — C2×#22559861 #30843586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559861 #30843586