Subgroups — C₂×#22559820 #30843506

Order 17152.

802 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863748	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863747	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863746	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863742	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863742	C₂	1	1	yes	maximal, characteristic
6	#22559820	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524043	C₂²	1	1	yes	characteristic
10	C₂×#14524043	C₂²	1	1	yes	characteristic
11	#16863748	—	2	2	no	—
12	#16863748	C₂²	1	4	yes	complemented
13	#16863747	—	2	2	no	—
14	#16863747	C₂²	1	4	yes	complemented
15	#16863746	—	2	2	no	—
16	#16863746	C₂²	1	4	yes	complemented
17	#16863742	—	2	2	no	—
18	#16863742	—	2	2	no	—
19	#16863742	C₂²	1	4	yes	—
20	#16863742	C₂²	1	4	yes	—
21	C₂×#14523968	—	2	2	no	—
22	C₂×#14523967	—	2	2	no	—
23	C₂×#14523966	—	2	2	no	—
24	C₂×#14523964	—	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	C₈×C₆₇⋊C₈	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	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
38	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
39	#14524043	—	2	1	no	—
40	#14524043	—	2	1	no	—
41	#14524043	—	2	1	no	—
42	#14524043	—	2	1	no	—
43	#14524043	—	2	1	no	—
44	#14524043	—	2	1	no	—
45	#14524043	C₂³	1	2	yes	—
46	#14524043	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₆₇	—	2	1	no	—
48	C₂×C₄⋊Dic₆₇	—	2	1	no	—
49	C₂×C₄⋊Dic₆₇	—	2	1	no	—
50	C₂×C₄⋊Dic₆₇	—	2	1	no	—
51	#14523968	—	2	8	no	—
52	#14523967	—	2	8	no	—
53	#14523966	—	2	8	no	—
54	#14523964	—	2	8	no	—
55	#14523964	—	2	8	no	—
56	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
57	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
58	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
59	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
60	C₂²×Q₈×C₆₇	—	2	1	no	—
61	C₂²×D₄×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	—	2	1	no	—
63	C₂²×D₄×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	C₂³	1	2	yes	—
69	#127×C₆₇	—	2	1	no	—
70	#127×C₆₇	—	2	1	no	—
71	#127×C₆₇	—	2	1	no	—
72	#127×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#102×C₆₇	—	2	8	no	—
77	C₄×C₈×C₆₇	D₄	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	2	yes	—
79	C₂×Dic₁₃₄	—	2	2	no	—
80	C₂×Dic₁₃₄	—	2	2	no	—
81	C₂×Dic₁₃₄	—	4	4	no	—
82	C₂×Dic₁₃₄	—	4	4	no	—
83	C₂×Dic₁₃₄	—	4	1	no	—
84	C₂×Dic₁₃₄	—	4	1	no	—
85	C₂²×Dic₆₇	—	4	1	no	—
86	C₂²×Dic₆₇	—	4	1	no	—
87	C₂×C₆₇⋊C₈	—	2	4	no	—
88	C₂×C₆₇⋊C₈	—	2	4	no	—
89	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
91	C₄⋊Dic₆₇	—	2	2	no	—
92	C₄⋊Dic₆₇	—	2	2	no	—
93	C₄⋊Dic₆₇	—	2	2	no	—
94	C₄⋊Dic₆₇	—	2	2	no	—
95	C₄⋊Dic₆₇	—	4	1	no	—
96	C₄⋊Dic₆₇	—	4	1	no	—
97	C₄⋊Dic₆₇	—	4	1	no	—
98	C₄⋊Dic₆₇	—	4	1	no	—
99	C₂⁴×C₆₇	—	4	1	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	4	1	no	—
102	C₂×Q₈×C₆₇	—	4	4	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	4	1	no	—
107	C₂×D₄×C₆₇	—	4	1	no	—
108	C₂×D₄×C₆₇	—	4	1	no	—
109	C₂×D₄×C₆₇	—	4	4	no	—
110	C₂×D₄×C₆₇	—	4	4	no	—
111	C₂×D₄×C₆₇	—	4	4	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	—	2	4	no	—
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	—	4	1	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
126	C₂³×C₆₇	—	4	2	no	—
127	C₂³×C₆₇	—	8	2	no	—
128	C₂³×C₆₇	—	8	2	no	—
129	C₂³×C₆₇	—	8	2	no	—
130	C₂³×C₆₇	#127	1	1	yes	characteristic
131	Q₈×C₆₇	—	4	4	no	—
132	Q₈×C₆₇	—	8	2	no	—
133	D₄×C₆₇	—	4	4	no	—
134	D₄×C₆₇	—	4	4	no	—
135	D₄×C₆₇	—	4	4	no	—
136	D₄×C₆₇	—	8	2	no	—
137	D₄×C₆₇	—	8	2	no	—
138	D₄×C₆₇	—	8	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	8	2	no	—
141	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
142	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#135	1	4	yes	—
145	C₂×C₄×C₆₇	#135	1	4	yes	—
146	C₂×C₄×C₆₇	#135	1	4	yes	—
147	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₆₇	—	4	2	no	—
151	C₂×Dic₆₇	—	4	2	no	—
152	C₂×Dic₆₇	—	8	2	no	—
153	C₂×Dic₆₇	—	8	2	no	—
154	Dic₁₃₄	—	4	4	no	—
155	Dic₁₃₄	—	4	4	no	—
156	Dic₁₃₄	—	8	2	no	—
157	Dic₁₃₄	—	8	2	no	—
158	C₈×C₆₇	—	2	4	no	—
159	C₆₇⋊C₈	—	2	4	no	—
160	C₆₇⋊C₈	—	2	4	no	—
161	C₂²×C₆₇	—	8	2	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	—	8	2	no	—
164	C₂²×C₆₇	—	16	4	no	—
165	C₂²×C₆₇	#495	1	2	yes	—
166	C₂²×C₆₇	#495	1	2	yes	—
167	C₂²×C₆₇	#495	1	2	yes	—
168	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
169	C₄×C₆₇	—	8	2	no	—
170	C₄×C₆₇	#495	1	2	yes	—
171	C₄×C₆₇	#495	1	2	yes	—
172	C₄×C₆₇	#495	1	2	yes	—
173	C₄×C₆₇	C₂×#135	1	2	yes	—
174	C₄×C₆₇	C₂×#135	1	2	yes	—
175	C₄×C₆₇	C₂×#135	1	2	yes	—
176	Dic₆₇	—	8	2	no	—
177	Dic₆₇	—	8	2	no	—
178	C₂×#1711	—	67	1	no	maximal, Sylow
179	C₂×C₆₇	—	16	2	no	—
180	C₂×C₆₇	#1711	1	4	yes	complemented
181	C₂×C₆₇	C₂×#495	1	1	yes	characteristic
182	C₂×C₆₇	C₂×#495	1	1	yes	characteristic
183	C₂×C₆₇	C₂×#495	1	1	yes	characteristic
184	C₂×#487	—	67	1	no	—
185	C₂×#487	—	67	1	no	—
186	C₂×#487	—	67	1	no	—
187	C₂×#486	—	67	1	no	—
188	C₂×#486	—	67	1	no	—
189	C₂×#486	D₆₇	1	1	yes	characteristic
190	#1711	—	67	8	no	—
191	C₂×C₈²	—	67	1	no	—
192	C₆₇	C₂×#1711	1	1	yes	characteristic, complemented, Sylow
193	C₂×C₄⋊Q₈	—	67	1	no	—
194	C₂×C₄⋊Q₈	—	67	1	no	—
195	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
196	C₂×#127	C₂×D₆₇	1	1	yes	characteristic
197	#487	—	67	4	no	—
198	#487	—	67	4	no	—
199	#487	—	67	4	no	—
200	#487	—	134	2	no	—
201	#487	—	134	2	no	—
202	#487	—	134	2	no	—
203	#486	—	2	2	no	—
204	#486	—	67	4	no	—
205	#486	—	67	4	no	—
206	#486	—	134	2	no	—
207	#486	—	134	2	no	—
208	#486	C₂×D₆₇	1	4	yes	—
209	C₂×#103	—	134	2	no	—
210	C₂×#103	—	134	2	no	—
211	C₂×#103	—	134	2	no	—
212	C₂×#102	—	2	2	no	—
213	C₂×#102	—	134	2	no	—
214	C₂×#102	—	134	2	no	—
215	C₂×C₄×C₈	—	67	1	no	—
216	C₂×C₄×C₈	—	67	1	no	—
217	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
218	C₈²	—	67	4	no	—
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×Q₈	—	134	1	no	—
221	C₂²×Q₈	—	134	1	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂²×D₄	—	2	1	no	—
224	C₂²×D₄	—	2	1	no	—
225	C₂²×C₈	—	67	2	no	—
226	C₂²×C₈	—	67	2	no	—
227	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	67	2	no	—
232	C₄⋊Q₈	—	67	2	no	—
233	C₄⋊Q₈	—	134	1	no	—
234	C₄⋊Q₈	—	134	1	no	—
235	C₄⋊Q₈	—	134	1	no	—
236	C₄⋊Q₈	—	134	1	no	—
237	C₄⋊Q₈	—	134	1	no	—
238	C₄⋊Q₈	—	134	1	no	—
239	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
240	#127	—	2	1	no	—
241	#127	—	2	1	no	—
242	#127	—	2	1	no	—
243	#127	C₂²×D₆₇	1	2	yes	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	134	1	no	—
247	C₂×C₄⋊C₄	—	134	1	no	—
248	C₂×C₄⋊C₄	—	134	1	no	—
249	C₂×C₄⋊C₄	—	134	1	no	—
250	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
251	#103	—	134	8	no	—
252	#103	—	134	8	no	—
253	#103	—	134	8	no	—
254	#102	—	2	8	no	—
255	#102	—	134	8	no	—
256	#102	—	134	8	no	—
257	C₄×C₈	—	67	2	no	—
258	C₄×C₈	—	67	2	no	—
259	C₄×C₈	—	67	2	no	—
260	C₄×C₈	—	67	2	no	—
261	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
262	C₄×C₈	C₂²×D₆₇	1	2	yes	—
263	C₂⁴	—	4	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	4	no	—
267	C₂×Q₈	—	268	1	no	—
268	C₂×Q₈	—	268	1	no	—
269	C₂×Q₈	—	268	4	no	—
270	C₂×Q₈	—	268	4	no	—
271	C₂×Q₈	—	134	2	no	—
272	C₂×Q₈	—	134	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	4	no	—
280	C₂×D₄	—	4	4	no	—
281	C₂×D₄	—	4	4	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	268	1	no	—
284	C₂²×C₄	—	268	1	no	—
285	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	67	4	no	—
290	C₂×C₈	—	67	4	no	—
291	C₂×C₈	—	134	4	no	—
292	C₂×C₈	—	134	4	no	—
293	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	268	1	no	—
299	C₄⋊C₄	—	268	1	no	—
300	C₄⋊C₄	—	268	1	no	—
301	C₄⋊C₄	—	268	1	no	—
302	C₄⋊C₄	—	134	2	no	—
303	C₄⋊C₄	—	134	2	no	—
304	C₄⋊C₄	—	134	2	no	—
305	C₄⋊C₄	—	134	2	no	—
306	C₄²	D₄×D₆₇	1	1	yes	characteristic
307	C₄²	D₄×D₆₇	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
309	C₄²	C₂³×D₆₇	1	1	yes	characteristic
310	C₂³	—	4	2	no	—
311	C₂³	—	8	2	no	—
312	C₂³	—	8	2	no	—
313	C₂³	—	8	2	no	—
314	C₂³	#14524039	1	1	yes	characteristic
315	Q₈	—	4	4	no	—
316	Q₈	—	8	2	no	—
317	Q₈	—	268	4	no	—
318	Q₈	—	268	4	no	—
319	Q₈	—	536	2	no	—
320	Q₈	—	536	2	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	4	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	268	2	no	—
330	C₂×C₄	—	268	2	no	—
331	C₂×C₄	—	536	2	no	—
332	C₂×C₄	—	536	2	no	—
333	C₂×C₄	#14524039	1	1	yes	characteristic
334	C₂×C₄	#14524039	1	1	yes	characteristic
335	C₂×C₄	#14524039	1	1	yes	characteristic
336	C₂×C₄	#14524062	1	4	yes	—
337	C₂×C₄	#14524063	1	4	yes	—
338	C₂×C₄	#14524064	1	4	yes	—
339	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
342	C₈	—	2	4	no	—
343	C₈	—	134	4	no	—
344	C₈	—	134	4	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	16	4	no	—
349	C₂²	#16863777	1	2	yes	—
350	C₂²	#16863778	1	2	yes	—
351	C₂²	#16863779	1	2	yes	—
352	C₂²	C₂×#14524039	1	1	yes	characteristic
353	C₄	—	8	2	no	—
354	C₄	—	536	2	no	—
355	C₄	—	536	2	no	—
356	C₄	#16863777	1	2	yes	—
357	C₄	#16863778	1	2	yes	—
358	C₄	#16863779	1	2	yes	—
359	C₄	C₂×#14524062	1	2	yes	—
360	C₄	C₂×#14524063	1	2	yes	—
361	C₄	C₂×#14524064	1	2	yes	—
362	C₂	—	16	2	no	—
363	C₂	#22559820	1	4	yes	direct product factor
364	C₂	C₂×#16863777	1	1	yes	characteristic
365	C₂	C₂×#16863778	1	1	yes	characteristic
366	C₂	C₂×#16863779	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×#1711
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×#16863748	C₂	semidirect
2. C₂×#16863747	C₂	semidirect
3. C₂×#16863746	C₂	semidirect
6. #22559820	C₂	direct (N × H)
7. C₂×C₈×C₆₇⋊C₈	C₂	semidirect
12. #16863748	C₂²	semidirect
14. #16863747	C₂²	semidirect
16. #16863746	C₂²	semidirect
28. C₈×C₆₇⋊C₈	C₂²	semidirect
180. C₂×C₆₇	#1711	semidirect
192. C₆₇	C₂×#1711	semidirect
363. C₂	#22559820	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×#16863748	2
C₂×#16863747	2
C₂×#16863746	2
C₂×#16863742	2
#22559820	2
C₂×C₈×C₆₇⋊C₈	2
C₂×#486×C₆₇	2
C₂×#1711	67

Subgroups — C2×#22559820 #30843506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559820 #30843506