Subgroups — C₂×#499×D₆₇ #31725421

Order 17152.

1966 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×D₆₇	C₂	1	2	yes	maximal
2	C₂×#16863798	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863797	C₂	1	2	yes	maximal, complemented
4	C₂×#16863796	C₂	1	2	yes	maximal, complemented
5	C₂×#16863795	C₂	1	2	yes	maximal, complemented
6	#499×D₆₇	C₂	1	16	yes	maximal, direct product factor
7	C₂×#107×D₆₇	C₂	1	2	yes	maximal
8	C₂×#106×D₆₇	C₂	1	2	yes	maximal
9	C₂×C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
13	C₂×#14524035	C₂²	1	2	yes	—
14	C₂×#14524034	C₂²	1	4	yes	—
15	C₂×#14524033	—	2	4	no	—
16	#125×D₆₇	—	2	8	no	—
17	#125×D₆₇	C₂²	1	16	yes	—
18	C₂×C₄⋊C₄×D₆₇	—	2	4	no	—
19	C₂×C₄⋊C₄×D₆₇	C₂²	1	4	yes	—
20	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
21	#16863798	C₂²	1	8	yes	complemented
22	#16863797	C₂²	1	16	yes	complemented
23	#16863796	C₂²	1	16	yes	complemented
24	#16863795	C₂²	1	16	yes	complemented
25	C₂×#14523977	C₂²	1	2	yes	—
26	C₂×#14523976	C₂²	1	4	yes	—
27	#107×D₆₇	C₂²	1	16	yes	—
28	C₂×#14523975	C₂²	1	2	yes	—
29	C₂×#14523974	C₂²	1	4	yes	—
30	#106×D₆₇	C₂²	1	16	yes	—
31	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₆₇	C₂²	1	8	yes	—
35	C₂×#125×C₆₇	C₂²	1	2	yes	—
36	#499×C₆₇	C₂²	1	8	yes	complemented
37	C₂×#107×C₆₇	C₂²	1	2	yes	—
38	C₂×#106×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₂²×C₄×D₆₇	—	2	4	no	—
41	C₂²×C₄×D₆₇	D₄	1	2	yes	—
42	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₆₇	Q₈	1	8	yes	—
46	C₂×C₈×D₆₇	Q₈	1	8	yes	—
47	C₂×C₈×D₆₇	C₂³	1	4	yes	—
48	C₂×C₈×D₆₇	C₂³	1	4	yes	—
49	#14524035	—	2	4	no	—
50	#14524035	C₂³	1	8	yes	—
51	#14524034	—	2	8	no	—
52	#14524034	C₂³	1	16	yes	—
53	#14524033	—	2	32	no	—
54	C₂×C₄⋊Dic₆₇	—	2	4	no	—
55	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
56	C₂×Dic₆₇⋊C₄	—	2	4	no	—
57	C₂×Dic₆₇⋊C₄	—	2	4	no	—
58	C₂×Dic₆₇⋊C₄	—	2	4	no	—
59	C₄⋊C₄×D₆₇	—	2	32	no	—
60	C₄⋊C₄×D₆₇	—	2	8	no	—
61	C₄⋊C₄×D₆₇	C₂³	1	16	yes	—
62	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
63	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
64	C₄²×D₆₇	D₄	1	4	yes	—
65	C₄²×D₆₇	C₂³	1	4	yes	—
66	#14523977	C₂³	1	8	yes	—
67	#14523976	C₂³	1	16	yes	—
68	#14523975	C₂³	1	8	yes	—
69	#14523974	C₂³	1	16	yes	—
70	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
71	C₈×Dic₆₇	C₂³	1	4	yes	—
72	C₈×Dic₆₇	C₂³	1	4	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#125×C₆₇	—	2	4	no	—
76	#125×C₆₇	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
80	#107×C₆₇	C₂³	1	8	yes	—
81	#106×C₆₇	C₂³	1	8	yes	—
82	C₄×C₈×C₆₇	C₂³	1	4	yes	—
83	C₂³×D₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×Dic₆₇	—	2	4	no	—
85	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
86	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×D₆₇	—	2	8	no	—
88	C₂×C₄×D₆₇	—	2	16	no	—
89	C₂×C₄×D₆₇	—	4	16	no	—
90	C₂×C₄×D₆₇	C₂×D₄	1	8	yes	—
91	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×D₆₇	C₂×Q₈	1	4	yes	—
93	C₂×C₄×D₆₇	C₂×Q₈	1	4	yes	—
94	C₂×C₄×D₆₇	C₂⁴	1	2	yes	—
95	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
96	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
97	C₂×C₆₇⋊C₈	C₂⁴	1	2	yes	—
98	C₂×C₆₇⋊C₈	C₂⁴	1	2	yes	—
99	C₈×D₆₇	C₂×Q₈	1	16	yes	—
100	C₈×D₆₇	C₂×Q₈	1	16	yes	—
101	C₄⋊Dic₆₇	—	2	4	no	—
102	C₄⋊Dic₆₇	—	2	16	no	—
103	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
104	Dic₆₇⋊C₄	—	2	16	no	—
105	Dic₆₇⋊C₄	—	2	16	no	—
106	Dic₆₇⋊C₄	—	2	16	no	—
107	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
108	C₄×Dic₆₇	C₂×D₄	1	8	yes	—
109	C₄×Dic₆₇	C₂⁴	1	2	yes	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
114	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
115	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
116	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
117	C₄⋊C₄×C₆₇	—	2	16	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
120	C₄²×C₆₇	C₂×D₄	1	2	yes	—
121	C₄²×C₆₇	C₂⁴	1	2	yes	—
122	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂²×D₆₇	C₄⋊Q₈	1	4	yes	—
124	C₂²×D₆₇	#135	1	8	yes	—
125	C₂²×D₆₇	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	8	no	—
128	C₂×C₄×C₆₇	—	4	8	no	—
129	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
134	C₂×Dic₆₇	—	2	4	no	—
135	C₂×Dic₆₇	—	2	8	no	—
136	C₂×Dic₆₇	—	4	8	no	—
137	C₂×Dic₆₇	C₄⋊Q₈	1	4	yes	—
138	C₂×Dic₆₇	C₂²×D₄	1	4	yes	—
139	C₂×Dic₆₇	C₂²×D₄	1	2	yes	—
140	C₄×D₆₇	—	2	16	no	—
141	C₄×D₆₇	—	4	32	no	—
142	C₄×D₆₇	C₄⋊Q₈	1	8	yes	—
143	C₄×D₆₇	C₂²×Q₈	1	4	yes	—
144	C₄×D₆₇	C₂²×Q₈	1	4	yes	—
145	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
146	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
147	C₆₇⋊C₈	C₂²×Q₈	1	4	yes	—
148	C₆₇⋊C₈	C₂²×Q₈	1	4	yes	—
149	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
150	C₂²×C₆₇	C₂×#135	1	4	yes	—
151	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
152	C₂×D₆₇	#499	1	16	yes	direct product factor
153	C₂×D₆₇	C₂×C₄⋊Q₈	1	4	yes	—
154	C₂×D₆₇	C₂×#135	1	8	yes	—
155	C₄×C₆₇	—	2	4	no	—
156	C₄×C₆₇	—	4	8	no	—
157	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
158	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
159	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
160	Dic₆₇	—	2	4	no	—
161	Dic₆₇	—	4	8	no	—
162	Dic₆₇	C₂×C₄⋊Q₈	1	4	yes	—
163	C₂²×#499	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	C₂×#499	1	4	yes	complemented
165	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₆₇	C₂²×#135	1	2	yes	—
167	D₆₇	C₂×#499	1	8	yes	direct product factor
168	C₂²×#125	—	67	2	no	—
169	C₂×#499	—	67	16	no	—
170	C₂×#499	—	67	2	no	—
171	C₂×#499	—	67	2	no	—
172	C₂×#499	—	67	1	no	—
173	C₂×#499	—	67	2	no	—
174	C₂×#499	D₆₇	1	1	yes	characteristic, direct product factor
175	C₂²×#107	—	67	2	no	—
176	C₂²×#106	—	67	2	no	—
177	C₂²×C₄×C₈	—	67	1	no	—
178	C₆₇	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₈	—	67	1	no	—
180	C₂³×C₈	—	67	1	no	—
181	C₂×#125	—	67	16	no	—
182	C₂×#125	—	67	4	no	—
183	C₂×#125	—	67	2	no	—
184	C₂×#125	—	134	4	no	—
185	C₂×#125	—	134	8	no	—
186	C₂×#125	C₂×D₆₇	1	2	yes	—
187	C₂²×C₄⋊C₄	—	67	4	no	—
188	C₂²×C₄⋊C₄	—	134	4	no	—
189	C₂²×C₄²	—	67	1	no	—
190	#499	—	67	16	no	—
191	#499	—	67	16	no	—
192	#499	—	67	16	no	—
193	#499	—	67	8	no	—
194	#499	C₂×D₆₇	1	8	yes	direct product factor
195	C₂×#107	—	67	2	no	—
196	C₂×#107	—	67	4	no	—
197	C₂×#107	—	67	16	no	—
198	C₂×#107	C₂×D₆₇	1	2	yes	—
199	C₂×#106	—	67	2	no	—
200	C₂×#106	—	67	4	no	—
201	C₂×#106	—	67	16	no	—
202	C₂×#106	C₂×D₆₇	1	2	yes	—
203	C₂×C₄×C₈	—	67	1	no	—
204	C₂×C₄×C₈	—	67	1	no	—
205	C₂×C₄×C₈	—	67	1	no	—
206	C₂×C₄×C₈	—	67	8	no	—
207	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
208	C₂³×C₄	—	67	1	no	—
209	C₂³×C₄	—	67	2	no	—
210	C₂³×C₄	—	134	4	no	—
211	C₂²×C₈	—	67	1	no	—
212	C₂²×C₈	—	67	1	no	—
213	C₂²×C₈	—	67	8	no	—
214	C₂²×C₈	—	67	8	no	—
215	C₂²×C₈	—	67	4	no	—
216	C₂²×C₈	—	67	4	no	—
217	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
219	#125	—	2	4	no	—
220	#125	—	67	16	no	—
221	#125	—	67	8	no	—
222	#125	—	134	32	no	—
223	#125	—	134	8	no	—
224	#125	—	134	4	no	—
225	#125	C₂²×D₆₇	1	8	yes	—
226	C₂×C₄⋊C₄	—	2	4	no	—
227	C₂×C₄⋊C₄	—	67	4	no	—
228	C₂×C₄⋊C₄	—	67	16	no	—
229	C₂×C₄⋊C₄	—	134	8	no	—
230	C₂×C₄⋊C₄	—	134	32	no	—
231	C₂×C₄⋊C₄	—	134	4	no	—
232	C₂×C₄⋊C₄	—	134	4	no	—
233	C₂×C₄⋊C₄	—	134	4	no	—
234	C₂×C₄⋊C₄	—	134	4	no	—
235	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
236	C₂×C₄²	—	67	2	no	—
237	C₂×C₄²	—	67	1	no	—
238	C₂×C₄²	—	67	4	no	—
239	C₂×C₄²	—	67	4	no	—
240	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
241	#107	—	67	8	no	—
242	#107	—	67	16	no	—
243	#107	C₂²×D₆₇	1	8	yes	—
244	#106	—	67	8	no	—
245	#106	—	67	16	no	—
246	#106	C₂²×D₆₇	1	8	yes	—
247	C₄×C₈	—	67	4	no	—
248	C₄×C₈	—	67	4	no	—
249	C₄×C₈	—	67	4	no	—
250	C₄×C₈	C₂²×D₆₇	1	4	yes	—
251	C₂⁴	—	67	1	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	268	16	no	—
254	C₂²×C₄	—	67	1	no	—
255	C₂²×C₄	—	67	2	no	—
256	C₂²×C₄	—	67	4	no	—
257	C₂²×C₄	—	67	4	no	—
258	C₂²×C₄	—	67	2	no	—
259	C₂²×C₄	—	67	2	no	—
260	C₂²×C₄	—	67	8	no	—
261	C₂²×C₄	—	134	16	no	—
262	C₂²×C₄	—	134	8	no	—
263	C₂²×C₄	—	134	4	no	—
264	C₂²×C₄	D₄×D₆₇	1	2	yes	—
265	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
266	C₂×C₈	—	67	4	no	—
267	C₂×C₈	—	67	4	no	—
268	C₂×C₈	—	67	2	no	—
269	C₂×C₈	—	67	2	no	—
270	C₂×C₈	—	67	16	no	—
271	C₂×C₈	—	67	16	no	—
272	C₂×C₈	Q₈×D₆₇	1	4	yes	—
273	C₂×C₈	Q₈×D₆₇	1	4	yes	—
274	C₂×C₈	C₂³×D₆₇	1	2	yes	—
275	C₂×C₈	C₂³×D₆₇	1	2	yes	—
276	C₄⋊C₄	—	2	16	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	67	8	no	—
279	C₄⋊C₄	—	134	16	no	—
280	C₄⋊C₄	—	134	16	no	—
281	C₄⋊C₄	—	134	16	no	—
282	C₄⋊C₄	—	134	4	no	—
283	C₄⋊C₄	—	134	16	no	—
284	C₄⋊C₄	C₂³×D₆₇	1	8	yes	—
285	C₄²	—	67	8	no	—
286	C₄²	—	67	2	no	—
287	C₄²	—	67	2	no	—
288	C₄²	D₄×D₆₇	1	2	yes	—
289	C₄²	C₂³×D₆₇	1	2	yes	—
290	C₂³	—	67	8	no	—
291	C₂³	—	67	4	no	—
292	C₂³	—	67	2	no	—
293	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	268	32	no	—
298	C₂×C₄	—	268	8	no	—
299	C₂×C₄	—	67	4	no	—
300	C₂×C₄	—	67	2	no	—
301	C₂×C₄	—	67	8	no	—
302	C₂×C₄	—	67	4	no	—
303	C₂×C₄	—	67	4	no	—
304	C₂×C₄	—	67	4	no	—
305	C₂×C₄	—	134	16	no	—
306	C₂×C₄	—	134	8	no	—
307	C₂×C₄	—	134	4	no	—
308	C₂×C₄	C₂×D₄×D₆₇	1	4	yes	—
309	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
311	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
312	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
313	C₈	—	67	4	no	—
314	C₈	—	67	4	no	—
315	C₈	C₂×Q₈×D₆₇	1	4	yes	—
316	C₈	C₂×Q₈×D₆₇	1	4	yes	—
317	C₂²	—	67	8	no	—
318	C₂²	—	67	4	no	—
319	C₂²	—	67	16	no	—
320	C₂²	C₄⋊Q₈×D₆₇	1	2	yes	—
321	C₂²	#135×D₆₇	1	4	yes	—
322	C₂²	C₂²×D₄×D₆₇	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	4	8	no	—
325	C₄	—	268	8	no	—
326	C₄	—	67	4	no	—
327	C₄	—	134	4	no	—
328	C₄	C₄⋊Q₈×D₆₇	1	2	yes	—
329	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
330	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
331	C₂	—	67	8	no	—
332	C₂	#499×D₆₇	1	4	yes	direct product factor
333	C₂	C₂×C₄⋊Q₈×D₆₇	1	1	yes	characteristic
334	C₂	C₂×#135×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂³×D₆₇	C₂×D₄
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	C₂²×#499
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. C₂×#16863798	C₂	semidirect
3. C₂×#16863797	C₂	semidirect
4. C₂×#16863796	C₂	semidirect
5. C₂×#16863795	C₂	semidirect
6. #499×D₆₇	C₂	direct (N × H)
10. C₂×#499×C₆₇	C₂	semidirect
21. #16863798	C₂²	semidirect
22. #16863797	C₂²	semidirect
23. #16863796	C₂²	semidirect
24. #16863795	C₂²	semidirect
36. #499×C₆₇	C₂²	semidirect
152. C₂×D₆₇	#499	direct (N × H)
164. C₂×C₆₇	C₂×#499	semidirect
167. D₆₇	C₂×#499	direct (N × H)
174. C₂×#499	D₆₇	direct (N × H)
178. C₆₇	C₂²×#499	semidirect
194. #499	C₂×D₆₇	direct (N × H)
332. C₂	#499×D₆₇	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×#125×D₆₇	2
C₂×#16863798	2
C₂×#16863797	2
C₂×#16863796	2
C₂×#16863795	2
#499×D₆₇	2
C₂×#107×D₆₇	2
C₂×#106×D₆₇	2
C₂×C₄×C₈×D₆₇	2
C₂×#499×C₆₇	2
C₂²×#499	67

Subgroups — C2×#499×D67 #31725421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#499×D₆₇ #31725421