Subgroups — #2241×D₆₇ #31011117

Order 17152.

622 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	#500×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#494×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22562438	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562437	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562436	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562435	C₂	1	4	yes	maximal, complemented
7	#358×D₆₇	C₂	1	4	yes	maximal
8	C₄×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₆₇	—	2	1	no	—
11	C₂×Q₁₆×D₆₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×D₆₇	—	2	1	no	—
13	C₄⋊Q₈×D₆₇	—	2	1	no	—
14	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
15	#16863801	C₂²	1	1	yes	characteristic
16	#16863800	—	2	1	no	—
17	#16863800	C₂²	1	2	yes	—
18	#16863799	—	2	1	no	—
19	#16863776	C₂²	1	1	yes	characteristic
20	#16863775	—	2	1	no	—
21	#16863774	—	2	1	no	—
22	#16863774	C₂²	1	2	yes	—
23	#107×D₆₇	—	2	1	no	—
24	#107×D₆₇	C₂²	1	2	yes	—
25	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
26	#16863201	C₂²	1	4	yes	—
27	#16863200	C₂²	1	4	yes	—
28	#16863199	C₂²	1	4	yes	—
29	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₆₇	C₂²	1	2	yes	—
31	#500×C₆₇	C₂²	1	1	yes	characteristic
32	#494×C₆₇	C₂²	1	1	yes	characteristic
33	#358×C₆₇	C₂²	1	4	yes	—
34	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₆₇	—	2	2	no	—
36	C₂×Q₈×D₆₇	—	4	1	no	—
37	C₂×Dic₂₆₈	—	2	1	no	—
38	C₂×Dic₂₆₈	C₂³	1	2	yes	—
39	C₂×#11761618	—	2	2	no	—
40	C₂×#11761618	—	2	2	no	—
41	Q₁₆×D₆₇	—	2	8	no	—
42	Q₁₆×D₆₇	—	4	4	no	—
43	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
44	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
45	#14524043	—	2	1	no	—
46	#14524043	—	2	1	no	—
47	#14524042	—	2	2	no	—
48	#14524042	—	4	1	no	—
49	#14524042	—	4	1	no	—
50	#14524042	—	4	1	no	—
51	#14524041	—	2	2	no	—
52	#14524041	—	4	1	no	—
53	C₄⋊C₄×D₆₇	—	2	2	no	—
54	C₄⋊C₄×D₆₇	—	4	1	no	—
55	C₄⋊C₄×D₆₇	—	4	1	no	—
56	C₄⋊C₄×D₆₇	—	4	1	no	—
57	C₄²×D₆₇	D₄	1	1	yes	characteristic
58	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
59	C₁₆×D₆₇	—	2	4	no	—
60	#14523977	—	2	1	no	—
61	#14523977	C₂³	1	2	yes	—
62	#14523976	—	2	2	no	—
63	#14523976	—	2	2	no	—
64	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₆₇	D₄	1	1	yes	characteristic
66	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₆₇	—	2	1	no	—
68	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₆₇	—	2	1	no	—
70	C₄⋊Q₈×C₆₇	—	2	1	no	—
71	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
72	#107×C₆₇	—	2	1	no	—
73	#107×C₆₇	C₂³	1	2	yes	—
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₃₄	—	2	2	no	—
76	C₂×Dic₁₃₄	—	4	1	no	—
77	C₂×Dic₁₃₄	—	4	2	no	—
78	C₂×Dic₁₃₄	—	4	2	no	—
79	Q₈×D₆₇	—	4	4	no	—
80	Q₈×D₆₇	—	4	4	no	—
81	Q₈×D₆₇	—	8	4	no	—
82	C₂×C₄×D₆₇	—	4	2	no	—
83	C₂×C₄×D₆₇	—	4	2	no	—
84	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
85	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
86	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
87	Dic₂₆₈	—	2	4	no	—
88	Dic₂₆₈	—	4	2	no	—
89	#11761618	—	4	4	no	—
90	#11761618	—	4	4	no	—
91	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₈×D₆₇	SD₁₆	1	4	yes	—
94	C₈×D₆₇	#41	1	4	yes	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	C₄⋊Dic₆₇	—	4	1	no	—
97	C₄⋊Dic₆₇	—	4	1	no	—
98	C₄⋊Dic₆₇	—	4	1	no	—
99	Dic₆₇⋊C₄	—	4	2	no	—
100	Dic₆₇⋊C₄	—	4	2	no	—
101	Dic₆₇⋊C₄	—	4	2	no	—
102	Dic₆₇⋊C₄	—	4	2	no	—
103	C₄×Dic₆₇	D₈	1	1	yes	characteristic
104	C₄×Dic₆₇	D₈	1	1	yes	characteristic
105	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
106	C₆₇⋊C₁₆	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	4	1	no	—
109	Q₁₆×C₆₇	—	2	4	no	—
110	Q₁₆×C₆₇	—	4	2	no	—
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₄⋊C₄×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₆₇	—	2	2	no	—
119	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₆₇	—	4	2	no	—
121	Q₈×C₆₇	—	4	2	no	—
122	Q₈×C₆₇	—	8	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₆₇	—	4	2	no	—
129	C₂×Dic₆₇	—	4	2	no	—
130	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
133	C₄×D₆₇	—	8	4	no	—
134	C₄×D₆₇	—	8	4	no	—
135	C₄×D₆₇	SD₃₂	1	4	yes	—
136	C₄×D₆₇	Q₃₂	1	4	yes	—
137	C₄×D₆₇	#124	1	2	yes	—
138	C₄×D₆₇	C₂×SD₁₆	1	2	yes	—
139	Dic₁₃₄	—	4	2	no	—
140	Dic₁₃₄	—	4	2	no	—
141	Dic₁₃₄	—	8	2	no	—
142	Dic₁₃₄	—	8	4	no	—
143	Dic₁₃₄	—	8	4	no	—
144	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
145	C₈×C₆₇	C₂×#41	1	2	yes	—
146	C₆₇⋊C₈	#124	1	2	yes	—
147	C₆₇⋊C₈	C₂×SD₁₆	1	2	yes	—
148	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₆₇	#486	1	2	yes	—
150	C₂×D₆₇	C₂×SD₃₂	1	2	yes	—
151	C₂×D₆₇	C₂×Q₃₂	1	2	yes	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
155	C₄×C₆₇	C₂×Q₃₂	1	2	yes	—
156	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
157	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
158	Dic₆₇	—	8	2	no	—
159	Dic₆₇	—	8	2	no	—
160	Dic₆₇	#486	1	2	yes	—
161	Dic₆₇	C₂×SD₃₂	1	2	yes	—
162	Dic₆₇	C₂×Q₃₂	1	2	yes	—
163	C₂×#2241	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	C₂×#486	1	1	yes	characteristic
165	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
166	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
167	D₆₇	#2241	1	4	yes	direct product factor
168	C₂×#500	—	67	1	no	—
169	C₂×#494	—	67	1	no	—
170	#2241	—	67	1	no	—
171	#2241	—	67	1	no	—
172	#2241	—	67	4	no	—
173	#2241	—	67	1	no	—
174	#2241	D₆₇	1	1	yes	characteristic, direct product factor
175	C₂×#358	—	67	4	no	—
176	C₂×C₄×C₁₆	—	67	1	no	—
177	C₆₇	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	67	2	no	—
179	C₂²×Q₁₆	—	134	1	no	—
180	C₂×C₄⋊Q₈	—	134	1	no	—
181	C₂×C₄⋊Q₈	—	134	1	no	—
182	C₂²×C₁₆	—	67	2	no	—
183	#500	—	67	2	no	—
184	#500	—	67	1	no	—
185	#500	—	134	1	no	—
186	#500	—	134	1	no	—
187	#500	C₂×D₆₇	1	1	yes	characteristic
188	#494	—	67	2	no	—
189	#494	—	67	1	no	—
190	#494	—	134	1	no	—
191	#494	—	134	1	no	—
192	#494	C₂×D₆₇	1	1	yes	characteristic
193	C₂×#107	—	67	2	no	—
194	C₂×#107	—	134	1	no	—
195	C₂×C₄×C₈	—	67	1	no	—
196	#358	—	67	4	no	—
197	#358	—	67	4	no	—
198	#358	—	67	4	no	—
199	#358	C₂×D₆₇	1	4	yes	—
200	C₄×C₁₆	—	67	2	no	—
201	C₄×C₁₆	—	67	1	no	—
202	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
203	C₂²×Q₈	—	268	1	no	—
204	C₂²×Q₈	—	134	2	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	268	4	no	—
207	C₂×Q₁₆	—	67	2	no	—
208	C₂×Q₁₆	—	134	8	no	—
209	C₂×Q₁₆	—	134	2	no	—
210	C₂×Q₁₆	—	134	2	no	—
211	C₂×Q₁₆	—	134	1	no	—
212	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
213	C₂²×C₈	—	67	1	no	—
214	C₂²×C₈	—	67	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	268	1	no	—
218	C₄⋊Q₈	—	268	1	no	—
219	C₄⋊Q₈	—	268	1	no	—
220	C₄⋊Q₈	—	268	1	no	—
221	C₄⋊Q₈	—	134	2	no	—
222	C₄⋊Q₈	—	134	2	no	—
223	C₄⋊Q₈	—	134	1	no	—
224	C₄⋊Q₈	—	134	1	no	—
225	C₂×C₄⋊C₄	—	268	1	no	—
226	C₂×C₄⋊C₄	—	268	1	no	—
227	C₂×C₄⋊C₄	—	268	1	no	—
228	C₂×C₄⋊C₄	—	134	2	no	—
229	C₂×C₄²	—	67	1	no	—
230	C₂×C₁₆	—	67	2	no	—
231	C₂×C₁₆	—	134	4	no	—
232	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
233	#107	—	2	1	no	—
234	#107	—	67	2	no	—
235	#107	—	134	2	no	—
236	#107	—	134	2	no	—
237	#107	—	134	1	no	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	C₄×C₈	—	67	1	no	—
240	C₄×C₈	—	67	1	no	—
241	C₄×C₈	—	67	1	no	—
242	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	268	2	no	—
246	C₂×Q₈	—	268	2	no	—
247	C₂×Q₈	—	268	1	no	—
248	C₂×Q₈	—	268	4	no	—
249	C₂×Q₈	—	268	4	no	—
250	C₂×Q₈	—	536	4	no	—
251	C₂×Q₈	—	134	2	no	—
252	C₂²×C₄	—	268	2	no	—
253	C₂²×C₄	—	268	2	no	—
254	C₂²×C₄	—	67	1	no	—
255	C₂²×C₄	—	67	1	no	—
256	C₂²×C₄	—	67	1	no	—
257	Q₁₆	—	2	4	no	—
258	Q₁₆	—	4	2	no	—
259	Q₁₆	—	268	4	no	—
260	Q₁₆	—	268	4	no	—
261	Q₁₆	—	268	2	no	—
262	Q₁₆	—	134	4	no	—
263	C₂×C₈	—	67	1	no	—
264	C₂×C₈	—	67	1	no	—
265	C₂×C₈	—	67	4	no	—
266	C₂×C₈	—	67	4	no	—
267	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	268	2	no	—
275	C₄⋊C₄	—	268	2	no	—
276	C₄⋊C₄	—	268	2	no	—
277	C₄⋊C₄	—	268	1	no	—
278	C₄⋊C₄	—	268	1	no	—
279	C₄⋊C₄	—	268	1	no	—
280	C₄⋊C₄	—	134	2	no	—
281	C₄²	—	67	1	no	—
282	C₄²	—	67	1	no	—
283	C₄²	—	67	1	no	—
284	C₄²	D₄×D₆₇	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	134	2	no	—
287	C₂³	—	67	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	268	2	no	—
292	Q₈	—	268	2	no	—
293	Q₈	—	536	4	no	—
294	Q₈	—	536	4	no	—
295	Q₈	—	536	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	268	2	no	—
299	C₂×C₄	—	268	2	no	—
300	C₂×C₄	—	536	4	no	—
301	C₂×C₄	—	536	4	no	—
302	C₂×C₄	—	67	1	no	—
303	C₂×C₄	—	67	1	no	—
304	C₂×C₄	—	67	1	no	—
305	C₂×C₄	—	67	4	no	—
306	C₂×C₄	—	67	4	no	—
307	C₂×C₄	—	67	2	no	—
308	C₂×C₄	—	67	2	no	—
309	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
310	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
312	C₈	—	67	2	no	—
313	C₈	—	67	2	no	—
314	C₈	SD₁₆×D₆₇	1	2	yes	—
315	C₈	#41×D₆₇	1	2	yes	—
316	C₂²	—	67	2	no	—
317	C₂²	—	67	2	no	—
318	C₂²	—	67	2	no	—
319	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	536	2	no	—
323	C₄	—	536	2	no	—
324	C₄	—	67	2	no	—
325	C₄	—	67	2	no	—
326	C₄	—	67	2	no	—
327	C₄	SD₃₂×D₆₇	1	2	yes	—
328	C₄	Q₃₂×D₆₇	1	2	yes	—
329	C₄	#124×D₆₇	1	1	yes	characteristic
330	C₄	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
331	C₂	—	67	4	no	—
332	C₂	#486×D₆₇	1	1	yes	characteristic
333	C₂	C₂×SD₃₂×D₆₇	1	1	yes	characteristic
334	C₂	C₂×Q₃₂×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2241	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₆₇	C₂×D₈
hypercenter	#2241	D₆₇
hypocenter	C₆₇	C₂×#2241
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
3. #22562438	C₂	semidirect
4. #22562437	C₂	semidirect
5. #22562436	C₂	semidirect
6. #22562435	C₂	semidirect
9. C₆₇×#2241	C₂	semidirect
167. D₆₇	#2241	direct (N × H)
174. #2241	D₆₇	direct (N × H)
177. C₆₇	C₂×#2241	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#500×D₆₇	2
#494×D₆₇	2
#22562438	2
#22562437	2
#22562436	2
#22562435	2
#358×D₆₇	2
C₄×C₁₆×D₆₇	2
C₆₇×#2241	2
C₂×#2241	67

Subgroups — #2241×D67 #31011117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2241×D₆₇ #31011117