Subgroups — #31009211

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863825	C₂	1	1	yes	maximal, characteristic
2	#22569079	C₂	1	2	yes	maximal, complemented
3	#22562073	C₂	1	2	yes	maximal, complemented
4	#22562070	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562067	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#366×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561918	C₂	1	2	yes	maximal, complemented
10	#22561804	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524062	C₂²	1	2	yes	—
13	#16863875	—	2	2	no	—
14	#16863825	C₄	1	8	yes	complemented
15	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
16	C₂×#14523987	C₂²	1	2	yes	—
17	C₂×#14523982	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
19	#16863496	—	2	2	no	—
20	#16863493	—	2	2	no	—
21	C₄×Dic₂₆₈	C₂²	1	2	yes	—
22	#16863490	C₂²	1	2	yes	—
23	D₈×Dic₆₇	C₂²	1	2	yes	—
24	#107×D₆₇	C₂²	1	2	yes	—
25	#16863394	—	2	2	no	—
26	#16863350	C₂²	1	1	yes	characteristic
27	#16863222	C₂²	1	1	yes	characteristic
28	#16863221	C₂²	1	2	yes	—
29	#16863201	C₂²	1	2	yes	—
30	#16863199	C₂²	1	2	yes	—
31	#16863197	C₂²	1	2	yes	—
32	#16863172	C₂²	1	1	yes	characteristic
33	#16863171	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₆₇	C₂²	1	2	yes	—
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#357×C₆₇	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#11761627	—	2	2	no	—
41	#14524062	—	2	4	no	—
42	#14524062	C₂×C₄	1	8	yes	complemented
43	C₂×Dic₂₆₈	C₂³	1	2	yes	—
44	C₂×#11761616	—	2	2	no	—
45	C₂×#11761611	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
47	C₄×Dic₁₃₄	—	2	2	no	—
48	#14524008	—	2	2	no	—
49	D₄×Dic₆₇	—	2	2	no	—
50	C₄×C₂²⋊D₆₇	—	4	2	no	—
51	(C₂²⋊D₆₇)⋊C₄	—	4	2	no	—
52	#14524000	D₄	1	1	yes	characteristic
53	#14523999	—	4	2	no	—
54	C₂×Dic₆₇⋊C₄	—	4	2	no	—
55	C₄⋊C₄×D₆₇	—	2	2	no	—
56	Dic₅₃₆	C₂×C₄	1	4	yes	complemented
57	#14523987	C₂×C₄	1	8	yes	complemented
58	#14523982	—	2	2	no	—
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₆₇	C₂×C₄	1	4	yes	complemented
62	#14523977	C₂³	1	2	yes	—
63	#14523976	—	2	2	no	—
64	#14523968	—	2	2	no	—
65	#14523966	—	2	2	no	—
66	#14523964	—	2	2	no	—
67	#14523954	D₄	1	1	yes	characteristic
68	#14523953	D₄	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₆₇	C₂³	1	2	yes	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	D₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	2	yes	—
76	#102×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	#11761627	—	2	8	no	—
79	#11761627	—	4	4	no	—
80	C₂×Dic₁₃₄	—	2	2	no	—
81	C₂×C₂²⋊D₆₇	—	4	2	no	—
82	C₂²×Dic₆₇	—	4	2	no	—
83	C₂²×Dic₆₇	—	4	2	no	—
84	C₂×C₄×D₆₇	—	4	2	no	—
85	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
86	Dic₂₆₈	—	2	2	no	—
87	Dic₂₆₈	C₂²×C₄	1	4	yes	—
88	#11761616	—	2	8	no	—
89	#11761611	—	2	2	no	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₈×D₆₇	C₂²×C₄	1	2	yes	—
93	C₈×D₆₇	#41	1	2	yes	—
94	C₄⋊Dic₆₇	—	2	2	no	—
95	Dic₆₇⋊C₄	—	4	2	no	—
96	Dic₆₇⋊C₄	—	4	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
99	Dic₆₇⋊C₄	D₈	1	1	yes	characteristic
100	C₂²⋊Dic₆₇	—	4	2	no	—
101	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
102	(C₂×D₆₇)⋊C₄	D₈	1	1	yes	characteristic
103	C₄×Dic₆₇	—	4	2	no	—
104	C₄×Dic₆₇	—	4	2	no	—
105	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	D₈×C₆₇	—	2	2	no	—
111	D₈×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	2	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂³×C₆₇	—	4	2	no	—
120	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
121	D₄×C₆₇	—	2	4	no	—
122	D₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	16	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	C₂²⋊D₆₇	—	4	8	no	—
129	C₂×Dic₆₇	—	4	2	no	—
130	C₂×Dic₆₇	—	4	8	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	—	16	2	no	—
133	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
136	C₄×D₆₇	—	16	2	no	—
137	C₄×D₆₇	C₄×D₄	1	2	yes	—
138	C₄×D₆₇	#135	1	2	yes	—
139	Dic₁₃₄	—	2	4	no	—
140	Dic₁₃₄	—	4	2	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₆₇⋊C₈	—	2	1	no	—
145	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
146	C₂²×C₆₇	—	4	4	no	—
147	C₂²×C₆₇	—	8	2	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₆₇	—	2	1	no	—
151	C₂×D₆₇	—	2	1	no	—
152	C₂×D₆₇	C₄×D₈	1	2	yes	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
158	Dic₆₇	—	2	1	no	—
159	Dic₆₇	—	2	1	no	—
160	Dic₆₇	—	4	4	no	—
161	Dic₆₇	—	8	2	no	—
162	Dic₆₇	C₄×D₈	1	2	yes	—
163	#33459	—	67	1	no	maximal, Sylow
164	C₂×C₆₇	—	8	4	no	—
165	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
166	C₂×C₆₇	#3415	1	1	yes	characteristic
167	C₂×C₆₇	#3417	1	1	yes	characteristic
168	D₆₇	—	2	2	no	—
169	C₂×#508	—	67	1	no	—
170	#2951	—	67	2	no	—
171	#2175	—	67	2	no	—
172	#2175	—	67	2	no	—
173	C₄×Q₃₂	—	67	1	no	—
174	C₄×Q₃₂	—	67	1	no	—
175	C₄×D₁₆	—	67	1	no	—
176	C₄×D₁₆	D₆₇	1	1	yes	characteristic, complemented
177	C₂×#366	—	67	1	no	—
178	#2139	—	67	2	no	—
179	#2108	—	67	1	no	—
180	C₆₇	#33459	1	1	yes	characteristic, complemented, Sylow
181	C₂×#135	—	67	2	no	—
182	#520	—	134	2	no	—
183	#508	—	67	8	no	—
184	C₂×Q₃₂	—	67	1	no	—
185	C₂×SD₃₂	—	67	2	no	—
186	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic
187	C₂×M₃₂	—	67	1	no	—
188	C₂²×C₁₆	—	67	1	no	—
189	#440	—	134	2	no	—
190	#440	—	134	2	no	—
191	C₄×Q₁₆	—	67	2	no	—
192	C₄×Q₁₆	—	67	2	no	—
193	C₄×D₈	—	67	2	no	—
194	C₄×D₈	C₂×D₆₇	1	2	yes	—
195	C₂×#107	—	67	2	no	—
196	#417	—	134	2	no	—
197	#405	—	67	1	no	—
198	#366	—	67	1	no	—
199	#366	—	67	2	no	—
200	#366	C₂×D₆₇	1	1	yes	characteristic
201	#358	—	67	2	no	—
202	#358	—	67	2	no	—
203	#357	—	67	2	no	—
204	#357	C₂×D₆₇	1	2	yes	—
205	#346	—	67	1	no	—
206	#346	—	67	1	no	—
207	C₄×C₁₆	—	67	1	no	—
208	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
209	C₂×#41	—	134	2	no	—
210	#135	—	67	8	no	—
211	#135	—	134	4	no	—
212	C₂×Q₁₆	—	67	2	no	—
213	C₂×SD₁₆	—	134	2	no	—
214	C₂×D₈	C₂²×D₆₇	1	2	yes	—
215	C₂×M₁₆	—	67	1	no	—
216	C₂²×C₈	—	67	1	no	—
217	C₄×Q₈	—	134	2	no	—
218	C₄×Q₈	—	134	2	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	268	2	no	—
221	C₄×D₄	—	268	2	no	—
222	C₄×D₄	—	134	2	no	—
223	#117	—	268	2	no	—
224	#117	—	67	1	no	—
225	C₂×C₄⋊C₄	—	268	2	no	—
226	C₂×C₄⋊C₄	—	134	2	no	—
227	Q₃₂	—	67	4	no	—
228	SD₃₂	—	67	8	no	—
229	D₁₆	C₄×D₆₇	1	4	yes	complemented
230	M₃₂	—	134	2	no	—
231	C₂×C₁₆	—	67	1	no	—
232	C₂×C₁₆	—	67	4	no	—
233	C₂×C₁₆	—	67	1	no	—
234	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
236	#107	—	67	2	no	—
237	#107	—	134	2	no	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	#103	—	134	2	no	—
240	#103	—	134	2	no	—
241	#102	—	2	2	no	—
242	#102	—	134	2	no	—
243	#97	—	67	1	no	—
244	#97	—	67	1	no	—
245	C₄×C₈	—	67	1	no	—
246	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
247	#41	—	268	4	no	—
248	#41	—	134	8	no	—
249	C₂×Q₈	—	134	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	268	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	268	2	no	—
254	C₂²×C₄	—	268	2	no	—
255	C₂²×C₄	—	268	2	no	—
256	C₂²×C₄	—	67	1	no	—
257	Q₁₆	—	67	4	no	—
258	Q₁₆	—	134	2	no	—
259	SD₁₆	—	134	8	no	—
260	D₈	—	2	2	no	—
261	D₈	C₂×C₄×D₆₇	1	4	yes	—
262	M₁₆	—	134	2	no	—
263	C₂×C₈	—	67	1	no	—
264	C₂×C₈	—	67	2	no	—
265	C₂×C₈	—	67	2	no	—
266	C₂×C₈	—	67	1	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	—	268	2	no	—
271	C₄⋊C₄	—	268	2	no	—
272	C₄⋊C₄	—	268	2	no	—
273	C₄⋊C₄	—	67	1	no	—
274	C₄⋊C₄	—	67	1	no	—
275	C₄⋊C₄	—	134	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	268	2	no	—
278	C₂²⋊C₄	—	67	1	no	—
279	C₂²⋊C₄	—	67	1	no	—
280	C₄²	—	268	2	no	—
281	C₄²	—	268	2	no	—
282	C₄²	—	67	1	no	—
283	C₄²	D₄×D₆₇	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	67	2	no	—
286	C₁₆	—	134	1	no	—
287	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
288	C₂³	—	4	2	no	—
289	C₂³	—	67	1	no	—
290	Q₈	—	268	2	no	—
291	Q₈	—	134	4	no	—
292	D₄	—	2	4	no	—
293	D₄	—	4	2	no	—
294	D₄	—	268	8	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	268	2	no	—
297	C₂×C₄	—	268	8	no	—
298	C₂×C₄	—	268	2	no	—
299	C₂×C₄	—	16	2	no	—
300	C₂×C₄	—	1072	2	no	—
301	C₂×C₄	—	1072	2	no	—
302	C₂×C₄	—	67	1	no	—
303	C₂×C₄	—	67	2	no	—
304	C₂×C₄	—	67	2	no	—
305	C₂×C₄	—	67	1	no	—
306	C₂×C₄	—	67	1	no	—
307	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
308	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	67	2	no	—
312	C₈	—	134	1	no	—
313	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
314	C₈	#41×D₆₇	1	1	yes	characteristic
315	C₂²	—	4	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	67	2	no	—
319	C₂²	—	134	1	no	—
320	C₂²	—	134	1	no	—
321	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	268	4	no	—
326	C₄	—	536	2	no	—
327	C₄	—	67	2	no	—
328	C₄	—	134	1	no	—
329	C₄	—	134	1	no	—
330	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
331	C₄	#135×D₆₇	1	1	yes	characteristic
332	C₂	—	8	4	no	—
333	C₂	—	134	2	no	—
334	C₂	C₄×D₈×D₆₇	1	1	yes	characteristic
335	C₂	#22573346	1	1	yes	characteristic
336	C₂	#22573359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×D₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#16863825	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×D₁₆	D₆₇
hypocenter	C₆₇	#33459
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
2. #22569079	C₂	semidirect
3. #22562073	C₂	semidirect
4. #22562070	C₂	semidirect
5. C₄×Dic₅₃₆	C₂	semidirect
6. #22562067	C₂	semidirect
7. D₁₆×Dic₆₇	C₂	semidirect
8. #366×D₆₇	C₂	semidirect
9. #22561918	C₂	semidirect
10. #22561804	C₂	semidirect
11. C₄×D₁₆×C₆₇	C₂	semidirect
14. #16863825	C₄	semidirect
42. #14524062	C₂×C₄	semidirect
56. Dic₅₃₆	C₂×C₄	semidirect
57. #14523987	C₂×C₄	semidirect
61. C₁₆×D₆₇	C₂×C₄	semidirect
72. D₁₆×C₆₇	C₂×C₄	semidirect
176. C₄×D₁₆	D₆₇	semidirect
180. C₆₇	#33459	semidirect
229. D₁₆	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863825	2
#22569079	2
#22562073	2
#22562070	2
C₄×Dic₅₃₆	2
#22562067	2
D₁₆×Dic₆₇	2
#366×D₆₇	2
#22561918	2
#22561804	2
C₄×D₁₆×C₆₇	2
#33459	67