Subgroups — #31009223

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 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	#22562085	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562081	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562079	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562064	C₂	1	2	yes	maximal, complemented
7	C₄×#14523987	C₂	1	2	yes	maximal, complemented
8	#367×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561918	C₂	1	2	yes	maximal, complemented
10	#22561807	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2177	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	#16863224	C₂²	1	1	yes	characteristic
28	#16863223	C₂²	1	2	yes	—
29	#16863201	C₂²	1	2	yes	—
30	#16863199	C₂²	1	2	yes	—
31	#16863197	C₂²	1	2	yes	—
32	#16863171	C₂²	1	1	yes	characteristic
33	C₄×C₆₇⋊C₁₆	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	#367×C₆₇	C₂²	1	1	yes	characteristic
38	#357×C₆₇	C₂²	1	2	yes	—
39	#346×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	2	yes	—
99	C₂²⋊Dic₆₇	—	4	2	no	—
100	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
101	C₄×Dic₆₇	—	4	2	no	—
102	C₄×Dic₆₇	—	4	2	no	—
103	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
104	C₆₇⋊C₁₆	—	2	1	no	—
105	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	4	2	no	—
108	D₈×C₆₇	—	2	2	no	—
109	D₈×C₆₇	C₂²×C₄	1	4	yes	—
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₂²⋊C₄×C₆₇	—	4	2	no	—
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₆₇	—	2	1	no	—
116	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂³×C₆₇	—	4	2	no	—
118	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
119	D₄×C₆₇	—	2	4	no	—
120	D₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	16	2	no	—
123	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₆₇	—	4	8	no	—
126	C₂×Dic₆₇	—	4	2	no	—
127	C₂×Dic₆₇	—	4	8	no	—
128	C₂×Dic₆₇	—	4	2	no	—
129	C₂×Dic₆₇	—	16	2	no	—
130	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
131	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
132	C₄×D₆₇	—	16	2	no	—
133	C₄×D₆₇	C₄×D₄	1	2	yes	—
134	C₄×D₆₇	#135	1	2	yes	—
135	Dic₁₃₄	—	2	4	no	—
136	Dic₁₃₄	—	4	2	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₆₇⋊C₈	—	2	1	no	—
141	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
142	C₂²×C₆₇	—	4	4	no	—
143	C₂²×C₆₇	—	8	2	no	—
144	C₂²×C₆₇	—	8	2	no	—
145	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₆₇	—	2	2	no	—
147	C₂×D₆₇	C₄×D₈	1	2	yes	—
148	C₄×C₆₇	—	2	2	no	—
149	C₄×C₆₇	—	8	2	no	—
150	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
152	Dic₆₇	—	2	2	no	—
153	Dic₆₇	—	4	4	no	—
154	Dic₆₇	—	8	2	no	—
155	Dic₆₇	C₄×D₈	1	2	yes	—
156	#33460	—	67	1	no	maximal, Sylow
157	C₂×C₆₇	—	8	4	no	—
158	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
159	C₂×C₆₇	#3416	1	2	yes	—
160	D₆₇	—	2	2	no	—
161	C₂×#508	—	67	1	no	—
162	#2951	—	67	2	no	—
163	#2177	—	67	1	no	—
164	#2177	D₆₇	1	1	yes	characteristic, complemented
165	#2176	—	67	1	no	—
166	#2176	—	67	1	no	—
167	C₄×SD₃₂	—	67	2	no	—
168	C₄×SD₃₂	—	67	2	no	—
169	C₂×#367	—	67	1	no	—
170	#2139	—	67	2	no	—
171	#2108	—	67	1	no	—
172	C₆₇	#33460	1	1	yes	characteristic, complemented, Sylow
173	C₂×#135	—	67	2	no	—
174	#520	—	134	2	no	—
175	#508	—	67	8	no	—
176	C₂×Q₃₂	—	67	1	no	—
177	C₂×SD₃₂	—	67	2	no	—
178	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic
179	C₂×M₃₂	—	67	1	no	—
180	C₂²×C₁₆	—	67	1	no	—
181	#440	—	134	2	no	—
182	#440	—	134	2	no	—
183	C₄×Q₁₆	—	67	2	no	—
184	C₄×Q₁₆	—	67	2	no	—
185	C₄×D₈	—	67	2	no	—
186	C₄×D₈	C₂×D₆₇	1	2	yes	—
187	C₂×#107	—	67	2	no	—
188	#417	—	134	2	no	—
189	#405	—	67	1	no	—
190	#367	—	67	1	no	—
191	#367	—	67	2	no	—
192	#367	C₂×D₆₇	1	1	yes	characteristic
193	#358	—	67	2	no	—
194	#358	—	67	2	no	—
195	#357	—	67	2	no	—
196	#357	C₂×D₆₇	1	2	yes	—
197	#346	—	67	1	no	—
198	#346	C₂×D₆₇	1	1	yes	characteristic
199	C₄×C₁₆	—	67	1	no	—
200	C₄×C₁₆	—	67	1	no	—
201	C₂×#41	—	134	2	no	—
202	#135	—	67	8	no	—
203	#135	—	134	4	no	—
204	C₂×Q₁₆	—	67	2	no	—
205	C₂×SD₁₆	—	134	2	no	—
206	C₂×D₈	C₂²×D₆₇	1	2	yes	—
207	C₂×M₁₆	—	67	1	no	—
208	C₂²×C₈	—	67	1	no	—
209	C₄×Q₈	—	134	2	no	—
210	C₄×Q₈	—	134	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	268	2	no	—
213	C₄×D₄	—	268	2	no	—
214	C₄×D₄	—	134	2	no	—
215	#117	—	268	2	no	—
216	#117	—	67	1	no	—
217	C₂×C₄⋊C₄	—	268	2	no	—
218	C₂×C₄⋊C₄	—	134	2	no	—
219	Q₃₂	—	67	4	no	—
220	SD₃₂	—	67	8	no	—
221	D₁₆	C₄×D₆₇	1	4	yes	complemented
222	M₃₂	—	134	2	no	—
223	C₂×C₁₆	—	67	1	no	—
224	C₂×C₁₆	—	67	4	no	—
225	C₂×C₁₆	—	67	1	no	—
226	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
228	#107	—	67	2	no	—
229	#107	—	134	2	no	—
230	#107	C₂²×D₆₇	1	2	yes	—
231	#103	—	134	2	no	—
232	#103	—	134	2	no	—
233	#102	—	2	2	no	—
234	#102	—	134	2	no	—
235	#97	—	67	1	no	—
236	#97	—	67	1	no	—
237	C₄×C₈	—	67	1	no	—
238	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
239	#41	—	268	4	no	—
240	#41	—	134	8	no	—
241	C₂×Q₈	—	134	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	268	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	268	2	no	—
246	C₂²×C₄	—	268	2	no	—
247	C₂²×C₄	—	268	2	no	—
248	C₂²×C₄	—	67	1	no	—
249	Q₁₆	—	67	4	no	—
250	Q₁₆	—	134	2	no	—
251	SD₁₆	—	134	8	no	—
252	D₈	—	2	2	no	—
253	D₈	C₂×C₄×D₆₇	1	4	yes	—
254	M₁₆	—	134	2	no	—
255	C₂×C₈	—	67	1	no	—
256	C₂×C₈	—	67	2	no	—
257	C₂×C₈	—	67	2	no	—
258	C₂×C₈	—	67	1	no	—
259	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	268	2	no	—
263	C₄⋊C₄	—	268	2	no	—
264	C₄⋊C₄	—	268	2	no	—
265	C₄⋊C₄	—	67	2	no	—
266	C₄⋊C₄	—	134	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	268	2	no	—
269	C₂²⋊C₄	—	67	2	no	—
270	C₄²	—	268	2	no	—
271	C₄²	—	268	2	no	—
272	C₄²	—	67	1	no	—
273	C₄²	D₄×D₆₇	1	1	yes	characteristic
274	C₁₆	—	2	1	no	—
275	C₁₆	—	67	2	no	—
276	C₁₆	—	134	1	no	—
277	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
278	C₂³	—	4	2	no	—
279	C₂³	—	67	1	no	—
280	Q₈	—	268	2	no	—
281	Q₈	—	134	4	no	—
282	D₄	—	2	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	268	8	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	268	2	no	—
287	C₂×C₄	—	268	8	no	—
288	C₂×C₄	—	268	2	no	—
289	C₂×C₄	—	16	2	no	—
290	C₂×C₄	—	1072	2	no	—
291	C₂×C₄	—	1072	2	no	—
292	C₂×C₄	—	67	1	no	—
293	C₂×C₄	—	67	2	no	—
294	C₂×C₄	—	67	2	no	—
295	C₂×C₄	—	67	2	no	—
296	C₂×C₄	D₈×D₆₇	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	67	2	no	—
300	C₈	—	134	1	no	—
301	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
302	C₈	#41×D₆₇	1	1	yes	characteristic
303	C₂²	—	4	4	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	67	2	no	—
307	C₂²	—	134	2	no	—
308	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	268	4	no	—
312	C₄	—	536	2	no	—
313	C₄	—	67	2	no	—
314	C₄	—	134	2	no	—
315	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
316	C₄	#135×D₆₇	1	1	yes	characteristic
317	C₂	—	8	4	no	—
318	C₂	—	134	2	no	—
319	C₂	C₄×D₈×D₆₇	1	1	yes	characteristic
320	C₂	#22573355	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2177	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#16863825	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2177	D₆₇
hypocenter	C₆₇	#33460
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #22569079	C₂	semidirect
3. #22562085	C₂	semidirect
4. #22562081	C₂	semidirect
5. #22562079	C₂	semidirect
6. #22562064	C₂	semidirect
7. C₄×#14523987	C₂	semidirect
8. #367×D₆₇	C₂	semidirect
9. #22561918	C₂	semidirect
10. #22561807	C₂	semidirect
11. C₆₇×#2177	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
164. #2177	D₆₇	semidirect
172. C₆₇	#33460	semidirect
221. 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
#22562085	2
#22562081	2
#22562079	2
#22562064	2
C₄×#14523987	2
#367×D₆₇	2
#22561918	2
#22561807	2
C₆₇×#2177	2
#33460	67