Subgroups — #30864685

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570721	C₂	1	1	yes	maximal, characteristic
2	#22569147	C₂	1	2	yes	maximal
3	#22561188	C₂	1	2	yes	maximal
4	#22560963	C₂	1	1	yes	maximal, characteristic
5	#22560960	C₂	1	1	yes	maximal, characteristic
6	#22560956	C₂	1	2	yes	maximal
7	#22560379	C₂	1	1	yes	maximal, characteristic
8	#22560093	C₂	1	1	yes	maximal, characteristic
9	#22560089	C₂	1	2	yes	maximal
10	#22559897	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1773	C₂	1	1	yes	maximal, characteristic
12	#16863949	C₂²	1	2	yes	—
13	#16863800	C₄	1	2	yes	complemented
14	#16863797	C₄	1	2	yes	complemented
15	#16863796	C₄	1	2	yes	complemented
16	#16863793	C₄	1	2	yes	complemented
17	#16863526	—	2	4	no	—
18	#16863524	—	2	4	no	—
19	C₂×#14523979	C₂²	1	1	yes	characteristic
20	C₂×#14523978	C₂²	1	2	yes	—
21	#16863434	C₂²	1	1	yes	characteristic
22	#16863432	C₂²	1	2	yes	—
23	#16863424	C₂²	1	2	yes	—
24	#16863423	C₂²	1	2	yes	—
25	#16863419	C₂²	1	2	yes	—
26	#16863351	C₂²	1	2	yes	—
27	C₂×#14523953	C₂²	1	1	yes	characteristic
28	C₂×#14523953	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
32	#16863242	C₂²	1	1	yes	characteristic
33	#16863157	C₂²	1	2	yes	—
34	#16863156	C₂²	1	2	yes	—
35	#16863155	C₂²	1	2	yes	—
36	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
37	#427×C₆₇	C₂²	1	1	yes	characteristic
38	#405×C₆₇	C₂²	1	2	yes	—
39	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
40	#339×C₆₇	C₂²	1	2	yes	—
41	C₂×#11761612	C₂³	1	2	yes	—
42	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
44	#14524042	C₂×C₄	1	4	yes	—
45	#14524034	C₂×C₄	1	4	yes	—
46	#14524020	—	2	4	no	—
47	C₄×Dic₁₃₄	—	2	4	no	—
48	#14524002	C₂³	1	2	yes	—
49	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
51	#14523979	—	2	2	no	—
52	#14523978	—	2	4	no	—
53	#14523977	C₂×C₄	1	2	yes	—
54	#14523976	C₂×C₄	1	4	yes	—
55	#14523975	C₂×C₄	1	2	yes	—
56	#14523974	C₂×C₄	1	4	yes	—
57	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
58	Dic₆₇⋊C₈	—	2	4	no	—
59	Dic₆₇⋊C₈	—	2	4	no	—
60	#14523953	—	2	2	no	—
61	#14523953	—	2	2	no	—
62	#14523953	D₄	1	4	yes	—
63	C₄×C₆₇⋊C₈	—	2	2	no	—
64	C₈×Dic₆₇	—	2	2	no	—
65	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₆₇	Q₈	1	4	yes	complemented
70	C₆₇⋊#95	—	2	2	no	—
71	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	#117×C₆₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#108×C₆₇	—	2	2	no	—
77	#107×C₆₇	C₂×C₄	1	2	yes	—
78	#106×C₆₇	C₂×C₄	1	2	yes	—
79	#97×C₆₇	—	2	2	no	—
80	#97×C₆₇	Q₈	1	4	yes	complemented
81	C₄×C₈×C₆₇	—	2	2	no	—
82	C₂×Dic₁₃₄	—	2	4	no	—
83	C₂²×Dic₆₇	—	2	1	no	—
84	C₂²×Dic₆₇	#41	1	2	yes	—
85	#11761612	—	2	2	no	—
86	#11761612	—	2	2	no	—
87	C₂×C₆₇⋊C₈	—	2	1	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
93	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
94	C₂×C₆₇⋊C₈	#41	1	2	yes	—
95	C₄⋊Dic₆₇	—	2	4	no	—
96	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
97	Dic₆₇⋊C₄	—	2	4	no	—
98	Dic₆₇⋊C₄	—	2	4	no	—
99	C₂²⋊Dic₆₇	—	2	2	no	—
100	C₄×Dic₆₇	—	2	2	no	—
101	C₄×Dic₆₇	—	2	1	no	—
102	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
105	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
108	M₁₆×C₆₇	—	2	2	no	—
109	M₁₆×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₆₇	#41	1	2	yes	—
118	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
119	C₂²⋊C₄×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	1	no	—
122	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
123	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₆₇	—	2	4	no	—
134	C₂×Dic₆₇	—	2	2	no	—
135	C₂×Dic₆₇	—	4	2	no	—
136	C₂×Dic₆₇	—	4	2	no	—
137	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
138	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
139	Dic₁₃₄	—	8	4	no	—
140	C₈×C₆₇	—	4	2	no	—
141	C₈×C₆₇	C₄×Q₈	1	2	yes	—
142	C₈×C₆₇	C₄×Q₈	1	2	yes	—
143	C₈×C₆₇	C₄⋊Q₈	1	4	yes	—
144	C₆₇⋊C₈	—	2	2	no	—
145	C₆₇⋊C₈	—	4	2	no	—
146	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
147	C₆₇⋊C₈	C₄×Q₈	1	2	yes	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂²×C₆₇	#533	1	1	yes	characteristic
151	C₂²×C₆₇	#533	1	1	yes	characteristic
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	4	no	—
154	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
155	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₄×C₆₇	#532	1	1	yes	characteristic
158	Dic₆₇	—	2	2	no	—
159	Dic₆₇	—	2	2	no	—
160	Dic₆₇	—	4	2	no	—
161	Dic₆₇	—	4	4	no	—
162	#21733	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	2	1	no	—
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₆₇	#2957	1	2	yes	—
167	#3160	—	67	1	no	—
168	#2961	—	67	2	no	—
169	#1995	—	67	2	no	—
170	#1946	—	67	1	no	—
171	#1946	—	67	1	no	—
172	#1945	—	67	2	no	—
173	#1848	—	67	1	no	—
174	#1773	—	67	1	no	—
175	#1773	D₆₇	1	1	yes	characteristic
176	#1772	—	67	2	no	—
177	#1726	—	67	1	no	—
178	C₆₇	#21733	1	1	yes	characteristic, complemented, Sylow
179	#533	—	67	2	no	—
180	#500	—	67	2	no	—
181	#499	—	67	2	no	—
182	#499	—	67	2	no	—
183	#498	—	67	2	no	—
184	#446	—	134	4	no	—
185	#446	—	134	4	no	—
186	C₂×#108	—	67	2	no	—
187	C₂×#108	—	67	1	no	—
188	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
189	#427	—	67	1	no	—
190	#427	—	67	2	no	—
191	#427	C₂×D₆₇	1	1	yes	characteristic
192	#424	—	67	2	no	—
193	#424	—	67	2	no	—
194	#423	—	67	2	no	—
195	#405	—	67	2	no	—
196	#405	C₂×D₆₇	1	2	yes	—
197	C₂×#97	—	67	1	no	—
198	C₂×#97	—	67	1	no	—
199	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
200	C₂×C₄×C₈	—	67	1	no	—
201	C₂×C₄×C₈	—	67	1	no	—
202	C₂×C₄×C₈	—	67	1	no	—
203	#376	—	67	1	no	—
204	#339	—	67	2	no	—
205	#339	—	67	2	no	—
206	#339	—	67	2	no	—
207	#339	C₂×D₆₇	1	2	yes	—
208	C₂×M₁₆	—	67	2	no	—
209	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
210	C₂²×C₈	—	67	1	no	—
211	C₂²×C₈	—	67	1	no	—
212	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
214	C₄⋊Q₈	—	67	4	no	—
215	#125	—	67	4	no	—
216	C₂²⋊Q₈	—	134	4	no	—
217	C₄×Q₈	—	134	4	no	—
218	#117	—	67	2	no	—
219	#117	C₂²×D₆₇	1	2	yes	—
220	C₂×C₄²	—	67	1	no	—
221	C₂×C₄²	—	67	1	no	—
222	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
223	#108	—	2	2	no	—
224	#108	—	134	4	no	—
225	#108	—	134	2	no	—
226	#107	—	67	2	no	—
227	#107	—	67	4	no	—
228	#107	C₄×D₆₇	1	2	yes	—
229	#106	—	67	2	no	—
230	#106	—	67	4	no	—
231	#106	C₄×D₆₇	1	2	yes	—
232	C₄⋊C₈	—	134	4	no	—
233	C₄⋊C₈	—	134	4	no	—
234	C₄⋊C₈	—	134	4	no	—
235	#97	—	2	2	no	—
236	#97	—	67	4	no	—
237	#97	—	134	2	no	—
238	#97	—	134	2	no	—
239	#97	Dic₁₃₄	1	4	yes	complemented
240	C₄×C₈	—	2	2	no	—
241	C₄×C₈	—	67	1	no	—
242	C₄×C₈	—	67	1	no	—
243	C₄×C₈	—	67	1	no	—
244	C₄×C₈	—	67	1	no	—
245	C₄×C₈	—	67	4	no	—
246	C₄×C₈	—	134	2	no	—
247	C₄×C₈	—	134	2	no	—
248	#95	—	134	2	no	—
249	C₂×Q₈	—	134	4	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	67	2	no	—
252	C₂²×C₄	—	134	1	no	—
253	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	134	2	no	—
257	M₁₆	—	134	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	67	1	no	—
260	C₂×C₈	—	67	1	no	—
261	C₂×C₈	—	67	2	no	—
262	C₂×C₈	—	67	2	no	—
263	C₂×C₈	—	67	2	no	—
264	C₂×C₈	—	134	2	no	—
265	C₂×C₈	—	134	1	no	—
266	C₂×C₈	—	134	1	no	—
267	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
269	C₂×C₈	D₄×D₆₇	1	2	yes	—
270	C₂×C₈	Q₈×D₆₇	1	2	yes	—
271	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
272	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
273	C₂×C₈	#11761628	1	2	yes	—
274	C₄⋊C₄	—	67	4	no	—
275	C₄⋊C₄	—	134	4	no	—
276	C₄⋊C₄	—	134	4	no	—
277	C₄⋊C₄	—	134	4	no	—
278	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	134	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	67	1	no	—
284	C₄²	—	67	1	no	—
285	C₄²	—	67	2	no	—
286	C₄²	—	67	2	no	—
287	C₄²	—	134	2	no	—
288	C₄²	—	134	1	no	—
289	C₄²	C₂×Dic₁₃₄	1	2	yes	—
290	C₂³	#41×D₆₇	1	1	yes	characteristic
291	Q₈	—	536	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	268	2	no	—
296	C₂×C₄	—	268	2	no	—
297	C₂×C₄	—	67	2	no	—
298	C₂×C₄	—	67	2	no	—
299	C₂×C₄	—	134	4	no	—
300	C₂×C₄	—	134	2	no	—
301	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
305	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
307	C₈	—	4	2	no	—
308	C₈	—	268	2	no	—
309	C₈	—	67	2	no	—
310	C₈	—	67	2	no	—
311	C₈	—	134	2	no	—
312	C₈	C₄×Dic₁₃₄	1	2	yes	—
313	C₈	C₄×Dic₁₃₄	1	2	yes	—
314	C₈	#14524042	1	4	yes	—
315	C₂²	—	2	2	no	—
316	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
317	C₂²	#16863949	1	1	yes	characteristic
318	C₂²	#16863950	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	268	2	no	—
322	C₄	—	268	4	no	—
323	C₄	—	134	2	no	—
324	C₄	—	134	2	no	—
325	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
326	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
327	C₄	C₂×#14524042	1	1	yes	characteristic
328	C₄	#16863943	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	C₄×#14524042	1	1	yes	characteristic
332	C₂	#22569123	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1773	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₂×#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1773	D₆₇
hypocenter	C₆₇	#21733
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1773

Complements

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

N	H ≅ G/N	type
13. #16863800	C₄	semidirect
14. #16863797	C₄	semidirect
15. #16863796	C₄	semidirect
16. #16863793	C₄	semidirect
69. C₈×Dic₆₇	Q₈	semidirect
80. #97×C₆₇	Q₈	semidirect
178. C₆₇	#21733	semidirect
239. #97	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570721	2
#22569147	2
#22561188	2
#22560963	2
#22560960	2
#22560956	2
#22560379	2
#22560093	2
#22560089	2
#22559897	2
C₆₇×#1773	2
#21733	67