Subgroups — #30755702

Order 17152.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570768	C₂	1	1	yes	maximal, characteristic
2	#22561186	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561037	C₂	1	1	yes	maximal, characteristic
4	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558665	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558661	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863953	C₂²	1	1	yes	characteristic
9	#16863946	C₂²	1	1	yes	characteristic
10	#16863794	C₄	1	1	yes	characteristic, complemented
11	#16863769	C₄	1	1	yes	characteristic, complemented
12	#16863747	C₄	1	1	yes	characteristic, complemented
13	#16863743	C₄	1	1	yes	characteristic, complemented
14	#16863625	—	2	2	no	—
15	C₄×#11761617	—	2	2	no	—
16	#16863420	C₂²	1	1	yes	characteristic
17	#16863420	C₂²	1	1	yes	characteristic
18	C₂×#14523971	—	2	2	no	—
19	#16863156	—	2	2	no	—
20	#16863147	—	2	2	no	—
21	#16863128	—	2	2	no	—
22	#16863125	—	2	2	no	—
23	C₂×C₄×C₈×C₆₇	—	2	2	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₄³×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#11761628	—	2	1	no	—
28	C₂×#11761617	—	2	2	no	—
29	C₂×#11761612	—	2	1	no	—
30	C₂×#11761612	—	2	1	no	—
31	#14524043	C₂×C₄	1	1	yes	characteristic
32	#14524043	C₂×C₄	1	1	yes	characteristic
33	#14524043	C₂×C₄	1	1	yes	characteristic
34	#14524040	C₂×C₄	1	1	yes	characteristic
35	#14524035	—	2	1	no	—
36	#14524031	—	2	1	no	—
37	#14524022	—	2	1	no	—
38	#14524022	—	2	1	no	—
39	#14524017	—	2	1	no	—
40	#14524017	—	2	1	no	—
41	C₄×Dic₁₃₄	—	2	1	no	—
42	C₄×Dic₁₃₄	—	2	1	no	—
43	C₄×D₂₆₈	—	2	1	no	—
44	C₄×D₂₆₈	—	2	1	no	—
45	#14524002	—	2	1	no	—
46	#14523975	—	2	2	no	—
47	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
50	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
51	#14523971	—	4	4	no	—
52	#14523968	—	2	2	no	—
53	#14523965	—	2	2	no	—
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	D₄	1	2	yes	—
56	C₂×C₄²×C₆₇	—	2	1	no	—
57	C₂×C₄²×C₆₇	—	2	1	no	—
58	C₂×C₄²×C₆₇	D₄	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₆₇	—	2	8	no	—
61	C₄×C₈×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	D₄	1	2	yes	—
65	C₄×C₈×C₆₇	D₄	1	2	yes	—
66	#11761628	—	4	2	no	—
67	#11761628	—	4	2	no	—
68	C₂×Dic₁₃₄	—	2	1	no	—
69	C₂×Dic₁₃₄	—	2	1	no	—
70	C₂×Dic₁₃₄	—	2	1	no	—
71	C₂×D₂₆₈	—	2	1	no	—
72	C₂×D₂₆₈	—	2	1	no	—
73	C₂×D₂₆₈	—	2	1	no	—
74	C₂×C₂²⋊D₆₇	—	4	1	no	—
75	C₂×C₄×D₆₇	—	4	1	no	—
76	#11761617	—	4	4	no	—
77	#11761612	—	4	2	no	—
78	#11761612	—	4	2	no	—
79	C₂×C₆₇⋊C₈	—	4	1	no	—
80	C₂×C₆₇⋊C₈	—	4	1	no	—
81	C₄⋊Dic₆₇	—	2	1	no	—
82	C₄⋊Dic₆₇	—	2	1	no	—
83	C₄⋊Dic₆₇	—	2	1	no	—
84	C₄⋊Dic₆₇	—	2	1	no	—
85	C₄⋊Dic₆₇	—	2	1	no	—
86	C₄⋊Dic₆₇	—	2	1	no	—
87	Dic₆₇⋊C₄	—	4	1	no	—
88	Dic₆₇⋊C₄	—	4	1	no	—
89	C₂²⋊Dic₆₇	—	4	1	no	—
90	(C₂×D₆₇)⋊C₄	—	4	1	no	—
91	(C₂×D₆₇)⋊C₄	—	4	1	no	—
92	C₄×Dic₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
97	C₂×C₈×C₆₇	—	2	4	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	4	no	—
100	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₆₇	—	2	4	no	—
103	C₄²×C₆₇	—	2	4	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	SD₁₆	1	4	yes	complemented
106	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₇	#41	1	1	yes	characteristic
110	C₂³×C₆₇	#121	1	1	yes	characteristic
111	C₂²×D₆₇	—	4	1	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₆₇	#135	1	2	yes	—
125	C₂²⋊D₆₇	—	8	2	no	—
126	C₂×Dic₆₇	—	4	1	no	—
127	C₂×Dic₆₇	—	4	1	no	—
128	C₂×Dic₆₇	—	4	1	no	—
129	D₂₆₈	—	4	2	no	—
130	D₂₆₈	—	4	2	no	—
131	D₂₆₈	—	4	2	no	—
132	C₄×D₆₇	—	8	1	no	—
133	C₄×D₆₇	—	8	1	no	—
134	Dic₁₃₄	—	4	2	no	—
135	Dic₁₃₄	—	4	2	no	—
136	Dic₁₃₄	—	4	2	no	—
137	C₈×C₆₇	—	2	4	no	—
138	C₈×C₆₇	—	2	2	no	—
139	C₈×C₆₇	#104	1	4	yes	complemented
140	C₆₇⋊C₈	—	8	1	no	—
141	C₆₇⋊C₈	—	8	1	no	—
142	C₂²×C₆₇	—	2	1	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	#390	1	1	yes	characteristic
145	C₂²×C₆₇	#465	1	2	yes	—
146	C₂×D₆₇	—	8	1	no	—
147	C₂×D₆₇	—	8	1	no	—
148	C₂×D₆₇	—	8	1	no	—
149	C₄×C₆₇	—	2	8	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	C₂×#104	1	2	yes	—
153	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
154	C₄×C₆₇	#443	1	2	yes	—
155	C₄×C₆₇	#492	1	1	yes	characteristic
156	C₄×C₆₇	#495	1	1	yes	characteristic
157	Dic₆₇	—	8	1	no	—
158	Dic₆₇	—	8	1	no	—
159	Dic₆₇	—	8	1	no	—
160	#9843	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	—	2	2	no	—
162	C₂×C₆₇	#1962	1	1	yes	characteristic
163	C₂×C₆₇	#1968	1	1	yes	characteristic
164	C₂×C₆₇	#1974	1	1	yes	characteristic
165	D₆₇	—	16	1	no	—
166	#3167	—	67	1	no	—
167	#1994	—	67	1	no	—
168	#1962	—	67	1	no	—
169	C₄²×C₈	D₆₇	1	1	yes	characteristic, complemented
170	#1566	—	67	1	no	—
171	#1493	—	67	1	no	—
172	#1492	—	67	1	no	—
173	C₆₇	#9843	1	1	yes	characteristic, complemented, Sylow
174	#533	—	67	1	no	—
175	#532	—	67	1	no	—
176	#498	—	67	1	no	—
177	#492	—	67	1	no	—
178	#487	—	67	1	no	—
179	#486	—	67	1	no	—
180	#465	—	134	2	no	—
181	C₄×SD₁₆	—	134	2	no	—
182	#423	—	67	1	no	—
183	#423	—	67	1	no	—
184	C₂×#104	—	134	2	no	—
185	C₂×C₄×C₈	—	2	2	no	—
186	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
188	C₄³	C₂×D₆₇	1	1	yes	characteristic
189	#339	—	134	2	no	—
190	#334	—	134	2	no	—
191	Q₈⋊C₈	—	134	2	no	—
192	D₄⋊C₈	—	134	2	no	—
193	C₂×#41	—	134	1	no	—
194	C₂×SD₁₆	—	134	2	no	—
195	C₂×M₁₆	—	134	1	no	—
196	C₂×M₁₆	—	134	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
199	C₄⋊Q₈	—	67	1	no	—
200	C₄⋊Q₈	—	67	1	no	—
201	C₄⋊Q₈	—	67	1	no	—
202	#127	—	67	1	no	—
203	#125	—	134	1	no	—
204	#124	—	134	1	no	—
205	C₂²⋊Q₈	—	134	1	no	—
206	C₂²⋊Q₈	—	134	1	no	—
207	#121	—	134	1	no	—
208	#121	—	134	1	no	—
209	C₄×Q₈	—	134	1	no	—
210	C₄×Q₈	—	134	1	no	—
211	C₄×D₄	—	134	1	no	—
212	C₄×D₄	—	134	1	no	—
213	#117	—	134	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	D₂₆₈	1	2	yes	—
217	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
218	#106	—	134	2	no	—
219	C₄⋊C₈	—	134	1	no	—
220	C₄⋊C₈	—	134	1	no	—
221	C₄⋊C₈	—	134	1	no	—
222	C₄⋊C₈	—	134	1	no	—
223	#104	—	268	4	no	—
224	#103	—	134	2	no	—
225	#102	—	134	2	no	—
226	C₄×C₈	—	2	8	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
229	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
230	C₄×C₈	D₂₆₈	1	2	yes	—
231	C₄×C₈	D₂₆₈	1	2	yes	—
232	#41	—	268	2	no	—
233	#41	—	268	2	no	—
234	C₂×Q₈	—	134	1	no	—
235	C₂×Q₈	—	134	1	no	—
236	C₂×Q₈	—	134	1	no	—
237	C₂×D₄	—	268	1	no	—
238	C₂×D₄	—	134	1	no	—
239	C₂×D₄	—	134	1	no	—
240	C₂×D₄	—	134	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	268	1	no	—
243	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
245	C₂²×C₄	#11761628	1	1	yes	characteristic
246	SD₁₆	—	268	4	no	—
247	M₁₆	—	268	2	no	—
248	M₁₆	—	268	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	268	1	no	—
253	C₂×C₈	—	268	1	no	—
254	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
255	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
256	C₄⋊C₄	—	268	1	no	—
257	C₄⋊C₄	—	268	1	no	—
258	C₄⋊C₄	—	134	1	no	—
259	C₄⋊C₄	—	134	1	no	—
260	C₄⋊C₄	—	134	1	no	—
261	C₄⋊C₄	—	134	1	no	—
262	C₄⋊C₄	—	134	1	no	—
263	C₄⋊C₄	—	134	1	no	—
264	C₂²⋊C₄	—	268	1	no	—
265	C₂²⋊C₄	—	268	1	no	—
266	C₂²⋊C₄	—	268	1	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	268	1	no	—
271	C₄²	#11761617	1	4	yes	complemented
272	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
273	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
274	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
275	C₄²	#11761628	1	1	yes	characteristic
276	C₂³	—	268	1	no	—
277	C₂³	#14524017	1	1	yes	characteristic
278	Q₈	—	268	2	no	—
279	Q₈	—	268	2	no	—
280	Q₈	—	268	2	no	—
281	D₄	—	268	2	no	—
282	D₄	—	268	2	no	—
283	D₄	—	268	2	no	—
284	D₄	—	536	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	268	1	no	—
291	C₂×C₄	—	268	1	no	—
292	C₂×C₄	—	268	1	no	—
293	C₂×C₄	—	536	1	no	—
294	C₂×C₄	—	536	1	no	—
295	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
297	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
298	C₂×C₄	#14524017	1	1	yes	characteristic
299	C₂×C₄	#14524031	1	1	yes	characteristic
300	C₂×C₄	#14524040	1	1	yes	characteristic
301	C₂×C₄	C₂×#11761617	1	2	yes	—
302	C₂×C₄	#14524065	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	2	2	no	—
305	C₈	—	536	1	no	—
306	C₈	—	536	1	no	—
307	C₈	#14523971	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	536	1	no	—
311	C₂²	—	536	1	no	—
312	C₂²	—	536	1	no	—
313	C₂²	#16863298	1	1	yes	characteristic
314	C₂²	#16863625	1	2	yes	—
315	C₄	—	2	8	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	536	1	no	—
319	C₄	—	536	1	no	—
320	C₄	—	536	1	no	—
321	C₄	C₂×#14523971	1	2	yes	—
322	C₄	C₄×#11761617	1	2	yes	—
323	C₄	#16863514	1	2	yes	—
324	C₄	#16863769	1	1	yes	characteristic
325	C₄	#16863780	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	1072	1	no	—
328	C₂	#22561037	1	1	yes	characteristic
329	C₂	#22561063	1	1	yes	characteristic
330	C₂	#22561097	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	#16863946	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#9843
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. #22561186	C₂	semidirect
4. #22558965	C₂	semidirect
5. #22558665	C₂	semidirect
7. C₄²×C₈×C₆₇	C₂	semidirect
10. #16863794	C₄	semidirect
11. #16863769	C₄	semidirect
12. #16863747	C₄	semidirect
13. #16863743	C₄	semidirect
105. C₄²×C₆₇	SD₁₆	semidirect
139. C₈×C₆₇	#104	semidirect
169. C₄²×C₈	D₆₇	semidirect
173. C₆₇	#9843	semidirect
271. C₄²	#11761617	semidirect
307. C₈	#14523971	semidirect

Maximal subgroups

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

subgroup	index
#22570768	2
#22561186	2
#22561037	2
#22558965	2
#22558665	2
#22558661	2
C₄²×C₈×C₆₇	2
#9843	67