Subgroups — #30827526

Order 17152.

656 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569339	C₂	1	2	yes	maximal, complemented
2	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
3	#22559093	C₂	1	2	yes	maximal, complemented
4	#22559086	C₂	1	4	yes	maximal, complemented
5	#22559015	C₂	1	2	yes	maximal, complemented
6	#22558646	C₂	1	2	yes	maximal, complemented
7	C₈×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₄	1	1	yes	characteristic
10	C₄×#11761628	C₂²	1	2	yes	—
11	#16863520	C₂²	1	4	yes	—
12	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic
13	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
14	C₄×#11761617	C₄	1	2	yes	—
15	C₄×#11761617	C₂²	1	2	yes	—
16	C₄×D₅₃₆	C₄	1	1	yes	characteristic
17	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
18	#16863473	C₂²	1	4	yes	—
19	#16863472	—	2	4	no	—
20	#16863449	C₂²	1	1	yes	characteristic
21	#16863448	—	2	2	no	—
22	#16863434	C₄	1	1	yes	characteristic
23	#16863418	—	2	2	no	—
24	#16863393	C₄	1	2	yes	—
25	C₄×#11761612	C₂²	1	1	yes	characteristic
26	#16863126	C₂²	1	4	yes	—
27	#16863123	C₂²	1	4	yes	—
28	#16863118	C₂²	1	2	yes	—
29	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
30	#423×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#335×C₆₇	C₂²	1	2	yes	—
33	#335×C₆₇	C₂²	1	2	yes	—
34	C₂×#11761628	C₂×C₄	1	2	yes	—
35	#14524065	—	2	4	no	—
36	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#11761617	C₂×C₄	1	2	yes	—
38	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
39	C₂×#11761612	—	2	1	no	—
40	C₂×#11761611	—	2	4	no	—
41	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₃₄	C₂³	1	2	yes	—
43	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
44	C₄×D₂₆₈	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₆₇	—	2	2	no	—
46	C₄×C₂²⋊D₆₇	—	2	2	no	—
47	#14524002	C₂×C₄	1	2	yes	—
48	#14524000	—	2	2	no	—
49	C₄²×D₆₇	—	2	2	no	—
50	#14523977	C₂×C₄	1	1	yes	characteristic
51	#14523977	C₂×C₄	1	1	yes	characteristic
52	#14523975	C₂×C₄	1	2	yes	—
53	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
54	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
55	Dic₆₇⋊C₈	—	2	4	no	—
56	#14523968	C₂×C₄	1	2	yes	—
57	#14523968	C₂×C₄	1	2	yes	—
58	#14523965	C₂×C₄	1	2	yes	—
59	#14523965	C₂×C₄	1	2	yes	—
60	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
61	(C₂×D₆₇)⋊C₈	—	2	4	no	—
62	#14523954	—	2	1	no	—
63	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
64	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
65	C₈×Dic₆₇	—	2	4	no	—
66	C₂×M₁₆×C₆₇	—	2	2	no	—
67	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₆₇	—	2	2	no	—
71	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	#11761628	—	2	4	no	—
77	#11761628	—	4	2	no	—
78	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
79	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
80	C₂×C₂²⋊D₆₇	—	2	2	no	—
81	C₂×C₄×D₆₇	—	2	2	no	—
82	C₂×C₄×D₆₇	—	2	2	no	—
83	C₂×C₄×D₆₇	—	2	2	no	—
84	Dic₂₆₈	M₁₆	1	4	yes	complemented
85	#11761617	—	2	4	no	—
86	D₅₃₆	—	2	2	no	—
87	#11761612	—	2	4	no	—
88	#11761611	—	8	4	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
91	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
93	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
94	Dic₆₇⋊C₄	—	2	2	no	—
95	Dic₆₇⋊C₄	—	2	2	no	—
96	C₂²⋊Dic₆₇	—	2	2	no	—
97	(C₂×D₆₇)⋊C₄	—	2	2	no	—
98	(C₂×D₆₇)⋊C₄	—	2	2	no	—
99	C₄×Dic₆₇	—	2	2	no	—
100	C₄×Dic₆₇	—	2	2	no	—
101	C₄×Dic₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
105	M₁₆×C₆₇	—	8	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	4	no	—
108	C₂×C₈×C₆₇	—	4	1	no	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	#41	1	1	yes	characteristic
116	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
130	C₂²⋊D₆₇	—	4	4	no	—
131	C₂×Dic₆₇	—	2	2	no	—
132	C₂×Dic₆₇	—	2	2	no	—
133	C₂×Dic₆₇	—	2	2	no	—
134	D₂₆₈	—	2	2	no	—
135	D₂₆₈	—	2	2	no	—
136	C₄×D₆₇	—	4	4	no	—
137	C₄×D₆₇	—	4	2	no	—
138	C₄×D₆₇	—	4	2	no	—
139	C₄×D₆₇	—	4	4	no	—
140	Dic₁₃₄	—	2	2	no	—
141	Dic₁₃₄	C₂×M₁₆	1	4	yes	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	—	2	1	no	—
144	C₈×C₆₇	—	2	1	no	—
145	C₈×C₆₇	—	4	4	no	—
146	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
147	C₆₇⋊C₈	—	2	4	no	—
148	C₆₇⋊C₈	—	2	2	no	—
149	C₆₇⋊C₈	—	2	2	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	—	2	1	no	—
153	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂×D₆₇	—	4	2	no	—
155	C₂×D₆₇	—	4	2	no	—
156	C₂×D₆₇	—	4	2	no	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	#443	1	2	yes	—
161	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
162	C₄×C₆₇	#568	1	1	yes	characteristic
163	C₄×C₆₇	#576	1	1	yes	characteristic
164	C₄×C₆₇	#578	1	1	yes	characteristic
165	Dic₆₇	—	2	4	no	—
166	Dic₆₇	—	4	2	no	—
167	Dic₆₇	—	4	2	no	—
168	#18059	—	67	1	no	maximal, Sylow
169	C₂×C₆₇	—	4	1	no	—
170	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
171	C₂×C₆₇	#2951	1	1	yes	characteristic
172	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
173	D₆₇	—	8	2	no	—
174	#2991	—	67	2	no	—
175	C₄×#135	—	67	1	no	—
176	#1589	—	67	2	no	—
177	#1588	—	67	4	no	—
178	C₈×Q₁₆	—	67	2	no	—
179	#1567	D₆₇	1	1	yes	characteristic, complemented
180	#1489	—	67	2	no	—
181	C₈×M₁₆	—	67	1	no	—
182	C₆₇	#18059	1	1	yes	characteristic, complemented, Sylow
183	C₂×#135	—	67	1	no	—
184	C₄×#41	—	67	2	no	—
185	C₈×Q₈	—	67	4	no	—
186	C₄×Q₁₆	—	67	1	no	—
187	C₄×Q₁₆	—	67	1	no	—
188	C₄×SD₁₆	—	67	2	no	—
189	C₄×SD₁₆	—	67	2	no	—
190	C₄×D₈	—	67	1	no	—
191	C₄×D₈	—	67	1	no	—
192	#436	—	67	4	no	—
193	#436	—	134	4	no	—
194	#431	—	67	1	no	—
195	#431	—	134	2	no	—
196	#427	—	67	1	no	—
197	#423	—	134	2	no	—
198	#423	C₂×D₆₇	1	2	yes	—
199	#416	—	67	2	no	—
200	C₄×M₁₆	—	67	1	no	—
201	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
202	#335	C₂×D₆₇	1	2	yes	—
203	#335	C₂×D₆₇	1	2	yes	—
204	Q₈⋊C₈	—	67	4	no	—
205	D₄⋊C₈	—	67	4	no	—
206	#322	—	67	2	no	—
207	C₈²	—	67	2	no	—
208	C₂×#41	—	67	2	no	—
209	#135	—	134	4	no	—
210	C₂×Q₁₆	—	67	1	no	—
211	C₂×SD₁₆	—	67	2	no	—
212	C₂×D₈	—	67	1	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	134	4	no	—
215	C₂×M₁₆	—	134	1	no	—
216	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
218	C₄×Q₈	—	67	2	no	—
219	C₄×Q₈	—	67	2	no	—
220	C₄×D₄	—	67	2	no	—
221	C₄×D₄	—	67	2	no	—
222	C₄×D₄	—	134	2	no	—
223	C₄×D₄	—	134	2	no	—
224	#117	—	67	2	no	—
225	#117	—	134	2	no	—
226	C₂×C₄²	—	134	2	no	—
227	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
228	#107	—	67	1	no	—
229	#107	—	67	1	no	—
230	#106	—	67	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	67	1	no	—
233	C₄⋊C₈	—	67	1	no	—
234	C₄⋊C₈	—	134	4	no	—
235	C₄⋊C₈	C₂²×D₆₇	1	4	yes	—
236	#103	—	67	2	no	—
237	#103	—	67	2	no	—
238	#102	—	67	2	no	—
239	#102	—	67	2	no	—
240	C₂²⋊C₈	—	67	2	no	—
241	C₂²⋊C₈	—	134	4	no	—
242	#97	—	134	1	no	—
243	C₄×C₈	—	67	2	no	—
244	C₄×C₈	—	67	2	no	—
245	C₄×C₈	—	134	4	no	—
246	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
247	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
250	#41	—	268	2	no	—
251	#41	—	134	4	no	—
252	C₂×Q₈	—	67	2	no	—
253	C₂×D₄	—	67	2	no	—
254	C₂×D₄	—	134	2	no	—
255	C₂²×C₄	—	134	2	no	—
256	C₂²×C₄	—	134	2	no	—
257	C₂²×C₄	—	134	2	no	—
258	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
259	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
260	C₂²×C₄	#11761629	1	1	yes	characteristic
261	Q₁₆	—	67	4	no	—
262	SD₁₆	—	134	4	no	—
263	D₈	—	134	2	no	—
264	M₁₆	—	8	2	no	—
265	M₁₆	—	536	4	no	—
266	M₁₆	—	134	4	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	67	2	no	—
271	C₂×C₈	—	67	2	no	—
272	C₂×C₈	—	134	2	no	—
273	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
275	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	67	2	no	—
278	C₄⋊C₄	—	67	2	no	—
279	C₄⋊C₄	—	134	2	no	—
280	C₄⋊C₄	—	134	2	no	—
281	C₂²⋊C₄	—	134	2	no	—
282	C₂²⋊C₄	—	134	2	no	—
283	C₂²⋊C₄	—	134	2	no	—
284	C₄²	—	134	2	no	—
285	C₄²	—	134	2	no	—
286	C₄²	—	134	2	no	—
287	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
288	C₄²	D₄×D₆₇	1	1	yes	characteristic
289	C₄²	#11761629	1	1	yes	characteristic
290	C₄²	C₂³×D₆₇	1	1	yes	characteristic
291	C₂³	—	134	2	no	—
292	C₂³	#14524005	1	1	yes	characteristic
293	Q₈	—	67	4	no	—
294	Q₈	—	134	2	no	—
295	D₄	—	268	4	no	—
296	D₄	—	134	2	no	—
297	D₄	—	134	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	268	4	no	—
304	C₂×C₄	—	268	4	no	—
305	C₂×C₄	—	268	2	no	—
306	C₂×C₄	—	268	2	no	—
307	C₂×C₄	—	134	2	no	—
308	C₂×C₄	—	134	2	no	—
309	C₂×C₄	—	134	2	no	—
310	C₂×C₄	#14524005	1	1	yes	characteristic
311	C₂×C₄	#14524005	1	1	yes	characteristic
312	C₂×C₄	#14524005	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	4	4	no	—
320	C₈	—	134	4	no	—
321	C₈	—	134	2	no	—
322	C₈	—	134	2	no	—
323	C₈	M₁₆×D₆₇	1	2	yes	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	268	2	no	—
328	C₂²	—	268	2	no	—
329	C₂²	—	268	2	no	—
330	C₂²	C₂×#14524005	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	268	2	no	—
335	C₄	—	268	2	no	—
336	C₄	—	134	4	no	—
337	C₄	#16863512	1	2	yes	—
338	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
339	C₄	#16864162	1	1	yes	characteristic
340	C₄	#16864197	1	1	yes	characteristic
341	C₄	#16864209	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	536	2	no	—
344	C₂	#22568944	1	1	yes	characteristic
345	C₂	#22569083	1	1	yes	characteristic
346	C₂	C₂×#16863512	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#18059
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
1. #22569339	C₂	semidirect
3. #22559093	C₂	semidirect
4. #22559086	C₂	semidirect
5. #22559015	C₂	semidirect
6. #22558646	C₂	semidirect
7. C₈×#11761612	C₂	semidirect
8. C₆₇×#1567	C₂	semidirect
84. Dic₂₆₈	M₁₆	semidirect
179. #1567	D₆₇	semidirect
182. C₆₇	#18059	semidirect

Maximal subgroups

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

subgroup	index
#22569339	2
C₄×#14524065	2
#22559093	2
#22559086	2
#22559015	2
#22558646	2
C₈×#11761612	2
C₆₇×#1567	2
#18059	67