Subgroups — #30825519

Order 17152.

678 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569292	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569232	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11761617	C₂	1	1	yes	maximal, characteristic
4	#22559065	C₂	1	4	yes	maximal, complemented
5	#22559044	C₂	1	4	yes	maximal, complemented
6	#22558603	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558599	C₂	1	1	yes	maximal, characteristic
8	#22558501	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11761617	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
13	#16863526	—	2	2	no	—
14	#16863523	C₂²	1	4	yes	complemented
15	C₄×#11761617	C₄	1	2	yes	—
16	C₄×#11761617	C₄	1	2	yes	—
17	C₄×#11761617	C₂²	1	2	yes	—
18	C₄×#11761617	C₂²	1	2	yes	—
19	C₈×D₂₆₈	—	2	2	no	—
20	#16863461	C₂²	1	4	yes	—
21	#16863450	—	2	2	no	—
22	#16863449	C₂²	1	1	yes	characteristic
23	#16863449	C₂²	1	1	yes	characteristic
24	C₂×#14523975	C₄	1	1	yes	characteristic
25	C₂×#14523968	C₄	1	1	yes	characteristic
26	C₂×#14523965	C₄	1	1	yes	characteristic
27	C₂²⋊C₈×D₆₇	—	2	2	no	—
28	#16863126	C₂²	1	4	yes	complemented
29	#16863123	C₂²	1	4	yes	—
30	#16863118	C₂²	1	2	yes	complemented
31	#16863118	C₂²	1	2	yes	complemented
32	#431×C₆₇	C₂²	1	1	yes	characteristic
33	#431×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#335×C₆₇	C₂²	1	2	yes	complemented
36	#334×C₆₇	C₂²	1	2	yes	complemented
37	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×D₆₇	—	2	1	no	—
40	C₂×#11761617	—	2	4	no	—
41	C₂×#11761617	C₂×C₄	1	2	yes	—
42	C₂×#11761617	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₆₇	—	2	4	no	—
44	C₄×Dic₁₃₄	—	2	1	no	—
45	C₄×Dic₁₃₄	—	2	1	no	—
46	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₃₄	C₂³	1	2	yes	—
48	C₄×D₂₆₈	—	2	1	no	—
49	C₄×D₂₆₈	—	2	1	no	—
50	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
51	C₄×D₂₆₈	C₂³	1	2	yes	—
52	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×Dic₆₇⋊C₄	—	2	1	no	—
55	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
56	C₂×C₄×Dic₆₇	—	2	1	no	—
57	#14523975	C₂×C₄	1	2	yes	—
58	#14523975	C₂×C₄	1	2	yes	—
59	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
60	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
61	Dic₆₇⋊C₈	—	2	4	no	—
62	#14523968	C₂×C₄	1	2	yes	—
63	#14523968	C₂×C₄	1	2	yes	—
64	#14523965	C₂×C₄	1	2	yes	—
65	#14523965	C₂×C₄	1	2	yes	—
66	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
68	(C₂×D₆₇)⋊C₈	—	2	4	no	—
69	#14523953	—	2	4	no	—
70	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
71	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
76	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
77	C₂²⋊C₈×C₆₇	—	2	1	no	—
78	C₂²⋊C₈×C₆₇	—	2	1	no	—
79	C₄×C₈×C₆₇	—	2	1	no	—
80	C₄×C₈×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×D₆₇	—	2	1	no	—
86	C₂×Dic₁₃₄	—	2	2	no	—
87	C₂×Dic₁₃₄	—	2	2	no	—
88	C₂×Dic₁₃₄	—	2	1	no	—
89	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
90	C₂×D₂₆₈	—	2	1	no	—
91	C₂×D₂₆₈	—	2	2	no	—
92	C₂×D₂₆₈	M₁₆	1	4	yes	—
93	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
94	C₂²×Dic₆₇	—	2	1	no	—
95	C₂²×Dic₆₇	—	2	1	no	—
96	C₂²×Dic₆₇	—	2	1	no	—
97	C₂×C₄×D₆₇	—	2	2	no	—
98	C₂×C₄×D₆₇	—	2	2	no	—
99	C₂×C₄×D₆₇	—	8	2	no	—
100	#11761617	—	2	4	no	—
101	#11761617	—	2	4	no	—
102	C₂×C₆₇⋊C₈	—	2	2	no	—
103	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
104	C₈×D₆₇	—	8	4	no	—
105	C₄⋊Dic₆₇	—	2	1	no	—
106	C₄⋊Dic₆₇	—	2	1	no	—
107	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
108	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
109	Dic₆₇⋊C₄	—	2	2	no	—
110	Dic₆₇⋊C₄	—	2	2	no	—
111	(C₂×D₆₇)⋊C₄	—	2	2	no	—
112	(C₂×D₆₇)⋊C₄	—	2	2	no	—
113	C₄×Dic₆₇	—	2	2	no	—
114	C₄×Dic₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	#41	1	1	yes	characteristic
129	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
131	C₂²×D₆₇	—	2	2	no	—
132	C₂²×D₆₇	—	2	4	no	—
133	C₂²×D₆₇	—	8	1	no	—
134	C₂²×D₆₇	—	8	1	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₆₇	#131	1	2	yes	—
142	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₆₇	—	2	2	no	—
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	4	2	no	—
149	C₂×Dic₆₇	—	8	1	no	—
150	C₂×Dic₆₇	—	8	1	no	—
151	D₂₆₈	—	2	2	no	—
152	D₂₆₈	—	2	4	no	—
153	D₂₆₈	C₂×M₁₆	1	4	yes	—
154	C₄×D₆₇	—	4	4	no	—
155	C₄×D₆₇	—	8	2	no	—
156	Dic₁₃₄	—	2	2	no	—
157	Dic₁₃₄	—	2	2	no	—
158	Dic₁₃₄	—	4	2	no	—
159	C₈×C₆₇	—	2	1	no	—
160	C₈×C₆₇	—	2	1	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	4	2	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₆₇⋊C₈	—	2	4	no	—
166	C₆₇⋊C₈	—	4	2	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	#435	1	2	yes	—
170	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₆₇	—	2	4	no	—
172	C₂×D₆₇	—	4	2	no	—
173	C₂×D₆₇	—	4	2	no	—
174	C₂×D₆₇	—	8	4	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	#435	1	2	yes	—
179	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
180	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
181	C₄×C₆₇	#577	1	2	yes	—
182	Dic₆₇	—	4	2	no	—
183	Dic₆₇	—	4	2	no	—
184	Dic₆₇	—	4	2	no	—
185	#17879	—	67	1	no	maximal, Sylow
186	C₂×C₆₇	—	2	2	no	—
187	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
188	C₂×C₆₇	#2949	1	1	yes	characteristic
189	C₂×C₆₇	#2957	1	1	yes	characteristic
190	D₆₇	—	4	4	no	—
191	#2983	—	67	1	no	—
192	#2972	—	67	1	no	—
193	C₂×C₄×SD₁₆	—	67	1	no	—
194	#1585	—	67	4	no	—
195	#1582	—	67	4	no	—
196	#1564	D₆₇	1	1	yes	characteristic, complemented
197	#1479	—	67	1	no	—
198	#1478	—	67	1	no	—
199	#1452	—	67	1	no	—
200	C₆₇	#17879	1	1	yes	characteristic, complemented, Sylow
201	C₂²×SD₁₆	—	67	1	no	—
202	C₂×C₄×Q₈	—	67	1	no	—
203	C₂×C₄×D₄	—	67	1	no	—
204	#446	—	67	4	no	—
205	#446	—	134	2	no	—
206	C₄×SD₁₆	—	67	2	no	—
207	C₄×SD₁₆	—	67	2	no	—
208	C₄×SD₁₆	—	67	2	no	—
209	C₄×SD₁₆	—	67	2	no	—
210	C₈×D₄	—	67	4	no	—
211	C₈×D₄	—	134	2	no	—
212	#432	—	134	2	no	—
213	#431	—	67	1	no	—
214	#431	—	67	1	no	—
215	#431	C₂×D₆₇	1	1	yes	characteristic
216	#431	C₂×D₆₇	1	1	yes	characteristic
217	C₂×#106	—	67	1	no	—
218	C₂×#103	—	67	1	no	—
219	C₂×#102	—	67	1	no	—
220	C₂×C₂²⋊C₈	—	134	2	no	—
221	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
222	#335	C₂×D₆₇	1	2	yes	complemented
223	#334	C₂×D₆₇	1	2	yes	complemented
224	Q₈⋊C₈	—	67	4	no	—
225	D₄⋊C₈	—	67	4	no	—
226	#322	—	67	2	no	—
227	#322	—	67	2	no	—
228	C₂²×Q₈	—	67	1	no	—
229	C₂²×D₄	—	67	1	no	—
230	C₂³×C₄	—	134	1	no	—
231	C₂×SD₁₆	—	67	2	no	—
232	C₂×SD₁₆	—	67	2	no	—
233	C₂×SD₁₆	—	134	4	no	—
234	C₂²×C₈	—	134	4	no	—
235	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
236	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
237	C₄×Q₈	—	67	2	no	—
238	C₄×Q₈	—	67	2	no	—
239	C₄×Q₈	—	134	1	no	—
240	C₄×Q₈	—	134	1	no	—
241	C₄×D₄	—	67	2	no	—
242	C₄×D₄	—	67	2	no	—
243	C₄×D₄	—	134	1	no	—
244	C₄×D₄	—	134	1	no	—
245	C₂×C₄⋊C₄	—	67	1	no	—
246	C₂×C₄⋊C₄	—	67	1	no	—
247	C₂×C₄⋊C₄	—	134	1	no	—
248	C₂×C₂²⋊C₄	—	134	1	no	—
249	C₂×C₄²	—	134	1	no	—
250	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
251	#106	—	67	2	no	—
252	#106	—	67	2	no	—
253	C₄⋊C₈	—	67	2	no	—
254	C₄⋊C₈	—	134	4	no	—
255	C₄⋊C₈	—	134	1	no	—
256	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
257	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
258	#103	—	67	2	no	—
259	#103	—	67	2	no	—
260	#102	—	67	2	no	—
261	#102	—	67	2	no	—
262	C₂²⋊C₈	—	2	1	no	—
263	C₂²⋊C₈	—	2	1	no	—
264	C₂²⋊C₈	—	67	2	no	—
265	C₂²⋊C₈	—	134	4	no	—
266	C₂²⋊C₈	—	134	1	no	—
267	#97	—	134	4	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	C₄×C₈	—	67	2	no	—
271	C₄×C₈	—	67	2	no	—
272	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
273	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
276	C₂⁴	—	134	1	no	—
277	C₂×Q₈	—	67	2	no	—
278	C₂×Q₈	—	134	2	no	—
279	C₂×Q₈	—	134	2	no	—
280	C₂×Q₈	—	134	1	no	—
281	C₂×D₄	—	67	4	no	—
282	C₂×D₄	—	67	2	no	—
283	C₂×D₄	—	134	1	no	—
284	C₂×D₄	—	134	2	no	—
285	C₂²×C₄	—	536	2	no	—
286	C₂²×C₄	—	134	1	no	—
287	C₂²×C₄	—	134	2	no	—
288	C₂²×C₄	—	134	1	no	—
289	C₂²×C₄	—	134	1	no	—
290	C₂²×C₄	—	134	2	no	—
291	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
293	C₂²×C₄	#11761629	1	1	yes	characteristic
294	SD₁₆	—	134	4	no	—
295	SD₁₆	—	134	4	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	536	4	no	—
301	C₂×C₈	—	67	4	no	—
302	C₂×C₈	—	134	2	no	—
303	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
307	C₄⋊C₄	—	67	2	no	—
308	C₄⋊C₄	—	67	2	no	—
309	C₄⋊C₄	—	134	1	no	—
310	C₄⋊C₄	—	134	1	no	—
311	C₄⋊C₄	—	134	2	no	—
312	C₄⋊C₄	—	134	2	no	—
313	C₂²⋊C₄	—	134	2	no	—
314	C₂²⋊C₄	—	134	2	no	—
315	C₄²	—	134	2	no	—
316	C₄²	—	134	2	no	—
317	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
318	C₄²	D₄×D₆₇	1	1	yes	characteristic
319	C₄²	#11761629	1	1	yes	characteristic
320	C₄²	C₂³×D₆₇	1	1	yes	characteristic
321	C₂³	—	536	1	no	—
322	C₂³	—	536	1	no	—
323	C₂³	—	134	2	no	—
324	C₂³	—	134	4	no	—
325	C₂³	#14524005	1	1	yes	characteristic
326	Q₈	—	268	2	no	—
327	Q₈	—	134	2	no	—
328	Q₈	—	134	2	no	—
329	D₄	—	67	4	no	—
330	D₄	—	134	2	no	—
331	D₄	—	134	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	268	2	no	—
335	C₂×C₄	—	268	4	no	—
336	C₂×C₄	—	536	2	no	—
337	C₂×C₄	—	536	1	no	—
338	C₂×C₄	—	536	1	no	—
339	C₂×C₄	—	134	2	no	—
340	C₂×C₄	—	134	2	no	—
341	C₂×C₄	—	134	2	no	—
342	C₂×C₄	#14524005	1	1	yes	characteristic
343	C₂×C₄	#14524005	1	1	yes	characteristic
344	C₂×C₄	#14524005	1	1	yes	characteristic
345	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
346	C₂×C₄	#14524049	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	4	2	no	—
355	C₈	—	4	2	no	—
356	C₈	—	268	2	no	—
357	C₈	—	134	4	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	268	2	no	—
361	C₂²	—	268	2	no	—
362	C₂²	—	536	4	no	—
363	C₂²	—	134	4	no	—
364	C₂²	#16863466	1	2	yes	—
365	C₂²	C₂×#14524005	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	268	2	no	—
370	C₄	—	268	2	no	—
371	C₄	—	268	2	no	—
372	C₄	#16863466	1	2	yes	—
373	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
374	C₄	C₂×#14524049	1	1	yes	characteristic
375	C₄	#16864203	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	268	4	no	—
378	C₂	C₂×#16863466	1	1	yes	characteristic
379	C₂	#22569067	1	1	yes	characteristic
380	C₂	#22569125	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×#1564	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#17879
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #22569292	C₂	semidirect
4. #22559065	C₂	semidirect
5. #22559044	C₂	semidirect
6. #22558603	C₂	semidirect
8. #22558501	C₂	semidirect
9. C₆₇×#1564	C₂	semidirect
14. #16863523	C₂²	semidirect
28. #16863126	C₂²	semidirect
30. #16863118	C₂²	semidirect
31. #16863118	C₂²	semidirect
35. #335×C₆₇	C₂²	semidirect
36. #334×C₆₇	C₂²	semidirect
196. #1564	D₆₇	semidirect
200. C₆₇	#17879	semidirect
222. #335	C₂×D₆₇	semidirect
223. #334	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569292	2
#22569232	2
C₂×C₄×#11761617	2
#22559065	2
#22559044	2
#22558603	2
#22558599	2
#22558501	2
C₆₇×#1564	2
#17879	67