Subgroups — #30826384

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570769	C₂	1	1	yes	maximal, characteristic
2	#22569144	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568920	C₂	1	1	yes	maximal, characteristic
4	#22559114	C₂	1	4	yes	maximal, complemented
5	#22559024	C₂	1	4	yes	maximal, complemented
6	#22558967	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558661	C₂	1	1	yes	maximal, characteristic
8	C₂×#16863118	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1493	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524051	—	2	1	no	—
11	#16863946	C₂²	1	1	yes	characteristic
12	#16863792	C₄	1	1	yes	characteristic
13	#16863770	C₄	1	1	yes	characteristic
14	#16863746	C₄	1	1	yes	characteristic
15	#16863744	C₄	1	1	yes	characteristic
16	#16863627	C₄	1	4	yes	—
17	#16863527	—	2	1	no	—
18	#16863527	C₂²	1	2	yes	—
19	Q₈×C₆₇⋊C₈	—	2	1	no	—
20	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
21	C₄×#11761615	C₂²	1	4	yes	—
22	#16863473	—	2	1	no	—
23	#16863473	C₂²	1	2	yes	—
24	#16863461	—	2	1	no	—
25	#16863461	C₂²	1	2	yes	—
26	#16863423	—	2	1	no	—
27	#16863420	C₂²	1	1	yes	characteristic
28	#16863364	—	2	1	no	—
29	#16863352	—	2	1	no	—
30	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
32	#16863148	C₂²	1	4	yes	—
33	#16863125	C₂²	1	4	yes	—
34	#16863118	C₂²	1	2	yes	—
35	#16863118	C₂²	1	2	yes	—
36	#533×C₆₇	C₂²	1	1	yes	characteristic
37	#423×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
40	C₂×#11761628	—	2	1	no	—
41	C₂×#11761615	C₂×C₄	1	4	yes	—
42	#14524051	—	4	4	no	—
43	C₂×#11761612	—	2	1	no	—
44	C₂×#11761611	—	2	2	no	—
45	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
47	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
48	C₂×C₈×D₆₇	—	2	2	no	—
49	#14524043	C₂×C₄	1	1	yes	characteristic
50	#14524040	C₂×C₄	1	1	yes	characteristic
51	#14524031	—	2	1	no	—
52	#14524017	—	2	1	no	—
53	#14524017	C₂×C₄	1	2	yes	—
54	C₄×D₂₆₈	—	2	1	no	—
55	C₄×D₂₆₈	C₂³	1	2	yes	—
56	#14523974	C₂×C₄	1	4	yes	—
57	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
58	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
59	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
60	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
61	#14523967	C₂×C₄	1	4	yes	—
62	#14523963	C₂×C₄	1	4	yes	—
63	(C₂×D₆₇)⋊C₈	—	2	2	no	—
64	(C₂×D₆₇)⋊C₈	—	2	2	no	—
65	#14523954	—	2	2	no	—
66	C₄×C₆₇⋊C₈	—	2	2	no	—
67	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
70	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
71	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
72	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₆₇	—	2	1	no	—
75	C₂²×C₈×C₆₇	—	2	1	no	—
76	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	#125×C₆₇	—	2	1	no	—
79	C₂²⋊Q₈×C₆₇	—	2	1	no	—
80	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
81	C₄×Q₈×C₆₇	—	2	1	no	—
82	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
83	#117×C₆₇	—	2	1	no	—
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₆₇	—	2	1	no	—
86	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
87	C₄×C₈×C₆₇	—	2	1	no	—
88	C₄×C₈×C₆₇	C₂³	1	2	yes	—
89	#11761628	—	4	2	no	—
90	#11761628	—	4	2	no	—
91	C₂×Dic₁₃₄	—	2	1	no	—
92	C₂×D₂₆₈	—	2	1	no	—
93	C₂×D₂₆₈	—	2	1	no	—
94	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
95	C₂×C₂²⋊D₆₇	—	2	2	no	—
96	C₂×C₄×D₆₇	—	2	2	no	—
97	#11761615	—	2	8	no	—
98	#11761612	—	4	2	no	—
99	#11761611	—	4	4	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	2	no	—
102	C₂×C₆₇⋊C₈	—	2	4	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
110	C₈×D₆₇	—	4	4	no	—
111	C₄⋊Dic₆₇	—	2	1	no	—
112	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
113	(C₂×D₆₇)⋊C₄	—	2	2	no	—
114	(C₂×D₆₇)⋊C₄	—	2	2	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
120	M₁₆×C₆₇	—	4	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂²⋊C₄×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₆₇	#41	1	1	yes	characteristic
135	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
136	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₆₇	—	2	2	no	—
138	Q₈×C₆₇	—	2	4	no	—
139	Q₈×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₆₇	#137	1	2	yes	—
149	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₆₇	—	4	4	no	—
153	C₂×Dic₆₇	—	2	2	no	—
154	D₂₆₈	—	2	4	no	—
155	D₂₆₈	—	4	2	no	—
156	D₂₆₈	—	4	2	no	—
157	C₄×D₆₇	—	4	2	no	—
158	C₄×D₆₇	—	4	2	no	—
159	Dic₁₃₄	—	4	2	no	—
160	C₈×C₆₇	—	4	2	no	—
161	C₈×C₆₇	—	4	2	no	—
162	C₆₇⋊C₈	—	2	2	no	—
163	C₆₇⋊C₈	—	2	1	no	—
164	C₆₇⋊C₈	—	2	1	no	—
165	C₆₇⋊C₈	—	2	1	no	—
166	C₆₇⋊C₈	—	2	1	no	—
167	C₆₇⋊C₈	—	4	2	no	—
168	C₆₇⋊C₈	#131	1	4	yes	complemented
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	#440	1	2	yes	—
172	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₆₇	—	4	2	no	—
174	C₂×D₆₇	—	4	2	no	—
175	C₂×D₆₇	—	4	2	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	#440	1	2	yes	—
181	C₄×C₆₇	#444	1	2	yes	—
182	C₄×C₆₇	C₂×#131	1	2	yes	—
183	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
185	Dic₆₇	—	4	2	no	—
186	#17960	—	67	1	no	maximal, Sylow
187	C₂×C₆₇	—	2	2	no	—
188	C₂×C₆₇	#2931	1	1	yes	characteristic
189	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
190	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
191	D₆₇	—	8	2	no	—
192	#3167	—	67	1	no	—
193	#2960	—	67	1	no	—
194	#2931	—	67	1	no	—
195	#1592	—	67	4	no	—
196	SD₁₆⋊C₈	—	67	4	no	—
197	#1567	—	67	1	no	—
198	#1493	D₆₇	1	1	yes	characteristic, complemented
199	#1492	—	67	1	no	—
200	C₂×#322	—	67	1	no	—
201	C₆₇	#17960	1	1	yes	characteristic, complemented, Sylow
202	C₂×#131	—	134	1	no	—
203	#533	C₂×D₆₇	1	1	yes	characteristic
204	#532	—	67	1	no	—
205	#498	—	67	1	no	—
206	#492	—	67	1	no	—
207	#487	—	67	1	no	—
208	#486	—	67	1	no	—
209	#465	—	67	4	no	—
210	#446	—	67	2	no	—
211	#446	—	134	1	no	—
212	C₈×Q₈	—	67	2	no	—
213	C₈×Q₈	—	134	1	no	—
214	C₄×SD₁₆	—	67	4	no	—
215	#436	—	67	2	no	—
216	#436	—	134	1	no	—
217	C₈×D₄	—	67	2	no	—
218	C₈×D₄	—	134	1	no	—
219	#424	—	134	1	no	—
220	#423	—	67	1	no	—
221	#423	C₂×D₆₇	1	1	yes	characteristic
222	#408	—	134	1	no	—
223	#405	—	134	1	no	—
224	C₂×C₄×C₈	—	67	1	no	—
225	C₂×C₄×C₈	—	67	1	no	—
226	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
227	#335	—	67	4	no	—
228	Q₈⋊C₈	C₂×D₆₇	1	4	yes	—
229	D₄⋊C₈	—	67	4	no	—
230	#322	—	67	2	no	—
231	#322	—	67	2	no	—
232	C₂×#41	—	134	1	no	—
233	C₂×SD₁₆	—	67	4	no	—
234	#131	—	268	4	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	134	2	no	—
237	C₂×M₁₆	—	134	1	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	67	2	no	—
240	C₂²×C₈	—	67	1	no	—
241	C₂²×C₈	—	67	1	no	—
242	C₂²×C₈	—	134	2	no	—
243	C₄⋊Q₈	—	67	1	no	—
244	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
245	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
246	#127	—	67	1	no	—
247	#125	—	2	1	no	—
248	#124	—	134	1	no	—
249	C₂²⋊Q₈	—	2	1	no	—
250	C₂²⋊Q₈	C₄×D₆₇	1	2	yes	—
251	#121	—	67	2	no	—
252	#121	—	134	1	no	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
255	C₄×D₄	—	67	2	no	—
256	C₄×D₄	—	134	1	no	—
257	#117	—	2	1	no	—
258	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
259	#106	—	67	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	67	2	no	—
262	C₄⋊C₈	—	134	2	no	—
263	C₄⋊C₈	—	134	2	no	—
264	C₄⋊C₈	—	134	1	no	—
265	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
266	#103	—	67	4	no	—
267	#102	—	67	4	no	—
268	C₂²⋊C₈	—	134	2	no	—
269	C₂²⋊C₈	—	134	2	no	—
270	#97	—	134	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	67	1	no	—
273	C₄×C₈	—	67	1	no	—
274	C₄×C₈	—	67	1	no	—
275	C₄×C₈	—	67	1	no	—
276	C₄×C₈	—	67	2	no	—
277	C₄×C₈	—	67	1	no	—
278	C₄×C₈	—	67	1	no	—
279	C₄×C₈	—	134	2	no	—
280	C₄×C₈	C₂²×D₆₇	1	2	yes	—
281	#41	—	268	2	no	—
282	#41	—	268	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	134	1	no	—
285	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
286	C₂×D₄	—	67	2	no	—
287	C₂×D₄	—	134	2	no	—
288	C₂×D₄	—	134	1	no	—
289	C₂×D₄	—	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	#11761627	1	1	yes	characteristic
294	SD₁₆	—	134	8	no	—
295	M₁₆	—	4	2	no	—
296	M₁₆	—	268	4	no	—
297	M₁₆	—	268	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	268	4	no	—
302	C₂×C₈	—	67	2	no	—
303	C₂×C₈	—	67	2	no	—
304	C₂×C₈	—	67	1	no	—
305	C₂×C₈	—	67	1	no	—
306	C₂×C₈	—	67	1	no	—
307	C₂×C₈	—	67	1	no	—
308	C₂×C₈	—	134	2	no	—
309	C₂×C₈	—	134	4	no	—
310	C₂×C₈	—	134	2	no	—
311	C₂×C₈	—	134	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	67	2	no	—
318	C₄⋊C₄	—	134	1	no	—
319	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	134	2	no	—
322	C₂²⋊C₄	—	134	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
325	C₄²	D₄×D₆₇	1	1	yes	characteristic
326	C₄²	#11761627	1	1	yes	characteristic
327	C₄²	C₂³×D₆₇	1	1	yes	characteristic
328	C₂³	—	134	2	no	—
329	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	268	2	no	—
333	D₄	—	268	4	no	—
334	D₄	—	268	2	no	—
335	D₄	—	268	2	no	—
336	D₄	—	134	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	268	2	no	—
342	C₂×C₄	—	268	2	no	—
343	C₂×C₄	—	134	2	no	—
344	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
345	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
346	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
347	C₂×C₄	#14524070	1	2	yes	—
348	C₂×C₄	#14524075	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
351	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	—	268	2	no	—
355	C₈	—	67	4	no	—
356	C₈	—	134	2	no	—
357	C₈	—	134	1	no	—
358	C₈	—	134	1	no	—
359	C₈	—	134	1	no	—
360	C₈	—	134	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	268	2	no	—
364	C₂²	—	268	2	no	—
365	C₂²	—	268	2	no	—
366	C₂²	#16863499	1	2	yes	—
367	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	268	2	no	—
373	C₄	#16863499	1	2	yes	—
374	C₄	#16863515	1	2	yes	—
375	C₄	#131×D₆₇	1	2	yes	—
376	C₄	C₂×#14524070	1	1	yes	characteristic
377	C₄	C₂×#14524075	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	536	2	no	—
380	C₂	#22568917	1	1	yes	characteristic
381	C₂	C₂×#16863499	1	1	yes	characteristic
382	C₂	C₂×#16863515	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×#1493	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#16863946	C₂²
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1493	D₆₇
hypocenter	C₆₇	#17960
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1493

Complements

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

N	H ≅ G/N	type
2. #22569144	C₂	semidirect
4. #22559114	C₂	semidirect
5. #22559024	C₂	semidirect
6. #22558967	C₂	semidirect
8. C₂×#16863118	C₂	semidirect
9. C₆₇×#1493	C₂	semidirect
168. C₆₇⋊C₈	#131	semidirect
198. #1493	D₆₇	semidirect
201. C₆₇	#17960	semidirect

Maximal subgroups

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

subgroup	index
#22570769	2
#22569144	2
#22568920	2
#22559114	2
#22559024	2
#22558967	2
#22558661	2
C₂×#16863118	2
C₆₇×#1493	2
#17960	67