Subgroups — #30860231

Order 17152.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568980	C₂	1	1	yes	maximal, characteristic
2	#22568859	C₂	1	2	yes	maximal, complemented
3	#22561097	C₂	1	1	yes	maximal, characteristic
4	#22561095	C₂	1	1	yes	maximal, characteristic
5	#22560958	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560921	C₂	1	2	yes	maximal, complemented
7	#22560919	C₂	1	2	yes	maximal, complemented
8	#108×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560018	C₂	1	2	yes	maximal, complemented
10	#22559895	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1945	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524065	C₂²	1	1	yes	characteristic
13	#16863864	C₂²	1	2	yes	—
14	#16863490	C₄	1	2	yes	complemented
15	#16863483	C₄	1	4	yes	complemented
16	#16863478	C₄	1	2	yes	complemented
17	#16863456	—	2	4	no	—
18	#16863452	—	2	4	no	—
19	C₂×#14523979	C₂²	1	1	yes	characteristic
20	#16863423	C₂²	1	2	yes	—
21	C₂×#14523971	C₂²	1	2	yes	—
22	C₂×#14523969	C₂²	1	2	yes	—
23	#16863390	C₂²	1	2	yes	—
24	#16863364	C₂²	1	2	yes	—
25	#16863362	C₂²	1	2	yes	—
26	M₁₆×Dic₆₇	C₂²	1	2	yes	—
27	C₂×#14523954	C₂²	1	1	yes	characteristic
28	C₂×#14523953	C₂²	1	1	yes	characteristic
29	C₂×#14523953	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	#16863160	C₂²	1	2	yes	—
34	#16863155	C₂²	1	2	yes	—
35	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
36	#427×C₆₇	C₂²	1	1	yes	characteristic
37	#424×C₆₇	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#339×C₆₇	C₂²	1	2	yes	—
40	C₂×#11761628	C₂³	1	2	yes	—
41	#14524065	—	2	2	no	—
42	#14524065	—	2	2	no	—
43	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
44	C₂×#11761617	C₂×C₄	1	2	yes	—
45	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11761612	C₂³	1	2	yes	—
47	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
49	#14524008	C₂×C₄	1	4	yes	—
50	#14524005	C₂×C₄	1	4	yes	—
51	(C₂²⋊D₆₇)⋊C₄	—	2	4	no	—
52	#14524000	—	2	2	no	—
53	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
55	C₄²×D₆₇	—	2	2	no	—
56	#14523979	—	2	2	no	—
57	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
58	#14523971	—	2	4	no	—
59	#14523969	—	2	4	no	—
60	#14523966	C₂×C₄	1	4	yes	—
61	#14523963	C₂×C₄	1	4	yes	—
62	(C₂×D₆₇)⋊C₈	—	2	4	no	—
63	(C₂×D₆₇)⋊C₈	—	2	4	no	—
64	#14523954	—	2	2	no	—
65	#14523953	—	2	2	no	—
66	#14523953	—	2	2	no	—
67	#14523953	D₄	1	2	yes	—
68	#14523953	D₄	1	2	yes	—
69	C₈×Dic₆₇	—	2	2	no	—
70	C₈×Dic₆₇	—	2	2	no	—
71	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
72	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
73	C₆₇⋊#95	—	2	2	no	—
74	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	#117×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#108×C₆₇	—	2	2	no	—
80	#107×C₆₇	C₂×C₄	1	2	yes	—
81	#106×C₆₇	C₂×C₄	1	2	yes	—
82	C₄⋊C₈×C₆₇	—	2	4	no	—
83	C₄×C₈×C₆₇	—	2	2	no	—
84	#11761628	—	2	2	no	—
85	#11761628	—	2	2	no	—
86	#11761628	—	2	2	no	—
87	#11761628	—	2	2	no	—
88	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
89	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
90	C₂×C₂²⋊D₆₇	—	2	2	no	—
91	C₂²×Dic₆₇	—	2	1	no	—
92	C₂²×Dic₆₇	#41	1	2	yes	—
93	C₂×C₄×D₆₇	—	2	2	no	—
94	C₂×C₄×D₆₇	—	2	4	no	—
95	Dic₂₆₈	—	2	2	no	—
96	#11761617	—	2	4	no	—
97	D₅₃₆	—	2	2	no	—
98	#11761612	—	2	2	no	—
99	#11761612	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	1	no	—
101	C₂×C₆₇⋊C₈	—	2	1	no	—
102	C₂×C₆₇⋊C₈	—	2	1	no	—
103	C₂×C₆₇⋊C₈	—	2	1	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
107	Dic₆₇⋊C₄	—	2	4	no	—
108	(C₂×D₆₇)⋊C₄	—	2	4	no	—
109	C₄×Dic₆₇	—	2	1	no	—
110	C₄×Dic₆₇	—	2	4	no	—
111	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
112	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄×Dic₆₇	#41	1	2	yes	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	2	1	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	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₆₇	#41	1	2	yes	—
126	C₂×C₈×C₆₇	#41	1	2	yes	—
127	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂²⋊C₄×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	1	no	—
131	C₄²×C₆₇	#41	1	2	yes	—
132	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
133	C₂²×D₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₂²⋊D₆₇	—	4	2	no	—
144	C₂²⋊D₆₇	—	4	2	no	—
145	C₂×Dic₆₇	—	2	2	no	—
146	C₂×Dic₆₇	—	2	2	no	—
147	C₂×Dic₆₇	—	4	2	no	—
148	C₂×Dic₆₇	—	4	2	no	—
149	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
150	C₂×Dic₆₇	C₄×D₄	1	2	yes	—
151	D₂₆₈	—	2	2	no	—
152	D₂₆₈	—	2	2	no	—
153	C₄×D₆₇	—	4	2	no	—
154	C₄×D₆₇	—	4	2	no	—
155	C₄×D₆₇	—	8	4	no	—
156	Dic₁₃₄	—	2	2	no	—
157	Dic₁₃₄	—	2	2	no	—
158	C₈×C₆₇	—	2	2	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	#117	1	2	yes	—
161	C₈×C₆₇	#117	1	2	yes	—
162	C₆₇⋊C₈	—	2	2	no	—
163	C₆₇⋊C₈	—	2	2	no	—
164	C₆₇⋊C₈	—	4	2	no	—
165	C₂²×C₆₇	—	2	2	no	—
166	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₆₇	#529	1	1	yes	characteristic
168	C₂²×C₆₇	#529	1	1	yes	characteristic
169	C₂×D₆₇	—	4	4	no	—
170	C₂×D₆₇	—	4	2	no	—
171	C₄×C₆₇	—	4	2	no	—
172	C₄×C₆₇	—	4	4	no	—
173	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
174	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
175	C₄×C₆₇	#532	1	1	yes	characteristic
176	C₄×C₆₇	#532	1	1	yes	characteristic
177	Dic₆₇	—	2	2	no	—
178	Dic₆₇	—	2	2	no	—
179	Dic₆₇	—	4	2	no	—
180	Dic₆₇	—	4	2	no	—
181	#21294	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	—	2	1	no	—
183	C₂×C₆₇	—	2	1	no	—
184	C₂×C₆₇	#2293	1	1	yes	characteristic
185	C₂×C₆₇	#2957	1	2	yes	—
186	D₆₇	—	8	2	no	—
187	C₄×#135	—	67	1	no	—
188	#2923	—	67	2	no	—
189	#1974	—	67	1	no	—
190	#1974	—	67	1	no	—
191	#1945	—	67	1	no	—
192	#1945	D₆₇	1	1	yes	characteristic, complemented
193	#1938	—	67	2	no	—
194	#1938	—	67	2	no	—
195	C₄×#108	—	67	1	no	—
196	#1759	—	67	2	no	—
197	#1726	—	67	1	no	—
198	C₆₇	#21294	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	—	67	1	no	—
200	C₄×#41	—	67	2	no	—
201	C₄×Q₁₆	—	67	2	no	—
202	C₄×SD₁₆	—	67	4	no	—
203	C₄×D₈	—	67	2	no	—
204	#433	—	134	4	no	—
205	#432	—	134	4	no	—
206	C₂×#108	—	67	1	no	—
207	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
208	#427	C₂×D₆₇	1	1	yes	characteristic
209	#424	—	67	2	no	—
210	#424	C₂×D₆₇	1	2	yes	—
211	C₂×#104	—	67	2	no	—
212	C₂×#104	—	67	2	no	—
213	#416	—	67	2	no	—
214	#408	—	67	2	no	—
215	#408	—	67	2	no	—
216	C₄×M₁₆	—	67	2	no	—
217	C₂×#97	—	67	1	no	—
218	C₂×#97	—	67	1	no	—
219	C₂×#97	—	67	1	no	—
220	C₂×C₄×C₈	—	67	1	no	—
221	C₂×C₄×C₈	—	67	1	no	—
222	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
223	#376	—	67	1	no	—
224	#341	—	67	2	no	—
225	#339	—	67	2	no	—
226	#339	C₂×D₆₇	1	2	yes	—
227	C₂×#41	—	67	2	no	—
228	#135	—	134	2	no	—
229	#135	—	134	2	no	—
230	C₂×Q₁₆	—	67	1	no	—
231	C₂×SD₁₆	—	67	2	no	—
232	C₂×D₈	—	67	1	no	—
233	C₂×M₁₆	—	67	2	no	—
234	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
235	C₂²×C₈	—	67	1	no	—
236	C₂²×C₈	—	67	1	no	—
237	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
239	C₄×Q₈	—	67	4	no	—
240	C₄×D₄	—	67	4	no	—
241	C₄×D₄	—	134	4	no	—
242	#117	—	134	2	no	—
243	#117	C₂²×D₆₇	1	2	yes	—
244	C₂×C₄²	—	67	1	no	—
245	C₂×C₄²	—	67	1	no	—
246	C₂×C₄²	—	134	2	no	—
247	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
248	#108	—	2	2	no	—
249	#108	—	134	2	no	—
250	#107	C₄×D₆₇	1	2	yes	—
251	#106	C₄×D₆₇	1	2	yes	—
252	C₄⋊C₈	—	2	4	no	—
253	C₄⋊C₈	—	134	4	no	—
254	#104	—	134	4	no	—
255	#104	—	134	4	no	—
256	#103	—	67	4	no	—
257	#102	—	67	4	no	—
258	C₂²⋊C₈	—	134	4	no	—
259	C₂²⋊C₈	—	134	4	no	—
260	#97	—	67	2	no	—
261	#97	—	67	2	no	—
262	#97	—	134	2	no	—
263	#97	—	134	2	no	—
264	#97	—	134	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	67	2	no	—
267	C₄×C₈	—	67	2	no	—
268	C₄×C₈	—	134	2	no	—
269	C₄×C₈	—	134	2	no	—
270	#95	—	134	2	no	—
271	#41	—	134	2	no	—
272	#41	—	134	2	no	—
273	#41	—	134	2	no	—
274	#41	—	134	2	no	—
275	C₂×Q₈	—	67	2	no	—
276	C₂×D₄	—	67	2	no	—
277	C₂×D₄	—	134	2	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	67	2	no	—
280	C₂²×C₄	—	134	4	no	—
281	C₂²×C₄	—	134	1	no	—
282	C₂²×C₄	—	134	2	no	—
283	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
284	Q₁₆	—	134	2	no	—
285	SD₁₆	—	134	4	no	—
286	D₈	—	134	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	134	2	no	—
290	M₁₆	—	134	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	67	2	no	—
295	C₂×C₈	—	67	2	no	—
296	C₂×C₈	—	134	2	no	—
297	C₂×C₈	—	134	1	no	—
298	C₂×C₈	—	134	1	no	—
299	C₂×C₈	—	134	1	no	—
300	C₂×C₈	—	134	1	no	—
301	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₂×C₈	D₄×D₆₇	1	2	yes	—
304	C₂×C₈	#11761627	1	2	yes	—
305	C₂×C₈	#11761629	1	2	yes	—
306	C₄⋊C₄	—	134	4	no	—
307	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	134	4	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	67	2	no	—
313	C₄²	—	67	1	no	—
314	C₄²	—	67	1	no	—
315	C₄²	—	67	2	no	—
316	C₄²	—	134	4	no	—
317	C₄²	—	134	1	no	—
318	C₄²	#11761629	1	2	yes	—
319	C₂³	—	134	2	no	—
320	C₂³	#41×D₆₇	1	1	yes	characteristic
321	Q₈	—	134	2	no	—
322	Q₈	—	134	2	no	—
323	D₄	—	268	2	no	—
324	D₄	—	268	2	no	—
325	D₄	—	134	2	no	—
326	D₄	—	134	2	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	268	2	no	—
331	C₂×C₄	—	268	2	no	—
332	C₂×C₄	—	268	2	no	—
333	C₂×C₄	—	268	2	no	—
334	C₂×C₄	—	536	4	no	—
335	C₂×C₄	—	67	2	no	—
336	C₂×C₄	—	67	2	no	—
337	C₂×C₄	—	134	2	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
342	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
343	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	268	2	no	—
348	C₈	—	134	2	no	—
349	C₈	—	134	2	no	—
350	C₈	#14524001	1	2	yes	—
351	C₈	#14524001	1	2	yes	—
352	C₂²	—	2	2	no	—
353	C₂²	—	268	2	no	—
354	C₂²	—	268	4	no	—
355	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
356	C₂²	#16863931	1	1	yes	characteristic
357	C₂²	#16863932	1	1	yes	characteristic
358	C₄	—	4	2	no	—
359	C₄	—	4	4	no	—
360	C₄	—	268	2	no	—
361	C₄	—	268	2	no	—
362	C₄	—	134	2	no	—
363	C₄	—	134	2	no	—
364	C₄	C₂×#14524001	1	1	yes	characteristic
365	C₄	C₄×#11761629	1	1	yes	characteristic
366	C₄	#16863941	1	1	yes	characteristic
367	C₄	#16863947	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	536	2	no	—
371	C₂	#22562667	1	1	yes	characteristic
372	C₂	#22569121	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×#1945	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1945	D₆₇
hypocenter	C₆₇	#21294
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1945

Complements

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

N	H ≅ G/N	type
2. #22568859	C₂	semidirect
5. #22560958	C₂	semidirect
6. #22560921	C₂	semidirect
7. #22560919	C₂	semidirect
8. #108×Dic₆₇	C₂	semidirect
9. #22560018	C₂	semidirect
10. #22559895	C₂	semidirect
11. C₆₇×#1945	C₂	semidirect
14. #16863490	C₄	semidirect
15. #16863483	C₄	semidirect
16. #16863478	C₄	semidirect
192. #1945	D₆₇	semidirect
198. C₆₇	#21294	semidirect

Maximal subgroups

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

subgroup	index
#22568980	2
#22568859	2
#22561097	2
#22561095	2
#22560958	2
#22560921	2
#22560919	2
#108×Dic₆₇	2
#22560018	2
#22559895	2
C₆₇×#1945	2
#21294	67