Subgroups — #30755149

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863743	C₂	1	1	yes	maximal, characteristic
2	#22561135	C₂	1	1	yes	maximal, characteristic
3	#22561131	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560076	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863143	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863139	C₂	1	1	yes	maximal, characteristic
7	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524043	C₂²	1	1	yes	characteristic
9	C₂×#14524040	C₂²	1	1	yes	characteristic
10	#16863743	—	2	2	no	—
11	#16863743	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#14523968	—	2	2	no	—
14	C₂×#14523965	—	2	2	no	—
15	C₂×#14523965	—	2	1	no	—
16	C₂×#14523965	—	2	1	no	—
17	C₂×#14523953	—	2	1	no	—
18	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
19	#16863150	—	2	1	no	—
20	#16863143	—	2	2	no	—
21	#16863139	—	2	2	no	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#334×C₆₇	—	2	2	no	—
26	C₂²×Dic₁₃₄	—	2	1	no	—
27	C₂²×D₂₆₈	—	2	1	no	—
28	C₂²×D₂₆₈	—	2	1	no	—
29	C₂²×D₂₆₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	1	no	—
31	#14524043	—	2	1	no	—
32	#14524043	—	2	1	no	—
33	#14524043	—	2	1	no	—
34	#14524043	C₂×C₄	1	2	yes	—
35	#14524040	—	2	1	no	—
36	#14524040	—	2	1	no	—
37	#14524040	—	2	1	no	—
38	#14524040	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊Dic₆₇	—	2	1	no	—
40	C₂×C₄⋊Dic₆₇	—	2	1	no	—
41	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
42	C₂×C₄×Dic₆₇	—	2	1	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
45	#14523968	—	4	4	no	—
46	#14523965	—	2	4	no	—
47	#14523965	—	2	4	no	—
48	#14523965	—	4	4	no	—
49	#14523953	—	4	2	no	—
50	C₆₇⋊#95	—	2	1	no	—
51	C₂²×C₈×C₆₇	—	2	1	no	—
52	C₂²×C₈×C₆₇	—	2	1	no	—
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₇	—	2	1	no	—
57	C₄⋊C₈×C₆₇	—	2	1	no	—
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₆₇	D₄	1	2	yes	—
62	C₂³×D₆₇	—	4	1	no	—
63	C₂×Dic₁₃₄	—	2	2	no	—
64	C₂×Dic₁₃₄	—	4	2	no	—
65	C₂×Dic₁₃₄	—	4	2	no	—
66	C₂×Dic₁₃₄	—	4	1	no	—
67	C₂×D₂₆₈	—	2	2	no	—
68	C₂×D₂₆₈	—	2	2	no	—
69	C₂×D₂₆₈	—	2	2	no	—
70	C₂×D₂₆₈	—	4	1	no	—
71	C₂×D₂₆₈	—	4	1	no	—
72	C₂×D₂₆₈	—	4	2	no	—
73	C₂×D₂₆₈	—	4	2	no	—
74	C₂×D₂₆₈	—	4	1	no	—
75	C₂×D₂₆₈	—	4	1	no	—
76	C₂×D₂₆₈	—	4	1	no	—
77	C₂×D₂₆₈	—	4	1	no	—
78	C₂×D₂₆₈	—	4	1	no	—
79	C₂×D₂₆₈	—	8	2	no	—
80	C₂²×Dic₆₇	—	2	2	no	—
81	C₂²×Dic₆₇	—	4	1	no	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	4	1	no	—
84	C₂×C₆₇⋊C₈	—	4	1	no	—
85	C₄⋊Dic₆₇	—	2	2	no	—
86	C₄⋊Dic₆₇	—	2	2	no	—
87	C₄⋊Dic₆₇	—	2	2	no	—
88	C₄⋊Dic₆₇	—	2	2	no	—
89	C₄⋊Dic₆₇	—	4	1	no	—
90	C₄⋊Dic₆₇	—	4	1	no	—
91	C₄×Dic₆₇	—	4	1	no	—
92	C₄×Dic₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
96	C₂×C₈×C₆₇	—	2	1	no	—
97	C₂×C₈×C₆₇	—	2	1	no	—
98	C₂×C₈×C₆₇	—	2	1	no	—
99	C₂×C₈×C₆₇	—	2	1	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	—	4	1	no	—
102	C₂×C₈×C₆₇	—	4	2	no	—
103	C₂×C₈×C₆₇	—	4	2	no	—
104	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
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	2	no	—
112	C₂²×D₆₇	—	8	1	no	—
113	C₂²×D₆₇	—	8	1	no	—
114	C₂²×D₆₇	—	8	2	no	—
115	C₂²×D₆₇	—	8	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	#135	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
129	C₂×Dic₆₇	—	4	2	no	—
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	—	4	2	no	—
133	C₂×Dic₆₇	—	8	2	no	—
134	D₂₆₈	—	4	2	no	—
135	D₂₆₈	—	4	2	no	—
136	D₂₆₈	—	4	4	no	—
137	D₂₆₈	—	8	2	no	—
138	D₂₆₈	—	8	2	no	—
139	D₂₆₈	—	8	1	no	—
140	D₂₆₈	—	8	1	no	—
141	D₂₆₈	—	8	1	no	—
142	D₂₆₈	—	8	1	no	—
143	Dic₁₃₄	—	4	4	no	—
144	Dic₁₃₄	—	8	1	no	—
145	Dic₁₃₄	—	8	1	no	—
146	C₈×C₆₇	—	2	2	no	—
147	C₈×C₆₇	—	2	2	no	—
148	C₈×C₆₇	—	8	1	no	—
149	C₈×C₆₇	—	8	1	no	—
150	C₆₇⋊C₈	—	4	2	no	—
151	C₂²×C₆₇	#390	1	1	yes	characteristic
152	C₂²×C₆₇	#463	1	1	yes	characteristic
153	C₂²×C₆₇	#464	1	1	yes	characteristic
154	C₂²×C₆₇	#469	1	2	yes	—
155	C₂²×C₆₇	#471	1	2	yes	—
156	C₂×D₆₇	—	8	2	no	—
157	C₂×D₆₇	—	8	2	no	—
158	C₂×D₆₇	—	8	2	no	—
159	C₂×D₆₇	—	16	2	no	—
160	C₂×D₆₇	—	16	2	no	—
161	C₄×C₆₇	—	2	1	no	—
162	C₄×C₆₇	—	2	1	no	—
163	C₄×C₆₇	#416	1	1	yes	characteristic
164	C₄×C₆₇	#417	1	1	yes	characteristic
165	C₄×C₆₇	#442	1	2	yes	—
166	C₄×C₆₇	#463	1	1	yes	characteristic
167	C₄×C₆₇	#464	1	1	yes	characteristic
168	C₄×C₆₇	#496	1	1	yes	characteristic
169	C₄×C₆₇	#496	1	1	yes	characteristic
170	Dic₆₇	—	8	2	no	—
171	Dic₆₇	—	8	2	no	—
172	#9746	—	67	1	no	maximal, Sylow
173	C₂×C₆₇	#1701	1	2	yes	—
174	C₂×C₆₇	#1702	1	2	yes	—
175	C₂×C₆₇	#1972	1	1	yes	characteristic
176	C₂×C₆₇	#1977	1	1	yes	characteristic
177	C₂×C₆₇	#1983	1	1	yes	characteristic
178	D₆₇	—	16	2	no	—
179	C₂×#486	—	67	1	no	—
180	#1982	—	67	1	no	—
181	#1981	—	67	1	no	—
182	#1770	—	67	1	no	—
183	C₂×#334	D₆₇	1	1	yes	characteristic, complemented
184	C₂×#332	—	67	1	no	—
185	C₂×#331	—	67	1	no	—
186	C₆₇	#9746	1	1	yes	characteristic, complemented, Sylow
187	C₂×C₄⋊Q₈	—	67	1	no	—
188	C₂×#127	—	67	1	no	—
189	#486	—	67	4	no	—
190	#486	—	134	2	no	—
191	C₂×C₄⋊C₈	—	67	1	no	—
192	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
194	C₂×#103	—	134	2	no	—
195	C₂×#102	—	134	2	no	—
196	C₂×#102	—	134	1	no	—
197	C₂×#102	—	134	1	no	—
198	C₂×#97	—	134	1	no	—
199	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
200	C₄×C₄⋊C₄	—	67	1	no	—
201	#336	—	134	1	no	—
202	#334	—	2	2	no	—
203	#332	—	134	2	no	—
204	#331	—	134	2	no	—
205	C₂²×Q₈	—	134	1	no	—
206	C₂²×D₄	—	134	1	no	—
207	C₂²×D₄	—	134	1	no	—
208	C₂²×D₄	—	134	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	134	1	no	—
212	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
213	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
214	C₄⋊Q₈	—	67	2	no	—
215	C₄⋊Q₈	—	134	1	no	—
216	C₄⋊Q₈	—	134	1	no	—
217	C₄⋊Q₈	—	134	1	no	—
218	#127	—	67	2	no	—
219	#127	—	134	1	no	—
220	#127	—	134	1	no	—
221	#127	—	134	1	no	—
222	C₂×C₄⋊C₄	—	67	2	no	—
223	C₂×C₄⋊C₄	—	134	1	no	—
224	C₂×C₄⋊C₄	—	134	1	no	—
225	C₂×C₄²	—	134	1	no	—
226	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	67	2	no	—
232	C₄⋊C₈	—	134	1	no	—
233	#103	—	268	4	no	—
234	#102	—	268	4	no	—
235	#102	—	134	4	no	—
236	#102	—	134	4	no	—
237	#97	—	268	2	no	—
238	C₄×C₈	C₄×D₆₇	1	2	yes	—
239	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
240	#95	—	134	1	no	—
241	C₂⁴	—	268	1	no	—
242	C₂×Q₈	—	268	1	no	—
243	C₂×Q₈	—	268	2	no	—
244	C₂×Q₈	—	268	2	no	—
245	C₂×Q₈	—	134	2	no	—
246	C₂×D₄	—	268	1	no	—
247	C₂×D₄	—	268	1	no	—
248	C₂×D₄	—	268	1	no	—
249	C₂×D₄	—	268	1	no	—
250	C₂×D₄	—	268	1	no	—
251	C₂×D₄	—	268	2	no	—
252	C₂×D₄	—	268	2	no	—
253	C₂×D₄	—	268	1	no	—
254	C₂×D₄	—	268	1	no	—
255	C₂×D₄	—	536	2	no	—
256	C₂×D₄	—	134	2	no	—
257	C₂×D₄	—	134	2	no	—
258	C₂×D₄	—	134	2	no	—
259	C₂²×C₄	—	268	1	no	—
260	C₂²×C₄	—	134	2	no	—
261	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
263	C₂²×C₄	#11761629	1	1	yes	characteristic
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	268	1	no	—
273	C₂×C₈	—	268	1	no	—
274	C₂×C₈	—	134	2	no	—
275	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
276	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
277	C₄⋊C₄	—	268	1	no	—
278	C₄⋊C₄	—	268	1	no	—
279	C₄⋊C₄	—	134	2	no	—
280	C₄⋊C₄	—	134	2	no	—
281	C₄⋊C₄	—	134	2	no	—
282	C₄⋊C₄	—	134	2	no	—
283	C₄²	—	268	1	no	—
284	C₄²	—	268	1	no	—
285	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
286	C₄²	D₄×D₆₇	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
288	C₄²	#11761629	1	1	yes	characteristic
289	C₂³	—	268	2	no	—
290	C₂³	—	536	1	no	—
291	C₂³	—	536	1	no	—
292	C₂³	—	536	2	no	—
293	C₂³	—	536	2	no	—
294	C₂³	#14524018	1	1	yes	characteristic
295	Q₈	—	268	4	no	—
296	Q₈	—	536	1	no	—
297	Q₈	—	536	1	no	—
298	D₄	—	268	4	no	—
299	D₄	—	268	2	no	—
300	D₄	—	268	2	no	—
301	D₄	—	536	2	no	—
302	D₄	—	536	2	no	—
303	D₄	—	536	1	no	—
304	D₄	—	536	1	no	—
305	D₄	—	536	1	no	—
306	D₄	—	536	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	268	2	no	—
309	C₂×C₄	—	268	2	no	—
310	C₂×C₄	—	268	2	no	—
311	C₂×C₄	—	268	2	no	—
312	C₂×C₄	—	536	2	no	—
313	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
314	C₂×C₄	#14524005	1	1	yes	characteristic
315	C₂×C₄	#14524005	1	1	yes	characteristic
316	C₂×C₄	#14524018	1	1	yes	characteristic
317	C₂×C₄	#14524032	1	1	yes	characteristic
318	C₂×C₄	#14524039	1	1	yes	characteristic
319	C₂×C₄	#14524065	1	1	yes	characteristic
320	C₂×C₄	#14524065	1	1	yes	characteristic
321	C₂×C₄	#14524070	1	2	yes	—
322	C₂×C₄	#14524070	1	2	yes	—
323	C₂×C₄	#14524071	1	1	yes	characteristic
324	C₂×C₄	#14524076	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	8	1	no	—
328	C₈	—	8	1	no	—
329	C₈	—	268	2	no	—
330	C₂²	—	536	2	no	—
331	C₂²	—	536	2	no	—
332	C₂²	—	536	2	no	—
333	C₂²	—	1072	2	no	—
334	C₂²	—	1072	2	no	—
335	C₂²	#16863297	1	1	yes	characteristic
336	C₂²	#16863612	1	1	yes	characteristic
337	C₂²	#16863619	1	1	yes	characteristic
338	C₂²	#16863650	1	2	yes	—
339	C₂²	#16863660	1	2	yes	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	536	2	no	—
343	C₄	—	536	2	no	—
344	C₄	#16863393	1	1	yes	characteristic
345	C₄	#16863398	1	1	yes	characteristic
346	C₄	#16863507	1	2	yes	—
347	C₄	#16863612	1	1	yes	characteristic
348	C₄	#16863619	1	1	yes	characteristic
349	C₄	#16863782	1	1	yes	characteristic
350	C₄	#16863782	1	1	yes	characteristic
351	C₂	—	1072	2	no	—
352	C₂	#22559771	1	2	yes	—
353	C₂	#22559778	1	2	yes	—
354	C₂	#22561083	1	1	yes	characteristic
355	C₂	#22561110	1	1	yes	characteristic
356	C₂	#22561139	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂×#14524040	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#9746
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #22561131	C₂	semidirect
4. #22560076	C₂	semidirect
5. C₂×#16863143	C₂	semidirect
7. C₂×#334×C₆₇	C₂	semidirect
11. #16863743	C₄	semidirect
183. C₂×#334	D₆₇	semidirect
186. C₆₇	#9746	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863743	2
#22561135	2
#22561131	2
#22560076	2
C₂×#16863143	2
C₂×#16863139	2
C₂×#334×C₆₇	2
#9746	67