Subgroups — #31009308

Order 17152.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863834	C₂	1	1	yes	maximal, characteristic
2	#22569127	C₂	1	1	yes	maximal, characteristic, complemented
3	#22569062	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562109	C₂	1	8	yes	maximal, complemented
5	#22561982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863210	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561946	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
11	#16864162	—	2	1	no	—
12	#16863870	—	2	1	no	—
13	#16863834	C₄	1	4	yes	complemented
14	#16863834	C₂²	1	4	yes	—
15	C₂×#14523988	C₄	1	2	yes	complemented
16	C₂×D₁₀₇₂	C₄	1	2	yes	complemented
17	#16863519	C₂²	1	4	yes	complemented
18	#16863514	—	2	2	no	—
19	#16863509	C₂²	1	4	yes	complemented
20	C₄×D₅₃₆	—	2	2	no	—
21	C₂×#14523979	C₂²	1	1	yes	characteristic
22	#16863434	C₂²	1	1	yes	characteristic
23	#16863414	—	2	1	no	—
24	#16863402	—	2	1	no	—
25	#16863219	C₂²	1	4	yes	complemented
26	#16863210	C₂²	1	4	yes	complemented
27	#16863207	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
30	#405×C₆₇	C₂²	1	1	yes	characteristic
31	#347×C₆₇	C₂²	1	4	yes	complemented
32	C₂×#11761628	—	2	1	no	—
33	C₂²×D₂₆₈	—	2	1	no	—
34	#14524065	—	2	1	no	—
35	#14524065	—	2	1	no	—
36	#14524065	C₂×C₄	1	2	yes	complemented
37	#14524065	C₂³	1	2	yes	—
38	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×#11761617	—	2	1	no	—
40	C₂×D₅₃₆	—	2	1	no	—
41	C₂×D₅₃₆	—	2	1	no	—
42	C₂×D₅₃₆	—	2	1	no	—
43	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
44	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
45	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₅₃₆	C₂³	1	2	yes	—
47	#14524051	—	4	2	no	—
48	#14524049	—	2	4	no	—
49	C₂×#11761612	—	2	1	no	—
50	M₁₆×D₆₇	—	4	2	no	—
51	C₄×D₂₆₈	—	2	4	no	—
52	#14524005	—	4	2	no	—
53	#14524002	—	2	1	no	—
54	C₂²⋊C₄×D₆₇	—	4	2	no	—
55	#14523988	C₂×C₄	1	8	yes	complemented
56	D₁₀₇₂	C₂×C₄	1	8	yes	complemented
57	#14523979	—	2	1	no	—
58	#14523979	C₂³	1	2	yes	—
59	#14523977	—	2	1	no	—
60	#14523975	C₂³	1	2	yes	—
61	#14523971	—	2	4	no	—
62	#14523965	—	2	4	no	—
63	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	#117×C₆₇	D₄	1	1	yes	characteristic
66	M₃₂×C₆₇	—	2	1	no	—
67	M₃₂×C₆₇	C₂×C₄	1	2	yes	complemented
68	M₃₂×C₆₇	C₂³	1	2	yes	—
69	M₃₂×C₆₇	C₂³	1	2	yes	—
70	C₂×C₁₆×C₆₇	—	2	1	no	—
71	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
72	#97×C₆₇	C₂³	1	2	yes	—
73	C₄×C₈×C₆₇	D₄	1	2	yes	—
74	C₂³×D₆₇	—	4	1	no	—
75	#11761628	—	2	2	no	—
76	#11761628	—	2	2	no	—
77	#11761628	—	4	1	no	—
78	#11761628	—	4	1	no	—
79	C₂×Dic₁₃₄	—	2	1	no	—
80	C₂×D₂₆₈	—	2	2	no	—
81	C₂×D₂₆₈	—	2	2	no	—
82	C₂×D₂₆₈	—	2	1	no	—
83	C₂×D₂₆₈	—	2	2	no	—
84	C₂×D₂₆₈	—	4	1	no	—
85	C₂×D₂₆₈	—	4	1	no	—
86	C₂×D₂₆₈	—	4	1	no	—
87	C₂×C₂²⋊D₆₇	—	4	1	no	—
88	C₂×C₄×D₆₇	—	4	1	no	—
89	C₂×C₄×D₆₇	—	4	4	no	—
90	Dic₂₆₈	—	2	1	no	—
91	Dic₂₆₈	C₂²×C₄	1	2	yes	—
92	#11761617	—	2	2	no	—
93	#11761617	—	2	2	no	—
94	D₅₃₆	—	2	2	no	—
95	D₅₃₆	—	2	2	no	—
96	D₅₃₆	—	2	1	no	—
97	D₅₃₆	—	2	2	no	—
98	D₅₃₆	C₂²×C₄	1	2	yes	—
99	D₅₃₆	C₂²×C₄	1	4	yes	—
100	#11761612	—	2	2	no	—
101	#11761612	—	4	1	no	—
102	#11761611	—	4	4	no	—
103	C₂×C₆₇⋊C₈	—	4	1	no	—
104	C₈×D₆₇	—	4	4	no	—
105	C₄⋊Dic₆₇	—	2	2	no	—
106	C₂²⋊Dic₆₇	—	4	1	no	—
107	(C₂×D₆₇)⋊C₄	—	4	4	no	—
108	C₄×Dic₆₇	—	4	1	no	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	M₁₆×C₆₇	—	2	1	no	—
111	M₁₆×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
117	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₂²⋊C₄×C₆₇	—	2	1	no	—
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₁₆×C₆₇	—	2	2	no	—
123	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₆₇	—	4	2	no	—
126	C₂²×D₆₇	—	4	1	no	—
127	C₂²×D₆₇	—	4	2	no	—
128	C₂²×D₆₇	—	4	2	no	—
129	C₂²×D₆₇	—	16	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₆₇	—	4	2	no	—
138	C₂²⋊D₆₇	—	4	2	no	—
139	C₂×Dic₆₇	—	4	1	no	—
140	C₂×Dic₆₇	—	4	2	no	—
141	D₂₆₈	—	2	4	no	—
142	D₂₆₈	—	2	2	no	—
143	D₂₆₈	—	4	2	no	—
144	D₂₆₈	—	4	2	no	—
145	D₂₆₈	—	4	2	no	—
146	D₂₆₈	—	4	1	no	—
147	C₄×D₆₇	—	4	2	no	—
148	C₄×D₆₇	—	4	2	no	—
149	C₄×D₆₇	—	16	4	no	—
150	Dic₁₃₄	—	2	2	no	—
151	Dic₁₃₄	—	4	1	no	—
152	C₈×C₆₇	—	2	2	no	—
153	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
154	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
155	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
156	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
157	C₆₇⋊C₈	—	4	2	no	—
158	C₂²×C₆₇	—	4	1	no	—
159	C₂²×C₆₇	#442	1	1	yes	characteristic
160	C₂²×C₆₇	#442	1	1	yes	characteristic
161	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×D₆₇	—	4	2	no	—
163	C₂×D₆₇	—	4	4	no	—
164	C₂×D₆₇	—	8	4	no	—
165	C₂×D₆₇	—	16	1	no	—
166	C₂×D₆₇	—	16	2	no	—
167	C₂×D₆₇	—	16	2	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	#442	1	1	yes	characteristic
170	C₄×C₆₇	#442	1	1	yes	characteristic
171	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
172	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	Dic₆₇	—	4	2	no	—
174	Dic₆₇	—	8	2	no	—
175	#33471	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	2	1	no	—
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	—	2	1	no	—
179	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
180	D₆₇	—	8	4	no	—
181	D₆₇	—	8	2	no	—
182	C₂×#509	—	67	1	no	—
183	#2957	—	67	1	no	—
184	#2948	—	67	1	no	—
185	AGL(1,ℤ/16)	—	67	8	no	—
186	#2155	—	67	1	no	—
187	C₂×#361	—	67	1	no	—
188	#2145	—	67	1	no	—
189	#2110	D₆₇	1	1	yes	characteristic, complemented
190	C₆₇	#33471	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	—	67	1	no	—
192	C₂²×D₈	—	67	1	no	—
193	#568	—	134	1	no	—
194	#518	—	134	1	no	—
195	#509	—	67	4	no	—
196	#509	—	67	4	no	—
197	C₂×SD₃₂	—	67	2	no	—
198	C₂×D₁₆	—	67	2	no	—
199	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
200	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
201	#444	—	67	4	no	—
202	#443	—	134	2	no	—
203	#442	—	67	4	no	—
204	C₄×D₈	—	134	2	no	—
205	C₂×#108	—	67	1	no	—
206	#427	—	67	1	no	—
207	#421	—	134	1	no	—
208	#418	—	134	1	no	—
209	#405	C₂×D₆₇	1	1	yes	characteristic
210	#365	—	67	4	no	—
211	#361	—	67	4	no	—
212	#360	—	67	4	no	—
213	#347	C₂×D₆₇	1	4	yes	complemented
214	C₂×#41	—	134	1	no	—
215	C₂²×D₄	—	134	1	no	—
216	#135	—	67	2	no	—
217	#135	—	67	2	no	—
218	#135	—	134	1	no	—
219	#135	—	134	1	no	—
220	C₂×Q₁₆	—	67	1	no	—
221	C₂×SD₁₆	—	134	1	no	—
222	C₂×D₈	—	67	2	no	—
223	C₂×D₈	—	67	2	no	—
224	C₂×D₈	—	67	1	no	—
225	C₂×D₈	—	67	2	no	—
226	C₂×D₈	—	134	1	no	—
227	C₂×D₈	—	134	1	no	—
228	C₂×D₈	—	134	1	no	—
229	#131	—	268	2	no	—
230	#131	—	134	4	no	—
231	C₂×M₁₆	—	268	2	no	—
232	C₂×M₁₆	—	134	1	no	—
233	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₄×D₄	—	268	2	no	—
236	C₄×D₄	—	134	4	no	—
237	#117	—	134	1	no	—
238	#117	D₂₆₈	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	268	2	no	—
240	SD₃₂	—	67	8	no	—
241	D₁₆	—	67	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	C₄×D₆₇	1	2	yes	complemented
244	M₃₂	C₂²×D₆₇	1	2	yes	—
245	M₃₂	C₂²×D₆₇	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	C₄×D₆₇	1	2	yes	complemented
248	#108	—	67	2	no	—
249	#108	—	134	1	no	—
250	#107	—	134	1	no	—
251	#106	—	67	2	no	—
252	#104	—	134	4	no	—
253	#102	—	134	4	no	—
254	#97	C₂²×D₆₇	1	2	yes	—
255	C₄×C₈	D₂₆₈	1	2	yes	—
256	C₂⁴	—	268	1	no	—
257	#41	—	268	1	no	—
258	#41	—	268	1	no	—
259	#41	—	134	2	no	—
260	#41	—	134	2	no	—
261	C₂×Q₈	—	134	1	no	—
262	C₂×D₄	—	268	1	no	—
263	C₂×D₄	—	268	1	no	—
264	C₂×D₄	—	268	1	no	—
265	C₂×D₄	—	268	1	no	—
266	C₂×D₄	—	134	2	no	—
267	C₂×D₄	—	134	2	no	—
268	C₂×D₄	—	134	1	no	—
269	C₂×D₄	—	134	2	no	—
270	C₂²×C₄	—	268	4	no	—
271	C₂²×C₄	—	268	1	no	—
272	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
273	Q₁₆	—	67	2	no	—
274	Q₁₆	—	134	1	no	—
275	SD₁₆	—	134	2	no	—
276	SD₁₆	—	134	2	no	—
277	D₈	—	67	2	no	—
278	D₈	—	67	4	no	—
279	D₈	—	134	2	no	—
280	D₈	—	134	2	no	—
281	D₈	—	134	1	no	—
282	D₈	—	134	2	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	268	4	no	—
286	M₁₆	—	268	1	no	—
287	M₁₆	—	134	2	no	—
288	C₂×C₈	—	268	4	no	—
289	C₂×C₈	—	268	1	no	—
290	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
292	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
293	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
294	C₂×C₈	#11761628	1	1	yes	characteristic
295	C₂×C₈	#11761628	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	134	2	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	268	4	no	—
301	C₂²⋊C₄	—	268	1	no	—
302	C₄²	—	268	1	no	—
303	C₄²	C₂×D₂₆₈	1	2	yes	—
304	C₁₆	—	2	2	no	—
305	C₁₆	C₂×C₄×D₆₇	1	4	yes	—
306	C₂³	—	268	2	no	—
307	C₂³	—	268	1	no	—
308	C₂³	—	268	2	no	—
309	C₂³	—	268	2	no	—
310	C₂³	—	1072	2	no	—
311	C₂³	#14524071	1	1	yes	characteristic
312	Q₈	—	268	1	no	—
313	Q₈	—	134	2	no	—
314	D₄	—	268	2	no	—
315	D₄	—	268	2	no	—
316	D₄	—	268	2	no	—
317	D₄	—	268	2	no	—
318	D₄	—	268	1	no	—
319	D₄	—	268	2	no	—
320	D₄	—	134	4	no	—
321	D₄	—	134	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	268	1	no	—
324	C₂×C₄	—	268	2	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	268	2	no	—
327	C₂×C₄	—	1072	4	no	—
328	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
329	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
330	C₂×C₄	#14524071	1	1	yes	characteristic
331	C₂×C₄	#14524071	1	1	yes	characteristic
332	C₂×C₄	#14524071	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	268	2	no	—
336	C₈	C₄×D₂₆₈	1	1	yes	characteristic
337	C₈	C₄×D₂₆₈	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
339	C₈	C₂×#11761628	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	268	2	no	—
342	C₂²	—	268	4	no	—
343	C₂²	—	536	4	no	—
344	C₂²	—	1072	2	no	—
345	C₂²	—	1072	2	no	—
346	C₂²	—	1072	1	no	—
347	C₂²	#16863509	1	1	yes	characteristic
348	C₂²	#16863509	1	1	yes	characteristic
349	C₂²	C₂×#14524071	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	268	2	no	—
352	C₄	—	536	2	no	—
353	C₄	#16863509	1	1	yes	characteristic
354	C₄	#16863509	1	1	yes	characteristic
355	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
356	C₄	C₂×#14524071	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	536	4	no	—
361	C₂	—	536	2	no	—
362	C₂	C₂×#16863509	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16863509
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2110	C₂
socle	C₂×C₆₇	C₂×#442
prime radical	C₂×#16863834	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2110	D₆₇
hypocenter	C₆₇	#33471
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
2. #22569127	C₂	semidirect
3. #22569062	C₂	semidirect
4. #22562109	C₂	semidirect
5. #22561982	C₂	semidirect
6. C₂×#16863210	C₂	semidirect
7. #22561946	C₂	semidirect
8. C₆₇×#2110	C₂	semidirect
13. #16863834	C₄	semidirect
15. C₂×#14523988	C₄	semidirect
16. C₂×D₁₀₇₂	C₄	semidirect
17. #16863519	C₂²	semidirect
19. #16863509	C₂²	semidirect
25. #16863219	C₂²	semidirect
26. #16863210	C₂²	semidirect
27. #16863207	C₂²	semidirect
31. #347×C₆₇	C₂²	semidirect
36. #14524065	C₂×C₄	semidirect
38. C₂×Dic₂₆₈	C₂×C₄	semidirect
45. C₂×D₅₃₆	C₂×C₄	semidirect
55. #14523988	C₂×C₄	semidirect
56. D₁₀₇₂	C₂×C₄	semidirect
67. M₃₂×C₆₇	C₂×C₄	semidirect
71. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
189. #2110	D₆₇	semidirect
190. C₆₇	#33471	semidirect
213. #347	C₂×D₆₇	semidirect
243. M₃₂	C₄×D₆₇	semidirect
247. C₂×C₁₆	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863834	2
#22569127	2
#22569062	2
#22562109	2
#22561982	2
C₂×#16863210	2
#22561946	2
C₆₇×#2110	2
#33471	67