Subgroups — #31009303

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×#16863831	C₂	1	1	yes	maximal, characteristic
2	#22569126	C₂	1	1	yes	maximal, characteristic, complemented
3	#22569060	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562104	C₂	1	8	yes	maximal, complemented
5	#22561981	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863210	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561813	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2145	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
10	#16864163	—	2	1	no	—
11	#16863870	—	2	1	no	—
12	#16863831	C₄	1	4	yes	complemented
13	#16863831	C₂²	1	4	yes	—
14	C₂×#14523986	C₄	1	2	yes	complemented
15	C₂×#14523984	C₄	1	2	yes	complemented
16	C₂×#14523983	C₂²	1	1	yes	characteristic
17	#16863518	C₂²	1	4	yes	complemented
18	#16863513	—	2	2	no	—
19	#16863507	C₂²	1	4	yes	complemented
20	#16863478	—	2	2	no	—
21	C₂×#14523979	C₂²	1	1	yes	characteristic
22	#16863413	—	2	1	no	—
23	#16863400	—	2	1	no	—
24	#16863351	C₂²	1	1	yes	characteristic
25	#16863218	C₂²	1	4	yes	complemented
26	#16863210	C₂²	1	4	yes	complemented
27	#16863173	C₂²	1	4	yes	complemented
28	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
30	#427×C₆₇	C₂²	1	1	yes	characteristic
31	#360×C₆₇	C₂²	1	4	yes	complemented
32	C₂²×D₂₆₈	—	2	1	no	—
33	C₂×D₅₃₆	—	2	1	no	—
34	C₂×D₅₃₆	—	2	1	no	—
35	C₂×D₅₃₆	—	2	1	no	—
36	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
37	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
38	C₂×D₅₃₆	C₂³	1	2	yes	—
39	#14524050	—	2	4	no	—
40	#14524050	—	4	2	no	—
41	C₂×#11761612	—	2	1	no	—
42	C₂×#11761612	—	4	2	no	—
43	C₂×#11761612	D₄	1	1	yes	characteristic
44	C₄×D₂₆₈	—	4	2	no	—
45	#14524005	—	2	4	no	—
46	#14524002	D₄	1	1	yes	characteristic
47	C₂²⋊C₄×D₆₇	—	4	2	no	—
48	#14523986	C₂×C₄	1	8	yes	complemented
49	#14523984	C₂×C₄	1	8	yes	complemented
50	#14523983	C₂×C₄	1	2	yes	complemented
51	#14523983	C₂³	1	2	yes	—
52	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
53	#14523979	—	2	1	no	—
54	#14523979	C₂³	1	2	yes	—
55	#14523970	—	2	4	no	—
56	#14523963	—	2	4	no	—
57	#14523953	C₂³	1	2	yes	—
58	C₈×Dic₆₇	D₄	1	2	yes	—
59	C₂×#41×C₆₇	—	2	1	no	—
60	#135×C₆₇	—	2	1	no	—
61	#135×C₆₇	—	2	1	no	—
62	#135×C₆₇	C₂×C₄	1	2	yes	complemented
63	#135×C₆₇	C₂³	1	2	yes	—
64	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
65	C₂×SD₁₆×C₆₇	—	2	1	no	—
66	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	#117×C₆₇	—	2	1	no	—
69	M₃₂×C₆₇	—	2	1	no	—
70	M₃₂×C₆₇	C₂³	1	2	yes	—
71	C₂×C₁₆×C₆₇	—	2	1	no	—
72	#107×C₆₇	—	2	1	no	—
73	#106×C₆₇	C₂³	1	2	yes	—
74	C₂³×D₆₇	—	4	1	no	—
75	C₂×D₂₆₈	—	2	2	no	—
76	C₂×D₂₆₈	—	2	2	no	—
77	C₂×D₂₆₈	—	2	2	no	—
78	C₂×D₂₆₈	—	4	1	no	—
79	C₂×D₂₆₈	—	4	1	no	—
80	C₂×D₂₆₈	—	4	1	no	—
81	C₂×C₄×D₆₇	—	4	4	no	—
82	D₅₃₆	—	2	2	no	—
83	D₅₃₆	—	2	2	no	—
84	D₅₃₆	—	2	2	no	—
85	D₅₃₆	C₂²×C₄	1	4	yes	—
86	#11761612	—	2	1	no	—
87	#11761612	—	2	1	no	—
88	#11761612	—	2	2	no	—
89	#11761612	—	4	4	no	—
90	#11761612	—	4	1	no	—
91	C₂×C₆₇⋊C₈	—	4	4	no	—
92	C₂×C₆₇⋊C₈	—	4	1	no	—
93	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
94	C₄⋊Dic₆₇	—	2	1	no	—
95	C₂²⋊Dic₆₇	—	2	1	no	—
96	(C₂×D₆₇)⋊C₄	—	4	4	no	—
97	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
98	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	—
99	#41×C₆₇	—	2	2	no	—
100	#41×C₆₇	—	2	2	no	—
101	#41×C₆₇	—	4	1	no	—
102	#41×C₆₇	—	4	1	no	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	2	1	no	—
105	C₂×D₄×C₆₇	—	4	1	no	—
106	C₂²×C₄×C₆₇	—	4	1	no	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₆₇	—	2	1	no	—
109	Q₁₆×C₆₇	C₂²×C₄	1	2	yes	—
110	SD₁₆×C₆₇	—	2	2	no	—
111	SD₁₆×C₆₇	—	2	2	no	—
112	D₈×C₆₇	—	2	1	no	—
113	D₈×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
118	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₂²⋊C₄×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	—	4	1	no	—
123	C₁₆×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	4	1	no	—
125	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂²×D₆₇	—	4	2	no	—
127	C₂²×D₆₇	—	4	2	no	—
128	C₂²×D₆₇	—	4	2	no	—
129	C₂²×D₆₇	—	16	2	no	—
130	Q₈×C₆₇	—	2	2	no	—
131	Q₈×C₆₇	—	4	1	no	—
132	D₄×C₆₇	—	2	2	no	—
133	D₄×C₆₇	—	4	2	no	—
134	D₄×C₆₇	—	4	2	no	—
135	D₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×Dic₆₇	—	2	2	no	—
147	D₂₆₈	—	2	4	no	—
148	D₂₆₈	—	4	2	no	—
149	D₂₆₈	—	4	2	no	—
150	D₂₆₈	—	4	2	no	—
151	C₄×D₆₇	—	16	4	no	—
152	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
153	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
154	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
155	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₆₇⋊C₈	—	2	2	no	—
157	C₆₇⋊C₈	—	4	2	no	—
158	C₂²×C₆₇	—	4	1	no	—
159	C₂²×C₆₇	—	4	2	no	—
160	C₂²×C₆₇	—	16	1	no	—
161	C₂²×C₆₇	#442	1	1	yes	characteristic
162	C₂²×C₆₇	#442	1	1	yes	characteristic
163	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×D₆₇	—	4	4	no	—
165	C₂×D₆₇	—	8	4	no	—
166	C₂×D₆₇	—	16	2	no	—
167	C₂×D₆₇	—	16	2	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	—	8	2	no	—
170	C₄×C₆₇	#442	1	1	yes	characteristic
171	C₄×C₆₇	#442	1	1	yes	characteristic
172	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	Dic₆₇	—	4	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	—	8	2	no	—
180	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
181	D₆₇	—	8	4	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	D₆₇	1	1	yes	characteristic, complemented
189	#2110	—	67	1	no	—
190	C₆₇	#33471	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	C₂×D₆₇	1	1	yes	characteristic
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	—	67	1	no	—
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	C₂×D₆₇	1	1	yes	characteristic
207	#421	—	134	1	no	—
208	#418	—	134	1	no	—
209	#405	—	67	1	no	—
210	#365	—	67	4	no	—
211	#361	—	67	4	no	—
212	#360	C₂×D₆₇	1	4	yes	complemented
213	#347	—	67	4	no	—
214	C₂×#41	—	2	1	no	—
215	C₂²×D₄	—	134	1	no	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	C₄×D₆₇	1	2	yes	complemented
219	#135	C₂²×D₆₇	1	2	yes	—
220	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	—	2	1	no	—
222	C₂×D₈	—	67	2	no	—
223	C₂×D₈	—	67	2	no	—
224	C₂×D₈	—	67	2	no	—
225	C₂×D₈	—	134	1	no	—
226	C₂×D₈	—	134	1	no	—
227	C₂×D₈	—	134	1	no	—
228	C₂×D₈	C₄×D₆₇	1	1	yes	characteristic, complemented
229	#131	—	268	2	no	—
230	#131	—	134	4	no	—
231	C₂×M₁₆	—	268	2	no	—
232	C₂×M₁₆	—	67	1	no	—
233	C₂×M₁₆	—	134	1	no	—
234	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₄×D₄	—	268	2	no	—
236	C₄×D₄	—	134	4	no	—
237	#117	—	2	1	no	—
238	#117	—	67	1	no	—
239	C₂×C₂²⋊C₄	—	268	2	no	—
240	SD₃₂	—	67	8	no	—
241	D₁₆	—	67	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	—	67	2	no	—
244	M₃₂	—	67	2	no	—
245	M₃₂	C₂²×D₆₇	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	—	67	2	no	—
248	#108	—	67	2	no	—
249	#108	—	134	1	no	—
250	#107	—	2	1	no	—
251	#106	C₂²×D₆₇	1	2	yes	—
252	#104	—	134	4	no	—
253	#102	—	134	4	no	—
254	#97	—	67	2	no	—
255	C₄×C₈	—	67	2	no	—
256	C₂⁴	—	268	1	no	—
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	#41	—	4	1	no	—
260	#41	—	4	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	268	1	no	—
265	C₂×D₄	—	268	1	no	—
266	C₂×D₄	—	268	1	no	—
267	C₂×D₄	—	134	2	no	—
268	C₂×D₄	—	134	2	no	—
269	C₂×D₄	—	134	2	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	268	4	no	—
272	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
273	Q₁₆	—	2	1	no	—
274	Q₁₆	C₂×C₄×D₆₇	1	2	yes	—
275	SD₁₆	—	2	2	no	—
276	SD₁₆	—	2	2	no	—
277	D₈	—	2	1	no	—
278	D₈	—	67	4	no	—
279	D₈	—	134	2	no	—
280	D₈	—	134	2	no	—
281	D₈	—	134	2	no	—
282	D₈	C₂×C₄×D₆₇	1	2	yes	—
283	M₁₆	—	268	1	no	—
284	M₁₆	—	268	4	no	—
285	M₁₆	—	134	2	no	—
286	M₁₆	—	134	1	no	—
287	M₁₆	—	134	1	no	—
288	C₂×C₈	—	268	1	no	—
289	C₂×C₈	—	268	4	no	—
290	C₂×C₈	—	67	2	no	—
291	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
293	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
294	C₂×C₈	#11761627	1	1	yes	characteristic
295	C₂×C₈	#11761627	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	134	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	268	4	no	—
301	C₂²⋊C₄	—	134	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	67	2	no	—
304	C₁₆	—	2	2	no	—
305	C₁₆	—	67	4	no	—
306	C₂³	—	4	1	no	—
307	C₂³	—	268	2	no	—
308	C₂³	—	268	2	no	—
309	C₂³	—	268	2	no	—
310	C₂³	—	1072	2	no	—
311	C₂³	#14524067	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	268	2	no	—
319	D₄	—	268	2	no	—
320	D₄	—	268	2	no	—
321	D₄	—	134	4	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	1072	4	no	—
327	C₂×C₄	—	134	2	no	—
328	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
329	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
330	C₂×C₄	#14524067	1	1	yes	characteristic
331	C₂×C₄	#14524070	1	1	yes	characteristic
332	C₂×C₄	#14524070	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
334	C₈	—	268	2	no	—
335	C₈	—	134	2	no	—
336	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
337	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
339	C₈	C₂×#11761627	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	268	4	no	—
343	C₂²	—	16	1	no	—
344	C₂²	—	536	4	no	—
345	C₂²	—	1072	2	no	—
346	C₂²	—	1072	2	no	—
347	C₂²	#16863505	1	1	yes	characteristic
348	C₂²	#16863505	1	1	yes	characteristic
349	C₂²	C₂×#14524067	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	—	268	2	no	—
353	C₄	#16863505	1	1	yes	characteristic
354	C₄	#16863505	1	1	yes	characteristic
355	C₄	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
356	C₄	C₂×#14524070	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	2	no	—
361	C₂	—	536	4	no	—
362	C₂	C₂×#16863505	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16863505
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2145	C₂
socle	C₂×C₆₇	C₂×#442
prime radical	C₂×#16863831	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2145	D₆₇
hypocenter	C₆₇	#33471
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
2. #22569126	C₂	semidirect
3. #22569060	C₂	semidirect
4. #22562104	C₂	semidirect
5. #22561981	C₂	semidirect
6. C₂×#16863210	C₂	semidirect
7. #22561813	C₂	semidirect
8. C₆₇×#2145	C₂	semidirect
12. #16863831	C₄	semidirect
14. C₂×#14523986	C₄	semidirect
15. C₂×#14523984	C₄	semidirect
17. #16863518	C₂²	semidirect
19. #16863507	C₂²	semidirect
25. #16863218	C₂²	semidirect
26. #16863210	C₂²	semidirect
27. #16863173	C₂²	semidirect
31. #360×C₆₇	C₂²	semidirect
48. #14523986	C₂×C₄	semidirect
49. #14523984	C₂×C₄	semidirect
50. #14523983	C₂×C₄	semidirect
52. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
62. #135×C₆₇	C₂×C₄	semidirect
64. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
66. C₂×D₈×C₆₇	C₂×C₄	semidirect
188. #2145	D₆₇	semidirect
190. C₆₇	#33471	semidirect
212. #360	C₂×D₆₇	semidirect
218. #135	C₄×D₆₇	semidirect
220. C₂×Q₁₆	C₄×D₆₇	semidirect
228. C₂×D₈	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×#16863831	2
#22569126	2
#22569060	2
#22562104	2
#22561981	2
C₂×#16863210	2
#22561813	2
C₆₇×#2145	2
#33471	67