Subgroups — #31012708

Order 17152.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572558	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562437	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562408	C₂	1	2	yes	maximal, complemented
5	#22562351	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562304	C₂	1	2	yes	maximal, complemented
7	#22562254	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562124	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#358×D₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₂²	1	2	yes	—
13	Q₈×D₂₆₈	—	2	1	no	—
14	#16864051	—	2	1	no	—
15	C₂×#14523987	C₂²	1	2	yes	—
16	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
17	#16863801	C₂²	1	1	yes	characteristic
18	#16863776	C₂²	1	1	yes	characteristic
19	#16863708	—	2	2	no	—
20	#16863676	—	2	1	no	—
21	#16863638	C₂²	1	2	yes	—
22	#16863629	C₂²	1	2	yes	—
23	#16863606	—	2	1	no	—
24	#16863553	—	2	2	no	—
25	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
26	#107×D₆₇	C₂²	1	2	yes	—
27	#16863223	C₂²	1	2	yes	—
28	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
29	#16863201	C₂²	1	2	yes	—
30	#16863201	C₂²	1	2	yes	—
31	#16863200	C₂²	1	2	yes	—
32	#16863199	C₂²	1	2	yes	—
33	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
37	#367×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	1	yes	characteristic
39	#357×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#11761628	—	4	1	no	—
42	C₂×Q₈×D₆₇	—	2	2	no	—
43	C₂×Dic₂₆₈	—	2	1	no	—
44	C₂×Dic₂₆₈	C₂³	1	2	yes	—
45	C₂×#11761618	—	2	2	no	—
46	Q₁₆×D₆₇	—	2	4	no	—
47	Q₁₆×D₆₇	—	4	2	no	—
48	C₂×C₈×D₆₇	C₂³	1	2	yes	—
49	#14524043	—	2	1	no	—
50	#14524043	—	2	1	no	—
51	#14524043	—	4	1	no	—
52	#14524025	—	4	2	no	—
53	#14524023	—	2	2	no	—
54	#14524023	—	4	2	no	—
55	#14524022	—	4	1	no	—
56	#14524014	—	2	2	no	—
57	C₄×D₂₆₈	—	4	1	no	—
58	C₄×D₂₆₈	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₆₇	—	2	2	no	—
60	#14523987	—	4	2	no	—
61	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
62	C₁₆×D₆₇	—	4	2	no	—
63	#14523977	—	2	1	no	—
64	#14523977	C₂³	1	2	yes	—
65	#14523976	—	2	2	no	—
66	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
67	#14523966	—	2	2	no	—
68	#14523964	—	2	2	no	—
69	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
70	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₇	—	2	1	no	—
73	C₄×D₄×C₆₇	—	2	1	no	—
74	SD₃₂×C₆₇	—	2	2	no	—
75	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
77	#107×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	C₂³	1	1	yes	characteristic
79	#103×C₆₇	—	2	1	no	—
80	#102×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	#11761628	—	8	4	no	—
83	C₂×Dic₁₃₄	—	2	2	no	—
84	C₂×Dic₁₃₄	—	4	1	no	—
85	C₂×Dic₁₃₄	—	4	2	no	—
86	Q₈×D₆₇	—	4	4	no	—
87	Q₈×D₆₇	—	8	2	no	—
88	C₂×D₂₆₈	D₈	1	1	yes	characteristic
89	C₂×C₂²⋊D₆₇	—	4	2	no	—
90	C₂×C₄×D₆₇	—	4	2	no	—
91	C₂×C₄×D₆₇	—	4	2	no	—
92	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
93	Dic₂₆₈	—	2	2	no	—
94	Dic₂₆₈	—	2	2	no	—
95	Dic₂₆₈	—	8	1	no	—
96	#11761618	—	4	4	no	—
97	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
98	C₈×D₆₇	—	2	2	no	—
99	C₈×D₆₇	—	2	2	no	—
100	C₄⋊Dic₆₇	—	2	2	no	—
101	C₄⋊Dic₆₇	—	4	1	no	—
102	C₄⋊Dic₆₇	—	4	1	no	—
103	C₄⋊Dic₆₇	—	4	1	no	—
104	C₄⋊Dic₆₇	—	4	2	no	—
105	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
106	Dic₆₇⋊C₄	—	4	2	no	—
107	C₂²⋊Dic₆₇	—	4	2	no	—
108	(C₂×D₆₇)⋊C₄	—	4	2	no	—
109	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
110	C₆₇⋊C₁₆	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	4	1	no	—
114	Q₁₆×C₆₇	—	2	1	no	—
115	Q₁₆×C₆₇	#41	1	2	yes	complemented
116	D₈×C₆₇	—	2	1	no	—
117	D₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	4	1	no	—
123	C₂²⋊C₄×C₆₇	—	4	1	no	—
124	C₄²×C₆₇	—	4	1	no	—
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₆₇	—	2	1	no	—
127	C₁₆×C₆₇	—	2	1	no	—
128	C₂³×C₆₇	—	4	1	no	—
129	C₂²×D₆₇	C₂×D₈	1	2	yes	—
130	Q₈×C₆₇	—	2	2	no	—
131	Q₈×C₆₇	—	4	1	no	—
132	D₄×C₆₇	—	4	1	no	—
133	D₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	8	2	no	—
138	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₆₇	—	8	4	no	—
142	C₂×Dic₆₇	—	4	2	no	—
143	C₂×Dic₆₇	—	4	2	no	—
144	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
145	D₂₆₈	SD₃₂	1	4	yes	complemented
146	C₄×D₆₇	—	2	2	no	—
147	C₄×D₆₇	—	2	2	no	—
148	C₄×D₆₇	—	8	4	no	—
149	C₄×D₆₇	—	8	4	no	—
150	Dic₁₃₄	—	4	2	no	—
151	Dic₁₃₄	—	4	2	no	—
152	Dic₁₃₄	—	8	4	no	—
153	Dic₁₃₄	—	8	2	no	—
154	C₈×C₆₇	—	2	1	no	—
155	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
157	C₆₇⋊C₈	—	2	2	no	—
158	C₂²×C₆₇	—	8	1	no	—
159	C₂²×C₆₇	—	8	1	no	—
160	C₂²×C₆₇	—	8	1	no	—
161	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₆₇	—	2	2	no	—
163	C₂×D₆₇	—	2	2	no	—
164	C₂×D₆₇	C₂×SD₃₂	1	4	yes	—
165	C₄×C₆₇	—	2	1	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	8	1	no	—
168	C₄×C₆₇	—	8	1	no	—
169	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
170	C₄×C₆₇	#546	1	1	yes	characteristic
171	C₄×C₆₇	#577	1	1	yes	characteristic
172	Dic₆₇	—	2	2	no	—
173	Dic₆₇	—	8	2	no	—
174	Dic₆₇	—	8	2	no	—
175	#33865	—	67	1	no	maximal, Sylow
176	C₂×C₆₇	—	16	1	no	—
177	C₂×C₆₇	#3294	1	1	yes	characteristic
178	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
179	C₂×C₆₇	#3417	1	1	yes	characteristic
180	D₆₇	—	2	4	no	—
181	#3334	—	67	1	no	—
182	D₄×Q₁₆	—	67	1	no	—
183	#2241	—	67	1	no	—
184	#2235	—	67	2	no	—
185	#2225	—	67	1	no	—
186	#2217	—	67	2	no	—
187	#2208	—	67	1	no	—
188	#2185	—	67	2	no	—
189	C₄×SD₃₂	D₆₇	1	1	yes	characteristic, complemented
190	D₄×C₁₆	—	67	1	no	—
191	C₂×#358	—	67	2	no	—
192	C₆₇	#33865	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₁₆	—	67	2	no	—
194	D₄×Q₈	—	134	1	no	—
195	#547	—	134	1	no	—
196	C₂×SD₃₂	—	67	2	no	—
197	C₂×SD₃₂	C₂×D₆₇	1	1	yes	characteristic
198	C₂²×C₁₆	—	67	2	no	—
199	#500	—	67	1	no	—
200	#494	—	67	1	no	—
201	#480	—	134	2	no	—
202	#474	—	134	1	no	—
203	#467	—	67	2	no	—
204	#466	—	67	2	no	—
205	#462	—	134	1	no	—
206	#451	—	134	2	no	—
207	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
208	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
209	C₈×D₄	—	67	1	no	—
210	C₂×#107	—	67	2	no	—
211	#367	—	67	2	no	—
212	#367	C₂×D₆₇	1	1	yes	characteristic
213	C₄⋊C₁₆	—	67	1	no	—
214	#358	—	67	2	no	—
215	#358	—	67	2	no	—
216	#358	—	67	2	no	—
217	#358	—	67	2	no	—
218	#358	C₂×D₆₇	1	1	yes	characteristic
219	#357	C₂×D₆₇	1	1	yes	characteristic
220	C₂²⋊C₁₆	—	67	2	no	—
221	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
222	C₂×#41	—	268	1	no	—
223	C₂²×Q₈	—	134	2	no	—
224	C₂×Q₁₆	—	268	2	no	—
225	C₂×Q₁₆	—	67	2	no	—
226	C₂×Q₁₆	—	134	2	no	—
227	C₂×Q₁₆	—	134	4	no	—
228	C₂×Q₁₆	—	134	1	no	—
229	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
230	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
231	C₂²×C₈	—	67	2	no	—
232	C₄⋊Q₈	—	268	1	no	—
233	C₄⋊Q₈	—	134	1	no	—
234	C₄⋊Q₈	—	134	1	no	—
235	#123	—	268	2	no	—
236	C₂²⋊Q₈	—	268	1	no	—
237	C₂²⋊Q₈	—	268	2	no	—
238	C₂²⋊Q₈	—	134	2	no	—
239	#121	—	134	2	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	2	1	no	—
242	C₄×D₄	—	268	1	no	—
243	C₄×D₄	—	67	1	no	—
244	C₂×C₄⋊C₄	—	134	2	no	—
245	SD₃₂	—	2	2	no	—
246	SD₃₂	—	268	2	no	—
247	C₂×C₁₆	—	268	2	no	—
248	C₂×C₁₆	—	67	2	no	—
249	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
251	#107	—	67	2	no	—
252	#107	—	134	2	no	—
253	#107	—	134	1	no	—
254	#107	C₂²×D₆₇	1	1	yes	characteristic
255	#107	C₂²×D₆₇	1	1	yes	characteristic
256	C₄⋊C₈	—	67	1	no	—
257	#103	—	2	1	no	—
258	#103	—	134	2	no	—
259	#102	—	2	1	no	—
260	#102	—	134	2	no	—
261	C₂²⋊C₈	—	67	2	no	—
262	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
263	#41	—	536	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	268	2	no	—
266	C₂×Q₈	—	268	1	no	—
267	C₂×Q₈	—	268	4	no	—
268	C₂×Q₈	—	536	2	no	—
269	C₂×Q₈	—	134	2	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	268	2	no	—
272	C₂×D₄	—	67	1	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	268	2	no	—
275	C₂²×C₄	—	268	2	no	—
276	C₂²×C₄	—	67	2	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	—	268	4	no	—
279	Q₁₆	—	536	1	no	—
280	Q₁₆	—	134	2	no	—
281	Q₁₆	—	134	2	no	—
282	Q₁₆	#11761628	1	2	yes	complemented
283	D₈	—	2	1	no	—
284	D₈	—	2	1	no	—
285	C₂×C₈	—	67	2	no	—
286	C₂×C₈	—	134	2	no	—
287	C₂×C₈	—	134	2	no	—
288	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	268	2	no	—
294	C₄⋊C₄	—	268	2	no	—
295	C₄⋊C₄	—	268	1	no	—
296	C₄⋊C₄	—	268	1	no	—
297	C₄⋊C₄	—	268	1	no	—
298	C₄⋊C₄	—	67	1	no	—
299	C₄⋊C₄	—	134	2	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	268	2	no	—
302	C₂²⋊C₄	—	268	2	no	—
303	C₂²⋊C₄	—	67	2	no	—
304	C₄²	—	4	1	no	—
305	C₄²	D₄×D₆₇	1	1	yes	characteristic
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	134	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	67	2	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	268	2	no	—
314	Q₈	—	268	2	no	—
315	Q₈	—	536	4	no	—
316	Q₈	—	536	2	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	536	4	no	—
320	D₄	—	67	4	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	268	2	no	—
327	C₂×C₄	—	536	4	no	—
328	C₂×C₄	—	536	4	no	—
329	C₂×C₄	—	67	2	no	—
330	C₂×C₄	—	134	2	no	—
331	C₂×C₄	—	134	2	no	—
332	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
333	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	134	2	no	—
337	C₈	C₂×#11761628	1	1	yes	characteristic
338	C₈	#14524089	1	1	yes	characteristic
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	67	4	no	—
343	C₂²	—	134	2	no	—
344	C₂²	—	134	2	no	—
345	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	8	1	no	—
349	C₄	—	8	1	no	—
350	C₄	—	536	2	no	—
351	C₄	—	536	2	no	—
352	C₄	—	134	2	no	—
353	C₄	SD₃₂×D₆₇	1	2	yes	—
354	C₄	#16864042	1	1	yes	characteristic
355	C₄	#16864203	1	1	yes	characteristic
356	C₂	—	16	1	no	—
357	C₂	—	134	4	no	—
358	C₂	#22572024	1	1	yes	characteristic
359	C₂	C₂×SD₃₂×D₆₇	1	1	yes	characteristic
360	C₂	#22573359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×SD₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22572558	C₂
hypercenter	C₄×SD₃₂	D₆₇
hypocenter	C₆₇	#33865
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₂₆₈	C₂	semidirect
3. #22562437	C₂	semidirect
4. #22562408	C₂	semidirect
5. #22562351	C₂	semidirect
6. #22562304	C₂	semidirect
7. #22562254	C₂	semidirect
8. #22562124	C₂	semidirect
9. C₁₆×D₂₆₈	C₂	semidirect
10. #358×D₆₇	C₂	semidirect
11. C₄×SD₃₂×C₆₇	C₂	semidirect
115. Q₁₆×C₆₇	#41	semidirect
145. D₂₆₈	SD₃₂	semidirect
189. C₄×SD₃₂	D₆₇	semidirect
192. C₆₇	#33865	semidirect
282. Q₁₆	#11761628	semidirect

Maximal subgroups

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

subgroup	index
#22572558	2
Q₁₆×D₂₆₈	2
#22562437	2
#22562408	2
#22562351	2
#22562304	2
#22562254	2
#22562124	2
C₁₆×D₂₆₈	2
#358×D₆₇	2
C₄×SD₃₂×C₆₇	2
#33865	67