Subgroups — #31012719

Order 17152.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572603	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#22562437	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562413	C₂	1	2	yes	maximal, complemented
5	#22562361	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562299	C₂	1	2	yes	maximal, complemented
7	#22562259	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562141	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₆₈	C₂	1	1	yes	maximal, characteristic
10	#358×D₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₂²	1	2	yes	—
13	#16864106	—	2	1	no	—
14	Q₈×D₂₆₈	—	2	1	no	—
15	C₂×#14523989	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	#16863730	—	2	2	no	—
20	#16863691	—	2	1	no	—
21	#16863638	C₂²	1	2	yes	—
22	#16863634	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	#16863221	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×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	1	yes	characteristic
39	#358×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₆₇	—	2	2	no	—
42	C₂×Dic₂₆₈	—	2	1	no	—
43	C₂×Dic₂₆₈	C₂³	1	2	yes	—
44	C₂×#11761618	—	2	2	no	—
45	Q₁₆×D₆₇	—	2	4	no	—
46	Q₁₆×D₆₇	—	4	2	no	—
47	C₂×C₈×D₆₇	C₂³	1	2	yes	—
48	#14524043	—	2	1	no	—
49	#14524043	—	2	1	no	—
50	#14524043	—	4	1	no	—
51	#14524035	—	4	1	no	—
52	#14524023	—	2	2	no	—
53	#14524023	—	4	2	no	—
54	#14524021	—	4	2	no	—
55	#14524019	—	2	2	no	—
56	C₄×D₂₆₈	—	4	1	no	—
57	C₄×D₂₆₈	—	4	1	no	—
58	C₄×D₂₆₈	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₆₇	—	2	2	no	—
60	#14523989	—	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	#14523967	—	2	2	no	—
68	#14523966	—	2	2	no	—
69	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
70	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₇	—	2	1	no	—
73	C₄×Q₈×C₆₇	—	2	1	no	—
74	Q₃₂×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	#103×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₃₄	—	2	2	no	—
83	C₂×Dic₁₃₄	—	4	2	no	—
84	C₂×Dic₁₃₄	—	4	1	no	—
85	Q₈×D₆₇	—	4	4	no	—
86	Q₈×D₆₇	—	8	2	no	—
87	C₂×D₂₆₈	D₈	1	1	yes	characteristic
88	C₂×C₄×D₆₇	—	4	2	no	—
89	C₂×C₄×D₆₇	—	4	2	no	—
90	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
91	Dic₂₆₈	—	2	2	no	—
92	Dic₂₆₈	—	2	2	no	—
93	Dic₂₆₈	—	8	1	no	—
94	#11761618	—	4	4	no	—
95	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₈×D₆₇	—	2	2	no	—
97	C₈×D₆₇	—	2	2	no	—
98	C₄⋊Dic₆₇	—	2	2	no	—
99	C₄⋊Dic₆₇	—	4	1	no	—
100	C₄⋊Dic₆₇	—	4	1	no	—
101	C₄⋊Dic₆₇	—	4	1	no	—
102	C₄⋊Dic₆₇	—	4	2	no	—
103	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
104	Dic₆₇⋊C₄	—	4	2	no	—
105	Dic₆₇⋊C₄	—	4	2	no	—
106	(C₂×D₆₇)⋊C₄	—	4	2	no	—
107	(C₂×D₆₇)⋊C₄	—	4	2	no	—
108	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
109	C₆₇⋊C₁₆	—	2	2	no	—
110	C₂×Q₈×C₆₇	—	2	1	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	Q₁₆×C₆₇	—	2	1	no	—
113	Q₁₆×C₆₇	—	2	1	no	—
114	Q₁₆×C₆₇	—	2	1	no	—
115	Q₁₆×C₆₇	#41	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	4	1	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	—	4	1	no	—
123	C₄²×C₆₇	—	4	1	no	—
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₆₇	—	2	1	no	—
126	C₁₆×C₆₇	—	2	1	no	—
127	C₂²×D₆₇	C₂×D₈	1	2	yes	—
128	Q₈×C₆₇	—	2	2	no	—
129	Q₈×C₆₇	—	4	1	no	—
130	Q₈×C₆₇	—	4	1	no	—
131	Q₈×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₆₇	—	4	2	no	—
140	C₂×Dic₆₇	—	4	2	no	—
141	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
142	D₂₆₈	Q₃₂	1	4	yes	complemented
143	C₄×D₆₇	—	2	2	no	—
144	C₄×D₆₇	—	2	2	no	—
145	C₄×D₆₇	—	8	4	no	—
146	C₄×D₆₇	—	16	2	no	—
147	Dic₁₃₄	—	4	2	no	—
148	Dic₁₃₄	—	4	2	no	—
149	Dic₁₃₄	—	8	4	no	—
150	Dic₁₃₄	—	16	1	no	—
151	C₈×C₆₇	—	2	1	no	—
152	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
153	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
154	C₆₇⋊C₈	—	2	2	no	—
155	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₆₇	—	2	2	no	—
157	C₂×D₆₇	—	2	2	no	—
158	C₂×D₆₇	C₂×Q₃₂	1	4	yes	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	4	2	no	—
161	C₄×C₆₇	—	8	1	no	—
162	C₄×C₆₇	—	8	1	no	—
163	C₄×C₆₇	—	8	1	no	—
164	C₄×C₆₇	C₂×Q₃₂	1	2	yes	—
165	C₄×C₆₇	#546	1	1	yes	characteristic
166	C₄×C₆₇	#577	1	1	yes	characteristic
167	Dic₆₇	—	2	2	no	—
168	Dic₆₇	—	8	2	no	—
169	Dic₆₇	—	8	2	no	—
170	#33866	—	67	1	no	maximal, Sylow
171	C₂×C₆₇	#3294	1	1	yes	characteristic
172	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
173	C₂×C₆₇	#3416	1	1	yes	characteristic
174	D₆₇	—	2	4	no	—
175	#3338	—	67	1	no	—
176	D₄×Q₁₆	—	67	1	no	—
177	#2241	—	67	1	no	—
178	#2236	—	67	2	no	—
179	#2227	—	67	1	no	—
180	#2216	—	67	2	no	—
181	#2209	—	67	1	no	—
182	#2188	—	67	2	no	—
183	C₄×Q₃₂	D₆₇	1	1	yes	characteristic, complemented
184	D₄×C₁₆	—	67	1	no	—
185	C₂×#358	—	67	2	no	—
186	C₆₇	#33866	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₁₆	—	67	2	no	—
188	#554	—	134	1	no	—
189	D₄×Q₈	—	134	1	no	—
190	C₂×Q₃₂	—	67	2	no	—
191	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
192	C₂²×C₁₆	—	67	2	no	—
193	#500	—	67	1	no	—
194	#494	—	67	1	no	—
195	#484	—	134	2	no	—
196	#477	—	134	1	no	—
197	#467	—	67	2	no	—
198	#467	—	67	2	no	—
199	#462	—	134	1	no	—
200	#451	—	134	2	no	—
201	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
202	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
203	C₈×D₄	—	67	1	no	—
204	C₂×#107	—	67	2	no	—
205	#366	—	67	2	no	—
206	#366	C₂×D₆₇	1	1	yes	characteristic
207	C₄⋊C₁₆	—	67	1	no	—
208	#358	—	67	2	no	—
209	#358	—	67	2	no	—
210	#358	—	67	2	no	—
211	#358	—	67	2	no	—
212	#358	C₂×D₆₇	1	1	yes	characteristic
213	#358	C₂×D₆₇	1	1	yes	characteristic
214	C₂²⋊C₁₆	—	67	2	no	—
215	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
216	C₂²×Q₈	—	134	2	no	—
217	C₂×Q₁₆	—	268	2	no	—
218	C₂×Q₁₆	—	67	2	no	—
219	C₂×Q₁₆	—	134	2	no	—
220	C₂×Q₁₆	—	134	4	no	—
221	C₂×Q₁₆	—	134	1	no	—
222	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
223	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
224	C₂²×C₈	—	67	2	no	—
225	C₄⋊Q₈	—	268	1	no	—
226	C₄⋊Q₈	—	134	1	no	—
227	C₄⋊Q₈	—	134	1	no	—
228	#125	—	268	1	no	—
229	C₂²⋊Q₈	—	268	2	no	—
230	C₂²⋊Q₈	—	268	2	no	—
231	C₂²⋊Q₈	—	134	2	no	—
232	C₂²⋊Q₈	—	134	2	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×D₄	—	268	1	no	—
236	C₄×D₄	—	268	1	no	—
237	C₄×D₄	—	67	1	no	—
238	C₂×C₄⋊C₄	—	134	2	no	—
239	Q₃₂	—	2	2	no	—
240	Q₃₂	—	268	2	no	—
241	C₂×C₁₆	—	268	2	no	—
242	C₂×C₁₆	—	67	2	no	—
243	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
244	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
245	#107	—	67	2	no	—
246	#107	—	134	2	no	—
247	#107	—	134	1	no	—
248	#107	C₂²×D₆₇	1	1	yes	characteristic
249	#107	C₂²×D₆₇	1	1	yes	characteristic
250	C₄⋊C₈	—	67	1	no	—
251	#103	—	2	1	no	—
252	#103	—	2	1	no	—
253	#103	—	134	2	no	—
254	#103	—	134	2	no	—
255	C₂²⋊C₈	—	67	2	no	—
256	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	268	2	no	—
260	C₂×Q₈	—	268	1	no	—
261	C₂×Q₈	—	268	4	no	—
262	C₂×Q₈	—	536	2	no	—
263	C₂×Q₈	—	134	2	no	—
264	C₂×D₄	—	67	1	no	—
265	C₂²×C₄	—	268	2	no	—
266	C₂²×C₄	—	268	2	no	—
267	C₂²×C₄	—	67	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	268	4	no	—
272	Q₁₆	—	536	1	no	—
273	Q₁₆	—	134	2	no	—
274	Q₁₆	—	134	2	no	—
275	Q₁₆	#11761628	1	2	yes	—
276	C₂×C₈	—	67	2	no	—
277	C₂×C₈	—	134	2	no	—
278	C₂×C₈	—	134	2	no	—
279	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	268	2	no	—
286	C₄⋊C₄	—	268	2	no	—
287	C₄⋊C₄	—	268	2	no	—
288	C₄⋊C₄	—	268	1	no	—
289	C₄⋊C₄	—	268	1	no	—
290	C₄⋊C₄	—	268	1	no	—
291	C₄⋊C₄	—	67	1	no	—
292	C₄⋊C₄	—	134	2	no	—
293	C₂²⋊C₄	—	268	2	no	—
294	C₂²⋊C₄	—	268	2	no	—
295	C₂²⋊C₄	—	67	2	no	—
296	C₄²	—	4	1	no	—
297	C₄²	—	4	1	no	—
298	C₄²	D₄×D₆₇	1	1	yes	characteristic
299	C₁₆	—	2	1	no	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	134	2	no	—
302	C₂³	—	67	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	268	2	no	—
308	Q₈	—	268	2	no	—
309	Q₈	—	536	4	no	—
310	Q₈	—	1072	1	no	—
311	D₄	—	67	4	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	268	2	no	—
317	C₂×C₄	—	268	2	no	—
318	C₂×C₄	—	536	4	no	—
319	C₂×C₄	—	1072	2	no	—
320	C₂×C₄	—	67	2	no	—
321	C₂×C₄	—	134	2	no	—
322	C₂×C₄	—	134	2	no	—
323	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
324	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	134	2	no	—
328	C₈	C₂×#11761628	1	1	yes	characteristic
329	C₈	#14524089	1	1	yes	characteristic
330	C₂²	—	67	4	no	—
331	C₂²	—	134	2	no	—
332	C₂²	—	134	2	no	—
333	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	—	536	2	no	—
340	C₄	—	536	2	no	—
341	C₄	—	134	2	no	—
342	C₄	Q₃₂×D₆₇	1	2	yes	—
343	C₄	#16864042	1	1	yes	characteristic
344	C₄	#16864203	1	1	yes	characteristic
345	C₂	—	134	4	no	—
346	C₂	#22572024	1	1	yes	characteristic
347	C₂	C₂×Q₃₂×D₆₇	1	1	yes	characteristic
348	C₂	#22573352	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×Q₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×D₂₆₈	D₈
hypercenter	C₄×Q₃₂	D₆₇
hypocenter	C₆₇	#33866
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #22562437	C₂	semidirect
4. #22562413	C₂	semidirect
5. #22562361	C₂	semidirect
6. #22562299	C₂	semidirect
7. #22562259	C₂	semidirect
8. #22562141	C₂	semidirect
10. #358×D₆₇	C₂	semidirect
11. C₄×Q₃₂×C₆₇	C₂	semidirect
142. D₂₆₈	Q₃₂	semidirect
183. C₄×Q₃₂	D₆₇	semidirect
186. C₆₇	#33866	semidirect

Maximal subgroups

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

subgroup	index
#22572603	2
Q₁₆×D₂₆₈	2
#22562437	2
#22562413	2
#22562361	2
#22562299	2
#22562259	2
#22562141	2
C₁₆×D₂₆₈	2
#358×D₆₇	2
C₄×Q₃₂×C₆₇	2
#33866	67