Subgroups — #30839126

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571159	C₂	1	1	yes	maximal, characteristic, complemented
2	#22571152	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863743	C₂	1	1	yes	maximal, characteristic
4	#22569143	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559778	C₂	1	4	yes	maximal, complemented
6	#22559654	C₂	1	4	yes	maximal, complemented
7	#22558893	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558876	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	#16863953	C₂²	1	1	yes	characteristic
11	C₂×#14524043	C₂²	1	1	yes	characteristic
12	C₂×#14524040	C₂²	1	1	yes	characteristic
13	#16863776	—	2	1	no	—
14	#16863769	—	2	1	no	—
15	#16863747	—	2	1	no	—
16	#16863743	—	2	1	no	—
17	#16863743	—	2	1	no	—
18	#16863743	—	2	1	no	—
19	#16863743	C₂²	1	2	yes	—
20	#16863743	C₂²	1	2	yes	—
21	#16863607	C₂²	1	4	yes	complemented
22	#16863597	C₂²	1	4	yes	—
23	#16863554	—	2	2	no	—
24	#16863549	—	2	2	no	—
25	#16863526	—	2	1	no	—
26	#16863526	C₂²	1	2	yes	complemented
27	C₈×Dic₁₃₄	—	2	1	no	—
28	C₈×Dic₁₃₄	C₂²	1	2	yes	complemented
29	#16863422	—	2	1	no	—
30	#16863420	C₂²	1	1	yes	characteristic
31	C₂×#14523965	—	2	1	no	—
32	C₂×#14523965	—	2	1	no	—
33	#16863351	—	2	1	no	—
34	#16863143	C₂²	1	4	yes	complemented
35	#16863139	C₂²	1	4	yes	—
36	#431×C₆₇	C₂²	1	1	yes	characteristic
37	#431×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#335×C₆₇	C₂²	1	2	yes	complemented
40	#334×C₆₇	C₂²	1	2	yes	complemented
41	C₂²×Dic₁₃₄	—	2	1	no	—
42	C₂²×D₂₆₈	—	2	1	no	—
43	C₂²×D₂₆₈	—	2	1	no	—
44	C₂²×D₂₆₈	—	2	1	no	—
45	C₂×Dic₂₆₈	—	2	4	no	—
46	C₂×#11761617	—	2	4	no	—
47	C₂×#11761612	—	2	1	no	—
48	#14524043	—	2	1	no	—
49	#14524043	—	2	1	no	—
50	#14524043	—	2	1	no	—
51	#14524043	D₄	1	1	yes	characteristic
52	#14524043	D₄	1	1	yes	characteristic
53	#14524043	C₂³	1	2	yes	—
54	#14524040	—	2	1	no	—
55	#14524040	—	2	1	no	—
56	#14524040	—	2	1	no	—
57	#14524040	C₂³	1	2	yes	—
58	#14524035	—	2	1	no	—
59	#14524022	—	2	1	no	—
60	#14524022	D₄	1	2	yes	—
61	C₄×Dic₁₃₄	—	2	1	no	—
62	C₄×Dic₁₃₄	C₂³	1	2	yes	—
63	#14524002	—	2	1	no	—
64	C₂×C₄⋊Dic₆₇	—	2	1	no	—
65	C₂×C₄⋊Dic₆₇	—	2	1	no	—
66	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
67	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
68	Dic₆₇⋊C₈	—	2	2	no	—
69	Dic₆₇⋊C₈	—	2	2	no	—
70	#14523968	—	2	4	no	—
71	#14523965	—	2	4	no	—
72	#14523965	—	2	2	no	—
73	#14523965	—	2	2	no	—
74	#14523965	—	2	2	no	—
75	#14523965	—	2	2	no	—
76	#14523953	—	2	2	no	—
77	C₈×Dic₆₇	—	2	2	no	—
78	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
82	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
83	C₂²⋊C₈×C₆₇	—	2	1	no	—
84	C₂²⋊C₈×C₆₇	—	2	1	no	—
85	C₄×C₈×C₆₇	—	2	1	no	—
86	C₄×C₈×C₆₇	—	2	1	no	—
87	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
88	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
89	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₂³×D₆₇	—	4	1	no	—
92	C₂×Dic₁₃₄	—	2	1	no	—
93	C₂×Dic₁₃₄	—	2	2	no	—
94	C₂×Dic₁₃₄	—	4	1	no	—
95	C₂×Dic₁₃₄	—	8	2	no	—
96	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
97	C₂×D₂₆₈	—	2	2	no	—
98	C₂×D₂₆₈	—	2	2	no	—
99	C₂×D₂₆₈	—	2	2	no	—
100	C₂×D₂₆₈	—	4	2	no	—
101	C₂×D₂₆₈	—	4	2	no	—
102	C₂×D₂₆₈	—	4	1	no	—
103	C₂×D₂₆₈	—	4	1	no	—
104	C₂×D₂₆₈	—	4	1	no	—
105	C₂×D₂₆₈	—	8	2	no	—
106	C₂×D₂₆₈	—	8	2	no	—
107	C₂²×Dic₆₇	—	4	1	no	—
108	Dic₂₆₈	—	8	4	no	—
109	#11761617	—	8	4	no	—
110	#11761612	—	4	2	no	—
111	C₂×C₆₇⋊C₈	—	2	2	no	—
112	C₄⋊Dic₆₇	—	2	1	no	—
113	C₄⋊Dic₆₇	—	2	1	no	—
114	C₄⋊Dic₆₇	—	2	1	no	—
115	C₄⋊Dic₆₇	—	2	2	no	—
116	C₄⋊Dic₆₇	—	2	2	no	—
117	C₄⋊Dic₆₇	—	4	1	no	—
118	C₄⋊Dic₆₇	—	4	1	no	—
119	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
120	Dic₆₇⋊C₄	—	2	2	no	—
121	Dic₆₇⋊C₄	—	2	2	no	—
122	C₂²⋊Dic₆₇	—	2	2	no	—
123	C₄×Dic₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₆₇	#41	1	1	yes	characteristic
138	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
139	C₂³×C₆₇	#121	1	1	yes	characteristic
140	C₂²×D₆₇	—	4	2	no	—
141	C₂²×D₆₇	—	8	2	no	—
142	C₂²×D₆₇	—	8	2	no	—
143	C₂²×D₆₇	—	8	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#135	1	2	yes	—
150	C₂×C₄×C₆₇	#137	1	2	yes	—
151	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₆₇	—	2	2	no	—
155	C₂×Dic₆₇	—	2	2	no	—
156	C₂×Dic₆₇	—	4	2	no	—
157	C₂×Dic₆₇	—	8	2	no	—
158	D₂₆₈	—	4	2	no	—
159	D₂₆₈	—	4	2	no	—
160	D₂₆₈	—	4	4	no	—
161	D₂₆₈	—	8	2	no	—
162	D₂₆₈	—	8	2	no	—
163	D₂₆₈	—	8	2	no	—
164	D₂₆₈	—	8	2	no	—
165	Dic₁₃₄	—	2	4	no	—
166	Dic₁₃₄	—	4	2	no	—
167	Dic₁₃₄	—	4	4	no	—
168	Dic₁₃₄	—	8	2	no	—
169	C₈×C₆₇	—	2	1	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	2	1	no	—
173	C₈×C₆₇	—	4	2	no	—
174	C₈×C₆₇	—	4	2	no	—
175	C₆₇⋊C₈	—	4	2	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	#464	1	2	yes	—
179	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
180	C₂×D₆₇	—	8	2	no	—
181	C₂×D₆₇	—	8	2	no	—
182	C₂×D₆₇	—	8	2	no	—
183	C₂×D₆₇	—	16	4	no	—
184	C₄×C₆₇	—	2	1	no	—
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	#464	1	2	yes	—
187	C₄×C₆₇	#471	1	2	yes	—
188	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
189	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
190	C₄×C₆₇	#576	1	2	yes	—
191	Dic₆₇	—	4	2	no	—
192	Dic₆₇	—	4	2	no	—
193	Dic₆₇	—	8	2	no	—
194	#19214	—	67	1	no	maximal, Sylow
195	C₂×C₆₇	—	2	2	no	—
196	C₂×C₆₇	C₂×#464	1	1	yes	characteristic
197	C₂×C₆₇	#3058	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#471	1	1	yes	characteristic
199	D₆₇	—	16	2	no	—
200	#3209	—	67	1	no	—
201	#3208	—	67	1	no	—
202	C₂×#486	—	67	1	no	—
203	#2960	—	67	1	no	—
204	#1702	—	67	4	no	—
205	#1682	—	67	4	no	—
206	#1564	D₆₇	1	1	yes	characteristic, complemented
207	#1549	—	67	1	no	—
208	#1545	—	67	1	no	—
209	C₆₇	#19214	1	1	yes	characteristic, complemented, Sylow
210	#533	—	67	1	no	—
211	C₂×C₄⋊Q₈	—	67	1	no	—
212	C₂×#127	—	67	1	no	—
213	#494	—	134	1	no	—
214	#492	—	134	1	no	—
215	#487	—	134	1	no	—
216	#486	—	67	2	no	—
217	#486	—	67	2	no	—
218	#486	—	134	1	no	—
219	#486	—	134	1	no	—
220	#486	—	134	1	no	—
221	#462	—	67	4	no	—
222	#460	—	67	4	no	—
223	#451	—	134	2	no	—
224	#450	—	134	2	no	—
225	#446	—	67	2	no	—
226	#446	—	134	1	no	—
227	C₈×Q₈	—	67	2	no	—
228	C₈×Q₈	—	134	1	no	—
229	#431	C₂×D₆₇	1	1	yes	characteristic
230	#431	C₂×D₆₇	1	1	yes	characteristic
231	#424	—	134	1	no	—
232	#423	—	67	1	no	—
233	C₂×#102	—	134	1	no	—
234	C₂×#102	—	134	1	no	—
235	#405	—	134	1	no	—
236	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
237	#335	C₂×D₆₇	1	2	yes	complemented
238	#334	C₂×D₆₇	1	2	yes	complemented
239	#332	—	67	4	no	—
240	#331	—	67	4	no	—
241	C₂²×Q₈	—	134	1	no	—
242	C₂²×D₄	—	134	1	no	—
243	C₂²×D₄	—	134	1	no	—
244	C₂²×D₄	—	134	1	no	—
245	C₂×Q₁₆	—	134	4	no	—
246	C₂×SD₁₆	—	134	4	no	—
247	C₂×M₁₆	—	134	1	no	—
248	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
249	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
250	C₄⋊Q₈	—	67	1	no	—
251	C₄⋊Q₈	—	67	1	no	—
252	C₄⋊Q₈	—	67	2	no	—
253	C₄⋊Q₈	—	134	1	no	—
254	C₄⋊Q₈	—	134	1	no	—
255	C₄⋊Q₈	—	134	1	no	—
256	#127	—	67	2	no	—
257	#127	—	134	1	no	—
258	#127	—	134	1	no	—
259	#127	—	134	1	no	—
260	#125	—	134	1	no	—
261	C₂²⋊Q₈	—	67	2	no	—
262	C₂²⋊Q₈	—	134	1	no	—
263	C₄×Q₈	—	67	2	no	—
264	C₄×Q₈	—	134	1	no	—
265	#117	—	134	1	no	—
266	C₂×C₄⋊C₄	—	134	1	no	—
267	C₂×C₄⋊C₄	—	134	1	no	—
268	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
269	C₄⋊C₈	—	67	2	no	—
270	C₄⋊C₈	—	134	2	no	—
271	C₄⋊C₈	—	134	2	no	—
272	C₄⋊C₈	—	134	1	no	—
273	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
274	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
275	#103	—	134	4	no	—
276	#102	—	134	4	no	—
277	#102	—	134	2	no	—
278	#102	—	134	2	no	—
279	#102	—	134	2	no	—
280	#102	—	134	2	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	#97	—	134	2	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	—	134	2	no	—
287	C₄×C₈	D₂₆₈	1	1	yes	characteristic
288	C₄×C₈	D₂₆₈	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₂⁴	—	268	1	no	—
292	C₂×Q₈	—	268	1	no	—
293	C₂×Q₈	—	536	2	no	—
294	C₂×Q₈	—	67	2	no	—
295	C₂×Q₈	—	134	2	no	—
296	C₂×Q₈	—	134	1	no	—
297	C₂×D₄	—	268	1	no	—
298	C₂×D₄	—	268	1	no	—
299	C₂×D₄	—	268	1	no	—
300	C₂×D₄	—	268	2	no	—
301	C₂×D₄	—	268	2	no	—
302	C₂×D₄	—	536	2	no	—
303	C₂×D₄	—	536	2	no	—
304	C₂×D₄	—	134	2	no	—
305	C₂×D₄	—	134	2	no	—
306	C₂×D₄	—	134	2	no	—
307	C₂²×C₄	—	268	1	no	—
308	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
310	C₂²×C₄	#11761629	1	1	yes	characteristic
311	Q₁₆	—	536	4	no	—
312	SD₁₆	—	536	4	no	—
313	M₁₆	—	268	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	134	2	no	—
319	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
320	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
321	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
322	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
323	C₄⋊C₄	—	268	1	no	—
324	C₄⋊C₄	—	268	1	no	—
325	C₄⋊C₄	—	67	2	no	—
326	C₄⋊C₄	—	134	2	no	—
327	C₄⋊C₄	—	134	2	no	—
328	C₄⋊C₄	—	134	1	no	—
329	C₄⋊C₄	—	134	1	no	—
330	C₄⋊C₄	—	134	2	no	—
331	C₄⋊C₄	—	134	1	no	—
332	C₄⋊C₄	—	134	2	no	—
333	C₂²⋊C₄	—	134	2	no	—
334	C₄²	—	134	2	no	—
335	C₄²	D₄×D₆₇	1	1	yes	characteristic
336	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
337	C₄²	#11761629	1	1	yes	characteristic
338	C₄²	C₂³×D₆₇	1	1	yes	characteristic
339	C₂³	—	268	2	no	—
340	C₂³	—	536	2	no	—
341	C₂³	—	536	2	no	—
342	C₂³	—	536	2	no	—
343	C₂³	#14524018	1	1	yes	characteristic
344	Q₈	—	268	2	no	—
345	Q₈	—	268	4	no	—
346	Q₈	—	536	2	no	—
347	Q₈	—	134	4	no	—
348	D₄	—	268	2	no	—
349	D₄	—	268	2	no	—
350	D₄	—	268	4	no	—
351	D₄	—	536	2	no	—
352	D₄	—	536	2	no	—
353	D₄	—	536	2	no	—
354	D₄	—	536	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	268	2	no	—
358	C₂×C₄	—	536	2	no	—
359	C₂×C₄	—	134	2	no	—
360	C₂×C₄	—	134	2	no	—
361	C₂×C₄	#14524018	1	1	yes	characteristic
362	C₂×C₄	#14524018	1	1	yes	characteristic
363	C₂×C₄	#14524018	1	1	yes	characteristic
364	C₂×C₄	#14524065	1	2	yes	—
365	C₂×C₄	#14524076	1	2	yes	—
366	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
368	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	4	2	no	—
374	C₈	—	4	2	no	—
375	C₈	—	268	2	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	536	2	no	—
379	C₂²	—	536	2	no	—
380	C₂²	—	536	2	no	—
381	C₂²	—	1072	4	no	—
382	C₂²	#16863619	1	2	yes	—
383	C₂²	C₂×#14524018	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	268	2	no	—
387	C₄	—	268	2	no	—
388	C₄	—	536	2	no	—
389	C₄	#16863619	1	2	yes	—
390	C₄	#16863660	1	2	yes	—
391	C₄	C₂×#14524065	1	1	yes	characteristic
392	C₄	C₂×#14524076	1	1	yes	characteristic
393	C₄	#16864198	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	1072	2	no	—
396	C₂	C₂×#16863619	1	1	yes	characteristic
397	C₂	#22569858	1	1	yes	characteristic
398	C₂	C₂×#16863660	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524018
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂×#14524040	C₂²
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#19214
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #22571159	C₂	semidirect
4. #22569143	C₂	semidirect
5. #22559778	C₂	semidirect
6. #22559654	C₂	semidirect
7. #22558893	C₂	semidirect
9. C₆₇×#1564	C₂	semidirect
21. #16863607	C₂²	semidirect
26. #16863526	C₂²	semidirect
28. C₈×Dic₁₃₄	C₂²	semidirect
34. #16863143	C₂²	semidirect
39. #335×C₆₇	C₂²	semidirect
40. #334×C₆₇	C₂²	semidirect
206. #1564	D₆₇	semidirect
209. C₆₇	#19214	semidirect
237. #335	C₂×D₆₇	semidirect
238. #334	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22571159	2
#22571152	2
C₂×#16863743	2
#22569143	2
#22559778	2
#22559654	2
#22558893	2
#22558876	2
C₆₇×#1564	2
#19214	67