Subgroups — #30748704

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524051	C₂	1	1	yes	maximal, characteristic
2	#22558955	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863146	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558597	C₂	1	1	yes	maximal, characteristic
5	#22558594	C₂	1	1	yes	maximal, characteristic
6	#22558060	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524051	C₄	1	1	yes	characteristic
9	C₂×#14524051	C₄	1	1	yes	characteristic
10	C₄×#11761628	C₂²	1	1	yes	characteristic
11	#16863449	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#16863354	C₄	1	1	yes	characteristic
14	#16863354	C₄	1	1	yes	characteristic
15	#16863353	—	2	1	no	—
16	#16863353	—	2	1	no	—
17	C₄×#11761611	—	2	1	no	—
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₆₇	—	2	1	no	—
20	#16863148	—	2	1	no	—
21	#16863146	—	2	1	no	—
22	#16863146	—	2	1	no	—
23	#16863146	—	2	1	no	—
24	#16863128	—	2	1	no	—
25	#16863126	—	2	1	no	—
26	#16863125	—	2	1	no	—
27	#16863123	—	2	1	no	—
28	#431×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
32	#14524051	—	2	2	no	—
33	#14524051	—	2	2	no	—
34	#14524051	—	2	2	no	—
35	#14524051	C₈	1	4	yes	complemented
36	C₂×#11761611	—	2	1	no	—
37	C₂×#11761611	—	2	1	no	—
38	C₂²×C₆₇⋊C₈	—	2	1	no	—
39	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₆₇	—	2	1	no	—
42	C₂×C₈×D₆₇	—	2	1	no	—
43	C₄×Dic₁₃₄	D₄	1	1	yes	characteristic
44	C₄×Dic₁₃₄	Q₈	1	1	yes	characteristic
45	C₄×D₂₆₈	D₄	1	1	yes	characteristic
46	C₄×D₂₆₈	Q₈	1	1	yes	characteristic
47	C₄×C₂²⋊D₆₇	—	2	1	no	—
48	C₄×C₂²⋊D₆₇	—	2	1	no	—
49	#14524002	C₂×C₄	1	1	yes	characteristic
50	#14524000	—	2	1	no	—
51	C₄²×D₆₇	—	2	1	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
53	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
55	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
56	#14523953	—	2	1	no	—
57	#14523953	—	2	1	no	—
58	C₄×C₆₇⋊C₈	—	2	1	no	—
59	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
60	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
61	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
63	C₈×Dic₆₇	—	2	1	no	—
64	C₈×Dic₆₇	—	2	1	no	—
65	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	—	2	1	no	—
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₆₇	—	2	1	no	—
70	C₄⋊C₈×C₆₇	—	2	1	no	—
71	C₄⋊C₈×C₆₇	—	2	1	no	—
72	C₂²⋊C₈×C₆₇	—	2	1	no	—
73	#97×C₆₇	—	2	1	no	—
74	C₄×C₈×C₆₇	—	2	1	no	—
75	C₄×C₈×C₆₇	—	2	1	no	—
76	#11761628	—	2	1	no	—
77	#11761628	—	2	1	no	—
78	#11761628	C₂×C₈	1	2	yes	—
79	#11761628	M₁₆	1	2	yes	—
80	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₂²⋊D₆₇	—	2	1	no	—
83	C₂×C₄×D₆₇	—	2	1	no	—
84	C₂×C₄×D₆₇	—	2	1	no	—
85	C₂×C₄×D₆₇	—	2	1	no	—
86	#11761611	—	2	4	no	—
87	#11761611	—	2	4	no	—
88	C₂×C₆₇⋊C₈	—	2	1	no	—
89	C₂×C₆₇⋊C₈	—	2	1	no	—
90	C₂×C₆₇⋊C₈	—	4	1	no	—
91	C₂×C₆₇⋊C₈	—	4	1	no	—
92	C₂×C₆₇⋊C₈	—	8	1	no	—
93	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
98	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
99	C₈×D₆₇	—	2	4	no	—
100	C₈×D₆₇	—	2	4	no	—
101	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
102	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
103	Dic₆₇⋊C₄	—	2	1	no	—
104	Dic₆₇⋊C₄	—	2	1	no	—
105	C₂²⋊Dic₆₇	—	2	1	no	—
106	(C₂×D₆₇)⋊C₄	—	2	1	no	—
107	(C₂×D₆₇)⋊C₄	—	2	1	no	—
108	C₄×Dic₆₇	—	2	1	no	—
109	C₄×Dic₆₇	—	2	1	no	—
110	C₄×Dic₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
114	M₁₆×C₆₇	—	2	1	no	—
115	M₁₆×C₆₇	—	2	1	no	—
116	M₁₆×C₆₇	C₂×C₈	1	2	yes	—
117	M₁₆×C₆₇	M₁₆	1	2	yes	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	—	4	1	no	—
121	C₂×C₈×C₆₇	—	4	1	no	—
122	C₂×C₈×C₆₇	—	8	1	no	—
123	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
127	C₂³×C₆₇	#117	1	1	yes	characteristic
128	C₂²×D₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
142	C₂²⋊D₆₇	—	2	2	no	—
143	C₂²⋊D₆₇	—	2	2	no	—
144	C₂×Dic₆₇	—	2	1	no	—
145	C₂×Dic₆₇	—	2	1	no	—
146	C₂×Dic₆₇	—	2	1	no	—
147	D₂₆₈	—	2	1	no	—
148	D₂₆₈	C₄⋊C₈	1	2	yes	—
149	C₄×D₆₇	—	2	2	no	—
150	C₄×D₆₇	—	2	2	no	—
151	C₄×D₆₇	—	4	2	no	—
152	C₄×D₆₇	—	4	1	no	—
153	C₄×D₆₇	—	4	1	no	—
154	Dic₁₃₄	—	2	1	no	—
155	Dic₁₃₄	C₄⋊C₈	1	2	yes	—
156	C₈×C₆₇	—	2	2	no	—
157	C₈×C₆₇	—	2	2	no	—
158	C₈×C₆₇	—	8	1	no	—
159	C₈×C₆₇	—	8	1	no	—
160	C₆₇⋊C₈	—	2	1	no	—
161	C₆₇⋊C₈	—	2	1	no	—
162	C₆₇⋊C₈	—	8	1	no	—
163	C₆₇⋊C₈	—	8	1	no	—
164	C₆₇⋊C₈	C₂²⋊C₈	1	2	yes	—
165	C₆₇⋊C₈	C₂²⋊C₈	1	2	yes	—
166	C₂²×C₆₇	—	2	1	no	—
167	C₂²×C₆₇	—	2	1	no	—
168	C₂²×C₆₇	#380	1	1	yes	characteristic
169	C₂²×C₆₇	#431	1	1	yes	characteristic
170	C₂²×C₆₇	#432	1	1	yes	characteristic
171	C₂×D₆₇	—	2	2	no	—
172	C₂×D₆₇	—	4	1	no	—
173	C₂×D₆₇	—	4	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
178	C₄×C₆₇	#407	1	1	yes	characteristic
179	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
180	C₄×C₆₇	#423	1	1	yes	characteristic
181	C₄×C₆₇	#471	1	1	yes	characteristic
182	C₄×C₆₇	#471	1	1	yes	characteristic
183	Dic₆₇	—	2	2	no	—
184	Dic₆₇	—	4	1	no	—
185	Dic₆₇	—	4	1	no	—
186	#8604	—	67	1	no	maximal, Sylow
187	C₂×C₆₇	—	2	1	no	—
188	C₂×C₆₇	—	2	1	no	—
189	C₂×C₆₇	#1797	1	1	yes	characteristic
190	C₂×C₆₇	#1814	1	1	yes	characteristic
191	C₂×C₆₇	#1815	1	1	yes	characteristic
192	D₆₇	—	4	2	no	—
193	C₄×#131	—	67	1	no	—
194	#1564	—	67	1	no	—
195	C₂×#334	—	67	1	no	—
196	#1477	—	67	1	no	—
197	#1477	—	67	1	no	—
198	#1283	—	67	1	no	—
199	#1283	D₆₇	1	1	yes	characteristic, complemented
200	C₆₇	#8604	1	1	yes	characteristic, complemented, Sylow
201	C₂×#131	—	67	1	no	—
202	C₂×#131	—	67	1	no	—
203	C₄×#41	—	67	1	no	—
204	#431	—	67	1	no	—
205	#431	C₂×D₆₇	1	1	yes	characteristic
206	C₂×C₄⋊C₈	—	67	1	no	—
207	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
208	#405	—	67	1	no	—
209	#405	—	67	1	no	—
210	#405	—	134	1	no	—
211	#405	—	134	1	no	—
212	C₄×M₁₆	—	134	1	no	—
213	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
214	C₂×C₄×C₈	—	67	1	no	—
215	C₂×C₄×C₈	—	134	1	no	—
216	#335	—	134	1	no	—
217	#334	—	134	1	no	—
218	#334	—	134	1	no	—
219	#334	—	134	1	no	—
220	Q₈⋊C₈	—	134	1	no	—
221	Q₈⋊C₈	—	134	1	no	—
222	D₄⋊C₈	—	134	1	no	—
223	D₄⋊C₈	—	134	1	no	—
224	C₂×#41	—	67	1	no	—
225	#131	—	67	4	no	—
226	#131	—	134	2	no	—
227	#131	—	134	2	no	—
228	#131	—	134	2	no	—
229	C₂×M₁₆	—	134	1	no	—
230	C₂×M₁₆	—	134	1	no	—
231	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
232	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
233	C₂²×C₈	—	2	1	no	—
234	C₂²×C₈	—	67	1	no	—
235	C₂²×C₈	—	67	1	no	—
236	C₂²×C₈	—	134	1	no	—
237	C₂²×C₈	—	134	1	no	—
238	C₂²×C₈	—	134	1	no	—
239	C₄×Q₈	—	67	1	no	—
240	C₄×Q₈	—	67	1	no	—
241	C₄×D₄	—	67	1	no	—
242	C₄×D₄	—	67	1	no	—
243	C₄×D₄	—	134	1	no	—
244	C₄×D₄	—	134	1	no	—
245	#117	—	67	1	no	—
246	#117	—	134	1	no	—
247	C₂×C₄²	—	134	1	no	—
248	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	134	1	no	—
253	C₄⋊C₈	—	134	1	no	—
254	C₄⋊C₈	—	134	1	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	C₂²⋊C₈	—	134	1	no	—
257	#97	—	2	1	no	—
258	#97	—	134	1	no	—
259	#97	—	134	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	—	67	1	no	—
263	C₄×C₈	—	67	1	no	—
264	C₄×C₈	—	67	1	no	—
265	C₄×C₈	—	67	1	no	—
266	C₄×C₈	—	134	1	no	—
267	C₄×C₈	—	134	1	no	—
268	C₄×C₈	—	134	1	no	—
269	#41	—	67	2	no	—
270	#41	—	67	2	no	—
271	#41	—	134	1	no	—
272	#41	—	134	1	no	—
273	C₂×Q₈	—	67	1	no	—
274	C₂×D₄	—	67	1	no	—
275	C₂×D₄	—	134	1	no	—
276	C₂²×C₄	—	134	1	no	—
277	C₂²×C₄	—	134	1	no	—
278	C₂²×C₄	—	134	1	no	—
279	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
280	C₂²×C₄	#11761628	1	1	yes	characteristic
281	C₂²×C₄	#11761628	1	1	yes	characteristic
282	M₁₆	—	2	1	no	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	134	4	no	—
285	M₁₆	—	134	4	no	—
286	M₁₆	C₈×D₆₇	1	2	yes	—
287	M₁₆	#11761611	1	2	yes	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	8	1	no	—
293	C₂×C₈	—	268	1	no	—
294	C₂×C₈	—	268	1	no	—
295	C₂×C₈	—	536	1	no	—
296	C₂×C₈	—	67	1	no	—
297	C₂×C₈	—	67	1	no	—
298	C₂×C₈	—	67	2	no	—
299	C₂×C₈	—	67	2	no	—
300	C₂×C₈	—	67	1	no	—
301	C₂×C₈	—	67	1	no	—
302	C₂×C₈	—	134	4	no	—
303	C₂×C₈	—	134	4	no	—
304	C₂×C₈	—	134	1	no	—
305	C₂×C₈	—	134	1	no	—
306	C₄⋊C₄	—	67	1	no	—
307	C₄⋊C₄	—	67	1	no	—
308	C₄⋊C₄	—	134	1	no	—
309	C₄⋊C₄	—	134	1	no	—
310	C₂²⋊C₄	—	134	1	no	—
311	C₂²⋊C₄	—	134	1	no	—
312	C₂²⋊C₄	—	134	1	no	—
313	C₄²	—	134	1	no	—
314	C₄²	—	134	1	no	—
315	C₄²	—	134	1	no	—
316	C₄²	D₄×D₆₇	1	1	yes	characteristic
317	C₄²	D₄×D₆₇	1	1	yes	characteristic
318	C₄²	D₄×D₆₇	1	1	yes	characteristic
319	C₄²	Q₈×D₆₇	1	1	yes	characteristic
320	C₂³	—	134	1	no	—
321	C₂³	#14524000	1	1	yes	characteristic
322	Q₈	—	67	2	no	—
323	Q₈	—	134	1	no	—
324	D₄	—	67	2	no	—
325	D₄	—	134	2	no	—
326	D₄	—	134	2	no	—
327	D₄	—	134	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	268	2	no	—
332	C₂×C₄	—	268	1	no	—
333	C₂×C₄	—	268	1	no	—
334	C₂×C₄	—	134	1	no	—
335	C₂×C₄	—	134	1	no	—
336	C₂×C₄	—	134	1	no	—
337	C₂×C₄	—	134	2	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
340	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
341	C₂×C₄	#14524016	1	1	yes	characteristic
342	C₂×C₄	#14524016	1	1	yes	characteristic
343	C₂×C₄	#14524021	1	1	yes	characteristic
344	C₂×C₄	#14524021	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
346	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
347	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₈	—	8	1	no	—
352	C₈	—	8	1	no	—
353	C₈	—	536	1	no	—
354	C₈	—	536	1	no	—
355	C₈	—	67	2	no	—
356	C₈	—	67	2	no	—
357	C₈	—	134	1	no	—
358	C₈	—	134	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	268	1	no	—
362	C₂²	—	268	1	no	—
363	C₂²	—	134	2	no	—
364	C₂²	#16863255	1	1	yes	characteristic
365	C₂²	#16863447	1	1	yes	characteristic
366	C₂²	#16863451	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	268	1	no	—
371	C₄	—	268	1	no	—
372	C₄	—	134	2	no	—
373	C₄	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
374	C₄	#16863358	1	1	yes	characteristic
375	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
376	C₄	#16863418	1	1	yes	characteristic
377	C₄	#16863658	1	1	yes	characteristic
378	C₄	#16863662	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	268	2	no	—
382	C₂	#22560188	1	1	yes	characteristic
383	C₂	#22560262	1	1	yes	characteristic
384	C₂	#22560267	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524016
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1283	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1283	D₆₇
hypocenter	C₆₇	#8604
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1283

Complements

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

N	H ≅ G/N	type
2. #22558955	C₂	semidirect
3. C₂×#16863146	C₂	semidirect
6. #22558060	C₂	semidirect
7. C₆₇×#1283	C₂	semidirect
35. #14524051	C₈	semidirect
199. #1283	D₆₇	semidirect
200. C₆₇	#8604	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524051	2
#22558955	2
C₂×#16863146	2
#22558597	2
#22558594	2
#22558060	2
C₆₇×#1283	2
#8604	67