Subgroups — #30754931

Order 17152.

610 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568567	C₂	1	1	yes	maximal, characteristic
2	#22560360	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560064	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863145	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863139	C₂	1	1	yes	maximal, characteristic
6	C₂×#16863139	C₂	1	1	yes	maximal, characteristic
7	C₂×#333×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524040	C₂²	1	1	yes	characteristic
9	C₂×#14524032	C₄	1	4	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₈	—	2	2	no	—
13	C₂×#14523953	—	2	1	no	—
14	#16863297	—	2	1	no	—
15	#16863297	—	2	1	no	—
16	#16863297	—	2	2	no	—
17	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
18	#16863150	—	2	1	no	—
19	#16863145	—	2	2	no	—
20	#16863139	—	2	2	no	—
21	#16863139	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#333×C₆₇	—	2	2	no	—
26	C₂²×D₂₆₈	—	2	1	no	—
27	C₂²×D₂₆₈	—	2	1	no	—
28	C₂²×D₂₆₈	C₂×C₄	1	2	yes	—
29	C₂²×C₆₇⋊C₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	1	no	—
31	#14524040	—	2	1	no	—
32	#14524040	—	2	1	no	—
33	#14524040	—	2	1	no	—
34	#14524040	—	2	1	no	—
35	#14524032	—	4	8	no	—
36	C₂×(C₂×D₆₇)⋊C₄	—	2	4	no	—
37	C₂×(C₂×D₆₇)⋊C₄	—	2	4	no	—
38	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
42	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
46	Dic₆₇⋊C₈	—	4	4	no	—
47	#14523953	—	4	2	no	—
48	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₆₇	—	2	1	no	—
50	C₂²×C₈×C₆₇	—	2	1	no	—
51	C₄⋊Q₈×C₆₇	—	2	1	no	—
52	C₄⋊Q₈×C₆₇	—	2	1	no	—
53	C₄⋊Q₈×C₆₇	—	2	1	no	—
54	C₄⋊Q₈×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
62	C₂³×D₆₇	—	2	2	no	—
63	C₂×D₂₆₈	—	2	2	no	—
64	C₂×D₂₆₈	—	2	2	no	—
65	C₂×D₂₆₈	—	4	4	no	—
66	C₂×D₂₆₈	—	4	1	no	—
67	C₂×D₂₆₈	—	4	1	no	—
68	C₂×D₂₆₈	—	4	1	no	—
69	C₂×D₂₆₈	—	4	1	no	—
70	C₂×D₂₆₈	—	4	1	no	—
71	C₂×D₂₆₈	—	4	1	no	—
72	C₂×D₂₆₈	—	4	1	no	—
73	C₂×D₂₆₈	—	4	2	no	—
74	C₂×D₂₆₈	—	4	1	no	—
75	C₂×D₂₆₈	—	4	2	no	—
76	C₂²×Dic₆₇	C₂²⋊C₄	1	2	yes	—
77	C₂²×Dic₆₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₆₇⋊C₈	—	2	2	no	—
79	C₂×C₆₇⋊C₈	—	4	1	no	—
80	C₂×C₆₇⋊C₈	—	4	1	no	—
81	C₂×C₆₇⋊C₈	—	4	1	no	—
82	C₂×C₆₇⋊C₈	—	4	2	no	—
83	(C₂×D₆₇)⋊C₄	—	4	8	no	—
84	(C₂×D₆₇)⋊C₄	—	4	8	no	—
85	C₄×Dic₆₇	—	2	1	no	—
86	C₄×Dic₆₇	—	2	1	no	—
87	C₄×Dic₆₇	—	2	1	no	—
88	C₄×Dic₆₇	—	2	2	no	—
89	C₄×Dic₆₇	—	2	2	no	—
90	C₄×Dic₆₇	—	2	1	no	—
91	C₄×Dic₆₇	—	2	4	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	4	4	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	—	4	1	no	—
102	C₂×C₈×C₆₇	—	4	1	no	—
103	C₂×C₈×C₆₇	—	4	2	no	—
104	C₄⋊C₄×C₆₇	—	4	1	no	—
105	C₄⋊C₄×C₆₇	—	4	1	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄⋊C₄×C₆₇	—	4	1	no	—
108	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₆₇	#41	1	1	yes	characteristic
111	C₄²×C₆₇	#41	1	1	yes	characteristic
112	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×D₆₇	—	4	2	no	—
114	C₂²×D₆₇	—	4	4	no	—
115	C₂²×D₆₇	—	4	2	no	—
116	C₂²×D₆₇	—	4	4	no	—
117	C₂²×D₆₇	—	4	2	no	—
118	Q₈×C₆₇	—	8	2	no	—
119	Q₈×C₆₇	—	8	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	4	no	—
123	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
130	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
131	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#137	1	2	yes	—
135	C₂×Dic₆₇	—	2	2	no	—
136	C₂×Dic₆₇	—	2	2	no	—
137	C₂×Dic₆₇	—	2	2	no	—
138	C₂×Dic₆₇	—	2	2	no	—
139	C₂×Dic₆₇	—	2	2	no	—
140	C₂×Dic₆₇	#100	1	4	yes	—
141	D₂₆₈	—	8	2	no	—
142	D₂₆₈	—	8	2	no	—
143	D₂₆₈	—	8	1	no	—
144	D₂₆₈	—	8	1	no	—
145	D₂₆₈	—	8	1	no	—
146	D₂₆₈	—	8	1	no	—
147	D₂₆₈	—	8	2	no	—
148	D₂₆₈	—	8	1	no	—
149	D₂₆₈	—	8	1	no	—
150	C₈×C₆₇	—	4	2	no	—
151	C₈×C₆₇	—	8	1	no	—
152	C₆₇⋊C₈	—	4	2	no	—
153	C₆₇⋊C₈	—	8	1	no	—
154	C₂²×C₆₇	#390	1	1	yes	characteristic
155	C₂²×C₆₇	C₂×#100	1	2	yes	—
156	C₂²×C₆₇	#416	1	1	yes	characteristic
157	C₂²×C₆₇	#417	1	2	yes	—
158	C₂²×C₆₇	#417	1	1	yes	characteristic
159	C₂×D₆₇	—	8	4	no	—
160	C₂×D₆₇	—	8	4	no	—
161	C₂×D₆₇	—	8	2	no	—
162	C₂×D₆₇	—	8	2	no	—
163	C₂×D₆₇	—	8	2	no	—
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	2	1	no	—
166	C₄×C₆₇	—	8	2	no	—
167	C₄×C₆₇	#463	1	1	yes	characteristic
168	C₄×C₆₇	#463	1	1	yes	characteristic
169	C₄×C₆₇	#464	1	1	yes	characteristic
170	C₄×C₆₇	#464	1	1	yes	characteristic
171	C₄×C₆₇	#489	1	2	yes	—
172	C₄×C₆₇	#496	1	1	yes	characteristic
173	C₄×C₆₇	#497	1	1	yes	characteristic
174	Dic₆₇	—	2	4	no	—
175	Dic₆₇	—	4	2	no	—
176	#9707	—	67	1	no	maximal, Sylow
177	C₂×C₆₇	#1551	1	2	yes	—
178	C₂×C₆₇	#1560	1	2	yes	—
179	C₂×C₆₇	#1959	1	1	yes	characteristic
180	C₂×C₆₇	#1965	1	1	yes	characteristic
181	C₂×C₆₇	#1983	1	1	yes	characteristic
182	D₆₇	—	16	2	no	—
183	#2858	—	67	1	no	—
184	#1842	—	67	1	no	—
185	#1767	—	67	1	no	—
186	C₂×#333	—	67	1	no	—
187	C₂×#333	D₆₇	1	1	yes	characteristic, complemented
188	C₂×#331	—	67	1	no	—
189	C₂×#331	—	67	1	no	—
190	C₆₇	#9707	1	1	yes	characteristic, complemented, Sylow
191	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
192	C₂×#127	—	67	1	no	—
193	C₂×#124	—	67	4	no	—
194	C₂×C₄⋊C₈	—	67	1	no	—
195	C₂×C₄⋊C₈	—	67	1	no	—
196	C₂×C₄⋊C₈	—	134	2	no	—
197	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
198	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
199	C₂×#97	—	134	1	no	—
200	#390	—	134	1	no	—
201	#390	—	134	1	no	—
202	#390	—	134	2	no	—
203	C₄³	—	67	1	no	—
204	#336	—	134	1	no	—
205	#333	—	2	2	no	—
206	#333	—	134	2	no	—
207	#331	—	134	2	no	—
208	#331	—	134	2	no	—
209	C₂²×Q₈	C₄×D₆₇	1	2	yes	—
210	C₂²×D₄	—	67	2	no	—
211	C₂²×D₄	—	134	1	no	—
212	C₂²×D₄	—	134	1	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	2	1	no	—
215	C₂²×C₈	—	134	1	no	—
216	C₂²×C₈	—	134	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	#127	—	134	1	no	—
222	#127	—	134	1	no	—
223	#127	—	134	1	no	—
224	#127	—	134	1	no	—
225	#124	—	268	8	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₂²⋊C₄	—	134	4	no	—
229	C₂×C₂²⋊C₄	—	134	4	no	—
230	C₂×C₄²	—	67	2	no	—
231	C₂×C₄²	—	67	1	no	—
232	C₂×C₄²	—	67	1	no	—
233	C₂×C₄²	—	67	2	no	—
234	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	268	4	no	—
239	C₄⋊C₈	—	67	2	no	—
240	C₄⋊C₈	—	134	1	no	—
241	C₄⋊C₈	—	134	1	no	—
242	C₄⋊C₈	—	134	1	no	—
243	C₄⋊C₈	C₂²⋊D₆₇	1	2	yes	—
244	#97	—	268	2	no	—
245	C₂⁴	—	134	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×D₄	—	268	4	no	—
250	C₂×D₄	—	268	1	no	—
251	C₂×D₄	—	268	1	no	—
252	C₂×D₄	—	268	1	no	—
253	C₂×D₄	—	268	1	no	—
254	C₂×D₄	—	268	1	no	—
255	C₂×D₄	—	268	1	no	—
256	C₂×D₄	—	268	2	no	—
257	C₂×D₄	—	268	1	no	—
258	C₂×D₄	—	268	2	no	—
259	C₂×D₄	—	268	1	no	—
260	C₂×D₄	—	134	2	no	—
261	C₂×D₄	—	134	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	67	2	no	—
264	C₂²×C₄	—	67	2	no	—
265	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	268	1	no	—
274	C₂×C₈	—	268	1	no	—
275	C₂×C₈	—	268	1	no	—
276	C₂×C₈	—	268	2	no	—
277	C₂×C₈	—	134	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	268	8	no	—
283	C₂²⋊C₄	—	268	8	no	—
284	C₄²	—	134	4	no	—
285	C₄²	—	134	1	no	—
286	C₄²	—	134	1	no	—
287	C₄²	—	134	1	no	—
288	C₄²	—	134	2	no	—
289	C₄²	—	134	1	no	—
290	C₄²	—	134	2	no	—
291	C₄²	D₄×D₆₇	1	1	yes	characteristic
292	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
293	C₄²	#11761627	1	1	yes	characteristic
294	C₄²	#11761628	1	1	yes	characteristic
295	C₂³	—	268	2	no	—
296	C₂³	—	268	4	no	—
297	C₂³	—	268	2	no	—
298	C₂³	—	268	2	no	—
299	C₂³	—	268	4	no	—
300	C₂³	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
301	Q₈	—	8	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	536	2	no	—
304	D₄	—	536	2	no	—
305	D₄	—	536	2	no	—
306	D₄	—	536	1	no	—
307	D₄	—	536	1	no	—
308	D₄	—	536	1	no	—
309	D₄	—	536	1	no	—
310	D₄	—	536	1	no	—
311	D₄	—	536	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	67	4	no	—
316	C₂×C₄	—	134	2	no	—
317	C₂×C₄	—	134	2	no	—
318	C₂×C₄	—	134	2	no	—
319	C₂×C₄	—	134	2	no	—
320	C₂×C₄	—	134	2	no	—
321	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
323	C₂×C₄	#14524015	1	1	yes	characteristic
324	C₂×C₄	#14524016	1	1	yes	characteristic
325	C₂×C₄	#14524029	1	1	yes	characteristic
326	C₂×C₄	#14524039	1	1	yes	characteristic
327	C₂×C₄	#14524064	1	1	yes	characteristic
328	C₂×C₄	#14524066	1	1	yes	characteristic
329	C₂×C₄	#14524070	1	2	yes	—
330	C₂×C₄	#14524070	1	1	yes	characteristic
331	C₂×C₄	#14524075	1	2	yes	—
332	C₂×C₄	#14524075	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	8	1	no	—
335	C₈	—	268	2	no	—
336	C₈	—	536	1	no	—
337	C₂²	—	536	2	no	—
338	C₂²	—	536	2	no	—
339	C₂²	—	536	4	no	—
340	C₂²	—	536	4	no	—
341	C₂²	—	536	2	no	—
342	C₂²	#16863296	1	1	yes	characteristic
343	C₂²	#100×D₆₇	1	2	yes	—
344	C₂²	#16863391	1	1	yes	characteristic
345	C₂²	#16863396	1	1	yes	characteristic
346	C₂²	#16863396	1	2	yes	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	268	2	no	—
351	C₄	—	134	4	no	—
352	C₄	#16863613	1	1	yes	characteristic
353	C₄	#16863614	1	1	yes	characteristic
354	C₄	#16863620	1	1	yes	characteristic
355	C₄	#16863621	1	1	yes	characteristic
356	C₄	#16863757	1	2	yes	—
357	C₄	#16863782	1	1	yes	characteristic
358	C₄	#16863789	1	1	yes	characteristic
359	C₂	—	1072	2	no	—
360	C₂	#22558903	1	2	yes	—
361	C₂	#22558943	1	2	yes	—
362	C₂	#22561022	1	1	yes	characteristic
363	C₂	#22561047	1	1	yes	characteristic
364	C₂	#22561138	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#333×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂×#14524040	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#333	D₆₇
hypocenter	C₆₇	#9707
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. #22560360	C₂	semidirect
3. #22560064	C₂	semidirect
4. C₂×#16863145	C₂	semidirect
7. C₂×#333×C₆₇	C₂	semidirect
187. C₂×#333	D₆₇	semidirect
190. C₆₇	#9707	semidirect

Maximal subgroups

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

subgroup	index
#22568567	2
#22560360	2
#22560064	2
C₂×#16863145	2
C₂×#16863139	2
C₂×#333×C₆₇	2
#9707	67