Subgroups — #30755158

Order 17152.

510 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863743	C₂	1	1	yes	maximal, characteristic
2	#22561157	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561136	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863149	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863144	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863139	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524043	C₂²	1	1	yes	characteristic
9	C₂×#14524040	C₂²	1	1	yes	characteristic
10	#16863743	—	2	2	no	—
11	#16863743	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14523965	—	2	1	no	—
15	C₂×#14523965	—	2	1	no	—
16	C₂×#14523963	—	2	2	no	—
17	#16863158	—	2	2	no	—
18	#16863149	—	2	2	no	—
19	#16863144	—	2	2	no	—
20	#16863139	—	2	2	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	#336×C₆₇	—	2	1	no	—
26	C₂²×Dic₁₃₄	—	2	1	no	—
27	C₂²×D₂₆₈	—	2	1	no	—
28	C₂²×D₂₆₈	—	2	1	no	—
29	C₂²×D₂₆₈	—	2	1	no	—
30	C₂²×C₆₇⋊C₈	—	2	1	no	—
31	C₂²×C₆₇⋊C₈	—	2	1	no	—
32	#14524043	—	2	1	no	—
33	#14524043	—	2	1	no	—
34	#14524043	—	2	1	no	—
35	#14524043	C₂×C₄	1	2	yes	—
36	#14524040	—	2	1	no	—
37	#14524040	—	2	1	no	—
38	#14524040	—	2	1	no	—
39	#14524040	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₆₇	—	2	1	no	—
41	C₂×C₄⋊Dic₆₇	—	2	1	no	—
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)	—	2	1	no	—
46	#14523965	—	2	4	no	—
47	#14523965	—	2	4	no	—
48	#14523963	—	4	4	no	—
49	C₂²×C₈×C₆₇	—	2	1	no	—
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	—	2	1	no	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₆₇	—	2	1	no	—
56	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
57	#97×C₆₇	—	4	2	no	—
58	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₆₇	D₄	1	2	yes	—
60	#95×C₆₇	—	2	1	no	—
61	C₂³×D₆₇	—	4	1	no	—
62	C₂×Dic₁₃₄	—	2	2	no	—
63	C₂×Dic₁₃₄	—	4	1	no	—
64	C₂×Dic₁₃₄	—	8	2	no	—
65	C₂×D₂₆₈	—	2	2	no	—
66	C₂×D₂₆₈	—	2	2	no	—
67	C₂×D₂₆₈	—	2	2	no	—
68	C₂×D₂₆₈	—	4	1	no	—
69	C₂×D₂₆₈	—	4	1	no	—
70	C₂×D₂₆₈	—	4	2	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	1	no	—
76	C₂×D₂₆₈	—	4	1	no	—
77	C₂×D₂₆₈	—	8	2	no	—
78	C₂²×Dic₆₇	—	4	1	no	—
79	C₂×C₆₇⋊C₈	—	4	1	no	—
80	C₂×C₆₇⋊C₈	—	4	1	no	—
81	C₂×C₆₇⋊C₈	—	4	2	no	—
82	C₂×C₆₇⋊C₈	—	4	2	no	—
83	C₄⋊Dic₆₇	—	2	2	no	—
84	C₄⋊Dic₆₇	—	2	2	no	—
85	C₄⋊Dic₆₇	—	4	1	no	—
86	C₄⋊Dic₆₇	—	4	1	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	4	1	no	—
97	C₂×C₈×C₆₇	—	4	1	no	—
98	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	—	4	1	no	—
103	C₄²×C₆₇	—	4	1	no	—
104	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₆₇	#41	1	1	yes	characteristic
108	C₂³×C₆₇	#121	1	1	yes	characteristic
109	C₂²×D₆₇	—	4	2	no	—
110	C₂²×D₆₇	—	8	1	no	—
111	C₂²×D₆₇	—	8	1	no	—
112	C₂²×D₆₇	—	8	2	no	—
113	C₂²×D₆₇	—	8	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
126	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#137	1	2	yes	—
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	8	2	no	—
132	D₂₆₈	—	4	2	no	—
133	D₂₆₈	—	4	2	no	—
134	D₂₆₈	—	4	4	no	—
135	D₂₆₈	—	8	2	no	—
136	D₂₆₈	—	8	2	no	—
137	D₂₆₈	—	8	1	no	—
138	D₂₆₈	—	8	1	no	—
139	D₂₆₈	—	8	1	no	—
140	D₂₆₈	—	8	1	no	—
141	Dic₁₃₄	—	8	2	no	—
142	Dic₁₃₄	—	8	2	no	—
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	—	2	2	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₆₇⋊C₈	—	8	1	no	—
147	C₆₇⋊C₈	—	8	1	no	—
148	C₂²×C₆₇	#390	1	1	yes	characteristic
149	C₂²×C₆₇	#463	1	1	yes	characteristic
150	C₂²×C₆₇	#464	1	1	yes	characteristic
151	C₂²×C₆₇	#468	1	2	yes	—
152	C₂²×C₆₇	#472	1	2	yes	—
153	C₂×D₆₇	—	8	2	no	—
154	C₂×D₆₇	—	8	2	no	—
155	C₂×D₆₇	—	8	2	no	—
156	C₂×D₆₇	—	16	2	no	—
157	C₂×D₆₇	—	16	2	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	8	2	no	—
161	C₄×C₆₇	#416	1	1	yes	characteristic
162	C₄×C₆₇	#417	1	1	yes	characteristic
163	C₄×C₆₇	#440	1	2	yes	—
164	C₄×C₆₇	#463	1	1	yes	characteristic
165	C₄×C₆₇	#464	1	1	yes	characteristic
166	C₄×C₆₇	#496	1	1	yes	characteristic
167	C₄×C₆₇	#497	1	1	yes	characteristic
168	Dic₆₇	—	8	2	no	—
169	#9747	—	67	1	no	maximal, Sylow
170	C₂×C₆₇	#1697	1	2	yes	—
171	C₂×C₆₇	#1698	1	2	yes	—
172	C₂×C₆₇	#1973	1	1	yes	characteristic
173	C₂×C₆₇	#1978	1	1	yes	characteristic
174	C₂×C₆₇	#1983	1	1	yes	characteristic
175	D₆₇	—	16	2	no	—
176	C₂×#486	—	67	1	no	—
177	#1987	—	67	1	no	—
178	#1982	—	67	1	no	—
179	#1770	D₆₇	1	1	yes	characteristic, complemented
180	C₂×#335	—	67	1	no	—
181	C₂×#333	—	67	1	no	—
182	C₂×#331	—	67	1	no	—
183	C₆₇	#9747	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄⋊Q₈	—	67	1	no	—
185	C₂×#127	—	67	1	no	—
186	#486	—	67	4	no	—
187	#486	—	134	2	no	—
188	C₂×C₄⋊C₈	—	67	1	no	—
189	C₂×C₄⋊C₈	—	67	1	no	—
190	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
191	C₂×#102	—	134	2	no	—
192	C₂×#102	—	134	1	no	—
193	C₂×#102	—	134	1	no	—
194	C₂×#97	—	2	1	no	—
195	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
196	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
197	#340	—	134	2	no	—
198	#336	—	2	1	no	—
199	#335	—	134	2	no	—
200	#333	—	134	2	no	—
201	#331	—	134	2	no	—
202	C₂²×Q₈	—	134	1	no	—
203	C₂²×D₄	—	134	1	no	—
204	C₂²×D₄	—	134	1	no	—
205	C₂²×D₄	—	134	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	134	1	no	—
208	C₂²×C₈	—	134	1	no	—
209	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
211	C₄⋊Q₈	—	67	2	no	—
212	C₄⋊Q₈	—	134	1	no	—
213	C₄⋊Q₈	—	134	1	no	—
214	C₄⋊Q₈	—	134	1	no	—
215	#127	—	67	2	no	—
216	#127	—	134	1	no	—
217	#127	—	134	1	no	—
218	#127	—	134	1	no	—
219	C₂×C₄⋊C₄	—	134	1	no	—
220	C₂×C₄⋊C₄	—	134	1	no	—
221	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	134	1	no	—
226	C₄⋊C₈	—	134	1	no	—
227	C₄⋊C₈	—	134	1	no	—
228	C₄⋊C₈	—	134	1	no	—
229	C₄⋊C₈	D₂₆₈	1	2	yes	—
230	#102	—	268	4	no	—
231	#102	—	134	4	no	—
232	#102	—	134	4	no	—
233	#97	—	4	2	no	—
234	C₄×C₈	C₄×D₆₇	1	2	yes	—
235	C₄×C₈	D₂₆₈	1	2	yes	—
236	#95	—	2	1	no	—
237	C₂⁴	—	268	1	no	—
238	C₂×Q₈	—	268	1	no	—
239	C₂×Q₈	—	536	2	no	—
240	C₂×Q₈	—	134	2	no	—
241	C₂×D₄	—	268	1	no	—
242	C₂×D₄	—	268	1	no	—
243	C₂×D₄	—	268	1	no	—
244	C₂×D₄	—	268	1	no	—
245	C₂×D₄	—	268	1	no	—
246	C₂×D₄	—	268	1	no	—
247	C₂×D₄	—	268	2	no	—
248	C₂×D₄	—	268	1	no	—
249	C₂×D₄	—	268	2	no	—
250	C₂×D₄	—	536	2	no	—
251	C₂×D₄	—	134	2	no	—
252	C₂×D₄	—	134	2	no	—
253	C₂×D₄	—	134	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	268	1	no	—
256	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
258	C₂²×C₄	#11761628	1	1	yes	characteristic
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	268	1	no	—
267	C₂×C₈	—	268	1	no	—
268	C₂×C₈	—	268	2	no	—
269	C₂×C₈	—	268	2	no	—
270	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
271	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	268	1	no	—
275	C₄⋊C₄	—	268	1	no	—
276	C₄⋊C₄	—	134	2	no	—
277	C₄⋊C₄	—	134	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
281	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
282	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
283	C₄²	#11761628	1	1	yes	characteristic
284	C₂³	—	268	2	no	—
285	C₂³	—	536	1	no	—
286	C₂³	—	536	1	no	—
287	C₂³	—	536	2	no	—
288	C₂³	—	536	2	no	—
289	C₂³	#14524017	1	1	yes	characteristic
290	Q₈	—	536	2	no	—
291	Q₈	—	536	2	no	—
292	D₄	—	268	4	no	—
293	D₄	—	268	2	no	—
294	D₄	—	268	2	no	—
295	D₄	—	536	2	no	—
296	D₄	—	536	2	no	—
297	D₄	—	536	1	no	—
298	D₄	—	536	1	no	—
299	D₄	—	536	1	no	—
300	D₄	—	536	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	268	2	no	—
306	C₂×C₄	—	536	2	no	—
307	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
308	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
309	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
310	C₂×C₄	#14524017	1	1	yes	characteristic
311	C₂×C₄	#14524031	1	1	yes	characteristic
312	C₂×C₄	#14524040	1	1	yes	characteristic
313	C₂×C₄	#14524063	1	1	yes	characteristic
314	C₂×C₄	#14524063	1	1	yes	characteristic
315	C₂×C₄	#14524069	1	1	yes	characteristic
316	C₂×C₄	#14524071	1	2	yes	—
317	C₂×C₄	#14524074	1	1	yes	characteristic
318	C₂×C₄	#14524076	1	2	yes	—
319	C₈	—	2	2	no	—
320	C₈	—	2	2	no	—
321	C₈	—	4	2	no	—
322	C₈	—	536	1	no	—
323	C₈	—	536	1	no	—
324	C₂²	—	536	2	no	—
325	C₂²	—	536	2	no	—
326	C₂²	—	536	2	no	—
327	C₂²	—	1072	2	no	—
328	C₂²	—	1072	2	no	—
329	C₂²	#16863298	1	1	yes	characteristic
330	C₂²	#16863611	1	1	yes	characteristic
331	C₂²	#16863618	1	1	yes	characteristic
332	C₂²	#16863642	1	2	yes	—
333	C₂²	#16863666	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	536	2	no	—
338	C₄	#16863390	1	1	yes	characteristic
339	C₄	#16863395	1	1	yes	characteristic
340	C₄	#16863498	1	2	yes	—
341	C₄	#16863611	1	1	yes	characteristic
342	C₄	#16863618	1	1	yes	characteristic
343	C₄	#16863785	1	1	yes	characteristic
344	C₄	#16863790	1	1	yes	characteristic
345	C₂	—	1072	2	no	—
346	C₂	#22559742	1	2	yes	—
347	C₂	#22559749	1	2	yes	—
348	C₂	#22561090	1	1	yes	characteristic
349	C₂	#22561119	1	1	yes	characteristic
350	C₂	#22561141	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1770	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂×#14524040	C₂²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1770	D₆₇
hypocenter	C₆₇	#9747
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #22561157	C₂	semidirect
4. C₂×#16863149	C₂	semidirect
5. C₂×#16863144	C₂	semidirect
7. C₆₇×#1770	C₂	semidirect
11. #16863743	C₄	semidirect
179. #1770	D₆₇	semidirect
183. C₆₇	#9747	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863743	2
#22561157	2
#22561136	2
C₂×#16863149	2
C₂×#16863144	2
C₂×#16863139	2
C₆₇×#1770	2
#9747	67