Subgroups — #30750160

Order 17152.

454 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568944	C₂	1	1	yes	maximal, characteristic
2	#22560985	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558967	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558635	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863123	C₂	1	1	yes	maximal, characteristic
6	#22558045	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₆₇	—	2	1	no	—
9	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
10	#16863473	—	2	1	no	—
11	#16863466	—	2	2	no	—
12	#16863461	—	2	1	no	—
13	C₂×#14523979	—	2	1	no	—
14	#16863420	C₂²	1	1	yes	characteristic
15	#16863359	—	2	1	no	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	#16863161	—	2	1	no	—
19	#16863148	—	2	2	no	—
20	#16863125	—	2	1	no	—
21	#16863125	—	2	1	no	—
22	#16863123	C₄	1	1	yes	characteristic
23	#16863123	C₄	1	1	yes	characteristic
24	#16863123	C₄	1	1	yes	characteristic
25	#16863123	C₄	1	1	yes	characteristic
26	#431×C₆₇	C₂²	1	1	yes	characteristic
27	#423×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×D₂₆₈	Q₈	1	1	yes	characteristic
30	C₂²×C₄×D₆₇	—	2	1	no	—
31	C₂×#11761612	—	2	1	no	—
32	C₂×#11761612	D₄	1	1	yes	characteristic
33	C₂×#11761612	D₄	1	1	yes	characteristic
34	C₂×#11761611	—	2	2	no	—
35	M₁₆×D₆₇	—	4	2	no	—
36	M₁₆×D₆₇	—	4	2	no	—
37	C₂²×C₆₇⋊C₈	—	2	1	no	—
38	C₂×C₈×D₆₇	—	2	2	no	—
39	C₄×D₂₆₈	—	2	1	no	—
40	C₄×D₂₆₈	—	2	1	no	—
41	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
42	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
43	C₄×D₂₆₈	D₄	1	1	yes	characteristic
44	C₄×D₂₆₈	D₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
46	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
47	#14523979	—	4	2	no	—
48	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
50	(C₂×D₆₇)⋊C₈	—	2	2	no	—
51	(C₂×D₆₇)⋊C₈	—	2	2	no	—
52	#14523954	—	2	1	no	—
53	C₄×C₆₇⋊C₈	—	2	1	no	—
54	C₄×C₆₇⋊C₈	—	2	1	no	—
55	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₆₇	—	2	1	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	—	2	1	no	—
62	C₄⋊C₈×C₆₇	—	2	1	no	—
63	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
66	C₂²⋊C₈×C₆₇	—	2	1	no	—
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₂³×D₆₇	—	2	1	no	—
69	C₂×D₂₆₈	—	2	2	no	—
70	C₂×D₂₆₈	—	2	1	no	—
71	C₂×D₂₆₈	—	4	1	no	—
72	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×Dic₆₇	—	2	1	no	—
75	C₂×C₄×D₆₇	—	2	1	no	—
76	C₂×C₄×D₆₇	—	2	1	no	—
77	C₂×C₄×D₆₇	—	2	1	no	—
78	C₂×C₄×D₆₇	—	2	1	no	—
79	C₂×C₄×D₆₇	—	4	1	no	—
80	C₂×C₄×D₆₇	—	4	1	no	—
81	C₂×C₄×D₆₇	—	4	1	no	—
82	C₂×C₄×D₆₇	—	4	1	no	—
83	#11761612	—	2	1	no	—
84	#11761612	—	2	1	no	—
85	#11761612	—	2	1	no	—
86	#11761612	—	4	1	no	—
87	#11761612	—	4	1	no	—
88	#11761612	D₈	1	2	yes	—
89	#11761611	—	4	4	no	—
90	C₂×C₆₇⋊C₈	—	2	1	no	—
91	C₂×C₆₇⋊C₈	—	2	1	no	—
92	C₂×C₆₇⋊C₈	—	2	1	no	—
93	C₂×C₆₇⋊C₈	—	2	1	no	—
94	C₂×C₆₇⋊C₈	—	4	2	no	—
95	C₂×C₆₇⋊C₈	—	4	1	no	—
96	C₈×D₆₇	—	4	4	no	—
97	C₄⋊Dic₆₇	—	2	1	no	—
98	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
100	(C₂×D₆₇)⋊C₄	—	2	1	no	—
101	(C₂×D₆₇)⋊C₄	—	2	1	no	—
102	(C₂×D₆₇)⋊C₄	—	2	1	no	—
103	(C₂×D₆₇)⋊C₄	—	2	1	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
107	M₁₆×C₆₇	—	4	1	no	—
108	M₁₆×C₆₇	—	4	1	no	—
109	C₂×C₈×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	2	1	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	4	1	no	—
113	C₂×C₈×C₆₇	—	4	1	no	—
114	C₂×C₈×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₇	#41	1	1	yes	characteristic
119	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂²×D₆₇	—	2	1	no	—
121	C₂²×D₆₇	—	2	1	no	—
122	C₂²×D₆₇	—	4	2	no	—
123	C₂²×D₆₇	—	4	1	no	—
124	C₂²×D₆₇	—	4	1	no	—
125	C₂²×D₆₇	—	8	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
137	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
139	C₂×Dic₆₇	—	2	1	no	—
140	C₂×Dic₆₇	—	2	1	no	—
141	C₂×Dic₆₇	—	4	1	no	—
142	C₂×Dic₆₇	—	4	1	no	—
143	D₂₆₈	—	2	1	no	—
144	D₂₆₈	—	2	1	no	—
145	D₂₆₈	—	2	1	no	—
146	D₂₆₈	—	4	2	no	—
147	D₂₆₈	#108	1	2	yes	complemented
148	C₄×D₆₇	—	4	1	no	—
149	C₄×D₆₇	—	4	1	no	—
150	C₄×D₆₇	—	4	1	no	—
151	C₄×D₆₇	—	4	1	no	—
152	C₄×D₆₇	—	4	1	no	—
153	C₄×D₆₇	—	4	1	no	—
154	C₄×D₆₇	—	4	1	no	—
155	C₄×D₆₇	—	4	1	no	—
156	C₈×C₆₇	—	4	1	no	—
157	C₈×C₆₇	—	4	1	no	—
158	C₈×C₆₇	—	4	2	no	—
159	C₈×C₆₇	—	8	1	no	—
160	C₆₇⋊C₈	—	2	2	no	—
161	C₆₇⋊C₈	—	2	2	no	—
162	C₆₇⋊C₈	—	4	1	no	—
163	C₆₇⋊C₈	—	4	1	no	—
164	C₆₇⋊C₈	—	8	1	no	—
165	C₂²×C₆₇	—	2	1	no	—
166	C₂²×C₆₇	—	2	1	no	—
167	C₂²×C₆₇	#385	1	1	yes	characteristic
168	C₂²×C₆₇	#474	1	1	yes	characteristic
169	C₂²×C₆₇	#478	1	1	yes	characteristic
170	C₂×D₆₇	—	2	2	no	—
171	C₂×D₆₇	—	4	1	no	—
172	C₂×D₆₇	—	4	1	no	—
173	C₂×D₆₇	—	4	1	no	—
174	C₂×D₆₇	—	4	1	no	—
175	C₂×D₆₇	—	4	1	no	—
176	C₂×D₆₇	—	8	2	no	—
177	C₂×D₆₇	—	8	2	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
182	C₄×C₆₇	#430	1	1	yes	characteristic
183	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
184	C₄×C₆₇	#440	1	1	yes	characteristic
185	C₄×C₆₇	#447	1	1	yes	characteristic
186	C₄×C₆₇	#452	1	1	yes	characteristic
187	Dic₆₇	—	4	1	no	—
188	Dic₆₇	—	4	1	no	—
189	#8906	—	67	1	no	maximal, Sylow
190	C₂×C₆₇	—	2	1	no	—
191	C₂×C₆₇	—	2	1	no	—
192	C₂×C₆₇	#1855	1	1	yes	characteristic
193	C₂×C₆₇	#1861	1	1	yes	characteristic
194	C₂×C₆₇	#1862	1	1	yes	characteristic
195	D₆₇	—	4	2	no	—
196	D₆₇	—	8	1	no	—
197	D₄×M₁₆	—	67	1	no	—
198	#1950	—	67	1	no	—
199	#1567	—	67	1	no	—
200	#1486	—	67	1	no	—
201	C₂×D₄⋊C₈	—	67	1	no	—
202	#1301	D₆₇	1	1	yes	characteristic, complemented
203	#1278	—	67	1	no	—
204	C₆₇	#8906	1	1	yes	characteristic, complemented, Sylow
205	C₂²×M₁₆	—	134	1	no	—
206	C₂×C₄×D₄	—	67	1	no	—
207	#436	—	134	1	no	—
208	#435	—	134	2	no	—
209	C₈×D₄	—	134	1	no	—
210	#431	C₂×D₆₇	1	1	yes	characteristic
211	C₂×#108	—	134	1	no	—
212	#423	—	67	1	no	—
213	#423	C₂×D₆₇	1	1	yes	characteristic
214	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
215	#407	—	134	1	no	—
216	C₄×M₁₆	—	67	1	no	—
217	C₂×C₄×C₈	—	67	1	no	—
218	#341	—	134	1	no	—
219	#335	—	134	2	no	—
220	D₄⋊C₈	—	67	1	no	—
221	D₄⋊C₈	—	67	1	no	—
222	D₄⋊C₈	—	67	1	no	—
223	D₄⋊C₈	—	67	1	no	—
224	D₄⋊C₈	—	134	1	no	—
225	D₄⋊C₈	—	134	1	no	—
226	C₂²×D₄	—	67	1	no	—
227	C₂³×C₄	—	134	1	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	268	2	no	—
230	C₂×M₁₆	—	268	2	no	—
231	C₂×M₁₆	—	67	1	no	—
232	C₂×M₁₆	—	67	1	no	—
233	C₂×M₁₆	—	134	2	no	—
234	C₂×M₁₆	—	134	1	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	134	2	no	—
237	C₂²×C₈	—	134	1	no	—
238	C₄×D₄	—	67	1	no	—
239	C₄×D₄	—	67	1	no	—
240	C₄×D₄	—	67	1	no	—
241	C₄×D₄	—	67	1	no	—
242	C₄×D₄	—	134	1	no	—
243	C₄×D₄	—	134	1	no	—
244	C₂×C₄⋊C₄	—	67	1	no	—
245	C₂×C₂²⋊C₄	—	134	1	no	—
246	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
247	#108	—	268	2	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	134	1	no	—
252	C₄⋊C₈	—	134	1	no	—
253	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
254	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
255	C₄⋊C₈	D₂₆₈	1	2	yes	—
256	C₂²⋊C₈	—	2	1	no	—
257	C₂²⋊C₈	—	134	2	no	—
258	C₂²⋊C₈	—	134	2	no	—
259	#97	—	134	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	67	1	no	—
262	C₄×C₈	—	67	1	no	—
263	C₄×C₈	—	134	1	no	—
264	C₄×C₈	—	134	1	no	—
265	C₂⁴	—	134	1	no	—
266	C₂×D₄	—	268	1	no	—
267	C₂×D₄	—	67	1	no	—
268	C₂×D₄	—	67	1	no	—
269	C₂×D₄	—	134	2	no	—
270	C₂×D₄	—	134	1	no	—
271	C₂²×C₄	—	268	1	no	—
272	C₂²×C₄	—	268	1	no	—
273	C₂²×C₄	—	268	1	no	—
274	C₂²×C₄	—	268	1	no	—
275	C₂²×C₄	—	134	1	no	—
276	C₂²×C₄	—	134	1	no	—
277	C₂²×C₄	—	134	1	no	—
278	C₂²×C₄	—	134	1	no	—
279	C₂²×C₄	—	134	1	no	—
280	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
281	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
282	C₂²×C₄	#11761628	1	1	yes	characteristic
283	M₁₆	—	4	1	no	—
284	M₁₆	—	4	1	no	—
285	M₁₆	—	268	4	no	—
286	M₁₆	—	268	1	no	—
287	M₁₆	—	268	1	no	—
288	M₁₆	—	67	2	no	—
289	M₁₆	—	134	1	no	—
290	M₁₆	—	134	1	no	—
291	M₁₆	—	134	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	4	1	no	—
296	C₂×C₈	—	4	1	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	268	4	no	—
299	C₂×C₈	—	268	2	no	—
300	C₂×C₈	—	268	1	no	—
301	C₂×C₈	—	134	1	no	—
302	C₂×C₈	—	134	1	no	—
303	C₂×C₈	—	134	1	no	—
304	C₂×C₈	—	134	1	no	—
305	C₄⋊C₄	—	67	1	no	—
306	C₄⋊C₄	—	67	1	no	—
307	C₄⋊C₄	—	134	1	no	—
308	C₂²⋊C₄	—	134	1	no	—
309	C₂²⋊C₄	—	134	1	no	—
310	C₂²⋊C₄	—	134	1	no	—
311	C₂²⋊C₄	—	134	1	no	—
312	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
313	C₄²	D₄×D₆₇	1	1	yes	characteristic
314	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
315	C₄²	#11761627	1	1	yes	characteristic
316	C₂³	—	268	2	no	—
317	C₂³	—	268	1	no	—
318	C₂³	—	268	1	no	—
319	C₂³	—	536	1	no	—
320	C₂³	—	134	1	no	—
321	C₂³	—	134	1	no	—
322	C₂³	#14524021	1	1	yes	characteristic
323	D₄	—	268	2	no	—
324	D₄	—	67	2	no	—
325	D₄	—	134	1	no	—
326	D₄	—	134	1	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	1	no	—
332	C₂×C₄	—	268	1	no	—
333	C₂×C₄	—	268	1	no	—
334	C₂×C₄	—	268	1	no	—
335	C₂×C₄	—	268	1	no	—
336	C₂×C₄	—	268	1	no	—
337	C₂×C₄	—	268	1	no	—
338	C₂×C₄	—	268	1	no	—
339	C₂×C₄	—	268	1	no	—
340	C₂×C₄	—	268	1	no	—
341	C₂×C₄	—	134	1	no	—
342	C₂×C₄	—	134	1	no	—
343	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
344	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
345	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
346	C₂×C₄	#14524013	1	1	yes	characteristic
347	C₂×C₄	#14524023	1	1	yes	characteristic
348	C₂×C₄	#14524025	1	1	yes	characteristic
349	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
350	C₂×C₄	#14524064	1	1	yes	characteristic
351	C₂×C₄	#14524070	1	1	yes	characteristic
352	C₂×C₄	#14524075	1	1	yes	characteristic
353	C₈	—	4	2	no	—
354	C₈	—	4	1	no	—
355	C₈	—	4	1	no	—
356	C₈	—	8	1	no	—
357	C₈	—	268	1	no	—
358	C₈	—	268	1	no	—
359	C₈	—	536	1	no	—
360	C₈	—	134	2	no	—
361	C₈	—	134	2	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	268	1	no	—
365	C₂²	—	268	1	no	—
366	C₂²	—	268	1	no	—
367	C₂²	—	268	1	no	—
368	C₂²	—	268	1	no	—
369	C₂²	—	536	2	no	—
370	C₂²	—	536	2	no	—
371	C₂²	—	134	2	no	—
372	C₂²	#16863270	1	1	yes	characteristic
373	C₂²	#16863675	1	1	yes	characteristic
374	C₂²	#16863695	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	268	1	no	—
379	C₄	—	268	1	no	—
380	C₄	#108×D₆₇	1	1	yes	characteristic
381	C₄	#16863443	1	1	yes	characteristic
382	C₄	#16863476	1	1	yes	characteristic
383	C₄	#16863499	1	1	yes	characteristic
384	C₄	#16863531	1	1	yes	characteristic
385	C₄	#16863561	1	1	yes	characteristic
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	268	2	no	—
389	C₂	—	536	1	no	—
390	C₂	#22560415	1	1	yes	characteristic
391	C₂	#22560452	1	1	yes	characteristic
392	C₂	#22560457	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524013
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1301	C₂
socle	C₂²×C₆₇	#385
prime radical	C₂²×D₂₆₈	Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1301	D₆₇
hypocenter	C₆₇	#8906
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #22560985	C₂	semidirect
3. #22558967	C₂	semidirect
6. #22558045	C₂	semidirect
7. C₆₇×#1301	C₂	semidirect
147. D₂₆₈	#108	semidirect
202. #1301	D₆₇	semidirect
204. C₆₇	#8906	semidirect

Maximal subgroups

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

subgroup	index
#22568944	2
#22560985	2
#22558967	2
#22558635	2
C₂×#16863123	2
#22558045	2
C₆₇×#1301	2
#8906	67