Subgroups — #30772302

Order 17152.

420 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863217	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863216	C₂	1	1	yes	maximal, characteristic, complemented
4	#22561819	C₂	1	1	yes	maximal, characteristic
5	C₈×(C₂×D₆₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#22558290	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	C₂×#14523983	C₂²	1	1	yes	characteristic
11	C₂×#14523982	—	2	1	no	—
12	M₃₂×D₆₇	—	2	2	no	—
13	M₃₂×D₆₇	—	2	2	no	—
14	C₂×C₁₆×D₆₇	—	2	1	no	—
15	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	C₄×(C₂×D₆₇)⋊C₄	C₄	1	1	yes	characteristic
18	#16863217	—	2	2	no	—
19	#16863216	—	2	2	no	—
20	(C₂×D₆₇)⋊C₁₆	—	2	1	no	—
21	(C₂×D₆₇)⋊C₁₆	—	2	1	no	—
22	#16863151	C₄	1	1	yes	characteristic
23	#16863150	C₄	1	1	yes	characteristic
24	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₆₇	—	2	1	no	—
31	C₂×C₈×D₆₇	—	2	1	no	—
32	C₂×C₈×D₆₇	—	2	1	no	—
33	C₂×C₈×D₆₇	—	2	1	no	—
34	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
35	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
36	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
38	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
39	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
40	#14523983	—	2	1	no	—
41	#14523983	—	2	1	no	—
42	#14523983	—	2	1	no	—
43	#14523983	D₄	1	2	yes	—
44	#14523982	—	2	4	no	—
45	C₂×C₆₇⋊C₁₆	—	2	1	no	—
46	C₂×C₆₇⋊C₁₆	—	2	1	no	—
47	C₁₆×D₆₇	—	2	4	no	—
48	(C₂×D₆₇)⋊C₈	C₈	1	2	yes	—
49	(C₂×D₆₇)⋊C₈	C₈	1	2	yes	—
50	C₈×Dic₆₇	—	2	1	no	—
51	C₈×Dic₆₇	—	2	1	no	—
52	C₆₇⋊#95	—	2	1	no	—
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₆₇	—	2	1	no	—
57	M₃₂×C₆₇	—	2	1	no	—
58	M₃₂×C₆₇	—	2	1	no	—
59	M₃₂×C₆₇	D₄	1	2	yes	—
60	C₂×C₁₆×C₆₇	—	2	1	no	—
61	C₂×C₁₆×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	—	2	1	no	—
63	C₄×C₈×C₆₇	—	2	1	no	—
64	C₂³×D₆₇	M₁₆	1	1	yes	characteristic
65	C₂²×Dic₆₇	—	2	1	no	—
66	C₂²×Dic₆₇	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₆₇	—	2	1	no	—
68	C₂×C₄×D₆₇	—	2	1	no	—
69	C₂×C₄×D₆₇	—	2	1	no	—
70	C₂×C₄×D₆₇	—	2	1	no	—
71	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
75	C₂×C₆₇⋊C₈	—	2	1	no	—
76	C₂×C₆₇⋊C₈	—	2	1	no	—
77	C₂×C₆₇⋊C₈	—	2	1	no	—
78	C₂×C₆₇⋊C₈	—	2	1	no	—
79	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
82	C₈×D₆₇	—	2	1	no	—
83	C₈×D₆₇	—	2	1	no	—
84	C₈×D₆₇	—	2	1	no	—
85	C₈×D₆₇	—	2	1	no	—
86	C₈×D₆₇	—	2	1	no	—
87	C₈×D₆₇	—	2	1	no	—
88	C₈×D₆₇	—	2	1	no	—
89	C₈×D₆₇	—	2	1	no	—
90	(C₂×D₆₇)⋊C₄	—	2	4	no	—
91	C₄×Dic₆₇	—	2	1	no	—
92	C₄×Dic₆₇	—	2	1	no	—
93	C₆₇⋊C₁₆	—	2	2	no	—
94	C₆₇⋊C₁₆	—	4	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	—	2	1	no	—
98	C₂×C₈×C₆₇	—	2	1	no	—
99	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
105	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
106	C₄²×C₆₇	—	2	1	no	—
107	C₄²×C₆₇	—	2	1	no	—
108	C₁₆×C₆₇	—	2	2	no	—
109	C₁₆×C₆₇	—	4	1	no	—
110	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂²×D₆₇	—	2	1	no	—
112	C₂²×D₆₇	—	2	1	no	—
113	C₂²×D₆₇	—	2	2	no	—
114	C₂²×D₆₇	—	4	1	no	—
115	C₂²×D₆₇	C₄⋊C₈	1	1	yes	characteristic
116	C₂²×D₆₇	C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
126	C₂×Dic₆₇	—	2	1	no	—
127	C₂×Dic₆₇	—	2	1	no	—
128	C₂×Dic₆₇	—	2	2	no	—
129	C₂×Dic₆₇	—	4	1	no	—
130	C₂×Dic₆₇	—	4	1	no	—
131	C₂×Dic₆₇	C₄⋊C₈	1	1	yes	characteristic
132	C₂×Dic₆₇	C₄⋊C₈	1	1	yes	characteristic
133	C₄×D₆₇	—	2	1	no	—
134	C₄×D₆₇	—	2	1	no	—
135	C₄×D₆₇	—	2	1	no	—
136	C₄×D₆₇	—	2	1	no	—
137	C₄×D₆₇	—	2	1	no	—
138	C₄×D₆₇	—	2	1	no	—
139	C₄×D₆₇	—	2	1	no	—
140	C₄×D₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	2	no	—
142	C₈×C₆₇	#120	1	1	yes	characteristic
143	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₆₇	#123	1	1	yes	characteristic
146	C₆₇⋊C₈	—	2	1	no	—
147	C₆₇⋊C₈	—	2	1	no	—
148	C₆₇⋊C₈	—	2	2	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂²×C₆₇	#435	1	1	yes	characteristic
153	C₂²×C₆₇	#435	1	1	yes	characteristic
154	C₂×D₆₇	—	2	1	no	—
155	C₂×D₆₇	—	2	1	no	—
156	C₂×D₆₇	—	2	1	no	—
157	C₂×D₆₇	—	2	1	no	—
158	C₂×D₆₇	—	2	1	no	—
159	C₂×D₆₇	—	4	2	no	—
160	C₂×D₆₇	—	4	2	no	—
161	C₂×D₆₇	#364	1	2	yes	complemented
162	C₄×C₆₇	—	4	2	no	—
163	C₄×C₆₇	#385	1	1	yes	characteristic
164	C₄×C₆₇	#424	1	1	yes	characteristic
165	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
166	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
167	Dic₆₇	—	2	1	no	—
168	Dic₆₇	—	2	1	no	—
169	Dic₆₇	—	4	2	no	—
170	#12914	—	67	1	no	maximal, Sylow
171	C₂×C₆₇	—	2	1	no	—
172	C₂×C₆₇	—	2	1	no	—
173	C₂×C₆₇	#1850	1	1	yes	characteristic
174	C₂×C₆₇	C₂×#364	1	1	yes	characteristic
175	C₂×C₆₇	#2153	1	1	yes	characteristic
176	D₆₇	—	2	2	no	—
177	D₆₇	—	4	1	no	—
178	C₂²×M₃₂	—	67	1	no	—
179	C₂×#364	—	67	1	no	—
180	C₂×#364	—	67	1	no	—
181	#2112	—	67	1	no	—
182	C₈×C₂²⋊C₄	—	67	1	no	—
183	#1376	—	67	1	no	—
184	#1376	D₆₇	1	1	yes	characteristic, complemented
185	C₆₇	#12914	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₈	—	67	1	no	—
187	C₂×M₃₂	—	67	1	no	—
188	C₂×M₃₂	—	67	1	no	—
189	C₂×M₃₂	—	134	1	no	—
190	C₂×M₃₂	—	134	2	no	—
191	C₂×M₃₂	—	134	2	no	—
192	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
193	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
194	C₂²×C₁₆	—	134	1	no	—
195	C₂×C₂²⋊C₈	—	67	1	no	—
196	C₂×C₄×C₈	—	67	1	no	—
197	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
198	C₄×C₂²⋊C₄	—	67	1	no	—
199	#364	—	134	2	no	—
200	#364	—	134	2	no	—
201	C₂²⋊C₁₆	—	134	1	no	—
202	C₂²⋊C₁₆	—	134	1	no	—
203	#336	—	67	1	no	—
204	#336	—	67	1	no	—
205	C₂³×C₄	—	67	1	no	—
206	C₂²×C₈	—	67	1	no	—
207	C₂²×C₈	—	67	1	no	—
208	C₂²×C₈	—	67	1	no	—
209	C₂²×C₈	—	67	1	no	—
210	C₂²×C₈	—	67	1	no	—
211	C₂²×C₈	—	67	1	no	—
212	C₂²×C₈	—	134	1	no	—
213	C₂²×C₈	—	134	1	no	—
214	C₂²×C₈	—	134	1	no	—
215	C₂²×C₈	—	134	1	no	—
216	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	134	1	no	—
219	C₂×C₄²	—	67	1	no	—
220	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	67	2	no	—
225	M₃₂	—	134	4	no	—
226	M₃₂	—	134	1	no	—
227	M₃₂	—	134	1	no	—
228	M₃₂	—	134	1	no	—
229	M₃₂	D₂₆₈	1	2	yes	—
230	C₂×C₁₆	—	2	1	no	—
231	C₂×C₁₆	—	2	1	no	—
232	C₂×C₁₆	—	134	4	no	—
233	C₂×C₁₆	—	134	1	no	—
234	C₂×C₁₆	—	134	1	no	—
235	C₂²⋊C₈	—	67	2	no	—
236	C₂²⋊C₈	—	67	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	134	1	no	—
240	C₄×C₈	—	134	1	no	—
241	#95	—	134	1	no	—
242	C₂⁴	—	67	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	67	1	no	—
245	C₂²×C₄	—	67	1	no	—
246	C₂²×C₄	—	67	1	no	—
247	C₂²×C₄	—	67	1	no	—
248	C₂²×C₄	—	67	1	no	—
249	C₂²×C₄	—	134	1	no	—
250	C₂²×C₄	—	134	1	no	—
251	C₂²×C₄	—	134	1	no	—
252	C₂²×C₄	—	134	1	no	—
253	C₂²×C₄	—	134	1	no	—
254	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	67	1	no	—
258	C₂×C₈	—	67	1	no	—
259	C₂×C₈	—	67	2	no	—
260	C₂×C₈	—	134	1	no	—
261	C₂×C₈	—	134	1	no	—
262	C₂×C₈	—	134	1	no	—
263	C₂×C₈	—	134	1	no	—
264	C₂×C₈	—	134	1	no	—
265	C₂×C₈	—	134	1	no	—
266	C₂×C₈	—	134	1	no	—
267	C₂×C₈	—	134	1	no	—
268	C₂×C₈	—	134	1	no	—
269	C₂×C₈	—	134	1	no	—
270	C₂×C₈	—	134	1	no	—
271	C₂×C₈	—	134	1	no	—
272	C₂×C₈	C₈×D₆₇	1	2	yes	—
273	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
274	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
275	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
276	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
277	C₂×C₈	#11761627	1	1	yes	characteristic
278	C₂×C₈	#11761628	1	1	yes	characteristic
279	C₂²⋊C₄	—	134	4	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	134	1	no	—
283	C₄²	—	134	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	268	1	no	—
287	C₁₆	—	134	2	no	—
288	C₂³	—	268	1	no	—
289	C₂³	—	67	1	no	—
290	C₂³	—	67	1	no	—
291	C₂³	—	134	2	no	—
292	C₂³	—	134	1	no	—
293	C₂³	—	134	1	no	—
294	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	268	1	no	—
299	C₂×C₄	—	268	1	no	—
300	C₂×C₄	—	67	1	no	—
301	C₂×C₄	—	67	1	no	—
302	C₂×C₄	—	134	2	no	—
303	C₂×C₄	—	134	1	no	—
304	C₂×C₄	—	134	1	no	—
305	C₂×C₄	—	134	1	no	—
306	C₂×C₄	—	134	1	no	—
307	C₂×C₄	—	134	1	no	—
308	C₂×C₄	—	134	1	no	—
309	C₂×C₄	—	134	1	no	—
310	C₂×C₄	—	134	1	no	—
311	C₂×C₄	—	134	1	no	—
312	C₂×C₄	—	134	1	no	—
313	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
314	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
315	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
317	C₂×C₄	#14524049	1	1	yes	characteristic
318	C₂×C₄	#14524051	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	134	1	no	—
321	C₈	—	134	1	no	—
322	C₈	—	134	2	no	—
323	C₈	#14524013	1	1	yes	characteristic
324	C₈	#14524021	1	1	yes	characteristic
325	C₈	#14524023	1	1	yes	characteristic
326	C₈	#14524025	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	268	2	no	—
330	C₂²	—	268	2	no	—
331	C₂²	—	67	2	no	—
332	C₂²	—	134	1	no	—
333	C₂²	—	134	1	no	—
334	C₂²	—	134	1	no	—
335	C₂²	—	134	1	no	—
336	C₂²	—	134	1	no	—
337	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
338	C₂²	#16863469	1	1	yes	characteristic
339	C₂²	#16863470	1	1	yes	characteristic
340	C₄	—	4	2	no	—
341	C₄	—	268	2	no	—
342	C₄	—	134	1	no	—
343	C₄	—	134	1	no	—
344	C₄	#16863270	1	1	yes	characteristic
345	C₄	#16863421	1	1	yes	characteristic
346	C₄	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
347	C₄	C₈×D₂₆₈	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	268	1	no	—
351	C₂	—	134	2	no	—
352	C₂	#22560388	1	1	yes	characteristic
353	C₂	#364×D₆₇	1	1	yes	characteristic
354	C₂	#22561973	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1376	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×D₆₇	M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1376	D₆₇
hypocenter	C₆₇	#12914
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
2. C₂×#16863217	C₂	semidirect
3. C₂×#16863216	C₂	semidirect
6. #22558290	C₂	semidirect
7. C₆₇×#1376	C₂	semidirect
161. C₂×D₆₇	#364	semidirect
184. #1376	D₆₇	semidirect
185. C₆₇	#12914	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₆₇	2
C₂×#16863217	2
C₂×#16863216	2
#22561819	2
C₈×(C₂×D₆₇)⋊C₄	2
#22558290	2
C₆₇×#1376	2
#12914	67