Subgroups — #30763934

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863164	C₂	1	2	yes	maximal
3	#22559930	C₂	1	2	yes	maximal
4	C₂×#16863159	C₂	1	1	yes	maximal, characteristic
5	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×#14523979	—	2	4	no	—
9	C₂×#14523979	—	2	2	no	—
10	#16863164	—	2	4	no	—
11	#16863159	—	2	2	no	—
12	#16863158	—	2	2	no	—
13	#16863158	—	2	2	no	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	#424×C₆₇	—	2	2	no	—
18	#424×C₆₇	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
20	C₂³×Dic₆₇	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	D₄	1	4	yes	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	2	no	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	Q₈	1	4	yes	—
34	#14523979	—	2	8	no	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₆₇	—	2	1	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₂²×C₈×C₆₇	—	2	1	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	C₂²×C₈×C₆₇	D₄	1	2	yes	—
46	#117×C₆₇	—	2	1	no	—
47	#117×C₆₇	—	2	1	no	—
48	#117×C₆₇	C₂×C₄	1	2	yes	—
49	#117×C₆₇	Q₈	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄²×C₆₇	—	2	1	no	—
54	C₄⋊C₈×C₆₇	—	2	8	no	—
55	C₂²×Dic₆₇	—	2	4	no	—
56	C₂²×Dic₆₇	—	4	2	no	—
57	C₂²×Dic₆₇	—	4	2	no	—
58	C₂²×Dic₆₇	—	8	2	no	—
59	C₂²×Dic₆₇	—	8	2	no	—
60	#11761612	—	2	4	no	—
61	#11761612	—	2	4	no	—
62	#11761612	—	4	2	no	—
63	#11761612	—	4	2	no	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂×C₆₇⋊C₈	—	4	2	no	—
67	C₄⋊Dic₆₇	—	2	4	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	4	2	no	—
70	C₄⋊Dic₆₇	—	4	2	no	—
71	C₂⁴×C₆₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	2	yes	—
84	M₁₆×C₆₇	—	2	4	no	—
85	M₁₆×C₆₇	—	4	1	no	—
86	M₁₆×C₆₇	—	4	1	no	—
87	C₂×C₈×C₆₇	—	2	4	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	4	2	no	—
93	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
94	C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₄⋊C₄×C₆₇	—	2	1	no	—
96	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
97	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
98	C₂²⋊C₄×C₆₇	—	2	2	no	—
99	C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	2	2	no	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	—	2	1	no	—
104	C₂³×C₆₇	—	2	1	no	—
105	C₂³×C₆₇	#117	1	1	yes	characteristic
106	C₂³×C₆₇	#121	1	1	yes	characteristic
107	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂³×C₆₇	#126	1	2	yes	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	4	1	no	—
114	C₂×C₄×C₆₇	—	4	1	no	—
115	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#121	1	2	yes	—
121	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#121	1	2	yes	—
123	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#123	1	2	yes	—
128	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#125	1	2	yes	—
130	C₂×C₄×C₆₇	#125	1	2	yes	—
131	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
133	C₂×Dic₆₇	—	4	4	no	—
134	C₂×Dic₆₇	—	8	4	no	—
135	C₂×Dic₆₇	—	8	2	no	—
136	C₂×Dic₆₇	—	8	2	no	—
137	C₂×Dic₆₇	—	8	4	no	—
138	C₈×C₆₇	—	2	4	no	—
139	C₈×C₆₇	—	4	2	no	—
140	C₆₇⋊C₈	—	4	4	no	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	#380	1	1	yes	characteristic
150	C₂²×C₆₇	#387	1	2	yes	—
151	C₂²×C₆₇	#395	1	1	yes	characteristic
152	C₂²×C₆₇	#396	1	2	yes	—
153	C₂²×C₆₇	#400	1	1	yes	characteristic
154	C₂²×C₆₇	#471	1	2	yes	—
155	C₂²×C₆₇	#472	1	1	yes	characteristic
156	C₂²×C₆₇	#473	1	1	yes	characteristic
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	—	4	2	no	—
159	C₄×C₆₇	#386	1	1	yes	characteristic
160	C₄×C₆₇	#389	1	2	yes	—
161	C₄×C₆₇	#391	1	1	yes	characteristic
162	C₄×C₆₇	#394	1	1	yes	characteristic
163	C₄×C₆₇	#398	1	1	yes	characteristic
164	C₄×C₆₇	#398	1	2	yes	—
165	Dic₆₇	—	8	4	no	—
166	#11242	—	67	1	no	maximal, Sylow
167	C₂×C₆₇	—	2	2	no	—
168	C₂×C₆₇	—	2	1	no	—
169	C₂×C₆₇	—	2	1	no	—
170	C₂×C₆₇	#1441	1	1	yes	characteristic
171	C₂×C₆₇	#1815	1	1	yes	characteristic
172	C₂×C₆₇	#1816	1	1	yes	characteristic
173	C₂×C₆₇	#2063	1	2	yes	—
174	C₂×C₆₇	#2064	1	2	yes	—
175	C₂²×#108	—	67	1	no	—
176	C₂×#424	D₆₇	1	1	yes	characteristic
177	C₂×#343	—	67	2	no	—
178	#1736	—	67	2	no	—
179	C₂×#340	—	67	1	no	—
180	C₆₇	#11242	1	1	yes	characteristic, complemented, Sylow
181	C₂²×M₁₆	—	67	2	no	—
182	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
183	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
184	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	—	67	2	no	—
186	C₂×#108	—	134	4	no	—
187	C₂×#108	—	134	2	no	—
188	#424	—	2	2	no	—
189	#424	Dic₆₇	1	4	yes	complemented
190	C₂×C₄⋊C₈	—	2	2	no	—
191	#343	—	134	4	no	—
192	#340	—	134	2	no	—
193	#340	—	134	2	no	—
194	#340	—	134	2	no	—
195	C₂³×C₄	—	134	2	no	—
196	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	67	4	no	—
201	C₂×M₁₆	—	134	2	no	—
202	C₂×M₁₆	—	134	2	no	—
203	C₂×M₁₆	—	134	2	no	—
204	C₂×M₁₆	—	134	2	no	—
205	C₂×M₁₆	—	134	2	no	—
206	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	134	2	no	—
211	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
212	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
213	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
214	#117	—	2	1	no	—
215	#117	—	2	1	no	—
216	#117	Dic₁₃₄	1	2	yes	—
217	#117	C₂×Dic₆₇	1	2	yes	—
218	C₂×C₄⋊C₄	—	67	4	no	—
219	C₂×C₄⋊C₄	—	134	2	no	—
220	C₂×C₄⋊C₄	—	134	2	no	—
221	C₂×C₄⋊C₄	—	134	2	no	—
222	C₂×C₄⋊C₄	—	134	2	no	—
223	C₂×C₄⋊C₄	—	134	2	no	—
224	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
225	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₄²	—	2	1	no	—
228	#108	—	134	8	no	—
229	C₄⋊C₈	—	2	8	no	—
230	C₂⁴	#11761628	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	268	2	no	—
234	C₂²×C₄	—	268	2	no	—
235	C₂²×C₄	—	536	2	no	—
236	C₂²×C₄	—	536	2	no	—
237	C₂²×C₄	—	134	4	no	—
238	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
239	C₂²×C₄	D₄×D₆₇	1	2	yes	—
240	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
243	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
244	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
245	C₂²×C₄	#11761627	1	2	yes	—
246	C₂²×C₄	#11761628	1	1	yes	characteristic
247	C₂²×C₄	#11761629	1	2	yes	—
248	M₁₆	—	2	4	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	268	2	no	—
252	M₁₆	—	268	2	no	—
253	M₁₆	—	134	4	no	—
254	M₁₆	—	134	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	268	2	no	—
262	C₂×C₈	—	268	2	no	—
263	C₂×C₈	—	134	4	no	—
264	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	268	2	no	—
268	C₄⋊C₄	—	268	2	no	—
269	C₄⋊C₄	—	134	4	no	—
270	C₄⋊C₄	—	134	4	no	—
271	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
272	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	#14524002	1	1	yes	characteristic
281	C₂³	#14524017	1	1	yes	characteristic
282	C₂³	#14524021	1	2	yes	—
283	C₂³	#14524022	1	1	yes	characteristic
284	C₂³	#14524038	1	2	yes	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	268	4	no	—
291	C₂×C₄	—	536	4	no	—
292	C₂×C₄	—	536	2	no	—
293	C₂×C₄	—	536	2	no	—
294	C₂×C₄	—	536	4	no	—
295	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
297	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
298	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
299	C₂×C₄	#14524011	1	1	yes	characteristic
300	C₂×C₄	#14524015	1	2	yes	—
301	C₂×C₄	#14524017	1	1	yes	characteristic
302	C₂×C₄	#14524018	1	2	yes	—
303	C₂×C₄	#14524019	1	2	yes	—
304	C₂×C₄	#14524022	1	1	yes	characteristic
305	C₂×C₄	#14524023	1	2	yes	—
306	C₂×C₄	#14524024	1	1	yes	characteristic
307	C₂×C₄	#14524028	1	2	yes	—
308	C₂×C₄	#14524032	1	1	yes	characteristic
309	C₂×C₄	#14524033	1	2	yes	—
310	C₂×C₄	#14524034	1	2	yes	—
311	C₂×C₄	#14524041	1	1	yes	characteristic
312	C₂×C₄	#14524042	1	2	yes	—
313	C₈	—	2	4	no	—
314	C₈	—	4	2	no	—
315	C₈	—	268	4	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#16863256	1	1	yes	characteristic
325	C₂²	#16863282	1	2	yes	—
326	C₂²	#16863316	1	1	yes	characteristic
327	C₂²	#16863318	1	2	yes	—
328	C₂²	#16863335	1	1	yes	characteristic
329	C₂²	#16863659	1	2	yes	—
330	C₂²	#16863666	1	1	yes	characteristic
331	C₂²	#16863671	1	1	yes	characteristic
332	C₄	—	4	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	536	4	no	—
335	C₄	#16863276	1	1	yes	characteristic
336	C₄	#16863295	1	2	yes	—
337	C₄	#16863303	1	1	yes	characteristic
338	C₄	#16863313	1	1	yes	characteristic
339	C₄	#16863326	1	1	yes	characteristic
340	C₄	#16863329	1	2	yes	—
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	#22558462	1	1	yes	characteristic
345	C₂	#22560268	1	1	yes	characteristic
346	C₂	#22560273	1	1	yes	characteristic
347	C₂	#22561568	1	2	yes	—
348	C₂	#22561575	1	2	yes	—

	subgroup	G/H
center	C₂³	#14524017
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#424×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#424	D₆₇
hypocenter	C₆₇	#11242
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
18. #424×C₆₇	C₄	semidirect
180. C₆₇	#11242	semidirect
189. #424	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523979	2
C₂×#16863164	2
#22559930	2
C₂×#16863159	2
C₂×#424×C₆₇	2
#11242	67