Subgroups — #30748539

Order 17152.

492 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
4	#22558050	C₂	1	2	yes	maximal
5	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
11	#16863151	—	2	1	no	—
12	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#391×C₆₇	C₄	1	1	yes	characteristic
15	#391×C₆₇	C₄	1	1	yes	characteristic
16	#389×C₆₇	C₄	1	1	yes	characteristic
17	#389×C₆₇	C₄	1	1	yes	characteristic
18	#387×C₆₇	—	2	1	no	—
19	#387×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	#376×C₆₇	—	2	1	no	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₄×C₆₇⋊C₈	—	2	2	no	—
33	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₆₇	—	2	1	no	—
35	C₄×Q₈×C₆₇	—	2	1	no	—
36	C₄×Q₈×C₆₇	D₄	1	2	yes	—
37	C₄×Q₈×C₆₇	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	—	2	1	no	—
45	C₂×C₄²×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	#95×C₆₇	—	2	1	no	—
54	#95×C₆₇	—	2	1	no	—
55	#95×C₆₇	—	2	1	no	—
56	#95×C₆₇	—	2	1	no	—
57	#95×C₆₇	—	2	1	no	—
58	#95×C₆₇	—	2	1	no	—
59	#95×C₆₇	—	2	1	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂×Q₈×C₆₇	—	2	1	no	—
67	C₂×Q₈×C₆₇	—	2	2	no	—
68	C₂×Q₈×C₆₇	—	2	2	no	—
69	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₄⋊C₄×C₆₇	—	4	2	no	—
88	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₆₇	—	2	2	no	—
90	C₄²×C₆₇	—	2	2	no	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	2	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	4	1	no	—
97	C₄²×C₆₇	—	4	1	no	—
98	C₄²×C₆₇	—	4	1	no	—
99	C₄²×C₆₇	SD₁₆	1	2	yes	—
100	C₄²×C₆₇	Q₁₆	1	2	yes	—
101	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
103	C₄²×C₆₇	#41	1	1	yes	characteristic
104	C₄²×C₆₇	#41	1	1	yes	characteristic
105	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	Q₈×C₆₇	—	2	4	no	—
107	Q₈×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	4	1	no	—
131	C₂×C₄×C₆₇	#103	1	2	yes	—
132	C₂×C₄×C₆₇	#103	1	2	yes	—
133	C₂×C₄×C₆₇	#104	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#131	1	2	yes	—
141	C₂×C₄×C₆₇	#131	1	2	yes	—
142	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
146	C₆₇⋊C₈	—	4	4	no	—
147	C₆₇⋊C₈	—	8	2	no	—
148	C₂²×C₆₇	#380	1	1	yes	characteristic
149	C₂²×C₆₇	#408	1	1	yes	characteristic
150	C₂²×C₆₇	#408	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
152	C₂²×C₆₇	#417	1	1	yes	characteristic
153	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
154	C₂²×C₆₇	#421	1	1	yes	characteristic
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	—	2	2	no	—
160	C₄×C₆₇	—	4	2	no	—
161	C₄×C₆₇	—	4	2	no	—
162	C₄×C₆₇	—	4	1	no	—
163	C₄×C₆₇	—	8	1	no	—
164	C₄×C₆₇	—	8	1	no	—
165	C₄×C₆₇	#424	1	2	yes	—
166	C₄×C₆₇	#433	1	2	yes	—
167	C₄×C₆₇	#460	1	1	yes	characteristic
168	C₄×C₆₇	#462	1	1	yes	characteristic
169	C₄×C₆₇	#475	1	1	yes	characteristic
170	C₄×C₆₇	#477	1	1	yes	characteristic
171	#8568	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	#1481	1	1	yes	characteristic
173	C₂×C₆₇	#1483	1	1	yes	characteristic
174	C₂×C₆₇	#1493	1	1	yes	characteristic
175	C₂×C₆₇	#1495	1	1	yes	characteristic
176	C₂×C₆₇	#1798	1	1	yes	characteristic
177	C₂×C₆₇	#1803	1	1	yes	characteristic
178	C₂×C₆₇	#1806	1	1	yes	characteristic
179	#2295	D₆₇	1	1	yes	characteristic
180	#1842	—	67	1	no	—
181	C₄×C₄⋊C₈	—	67	1	no	—
182	C₂×Q₈⋊C₈	—	67	2	no	—
183	#1279	—	67	2	no	—
184	C₆₇	#8568	1	1	yes	characteristic, complemented, Sylow
185	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	67	2	no	—
187	C₂×C₄⋊C₈	—	134	1	no	—
188	C₂×C₄⋊C₈	—	134	1	no	—
189	C₂×C₄⋊C₈	—	134	1	no	—
190	C₂×C₄×C₈	—	67	2	no	—
191	#391	Dic₆₇	1	1	yes	characteristic
192	#391	Dic₆₇	1	1	yes	characteristic
193	#389	Dic₆₇	1	1	yes	characteristic
194	#389	Dic₆₇	1	1	yes	characteristic
195	#387	—	2	1	no	—
196	#387	—	2	1	no	—
197	C₄×C₄⋊C₄	—	2	1	no	—
198	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
199	#376	—	2	1	no	—
200	C₄³	C₂×D₆₇	1	1	yes	characteristic
201	#336	—	134	1	no	—
202	Q₈⋊C₈	—	134	4	no	—
203	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
204	C₂²×C₈	—	134	2	no	—
205	C₂²×C₈	—	134	2	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	Dic₁₃₄	1	2	yes	—
209	C₄×Q₈	D₂₆₈	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
225	C₄⋊C₈	—	268	2	no	—
226	C₄⋊C₈	—	268	2	no	—
227	C₄⋊C₈	—	134	4	no	—
228	C₄⋊C₈	—	134	2	no	—
229	C₄⋊C₈	—	134	2	no	—
230	C₄×C₈	—	134	2	no	—
231	C₄×C₈	—	134	2	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	#95	—	2	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
254	C₂×C₈	—	268	2	no	—
255	C₂×C₈	—	268	2	no	—
256	C₂×C₈	—	268	2	no	—
257	C₂×C₈	—	268	2	no	—
258	C₂×C₈	—	268	2	no	—
259	C₂×C₈	—	134	4	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	#11761615	1	2	yes	—
279	C₄²	#11761618	1	2	yes	—
280	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
281	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
282	C₄²	#11761628	1	1	yes	characteristic
283	C₄²	#11761628	1	1	yes	characteristic
284	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	4	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	#14523967	1	2	yes	—
311	C₂×C₄	#14523967	1	2	yes	—
312	C₂×C₄	#14523970	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
314	C₂×C₄	#14524002	1	1	yes	characteristic
315	C₂×C₄	#14524017	1	1	yes	characteristic
316	C₂×C₄	#14524017	1	1	yes	characteristic
317	C₂×C₄	#14524022	1	1	yes	characteristic
318	C₂×C₄	#14524022	1	1	yes	characteristic
319	C₂×C₄	#14524050	1	2	yes	—
320	C₂×C₄	#14524050	1	2	yes	—
321	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
322	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
323	C₂×C₄	#14524068	1	1	yes	characteristic
324	C₂×C₄	#14524073	1	1	yes	characteristic
325	C₈	—	268	4	no	—
326	C₈	—	536	2	no	—
327	C₂²	#16863256	1	1	yes	characteristic
328	C₂²	#16863363	1	1	yes	characteristic
329	C₂²	#16863363	1	1	yes	characteristic
330	C₂²	C₂×#14523967	1	1	yes	characteristic
331	C₂²	#16863397	1	1	yes	characteristic
332	C₂²	C₂×#14523970	1	1	yes	characteristic
333	C₂²	#16863413	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	1	no	—
344	C₄	#16863423	1	2	yes	—
345	C₄	#16863457	1	2	yes	—
346	C₄	#16863596	1	1	yes	characteristic
347	C₄	#16863606	1	1	yes	characteristic
348	C₄	#16863681	1	1	yes	characteristic
349	C₄	#16863691	1	1	yes	characteristic
350	C₂	#22558612	1	1	yes	characteristic
351	C₂	#22558622	1	1	yes	characteristic
352	C₂	#22558664	1	1	yes	characteristic
353	C₂	#22558674	1	1	yes	characteristic
354	C₂	#22560193	1	1	yes	characteristic
355	C₂	#22560214	1	1	yes	characteristic
356	C₂	#22560229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2295	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2295	D₆₇
hypocenter	C₆₇	#8568
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
184. C₆₇	#8568	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
#22558050	2
C₆₇×#2295	2
#8568	67