Subgroups — #30748648

Order 17152.

496 conjugacy classes of subgroups.

The subgroup lattice has 318 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	#22560326	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863142	C₂	1	2	yes	maximal
4	#22558092	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₆₇	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)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
10	#16863151	—	2	1	no	—
11	#16863151	—	2	1	no	—
12	#16863142	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
15	#391×C₆₇	—	2	1	no	—
16	#389×C₆₇	—	2	1	no	—
17	#389×C₆₇	C₄	1	2	yes	—
18	#384×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
30	C₂²×Q₈×C₆₇	—	2	1	no	—
31	C₄⋊Q₈×C₆₇	—	2	1	no	—
32	C₄⋊Q₈×C₆₇	—	2	1	no	—
33	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
34	C₄⋊Q₈×C₆₇	Q₈	1	2	yes	—
35	C₄×Q₈×C₆₇	—	4	4	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	#95×C₆₇	—	2	2	no	—
51	#95×C₆₇	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	—	2	2	no	—
60	C₂×Q₈×C₆₇	—	4	2	no	—
61	C₂×Q₈×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	2	2	no	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	4	2	no	—
75	C₄⋊C₄×C₆₇	—	4	4	no	—
76	C₄⋊C₄×C₆₇	—	4	1	no	—
77	C₄⋊C₄×C₆₇	—	4	2	no	—
78	C₄⋊C₄×C₆₇	—	4	1	no	—
79	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	—	2	2	no	—
83	C₄²×C₆₇	—	2	1	no	—
84	C₄²×C₆₇	—	2	1	no	—
85	C₄²×C₆₇	—	4	1	no	—
86	C₄²×C₆₇	—	4	4	no	—
87	C₄²×C₆₇	SD₁₆	1	2	yes	—
88	C₄²×C₆₇	Q₁₆	1	2	yes	—
89	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₆₇	#41	1	1	yes	characteristic
92	C₄²×C₆₇	#41	1	1	yes	characteristic
93	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₆₇	—	4	4	no	—
95	Q₈×C₆₇	—	4	4	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	1	no	—
101	C₂×C₄×C₆₇	—	2	1	no	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	1	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	#101	1	2	yes	—
114	C₂×C₄×C₆₇	#103	1	2	yes	—
115	C₂×C₄×C₆₇	#103	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#135	1	2	yes	—
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
128	C₆₇⋊C₈	—	8	2	no	—
129	C₆₇⋊C₈	—	8	2	no	—
130	C₂²×C₆₇	#380	1	1	yes	characteristic
131	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
132	C₂²×C₆₇	#413	1	1	yes	characteristic
133	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
134	C₂²×C₆₇	#416	1	1	yes	characteristic
135	C₂²×C₆₇	#417	1	1	yes	characteristic
136	C₂²×C₆₇	#418	1	1	yes	characteristic
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	—	2	1	no	—
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	2	2	no	—
142	C₄×C₆₇	—	4	4	no	—
143	C₄×C₆₇	—	4	1	no	—
144	C₄×C₆₇	—	8	2	no	—
145	C₄×C₆₇	#460	1	1	yes	characteristic
146	C₄×C₆₇	#462	1	1	yes	characteristic
147	C₄×C₆₇	#463	1	2	yes	—
148	C₄×C₆₇	#475	1	1	yes	characteristic
149	C₄×C₆₇	#477	1	1	yes	characteristic
150	C₄×C₆₇	#478	1	2	yes	—
151	#8591	—	67	1	no	maximal, Sylow
152	C₂×C₆₇	#1542	1	1	yes	characteristic
153	C₂×C₆₇	#1549	1	1	yes	characteristic
154	C₂×C₆₇	#1552	1	1	yes	characteristic
155	C₂×C₆₇	#1557	1	1	yes	characteristic
156	C₂×C₆₇	#1801	1	1	yes	characteristic
157	C₂×C₆₇	#1803	1	1	yes	characteristic
158	C₂×C₆₇	#1804	1	1	yes	characteristic
159	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
160	#1842	—	67	1	no	—
161	#1831	—	67	1	no	—
162	C₂×#332	—	67	2	no	—
163	#1296	—	67	2	no	—
164	C₆₇	#8591	1	1	yes	characteristic, complemented, Sylow
165	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
166	C₂×C₄×Q₈	—	2	1	no	—
167	C₂×C₄⋊C₈	—	67	2	no	—
168	C₂×C₄⋊C₈	—	67	2	no	—
169	C₂×C₄⋊C₈	—	134	1	no	—
170	C₂×C₄⋊C₈	—	134	1	no	—
171	#391	—	2	1	no	—
172	#389	—	2	1	no	—
173	#389	Dic₆₇	1	2	yes	—
174	#384	C₂×D₆₇	1	1	yes	characteristic
175	C₄×C₄⋊C₄	—	2	1	no	—
176	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
177	C₄³	C₂×D₆₇	1	1	yes	characteristic
178	#336	—	134	1	no	—
179	#336	—	134	1	no	—
180	#332	—	134	4	no	—
181	C₂²×Q₈	—	2	1	no	—
182	C₂²×C₈	—	134	2	no	—
183	C₂²×C₈	—	134	2	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	Dic₁₃₄	1	2	yes	—
187	C₄⋊Q₈	D₂₆₈	1	2	yes	—
188	C₄×Q₈	—	4	4	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
203	C₄⋊C₈	—	268	2	no	—
204	C₄⋊C₈	—	268	2	no	—
205	C₄⋊C₈	—	134	2	no	—
206	C₄⋊C₈	—	134	2	no	—
207	C₄⋊C₈	—	134	2	no	—
208	C₄⋊C₈	—	134	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
224	C₂×C₈	—	268	2	no	—
225	C₂×C₈	—	268	2	no	—
226	C₂×C₈	—	268	2	no	—
227	C₂×C₈	—	268	2	no	—
228	C₂×C₈	—	268	2	no	—
229	C₂×C₈	—	268	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	4	no	—
246	C₄²	#11761615	1	2	yes	—
247	C₄²	#11761618	1	2	yes	—
248	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
249	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
250	C₄²	#11761628	1	1	yes	characteristic
251	C₄²	#11761628	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	#14523962	1	2	yes	—
273	C₂×C₄	#14523967	1	2	yes	—
274	C₂×C₄	#14523967	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
276	C₂×C₄	#14524002	1	1	yes	characteristic
277	C₂×C₄	#14524017	1	1	yes	characteristic
278	C₂×C₄	#14524017	1	1	yes	characteristic
279	C₂×C₄	#14524022	1	1	yes	characteristic
280	C₂×C₄	#14524022	1	1	yes	characteristic
281	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
282	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
283	C₂×C₄	#14524063	1	2	yes	—
284	C₂×C₄	#14524068	1	1	yes	characteristic
285	C₂×C₄	#14524069	1	2	yes	—
286	C₂×C₄	#14524073	1	1	yes	characteristic
287	C₈	—	536	2	no	—
288	C₈	—	536	2	no	—
289	C₂²	#16863256	1	1	yes	characteristic
290	C₂²	C₂×#14523962	1	1	yes	characteristic
291	C₂²	#16863380	1	1	yes	characteristic
292	C₂²	C₂×#14523967	1	1	yes	characteristic
293	C₂²	#16863392	1	1	yes	characteristic
294	C₂²	#16863397	1	1	yes	characteristic
295	C₂²	#16863401	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	2	no	—
304	C₄	#16863596	1	1	yes	characteristic
305	C₄	#16863606	1	1	yes	characteristic
306	C₄	#16863611	1	2	yes	—
307	C₄	#16863681	1	1	yes	characteristic
308	C₄	#16863691	1	1	yes	characteristic
309	C₄	#16863696	1	2	yes	—
310	C₂	#22558866	1	1	yes	characteristic
311	C₂	#22558894	1	1	yes	characteristic
312	C₂	#22558909	1	1	yes	characteristic
313	C₂	#22558931	1	1	yes	characteristic
314	C₂	#22560205	1	1	yes	characteristic
315	C₂	#22560214	1	1	yes	characteristic
316	C₂	#22560219	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#8591
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
164. C₆₇	#8591	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
#22560326	2
C₂×#16863142	2
#22558092	2
C₄×C₄⋊Q₈×C₆₇	2
#8591	67