Subgroups — #30744124

Order 17152.

402 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560352	C₂	1	1	yes	maximal, characteristic
2	#22560326	C₂	1	1	yes	maximal, characteristic
3	#22558035	C₂	1	2	yes	maximal
4	#22558032	C₂	1	2	yes	maximal
5	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	#16863151	—	2	1	no	—
8	#16863151	—	2	1	no	—
9	#16863151	—	2	1	no	—
10	#16863151	C₂²	1	2	yes	—
11	#16863151	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
13	#391×C₆₇	C₄	1	1	yes	characteristic
14	#391×C₆₇	C₂²	1	1	yes	characteristic
15	#389×C₆₇	C₄	1	1	yes	characteristic
16	#389×C₆₇	C₂²	1	1	yes	characteristic
17	#387×C₆₇	—	2	1	no	—
18	#387×C₆₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
21	#376×C₆₇	—	2	1	no	—
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
26	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Q₈×C₆₇	—	2	2	no	—
28	C₄×Q₈×C₆₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	—	2	1	no	—
36	C₂×C₄²×C₆₇	—	2	1	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	#95×C₆₇	—	2	1	no	—
45	#95×C₆₇	—	2	1	no	—
46	#95×C₆₇	—	2	1	no	—
47	#95×C₆₇	—	2	1	no	—
48	#95×C₆₇	—	2	1	no	—
49	#95×C₆₇	—	2	1	no	—
50	#95×C₆₇	—	2	1	no	—
51	C₂×C₆₇⋊C₈	—	4	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×Q₈×C₆₇	—	2	1	no	—
58	C₂×Q₈×C₆₇	—	4	1	no	—
59	C₂×Q₈×C₆₇	—	4	1	no	—
60	C₂×Q₈×C₆₇	M₁₆	1	2	yes	—
61	C₂²×C₄×C₆₇	—	2	1	no	—
62	C₂²×C₄×C₆₇	—	2	1	no	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₄⋊C₄×C₆₇	—	4	1	no	—
75	C₄⋊C₄×C₆₇	—	4	1	no	—
76	C₄⋊C₄×C₆₇	—	4	2	no	—
77	C₄⋊C₄×C₆₇	—	4	2	no	—
78	C₄⋊C₄×C₆₇	M₁₆	1	2	yes	—
79	C₄²×C₆₇	—	2	4	no	—
80	C₄²×C₆₇	—	2	1	no	—
81	C₄²×C₆₇	—	2	1	no	—
82	C₄²×C₆₇	—	2	2	no	—
83	C₄²×C₆₇	—	4	1	no	—
84	C₄²×C₆₇	—	4	1	no	—
85	C₄²×C₆₇	—	4	1	no	—
86	C₄²×C₆₇	—	4	1	no	—
87	C₄²×C₆₇	—	4	2	no	—
88	C₄²×C₆₇	SD₁₆	1	2	yes	—
89	C₄²×C₆₇	Q₁₆	1	2	yes	—
90	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₆₇	—	4	2	no	—
92	Q₈×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	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	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	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	—	4	1	no	—
112	C₂×C₄×C₆₇	—	4	1	no	—
113	C₂×C₄×C₆₇	—	4	1	no	—
114	C₂×C₄×C₆₇	—	4	1	no	—
115	C₂×C₄×C₆₇	—	4	1	no	—
116	C₂×C₄×C₆₇	—	4	1	no	—
117	C₂×C₄×C₆₇	—	4	1	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	#103	1	2	yes	—
122	C₂×C₄×C₆₇	#103	1	2	yes	—
123	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#135	1	2	yes	—
128	C₆₇⋊C₈	—	8	2	no	—
129	C₆₇⋊C₈	—	8	2	no	—
130	C₂²×C₆₇	#407	1	1	yes	characteristic
131	C₂²×C₆₇	#410	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	#421	1	1	yes	characteristic
136	C₂²×C₆₇	#421	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	1	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	2	no	—
144	C₄×C₆₇	—	4	2	no	—
145	C₄×C₆₇	—	4	1	no	—
146	C₄×C₆₇	—	4	2	no	—
147	C₄×C₆₇	—	4	1	no	—
148	C₄×C₆₇	—	8	1	no	—
149	C₄×C₆₇	—	8	1	no	—
150	#7644	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	#1464	1	1	yes	characteristic
152	C₂×C₆₇	#1487	1	1	yes	characteristic
153	C₂×C₆₇	#1488	1	1	yes	characteristic
154	C₂×C₆₇	#1518	1	1	yes	characteristic
155	C₂×C₆₇	#1521	1	1	yes	characteristic
156	C₂×C₆₇	#1535	1	1	yes	characteristic
157	C₂×C₆₇	#1542	1	1	yes	characteristic
158	#2295	D₆₇	1	1	yes	characteristic
159	#1839	—	67	1	no	—
160	#1831	—	67	1	no	—
161	#1272	—	67	2	no	—
162	#1271	—	67	2	no	—
163	C₆₇	#7644	1	1	yes	characteristic, complemented, Sylow
164	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
165	C₂×C₄⋊C₈	—	134	1	no	—
166	#391	Dic₆₇	1	1	yes	characteristic
167	#391	C₂×D₆₇	1	1	yes	characteristic
168	#389	Dic₆₇	1	1	yes	characteristic
169	#389	C₂×D₆₇	1	1	yes	characteristic
170	#387	—	2	1	no	—
171	#387	—	2	1	no	—
172	C₄×C₄⋊C₄	—	2	1	no	—
173	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
174	#376	—	2	1	no	—
175	C₄³	C₂×D₆₇	1	1	yes	characteristic
176	#336	—	67	2	no	—
177	#336	—	67	2	no	—
178	#336	—	134	1	no	—
179	#336	—	134	1	no	—
180	#336	—	134	1	no	—
181	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
182	C₂²×C₈	—	134	2	no	—
183	C₂²×C₈	—	134	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
195	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₆₇	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	C₄⋊C₈	—	268	2	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	4	1	no	—
212	C₂×Q₈	#11761612	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	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	4	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	#11761612	1	2	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₄²	#11761615	1	2	yes	—
247	C₄²	#11761618	1	2	yes	—
248	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	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	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	#14523967	1	2	yes	—
280	C₂×C₄	#14523967	1	2	yes	—
281	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
282	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
283	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
284	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
285	C₂×C₄	#14524063	1	2	yes	—
286	C₈	—	536	2	no	—
287	C₈	—	536	2	no	—
288	C₂²	#16863360	1	1	yes	characteristic
289	C₂²	#16863370	1	1	yes	characteristic
290	C₂²	#16863380	1	1	yes	characteristic
291	C₂²	C₂×#14523967	1	1	yes	characteristic
292	C₂²	#16863392	1	1	yes	characteristic
293	C₂²	#16863413	1	1	yes	characteristic
294	C₂²	#16863413	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	#22558543	1	1	yes	characteristic
309	C₂	#22558638	1	1	yes	characteristic
310	C₂	#22558643	1	1	yes	characteristic
311	C₂	#22558773	1	1	yes	characteristic
312	C₂	#22558786	1	1	yes	characteristic
313	C₂	#22558840	1	1	yes	characteristic
314	C₂	#22558866	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
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	#7644
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
163. C₆₇	#7644	semidirect

Maximal subgroups

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

subgroup	index
#22560352	2
#22560326	2
#22558035	2
#22558032	2
C₆₇×#2295	2
#7644	67