Subgroups — #30744004

Order 17152.

428 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560326	C₂	1	1	yes	maximal, characteristic
2	#22560062	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	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×#14523954	—	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)	—	2	4	no	—
26	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
27	#14523954	—	4	2	no	—
28	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₆₇	—	2	2	no	—
30	C₄×Q₈×C₆₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	—	2	1	no	—
38	C₂×C₄²×C₆₇	—	2	1	no	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	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	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
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	#95×C₆₇	—	2	1	no	—
52	#95×C₆₇	—	2	1	no	—
53	C₂×C₆₇⋊C₈	—	2	4	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×C₆₇⋊C₈	—	4	2	no	—
59	C₂×Q₈×C₆₇	—	2	1	no	—
60	C₂×Q₈×C₆₇	—	4	1	no	—
61	C₂×Q₈×C₆₇	—	4	1	no	—
62	C₂×Q₈×C₆₇	M₁₆	1	2	yes	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	1	no	—
65	C₂²×C₄×C₆₇	—	2	1	no	—
66	C₂²×C₄×C₆₇	—	2	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₄⋊C₄×C₆₇	—	4	1	no	—
77	C₄⋊C₄×C₆₇	—	4	1	no	—
78	C₄⋊C₄×C₆₇	—	4	2	no	—
79	C₄⋊C₄×C₆₇	—	4	1	no	—
80	C₄⋊C₄×C₆₇	—	4	1	no	—
81	C₄⋊C₄×C₆₇	M₁₆	1	2	yes	—
82	C₄²×C₆₇	—	2	4	no	—
83	C₄²×C₆₇	—	2	2	no	—
84	C₄²×C₆₇	—	2	1	no	—
85	C₄²×C₆₇	—	2	1	no	—
86	C₄²×C₆₇	—	2	1	no	—
87	C₄²×C₆₇	—	2	1	no	—
88	C₄²×C₆₇	—	4	1	no	—
89	C₄²×C₆₇	—	4	1	no	—
90	C₄²×C₆₇	—	4	1	no	—
91	C₄²×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	SD₁₆	1	2	yes	—
93	C₄²×C₆₇	Q₁₆	1	2	yes	—
94	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈×C₆₇	—	4	2	no	—
96	Q₈×C₆₇	—	4	2	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	1	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	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	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	—	4	1	no	—
122	C₂×C₄×C₆₇	—	4	1	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	#103	1	2	yes	—
125	C₂×C₄×C₆₇	#103	1	2	yes	—
126	C₂×C₄×C₆₇	#104	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
133	C₆₇⋊C₈	—	4	4	no	—
134	C₆₇⋊C₈	—	8	2	no	—
135	C₂²×C₆₇	#407	1	1	yes	characteristic
136	C₂²×C₆₇	#410	1	1	yes	characteristic
137	C₂²×C₆₇	#413	1	1	yes	characteristic
138	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
139	C₂²×C₆₇	#417	1	1	yes	characteristic
140	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
141	C₂²×C₆₇	#421	1	1	yes	characteristic
142	C₄×C₆₇	—	2	2	no	—
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	1	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	—	8	1	no	—
155	C₄×C₆₇	—	8	1	no	—
156	#7615	—	67	1	no	maximal, Sylow
157	C₂×C₆₇	#1464	1	1	yes	characteristic
158	C₂×C₆₇	#1487	1	1	yes	characteristic
159	C₂×C₆₇	#1489	1	1	yes	characteristic
160	C₂×C₆₇	#1503	1	1	yes	characteristic
161	C₂×C₆₇	#1512	1	1	yes	characteristic
162	C₂×C₆₇	#1532	1	1	yes	characteristic
163	C₂×C₆₇	#1550	1	1	yes	characteristic
164	#2295	D₆₇	1	1	yes	characteristic
165	#1831	—	67	1	no	—
166	#1767	—	67	1	no	—
167	#1272	—	67	2	no	—
168	#1271	—	67	2	no	—
169	C₆₇	#7615	1	1	yes	characteristic, complemented, Sylow
170	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
171	C₂×C₄⋊C₈	—	134	1	no	—
172	C₂×C₄⋊C₈	—	134	1	no	—
173	C₂×#97	—	134	1	no	—
174	#391	Dic₆₇	1	1	yes	characteristic
175	#391	C₂×D₆₇	1	1	yes	characteristic
176	#389	Dic₆₇	1	1	yes	characteristic
177	#389	C₂×D₆₇	1	1	yes	characteristic
178	#387	—	2	1	no	—
179	#387	—	2	1	no	—
180	C₄×C₄⋊C₄	—	2	1	no	—
181	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
182	#376	—	2	1	no	—
183	C₄³	C₂×D₆₇	1	1	yes	characteristic
184	#336	—	67	2	no	—
185	#336	—	67	2	no	—
186	#336	—	134	1	no	—
187	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
188	C₂²×C₈	—	134	2	no	—
189	C₂²×C₈	—	134	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
201	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
207	C₄⋊C₈	—	268	2	no	—
208	C₄⋊C₈	—	134	4	no	—
209	#97	—	268	2	no	—
210	#95	—	2	1	no	—
211	#95	—	2	1	no	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	C₂×Q₈	—	2	1	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	#11761612	1	2	yes	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
232	C₂×C₈	—	268	2	no	—
233	C₂×C₈	—	268	2	no	—
234	C₂×C₈	—	268	2	no	—
235	C₂×C₈	—	268	2	no	—
236	C₂×C₈	—	268	2	no	—
237	C₂×C₈	—	134	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	#11761612	1	2	yes	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	—	4	1	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	2	no	—
256	C₄²	#11761615	1	2	yes	—
257	C₄²	#11761618	1	2	yes	—
258	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	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	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	#14523967	1	2	yes	—
289	C₂×C₄	#14523967	1	2	yes	—
290	C₂×C₄	#14523970	1	2	yes	—
291	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
292	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
293	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
294	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
295	C₂×C₄	#14524068	1	1	yes	characteristic
296	C₂×C₄	#14524073	1	1	yes	characteristic
297	C₈	—	268	4	no	—
298	C₈	—	536	2	no	—
299	C₂²	#16863360	1	1	yes	characteristic
300	C₂²	#16863370	1	1	yes	characteristic
301	C₂²	#16863380	1	1	yes	characteristic
302	C₂²	C₂×#14523967	1	1	yes	characteristic
303	C₂²	#16863397	1	1	yes	characteristic
304	C₂²	C₂×#14523970	1	1	yes	characteristic
305	C₂²	#16863413	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₂	#22558543	1	1	yes	characteristic
321	C₂	#22558638	1	1	yes	characteristic
322	C₂	#22558648	1	1	yes	characteristic
323	C₂	#22558708	1	1	yes	characteristic
324	C₂	#22558747	1	1	yes	characteristic
325	C₂	#22558829	1	1	yes	characteristic
326	C₂	#22558899	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	#7615
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
169. C₆₇	#7615	semidirect

Maximal subgroups

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

subgroup	index
#22560326	2
#22560062	2
#22558035	2
#22558032	2
C₆₇×#2295	2
#7615	67