Subgroups — #31008927

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	#22562013	C₂	1	4	yes	maximal, complemented
3	#22562009	C₂	1	4	yes	maximal, complemented
4	#22561969	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561967	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561823	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2113	C₂	1	1	yes	maximal, characteristic
10	C₂×#14524050	C₄	1	1	yes	characteristic
11	#41×Dic₆₇	C₄	1	1	yes	characteristic
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#16863354	C₄	1	2	yes	—
15	#16863354	C₄	1	1	yes	characteristic
16	#16863351	C₂²	1	1	yes	characteristic
17	#16863351	C₂²	1	2	yes	—
18	M₁₆×Dic₆₇	C₄	1	1	yes	characteristic
19	M₁₆×Dic₆₇	C₄	1	2	yes	—
20	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
22	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
23	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
24	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
25	#16863171	C₂²	1	2	yes	—
26	#16863171	C₂²	1	2	yes	—
27	#16863171	C₂²	1	2	yes	—
28	C₁₆×Dic₆₇	C₂²	1	2	yes	—
29	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
33	#14524050	—	2	4	no	—
34	C₂×#11761612	C₂×C₄	1	2	yes	—
35	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
38	Q₈×Dic₆₇	C₈	1	2	yes	—
39	D₄×Dic₆₇	C₈	1	4	yes	—
40	D₄×Dic₆₇	C₈	1	2	yes	—
41	#14524002	C₂×C₄	1	2	yes	—
42	#14524002	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
45	#14523983	—	2	2	no	—
46	C₂×C₆₇⋊C₁₆	—	2	2	no	—
47	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
49	#14523953	C₂×C₄	1	2	yes	—
50	#14523953	C₂×C₄	1	2	yes	—
51	#14523953	C₂×C₄	1	2	yes	—
52	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
53	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
55	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
57	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
58	C₈×Dic₆₇	C₂³	1	2	yes	—
59	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
60	#131×C₆₇	—	2	2	no	—
61	#131×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	M₃₂×C₆₇	—	2	2	no	—
67	C₂×C₁₆×C₆₇	—	2	2	no	—
68	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
70	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
71	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
72	#11761612	—	2	2	no	—
73	#11761612	—	2	4	no	—
74	C₂×C₆₇⋊C₈	—	2	4	no	—
75	C₂×C₆₇⋊C₈	M₁₆	1	4	yes	—
76	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
77	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
80	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
81	C₂²⋊Dic₆₇	C₂×C₈	1	4	yes	—
82	C₂²⋊Dic₆₇	C₂×C₈	1	2	yes	—
83	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
84	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
85	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
88	C₆₇⋊C₁₆	—	2	2	no	—
89	C₆₇⋊C₁₆	—	2	2	no	—
90	#41×C₆₇	—	2	2	no	—
91	#41×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
93	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
94	C₂×D₄×C₆₇	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	M₁₆×C₆₇	—	2	2	no	—
98	M₁₆×C₆₇	—	2	4	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	#41	1	2	yes	—
104	C₂×C₈×C₆₇	#41	1	2	yes	—
105	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
106	C₁₆×C₆₇	—	2	2	no	—
107	C₁₆×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
109	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
110	Q₈×C₆₇	—	2	2	no	—
111	D₄×C₆₇	—	2	4	no	—
112	D₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	#117	1	2	yes	—
116	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₆₇	—	4	2	no	—
121	C₂×Dic₆₇	—	4	4	no	—
122	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
123	C₂×Dic₆₇	C₂²×C₈	1	4	yes	—
124	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
125	C₈×C₆₇	—	2	2	no	—
126	C₈×C₆₇	#117	1	2	yes	—
127	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
128	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₆₇⋊C₈	—	2	2	no	—
130	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
131	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	#431	1	2	yes	—
136	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	#431	1	2	yes	—
139	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
140	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
141	Dic₆₇	—	2	2	no	—
142	Dic₆₇	—	2	2	no	—
143	Dic₆₇	—	2	4	no	—
144	#33421	—	67	1	no	maximal, Sylow
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	4	2	no	—
147	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
148	C₂×C₆₇	#3407	1	2	yes	—
149	C₄×#131	—	67	1	no	—
150	#2164	—	67	4	no	—
151	#2163	—	67	4	no	—
152	#2151	—	67	1	no	—
153	#2151	—	67	1	no	—
154	#2113	—	67	1	no	—
155	#2113	D₆₇	1	1	yes	characteristic
156	C₄×M₃₂	—	67	1	no	—
157	C₂×#346	—	67	1	no	—
158	C₆₇	#33421	1	1	yes	characteristic, complemented, Sylow
159	C₂×#131	—	67	1	no	—
160	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
161	C₄×#41	—	67	1	no	—
162	C₂×M₃₂	—	67	1	no	—
163	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
164	C₂²×C₁₆	—	67	1	no	—
165	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
166	#405	—	67	1	no	—
167	#405	—	67	2	no	—
168	#405	—	67	1	no	—
169	#405	—	67	2	no	—
170	C₄×M₁₆	—	67	2	no	—
171	C₄×M₁₆	—	67	1	no	—
172	C₂×C₄×C₈	—	67	1	no	—
173	C₂×C₄×C₈	—	67	2	no	—
174	C₄⋊C₁₆	—	67	4	no	—
175	C₄⋊C₁₆	—	67	4	no	—
176	C₂²⋊C₁₆	—	67	4	no	—
177	C₂²⋊C₁₆	C₂×D₆₇	1	4	yes	—
178	#346	—	67	2	no	—
179	#346	—	67	2	no	—
180	#346	—	67	2	no	—
181	C₄×C₁₆	—	67	2	no	—
182	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
183	#131	—	2	2	no	—
184	#131	—	2	2	no	—
185	#131	—	134	4	no	—
186	C₂×M₁₆	—	67	2	no	—
187	C₂×M₁₆	—	67	1	no	—
188	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
189	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
190	C₂²×C₈	—	67	1	no	—
191	C₂²×C₈	—	67	2	no	—
192	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
193	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
194	C₄×Q₈	—	67	2	no	—
195	C₄×D₄	—	67	4	no	—
196	C₄×D₄	—	67	2	no	—
197	#117	—	67	2	no	—
198	#117	—	67	1	no	—
199	C₂×C₄²	—	67	2	no	—
200	C₂×C₄²	—	67	1	no	—
201	M₃₂	—	2	2	no	—
202	M₃₂	—	134	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	67	2	no	—
205	C₂×C₁₆	—	67	2	no	—
206	C₂×C₁₆	—	134	2	no	—
207	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
208	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
209	#97	—	67	2	no	—
210	#97	—	67	2	no	—
211	#97	—	67	2	no	—
212	C₄×C₈	—	67	1	no	—
213	C₄×C₈	—	67	2	no	—
214	C₄×C₈	—	67	2	no	—
215	C₄×C₈	—	67	1	no	—
216	C₄×C₈	—	67	1	no	—
217	C₄×C₈	—	67	1	no	—
218	C₄×C₈	—	67	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
222	C₂×D₄	C₈×D₆₇	1	2	yes	—
223	C₂×D₄	C₈×D₆₇	1	1	yes	characteristic
224	C₂²×C₄	—	67	4	no	—
225	C₂²×C₄	—	67	2	no	—
226	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	134	4	no	—
231	M₁₆	—	134	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	67	4	no	—
235	C₂×C₈	—	67	1	no	—
236	C₂×C₈	—	67	1	no	—
237	C₂×C₈	—	67	2	no	—
238	C₂×C₈	—	134	4	no	—
239	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
241	C₂×C₈	#11761627	1	2	yes	—
242	C₂×C₈	#11761627	1	2	yes	—
243	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
244	C₄⋊C₄	—	67	4	no	—
245	C₄⋊C₄	—	67	2	no	—
246	C₂²⋊C₄	—	67	4	no	—
247	C₂²⋊C₄	—	67	2	no	—
248	C₄²	—	67	4	no	—
249	C₄²	—	67	2	no	—
250	C₄²	—	67	1	no	—
251	C₄²	—	67	1	no	—
252	C₄²	—	67	2	no	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	134	2	no	—
256	C₁₆	—	134	2	no	—
257	C₂³	C₂×C₈×D₆₇	1	2	yes	—
258	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
259	Q₈	—	2	2	no	—
260	D₄	—	2	4	no	—
261	D₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	268	2	no	—
265	C₂×C₄	—	268	4	no	—
266	C₂×C₄	—	67	2	no	—
267	C₂×C₄	—	67	4	no	—
268	C₂×C₄	—	67	2	no	—
269	C₂×C₄	#14523999	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	67	2	no	—
276	C₈	—	67	2	no	—
277	C₈	—	134	2	no	—
278	C₈	#14523999	1	2	yes	—
279	C₈	C₂×#11761627	1	1	yes	characteristic
280	C₈	C₂×#11761627	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#16863446	1	2	yes	—
285	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	134	2	no	—
288	C₄	—	134	2	no	—
289	C₄	—	134	4	no	—
290	C₄	#16863446	1	2	yes	—
291	C₄	C₂×#14523999	1	1	yes	characteristic
292	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	C₂×#16863446	1	1	yes	characteristic
296	C₂	#22573310	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2113	C₂
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2113	D₆₇
hypocenter	C₆₇	#33421
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #22562013	C₂	semidirect
3. #22562009	C₂	semidirect
4. #22561969	C₂	semidirect
5. #22561967	C₂	semidirect
7. M₃₂×Dic₆₇	C₂	semidirect
8. C₂×#16863171	C₂	semidirect
158. C₆₇	#33421	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₆₇	2
#22562013	2
#22562009	2
#22561969	2
#22561967	2
#22561823	2
M₃₂×Dic₆₇	2
C₂×#16863171	2
C₆₇×#2113	2
#33421	67