Subgroups — #30771069

Order 17152.

438 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863514	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863202	C₂	1	1	yes	maximal, characteristic
3	#22561914	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558304	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558302	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524065	C₂²	1	1	yes	characteristic
9	C₂×#14524049	—	2	1	no	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16863514	—	2	2	no	—
13	#16863514	C₄	1	4	yes	complemented
14	C₂×#14523979	C₂²	1	1	yes	characteristic
15	C₂×#14523971	—	2	1	no	—
16	#16863202	C₄	1	4	yes	complemented
17	#16863199	C₄	1	2	yes	complemented
18	#16863196	C₄	1	2	yes	complemented
19	C₄×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
21	#427×C₆₇	C₂²	1	1	yes	characteristic
22	#405×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#339×C₆₇	—	2	1	no	—
25	C₂×#11761628	—	2	1	no	—
26	#14524065	—	2	1	no	—
27	#14524065	—	2	1	no	—
28	#14524065	C₂×C₄	1	2	yes	—
29	#14524065	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
31	C₂×#11761617	—	2	1	no	—
32	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
33	#14524049	—	2	4	no	—
34	#14524049	—	4	2	no	—
35	C₂×#11761612	—	2	1	no	—
36	C₂×#11761611	—	4	1	no	—
37	C₂×C₈×D₆₇	—	4	1	no	—
38	C₂×C₆₇⋊C₁₆	—	2	1	no	—
39	C₂×C₆₇⋊C₁₆	—	2	1	no	—
40	C₂×C₆₇⋊C₁₆	—	2	1	no	—
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
44	#14523979	—	2	1	no	—
45	#14523979	C₂×C₄	1	2	yes	—
46	#14523971	—	2	4	no	—
47	C₂×M₁₆×C₆₇	—	2	1	no	—
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	#117×C₆₇	—	2	1	no	—
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	#108×C₆₇	—	2	1	no	—
53	#108×C₆₇	C₂×C₄	1	2	yes	—
54	#107×C₆₇	C₂×C₄	1	2	yes	—
55	#106×C₆₇	—	2	1	no	—
56	#97×C₆₇	—	4	1	no	—
57	C₄×C₈×C₆₇	—	4	1	no	—
58	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₆₇	D₄	1	2	yes	—
60	#11761628	—	2	2	no	—
61	#11761628	—	2	2	no	—
62	#11761628	—	4	1	no	—
63	#11761628	—	4	1	no	—
64	C₂×Dic₁₃₄	—	2	1	no	—
65	C₂×D₂₆₈	—	2	1	no	—
66	C₂×C₂²⋊D₆₇	—	4	1	no	—
67	C₂×C₄×D₆₇	—	4	1	no	—
68	Dic₂₆₈	—	2	1	no	—
69	Dic₂₆₈	C₄²	1	2	yes	—
70	#11761617	—	2	2	no	—
71	#11761617	—	2	2	no	—
72	D₅₃₆	—	2	1	no	—
73	D₅₃₆	C₄²	1	2	yes	—
74	#11761612	—	2	2	no	—
75	#11761612	—	4	1	no	—
76	#11761611	—	4	4	no	—
77	C₂×C₆₇⋊C₈	—	4	1	no	—
78	C₈×D₆₇	—	4	4	no	—
79	C₆₇⋊C₁₆	—	2	2	no	—
80	C₆₇⋊C₁₆	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	1	no	—
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	M₁₆×C₆₇	—	2	2	no	—
84	M₁₆×C₆₇	—	4	1	no	—
85	C₂×C₈×C₆₇	—	4	1	no	—
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
88	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
89	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
90	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
95	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₂²⋊C₄×C₆₇	—	4	1	no	—
98	C₄²×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	—	4	1	no	—
100	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
101	C₂³×C₆₇	#135	1	1	yes	characteristic
102	C₂²×D₆₇	—	4	1	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
112	C₂²⋊D₆₇	—	4	2	no	—
113	C₂²⋊D₆₇	—	4	2	no	—
114	C₂×Dic₆₇	—	4	1	no	—
115	D₂₆₈	—	2	2	no	—
116	D₂₆₈	—	4	1	no	—
117	C₄×D₆₇	—	4	2	no	—
118	C₄×D₆₇	—	4	2	no	—
119	Dic₁₃₄	—	2	2	no	—
120	Dic₁₃₄	—	4	1	no	—
121	C₈×C₆₇	—	4	2	no	—
122	C₈×C₆₇	#102	1	2	yes	—
123	C₈×C₆₇	#102	1	2	yes	—
124	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
125	C₈×C₆₇	#117	1	1	yes	characteristic
126	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
127	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
128	C₆₇⋊C₈	—	4	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	#416	1	1	yes	characteristic
131	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
132	C₂²×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
133	C₂×D₆₇	—	4	2	no	—
134	C₂×D₆₇	—	8	2	no	—
135	C₄×C₆₇	—	2	4	no	—
136	C₄×C₆₇	—	8	2	no	—
137	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
139	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
140	C₄×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
141	Dic₆₇	—	4	2	no	—
142	#12641	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	—	2	1	no	—
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	C₄×#102	1	1	yes	characteristic
146	C₂×C₆₇	#2178	1	2	yes	—
147	D₆₇	—	8	2	no	—
148	C₂×#443	—	67	1	no	—
149	C₂×#359	—	67	1	no	—
150	#2138	—	67	1	no	—
151	C₂×C₄×C₁₆	—	67	1	no	—
152	#1772	D₆₇	1	1	yes	characteristic, complemented
153	#1382	—	67	1	no	—
154	#1381	—	67	1	no	—
155	C₆₇	#12641	1	1	yes	characteristic, complemented, Sylow
156	C₂×#135	—	67	1	no	—
157	C₂×#131	—	134	1	no	—
158	C₂²×C₁₆	—	67	1	no	—
159	C₂²×C₁₆	—	67	1	no	—
160	#443	—	67	4	no	—
161	#443	—	134	2	no	—
162	C₂×#108	—	67	1	no	—
163	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
164	#427	C₂×D₆₇	1	1	yes	characteristic
165	C₂×#104	—	134	1	no	—
166	#405	—	2	1	no	—
167	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
168	#359	—	67	4	no	—
169	#358	—	67	2	no	—
170	#357	—	67	2	no	—
171	C₄×C₁₆	—	134	2	no	—
172	#339	—	2	1	no	—
173	C₂×#41	—	134	1	no	—
174	#135	—	67	2	no	—
175	#135	—	67	2	no	—
176	#135	—	134	1	no	—
177	#135	—	134	1	no	—
178	C₂×Q₁₆	—	67	1	no	—
179	C₂×SD₁₆	—	134	1	no	—
180	C₂×D₈	—	67	1	no	—
181	#131	—	268	2	no	—
182	#131	—	134	4	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	268	1	no	—
185	C₂×M₁₆	—	134	1	no	—
186	C₂²×C₈	—	268	1	no	—
187	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	#117	—	2	1	no	—
190	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂×C₁₆	—	67	2	no	—
192	C₂×C₁₆	—	67	2	no	—
193	C₂×C₁₆	—	134	1	no	—
194	C₂×C₁₆	—	134	1	no	—
195	C₂×C₁₆	—	134	1	no	—
196	C₂×C₁₆	—	134	1	no	—
197	#108	—	2	1	no	—
198	#108	—	67	2	no	—
199	#108	—	134	1	no	—
200	#108	C₄×D₆₇	1	2	yes	—
201	#107	C₄×D₆₇	1	2	yes	—
202	#106	—	2	1	no	—
203	#104	—	134	4	no	—
204	#97	—	4	1	no	—
205	C₄×C₈	—	4	1	no	—
206	C₄×C₈	C₄×D₆₇	1	2	yes	—
207	C₄×C₈	D₂₆₈	1	2	yes	—
208	#41	—	268	1	no	—
209	#41	—	268	1	no	—
210	#41	—	134	2	no	—
211	#41	—	134	2	no	—
212	C₂×Q₈	—	134	1	no	—
213	C₂×D₄	—	268	1	no	—
214	C₂×D₄	—	134	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	268	1	no	—
217	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
218	Q₁₆	—	67	2	no	—
219	Q₁₆	—	134	1	no	—
220	SD₁₆	—	134	2	no	—
221	SD₁₆	—	134	2	no	—
222	D₈	—	67	2	no	—
223	D₈	—	134	1	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	268	1	no	—
227	M₁₆	—	268	4	no	—
228	M₁₆	—	134	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	268	1	no	—
231	C₂×C₈	—	268	4	no	—
232	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
233	C₂×C₈	#11761613	1	1	yes	characteristic
234	C₂×C₈	#11761613	1	1	yes	characteristic
235	C₂×C₈	#11761615	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
239	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
240	C₂×C₈	#11761628	1	1	yes	characteristic
241	C₂×C₈	#11761628	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	4	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
247	C₁₆	—	134	2	no	—
248	C₁₆	—	134	2	no	—
249	C₂³	—	268	1	no	—
250	C₂³	#14524063	1	1	yes	characteristic
251	Q₈	—	268	1	no	—
252	Q₈	—	134	2	no	—
253	D₄	—	268	2	no	—
254	D₄	—	268	2	no	—
255	D₄	—	268	1	no	—
256	D₄	—	134	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	268	1	no	—
261	C₂×C₄	—	268	2	no	—
262	C₂×C₄	—	268	2	no	—
263	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
267	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
268	C₂×C₄	#14524063	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	268	2	no	—
271	C₈	#14523963	1	2	yes	—
272	C₈	#14523963	1	2	yes	—
273	C₈	C₄²×D₆₇	1	1	yes	characteristic
274	C₈	#14524000	1	1	yes	characteristic
275	C₈	C₄×D₂₆₈	1	1	yes	characteristic
276	C₈	C₄×D₂₆₈	1	1	yes	characteristic
277	C₂²	—	2	2	no	—
278	C₂²	—	268	2	no	—
279	C₂²	—	536	2	no	—
280	C₂²	#16863390	1	1	yes	characteristic
281	C₂²	C₄×#11761613	1	1	yes	characteristic
282	C₂²	C₄×#11761615	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	8	2	no	—
285	C₄	—	268	2	no	—
286	C₄	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
287	C₄	C₂×#14523963	1	1	yes	characteristic
288	C₄	C₄×#11761613	1	1	yes	characteristic
289	C₄	C₄×#11761615	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	536	2	no	—
293	C₂	C₄×#14523963	1	1	yes	characteristic
294	C₂	#22562088	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#11761613
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1772	C₂
socle	C₂²×C₆₇	C₄×D₈
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1772	D₆₇
hypocenter	C₆₇	#12641
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
4. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
5. #22558304	C₂	semidirect
6. #22558302	C₂	semidirect
7. C₆₇×#1772	C₂	semidirect
13. #16863514	C₄	semidirect
16. #16863202	C₄	semidirect
17. #16863199	C₄	semidirect
18. #16863196	C₄	semidirect
152. #1772	D₆₇	semidirect
155. C₆₇	#12641	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863514	2
C₂×#16863202	2
#22561914	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558304	2
#22558302	2
C₆₇×#1772	2
#12641	67