Subgroups — #30874304

Order 17152.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 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	#22569092	C₂	1	2	yes	maximal
3	#22568800	C₂	1	2	yes	maximal, complemented
4	#22560987	C₂	1	2	yes	maximal, complemented
5	#22560434	C₂	1	4	yes	maximal, complemented
6	#22560269	C₂	1	2	yes	maximal, complemented
7	#22560088	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	C₂×#14524049	C₂²	1	2	yes	—
11	#16863877	C₂²	1	4	yes	—
12	#16863514	C₄	1	8	yes	complemented
13	#16863507	—	2	2	no	—
14	#16863502	—	2	2	no	—
15	#16863483	—	2	4	no	—
16	C₂×#14523979	C₂²	1	1	yes	characteristic
17	C₂×#14523979	C₂²	1	1	yes	characteristic
18	#16863436	C₂²	1	4	yes	—
19	#16863435	—	2	4	no	—
20	#16863434	C₂²	1	1	yes	characteristic
21	#16863431	—	2	2	no	—
22	#106×D₆₇	—	2	2	no	—
23	C₂×#14523971	C₂²	1	2	yes	—
24	#16863395	C₂²	1	4	yes	—
25	#16863351	C₂²	1	2	yes	—
26	#16863165	C₂²	1	4	yes	—
27	#16863156	C₂²	1	2	yes	—
28	C₂×#106×C₆₇	C₂²	1	2	yes	—
29	C₂×#106×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#384×C₆₇	C₂²	1	2	yes	—
32	C₂×#11761628	C₂³	1	2	yes	—
33	#14524065	C₂×C₄	1	4	yes	—
34	#14524065	Q₈	1	4	yes	—
35	C₂×Dic₂₆₈	D₄	1	1	yes	characteristic
36	C₂×#11761617	D₄	1	2	yes	—
37	C₂×D₅₃₆	D₄	1	1	yes	characteristic
38	#14524049	—	2	4	no	—
39	#14524049	C₂×C₄	1	8	yes	complemented
40	C₂×#11761612	C₂³	1	2	yes	—
41	C₂×#11761612	C₂³	1	2	yes	—
42	C₂×#11761611	—	2	2	no	—
43	C₂×C₈×D₆₇	—	2	2	no	—
44	#14524008	—	2	4	no	—
45	#14524005	—	2	4	no	—
46	C₄×C₂²⋊D₆₇	—	4	4	no	—
47	#14524002	C₂³	1	2	yes	—
48	#14524001	—	2	4	no	—
49	C₄⋊C₄×D₆₇	—	2	4	no	—
50	#14523979	—	2	2	no	—
51	#14523979	C₂×C₄	1	4	yes	complemented
52	#14523977	D₄	1	2	yes	—
53	#14523976	—	2	4	no	—
54	#14523975	D₄	1	2	yes	—
55	#14523974	—	2	4	no	—
56	#14523971	C₂×C₄	1	8	yes	complemented
57	#14523966	—	2	4	no	—
58	#14523963	—	2	4	no	—
59	#14523953	—	2	2	no	—
60	C₈×Dic₆₇	—	2	2	no	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	#106×C₆₇	—	2	2	no	—
67	#106×C₆₇	—	2	2	no	—
68	#106×C₆₇	—	2	2	no	—
69	#106×C₆₇	—	2	2	no	—
70	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
71	#11761628	—	2	2	no	—
72	#11761628	—	2	2	no	—
73	#11761628	C₂²×C₄	1	4	yes	—
74	#11761628	C₂×Q₈	1	4	yes	—
75	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
76	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
77	C₂×C₂²⋊D₆₇	—	2	2	no	—
78	C₂×C₄×D₆₇	—	2	2	no	—
79	C₂×C₄×D₆₇	—	4	4	no	—
80	Dic₂₆₈	C₄⋊C₄	1	4	yes	complemented
81	#11761617	C₄⋊C₄	1	8	yes	complemented
82	D₅₃₆	C₄⋊C₄	1	4	yes	complemented
83	#11761612	—	2	2	no	—
84	#11761612	—	2	2	no	—
85	#11761612	—	2	2	no	—
86	#11761612	C₂²×C₄	1	4	yes	—
87	#11761611	—	2	8	no	—
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₈×D₆₇	—	2	8	no	—
91	C₄⋊Dic₆₇	C₂×D₄	1	4	yes	—
92	Dic₆₇⋊C₄	—	4	4	no	—
93	C₂²⋊Dic₆₇	—	2	2	no	—
94	(C₂×D₆₇)⋊C₄	—	4	4	no	—
95	C₄×Dic₆₇	—	2	2	no	—
96	C₄×Dic₆₇	—	4	4	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₆₇	#41	1	2	yes	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	8	4	no	—
111	C₄²×C₆₇	—	2	1	no	—
112	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×D₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	8	4	no	—
119	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂²⋊D₆₇	—	2	4	no	—
126	C₂²⋊D₆₇	—	2	4	no	—
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	2	4	no	—
129	D₂₆₈	—	2	2	no	—
130	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
131	C₄×D₆₇	—	2	4	no	—
132	C₄×D₆₇	—	2	4	no	—
133	C₄×D₆₇	—	16	4	no	—
134	Dic₁₃₄	—	2	2	no	—
135	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₆₇	C₄×D₄	1	2	yes	—
139	C₈×C₆₇	C₄×D₄	1	2	yes	—
140	C₆₇⋊C₈	—	2	4	no	—
141	C₆₇⋊C₈	—	4	2	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	#520	1	1	yes	characteristic
144	C₂²×C₆₇	D₄²	1	1	yes	characteristic
145	C₂²×C₆₇	#554	1	1	yes	characteristic
146	C₂×D₆₇	—	2	4	no	—
147	C₂×D₆₇	—	4	4	no	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	8	4	no	—
150	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₆₇	#547	1	1	yes	characteristic
153	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
154	Dic₆₇	—	2	4	no	—
155	Dic₆₇	—	4	4	no	—
156	#22689	—	67	1	no	maximal, Sylow
157	C₂×C₆₇	—	2	1	no	—
158	C₂×C₆₇	—	2	1	no	—
159	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₆₇	#3291	1	2	yes	—
161	D₆₇	—	4	4	no	—
162	C₂×#443	—	67	1	no	—
163	#2952	—	67	2	no	—
164	#2912	—	67	2	no	—
165	#1951	—	67	2	no	—
166	#1859	—	67	4	no	—
167	#1844	D₆₇	1	1	yes	characteristic, complemented
168	#1815	—	67	2	no	—
169	#1772	—	67	1	no	—
170	C₆₇	#22689	1	1	yes	characteristic, complemented, Sylow
171	C₂×#135	—	67	1	no	—
172	C₂×#131	—	67	2	no	—
173	#520	—	67	4	no	—
174	#443	—	67	8	no	—
175	#442	—	134	2	no	—
176	#441	—	134	2	no	—
177	C₄×SD₁₆	—	134	4	no	—
178	C₂×#108	—	67	1	no	—
179	C₂×#108	—	67	1	no	—
180	#428	—	67	4	no	—
181	#428	—	134	4	no	—
182	#427	—	67	1	no	—
183	#427	—	134	2	no	—
184	C₂×#106	—	134	2	no	—
185	C₂×#106	C₂×D₆₇	1	2	yes	—
186	C₂×#106	C₂×D₆₇	1	2	yes	—
187	C₂×#104	—	67	2	no	—
188	#417	—	67	4	no	—
189	#405	—	67	2	no	—
190	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
191	#384	C₂×D₆₇	1	2	yes	—
192	#343	—	67	4	no	—
193	#339	—	67	2	no	—
194	C₂×#41	—	67	2	no	—
195	#135	—	67	4	no	—
196	#135	—	67	4	no	—
197	C₂×Q₁₆	—	67	1	no	—
198	C₂×SD₁₆	—	67	2	no	—
199	C₂×D₈	—	67	1	no	—
200	#131	—	67	8	no	—
201	#131	—	134	4	no	—
202	C₂×M₁₆	—	67	2	no	—
203	C₂×M₁₆	—	67	2	no	—
204	C₂×M₁₆	—	134	2	no	—
205	C₂²×C₈	—	134	2	no	—
206	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
208	C₄×Q₈	—	134	4	no	—
209	C₄×D₄	—	268	4	no	—
210	C₄×D₄	—	134	4	no	—
211	#117	—	67	2	no	—
212	#117	—	134	4	no	—
213	C₂×C₄⋊C₄	—	4	2	no	—
214	C₂×C₄⋊C₄	—	134	4	no	—
215	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
216	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
217	#108	—	67	4	no	—
218	#108	—	134	2	no	—
219	#107	—	67	2	no	—
220	#107	—	134	4	no	—
221	#106	—	2	2	no	—
222	#106	—	2	2	no	—
223	#106	—	2	2	no	—
224	#106	—	2	2	no	—
225	#106	—	67	2	no	—
226	#106	—	134	4	no	—
227	#104	—	67	8	no	—
228	#103	—	134	4	no	—
229	#102	—	134	4	no	—
230	#97	—	134	2	no	—
231	C₄×C₈	—	134	2	no	—
232	C₄×C₈	C₄×D₆₇	1	4	yes	complemented
233	#41	—	67	4	no	—
234	#41	—	67	4	no	—
235	#41	—	134	2	no	—
236	#41	—	134	2	no	—
237	C₂×Q₈	—	67	2	no	—
238	C₂×D₄	—	67	2	no	—
239	C₂×D₄	—	134	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	268	4	no	—
243	C₂²×C₄	—	134	2	no	—
244	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
245	Q₁₆	—	67	4	no	—
246	SD₁₆	—	67	8	no	—
247	D₈	—	67	4	no	—
248	M₁₆	—	67	4	no	—
249	M₁₆	—	134	2	no	—
250	M₁₆	—	134	2	no	—
251	M₁₆	—	134	2	no	—
252	M₁₆	—	134	8	no	—
253	C₂×C₈	—	134	2	no	—
254	C₂×C₈	—	134	2	no	—
255	C₂×C₈	—	134	8	no	—
256	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
258	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
259	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
260	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
261	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
262	C₂×C₈	Q₈×D₆₇	1	2	yes	—
263	C₂×C₈	#11761629	1	2	yes	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	8	4	no	—
267	C₄⋊C₄	—	268	4	no	—
268	C₄⋊C₄	—	67	4	no	—
269	C₂²⋊C₄	—	268	4	no	—
270	C₂²⋊C₄	—	134	2	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	268	4	no	—
273	C₄²	—	134	2	no	—
274	C₄²	C₂×C₄×D₆₇	1	2	yes	—
275	C₂³	—	134	2	no	—
276	C₂³	#14524088	1	1	yes	characteristic
277	Q₈	—	67	4	no	—
278	Q₈	—	134	2	no	—
279	D₄	—	67	4	no	—
280	D₄	—	134	4	no	—
281	D₄	—	134	4	no	—
282	D₄	—	134	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	1072	4	no	—
288	C₂×C₄	—	134	4	no	—
289	C₂×C₄	—	134	2	no	—
290	C₂×C₄	—	134	4	no	—
291	C₂×C₄	—	134	4	no	—
292	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
296	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
297	C₂×C₄	#14524091	1	1	yes	characteristic
298	C₈	—	268	2	no	—
299	C₈	—	134	4	no	—
300	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
301	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
302	C₈	#14524005	1	2	yes	—
303	C₈	#14524005	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	268	4	no	—
306	C₂²	—	134	4	no	—
307	C₂²	#16863877	1	1	yes	characteristic
308	C₂²	#16864031	1	1	yes	characteristic
309	C₂²	#16864107	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	8	4	no	—
312	C₄	—	268	4	no	—
313	C₄	—	134	4	no	—
314	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
315	C₄	C₂×#14524005	1	1	yes	characteristic
316	C₄	#16864049	1	1	yes	characteristic
317	C₄	#16864061	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	268	4	no	—
321	C₂	#22563109	1	1	yes	characteristic
322	C₂	#22572002	1	2	yes	—

	subgroup	G/H
center	C₂²	#16864031
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1844	C₂
socle	C₂²×C₆₇	D₄²
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1844	D₆₇
hypocenter	C₆₇	#22689
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1844

Complements

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

N	H ≅ G/N	type
3. #22568800	C₂	semidirect
4. #22560987	C₂	semidirect
5. #22560434	C₂	semidirect
6. #22560269	C₂	semidirect
7. #22560088	C₂	semidirect
8. C₆₇×#1844	C₂	semidirect
12. #16863514	C₄	semidirect
39. #14524049	C₂×C₄	semidirect
51. #14523979	C₂×C₄	semidirect
56. #14523971	C₂×C₄	semidirect
70. C₄×C₈×C₆₇	C₂×C₄	semidirect
80. Dic₂₆₈	C₄⋊C₄	semidirect
81. #11761617	C₄⋊C₄	semidirect
82. D₅₃₆	C₄⋊C₄	semidirect
167. #1844	D₆₇	semidirect
170. C₆₇	#22689	semidirect
232. C₄×C₈	C₄×D₆₇	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
#22569092	2
#22568800	2
#22560987	2
#22560434	2
#22560269	2
#22560088	2
C₆₇×#1844	2
#22689	67