Subgroups — #30874315

Order 17152.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 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	#22569086	C₂	1	2	yes	maximal
3	#22568801	C₂	1	2	yes	maximal, complemented
4	#22560994	C₂	1	2	yes	maximal, complemented
5	#22560443	C₂	1	4	yes	maximal, complemented
6	#22560373	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560265	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	C₂×#14524049	C₂²	1	2	yes	—
11	#16863876	C₂²	1	4	yes	—
12	#16863514	C₄	1	8	yes	complemented
13	#16863498	—	2	4	no	—
14	C₄×Dic₂₆₈	—	2	2	no	—
15	C₄×D₅₃₆	—	2	2	no	—
16	C₂×#14523979	C₂²	1	1	yes	characteristic
17	#16863439	C₂²	1	4	yes	—
18	#16863437	—	2	4	no	—
19	#16863431	—	2	2	no	—
20	C₂×#14523977	C₂²	1	2	yes	—
21	C₂×#14523975	C₂²	1	2	yes	—
22	#106×D₆₇	—	2	2	no	—
23	C₂×#14523971	C₂²	1	2	yes	—
24	#16863398	C₂²	1	4	yes	—
25	#16863262	C₂²	1	2	yes	—
26	#16863164	C₂²	1	4	yes	—
27	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
28	#427×C₆₇	C₂²	1	1	yes	characteristic
29	#405×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#339×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×#11761611	—	2	2	no	—
42	C₂×C₈×D₆₇	—	2	2	no	—
43	C₄×Dic₁₃₄	—	2	4	no	—
44	C₄×D₂₆₈	—	2	4	no	—
45	(C₂²⋊D₆₇)⋊C₄	—	4	4	no	—
46	#14524001	—	2	4	no	—
47	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
48	C₄⋊C₄×D₆₇	—	2	4	no	—
49	#14523979	C₂×C₄	1	4	yes	complemented
50	#14523977	—	2	2	no	—
51	#14523977	—	2	2	no	—
52	#14523976	—	2	4	no	—
53	#14523975	—	2	2	no	—
54	#14523975	—	2	2	no	—
55	#14523974	—	2	4	no	—
56	#14523971	C₂×C₄	1	8	yes	complemented
57	#14523968	—	2	4	no	—
58	#14523965	—	2	4	no	—
59	C₆₇⋊#95	—	4	2	no	—
60	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	#117×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	#108×C₆₇	—	2	2	no	—
66	#107×C₆₇	D₄	1	2	yes	—
67	#106×C₆₇	D₄	1	1	yes	characteristic
68	#106×C₆₇	D₄	1	1	yes	characteristic
69	#97×C₆₇	—	2	2	no	—
70	C₄×C₈×C₆₇	—	2	2	no	—
71	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
72	#11761628	—	2	2	no	—
73	#11761628	—	2	2	no	—
74	#11761628	C₂²×C₄	1	4	yes	—
75	#11761628	C₂×Q₈	1	4	yes	—
76	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
77	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
78	C₂×C₂²⋊D₆₇	—	2	2	no	—
79	C₂²×Dic₆₇	—	2	4	no	—
80	C₂×C₄×D₆₇	—	2	2	no	—
81	C₂×C₄×D₆₇	—	4	4	no	—
82	Dic₂₆₈	C₄⋊C₄	1	4	yes	—
83	#11761617	C₄⋊C₄	1	8	yes	—
84	D₅₃₆	C₄⋊C₄	1	4	yes	—
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	C₈×D₆₇	—	2	8	no	—
90	C₄⋊Dic₆₇	—	2	4	no	—
91	C₄⋊Dic₆₇	—	2	4	no	—
92	Dic₆₇⋊C₄	—	4	4	no	—
93	(C₂×D₆₇)⋊C₄	—	4	4	no	—
94	C₄×Dic₆₇	—	4	4	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₆₇	—	2	2	no	—
98	M₁₆×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
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	C₂×D₄	1	2	yes	—
109	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	2	1	no	—
113	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×D₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂²⋊D₆₇	—	2	4	no	—
127	C₂²⋊D₆₇	—	2	4	no	—
128	C₂×Dic₆₇	—	2	2	no	—
129	C₂×Dic₆₇	—	4	4	no	—
130	C₂×Dic₆₇	—	8	4	no	—
131	D₂₆₈	—	2	2	no	—
132	D₂₆₈	C₂×C₄⋊C₄	1	4	yes	—
133	C₄×D₆₇	—	2	4	no	—
134	C₄×D₆₇	—	2	4	no	—
135	C₄×D₆₇	—	16	4	no	—
136	Dic₁₃₄	—	2	2	no	—
137	Dic₁₃₄	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₆₇	—	4	2	no	—
139	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₈×C₆₇	C₄×D₄	1	2	yes	—
142	C₈×C₆₇	C₄×D₄	1	2	yes	—
143	C₆₇⋊C₈	—	2	4	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	#520	1	1	yes	characteristic
146	C₂²×C₆₇	D₄²	1	1	yes	characteristic
147	C₂²×C₆₇	#554	1	1	yes	characteristic
148	C₂×D₆₇	—	2	4	no	—
149	C₂×D₆₇	—	4	4	no	—
150	C₄×C₆₇	—	4	2	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₆₇	#547	1	1	yes	characteristic
156	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
157	Dic₆₇	—	2	4	no	—
158	Dic₆₇	—	8	4	no	—
159	#22690	—	67	1	no	maximal, Sylow
160	C₂×C₆₇	—	2	1	no	—
161	C₂×C₆₇	—	2	1	no	—
162	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₆₇	#3290	1	2	yes	—
164	D₆₇	—	4	4	no	—
165	C₂×#443	—	67	1	no	—
166	#2951	—	67	2	no	—
167	#2912	—	67	2	no	—
168	#1952	—	67	2	no	—
169	#1860	—	67	4	no	—
170	#1846	—	67	1	no	—
171	#1815	—	67	2	no	—
172	#1772	D₆₇	1	1	yes	characteristic, complemented
173	C₆₇	#22690	1	1	yes	characteristic, complemented, Sylow
174	C₂×#135	—	67	1	no	—
175	C₂×#131	—	67	2	no	—
176	#520	—	67	4	no	—
177	#443	—	67	8	no	—
178	#440	—	134	4	no	—
179	C₄×Q₁₆	—	134	2	no	—
180	C₄×D₈	—	134	2	no	—
181	C₂×#108	—	67	1	no	—
182	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
183	#428	—	67	4	no	—
184	#428	—	134	4	no	—
185	#427	—	134	2	no	—
186	#427	C₂×D₆₇	1	1	yes	characteristic
187	C₂×#107	—	67	2	no	—
188	C₂×#106	—	67	2	no	—
189	C₂×#106	—	134	2	no	—
190	C₂×#104	—	67	2	no	—
191	#417	—	67	4	no	—
192	#405	C₂×D₆₇	1	2	yes	—
193	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
194	#382	—	67	2	no	—
195	#343	—	67	4	no	—
196	#339	C₂×D₆₇	1	2	yes	—
197	C₂×#41	—	67	2	no	—
198	#135	—	67	4	no	—
199	#135	—	67	4	no	—
200	C₂×Q₁₆	—	67	1	no	—
201	C₂×SD₁₆	—	67	2	no	—
202	C₂×D₈	—	67	1	no	—
203	#131	—	67	8	no	—
204	#131	—	134	4	no	—
205	C₂×M₁₆	—	67	2	no	—
206	C₂×M₁₆	—	134	2	no	—
207	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
208	C₂²×C₈	—	134	2	no	—
209	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
211	C₄×Q₈	—	134	4	no	—
212	C₄×D₄	—	268	4	no	—
213	C₄×D₄	—	134	4	no	—
214	#117	—	134	4	no	—
215	#117	C₂²×D₆₇	1	2	yes	—
216	C₂×C₄⋊C₄	—	67	4	no	—
217	C₂×C₄⋊C₄	—	134	4	no	—
218	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
219	#108	—	2	2	no	—
220	#108	—	67	4	no	—
221	#107	—	134	2	no	—
222	#107	—	134	2	no	—
223	#107	—	134	4	no	—
224	#107	D₂₆₈	1	2	yes	—
225	#106	—	134	2	no	—
226	#106	—	134	2	no	—
227	#106	—	134	4	no	—
228	#106	D₂₆₈	1	1	yes	characteristic
229	#106	D₂₆₈	1	1	yes	characteristic
230	#104	—	67	8	no	—
231	#103	—	134	4	no	—
232	#102	—	134	4	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	C₄×D₆₇	1	4	yes	complemented
236	#95	—	268	2	no	—
237	#41	—	67	4	no	—
238	#41	—	67	4	no	—
239	#41	—	134	2	no	—
240	#41	—	134	2	no	—
241	C₂×Q₈	—	67	2	no	—
242	C₂×D₄	—	67	2	no	—
243	C₂×D₄	—	134	2	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	268	4	no	—
246	C₂²×C₄	—	134	4	no	—
247	C₂²×C₄	—	134	2	no	—
248	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
249	Q₁₆	—	67	4	no	—
250	SD₁₆	—	67	8	no	—
251	D₈	—	67	4	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	67	4	no	—
255	M₁₆	—	134	2	no	—
256	M₁₆	—	134	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	134	2	no	—
259	C₂×C₈	—	134	8	no	—
260	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
262	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
263	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
264	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
265	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
266	C₂×C₈	Q₈×D₆₇	1	2	yes	—
267	C₂×C₈	#11761628	1	2	yes	—
268	C₄⋊C₄	—	268	4	no	—
269	C₄⋊C₄	—	134	4	no	—
270	C₄⋊C₄	—	134	4	no	—
271	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
272	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	268	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	268	4	no	—
278	C₄²	C₂×C₄×D₆₇	1	2	yes	—
279	C₂³	—	134	2	no	—
280	C₂³	#14524089	1	1	yes	characteristic
281	Q₈	—	67	4	no	—
282	Q₈	—	134	2	no	—
283	D₄	—	67	4	no	—
284	D₄	—	134	4	no	—
285	D₄	—	134	4	no	—
286	D₄	—	134	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	268	4	no	—
292	C₂×C₄	—	536	4	no	—
293	C₂×C₄	—	1072	4	no	—
294	C₂×C₄	—	134	2	no	—
295	C₂×C₄	—	134	4	no	—
296	C₂×C₄	—	134	4	no	—
297	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
302	C₂×C₄	#14524090	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	—	134	4	no	—
305	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
306	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
307	C₈	C₄×D₂₆₈	1	2	yes	—
308	C₈	C₄×D₂₆₈	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	268	4	no	—
311	C₂²	—	134	4	no	—
312	C₂²	#16863876	1	1	yes	characteristic
313	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
314	C₂²	#16864106	1	1	yes	characteristic
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	536	4	no	—
319	C₄	—	134	4	no	—
320	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
321	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
322	C₄	#16864051	1	1	yes	characteristic
323	C₄	Q₈×D₂₆₈	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	268	4	no	—
327	C₂	C₄⋊C₄×D₂₆₈	1	1	yes	characteristic
328	C₂	#22571996	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1772	C₂
socle	C₂²×C₆₇	D₄²
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1772	D₆₇
hypocenter	C₆₇	#22690
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #22568801	C₂	semidirect
4. #22560994	C₂	semidirect
5. #22560443	C₂	semidirect
6. #22560373	C₂	semidirect
7. #22560265	C₂	semidirect
8. C₆₇×#1772	C₂	semidirect
12. #16863514	C₄	semidirect
39. #14524049	C₂×C₄	semidirect
49. #14523979	C₂×C₄	semidirect
56. #14523971	C₂×C₄	semidirect
71. C₄×C₈×C₆₇	C₂×C₄	semidirect
172. #1772	D₆₇	semidirect
173. C₆₇	#22690	semidirect
235. 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
#22569086	2
#22568801	2
#22560994	2
#22560443	2
#22560373	2
#22560265	2
C₆₇×#1772	2
#22690	67