Subgroups — #30754987

Order 17152.

430 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570795	C₂	1	1	yes	maximal, characteristic
2	#22561153	C₂	1	2	yes	maximal
3	C₄×#14523977	C₂	1	1	yes	maximal, characteristic
4	#22558709	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	2	yes	—
7	#16863801	C₄	1	1	yes	characteristic, complemented
8	#16863776	C₄	1	1	yes	characteristic, complemented
9	#16863747	C₄	1	2	yes	complemented
10	#16863637	—	2	1	no	—
11	#16863637	—	2	1	no	—
12	C₄×Dic₂₆₈	—	2	1	no	—
13	C₄×Dic₂₆₈	—	2	1	no	—
14	C₂×#14523977	—	2	1	no	—
15	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
16	#16863159	—	2	1	no	—
17	#16863157	—	2	4	no	—
18	#16863134	—	2	4	no	—
19	#431×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#322×C₆₇	—	2	1	no	—
23	C₈²×C₆₇	—	2	1	no	—
24	C₂×Dic₂₆₈	—	2	1	no	—
25	C₂×Dic₂₆₈	—	2	1	no	—
26	#14524043	C₂×C₄	1	2	yes	—
27	#14524043	C₂×C₄	1	2	yes	—
28	#14524035	—	2	2	no	—
29	#14524022	—	2	2	no	—
30	#14524022	—	2	2	no	—
31	C₄×Dic₁₃₄	—	2	2	no	—
32	C₄×Dic₁₃₄	—	2	2	no	—
33	#14524002	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄×Dic₆₇	—	2	2	no	—
36	#14523977	—	2	4	no	—
37	#14523977	—	2	1	no	—
38	#14523977	—	2	1	no	—
39	#14523968	—	2	2	no	—
40	#14523968	—	2	2	no	—
41	C₆₇⋊#95	—	2	2	no	—
42	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₆₇	—	2	1	no	—
47	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
48	#97×C₆₇	—	2	1	no	—
49	C₄×C₈×C₆₇	—	2	1	no	—
50	C₄×C₈×C₆₇	—	2	1	no	—
51	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂×Dic₁₃₄	—	2	2	no	—
56	C₂×Dic₁₃₄	—	2	2	no	—
57	C₂²×Dic₆₇	—	4	2	no	—
58	Dic₂₆₈	—	4	2	no	—
59	Dic₂₆₈	—	4	2	no	—
60	C₄⋊Dic₆₇	—	2	4	no	—
61	C₄⋊Dic₆₇	—	2	2	no	—
62	C₄⋊Dic₆₇	—	2	2	no	—
63	C₄⋊Dic₆₇	—	2	2	no	—
64	C₄⋊Dic₆₇	—	2	2	no	—
65	C₄⋊Dic₆₇	—	4	2	no	—
66	Dic₆₇⋊C₄	—	4	2	no	—
67	Dic₆₇⋊C₄	—	4	2	no	—
68	C₂²⋊Dic₆₇	—	4	2	no	—
69	C₄×Dic₆₇	—	4	2	no	—
70	C₄×Dic₆₇	—	4	4	no	—
71	C₂²×C₄×C₆₇	C₂×D₄	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	M₁₆×C₆₇	—	2	2	no	—
75	M₁₆×C₆₇	Q₁₆	1	4	yes	complemented
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	4	1	no	—
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₆₇	#41	1	1	yes	characteristic
87	C₄²×C₆₇	#41	1	1	yes	characteristic
88	C₂³×C₆₇	#127	1	1	yes	characteristic
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	4	1	no	—
92	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₆₇	#135	1	2	yes	—
100	C₂×Dic₆₇	—	4	4	no	—
101	C₂×Dic₆₇	—	4	2	no	—
102	C₂×Dic₆₇	—	4	2	no	—
103	C₂×Dic₆₇	—	8	2	no	—
104	C₂×Dic₆₇	—	8	2	no	—
105	Dic₁₃₄	—	4	2	no	—
106	Dic₁₃₄	—	4	2	no	—
107	Dic₁₃₄	—	8	2	no	—
108	C₈×C₆₇	—	2	1	no	—
109	C₈×C₆₇	—	2	1	no	—
110	C₈×C₆₇	—	2	1	no	—
111	C₈×C₆₇	—	2	4	no	—
112	C₈×C₆₇	—	4	2	no	—
113	C₈×C₆₇	#104	1	2	yes	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	1	no	—
116	C₂²×C₆₇	#390	1	1	yes	characteristic
117	C₂²×C₆₇	#494	1	1	yes	characteristic
118	C₂²×C₆₇	#495	1	1	yes	characteristic
119	C₄×C₆₇	—	2	1	no	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
123	C₄×C₆₇	#421	1	1	yes	characteristic
124	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
125	C₄×C₆₇	#467	1	2	yes	—
126	Dic₆₇	—	8	4	no	—
127	Dic₆₇	—	8	2	no	—
128	Dic₆₇	—	8	2	no	—
129	#9718	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	—	2	1	no	—
131	C₂×C₆₇	—	2	1	no	—
132	C₂×C₆₇	#1969	1	1	yes	characteristic
133	C₂×C₆₇	#1976	1	1	yes	characteristic
134	C₂×C₆₇	#1980	1	1	yes	characteristic
135	#3170	—	67	1	no	—
136	#1986	—	67	2	no	—
137	C₄×#107	—	67	1	no	—
138	#1503	—	67	2	no	—
139	C₈×M₁₆	D₆₇	1	1	yes	characteristic
140	C₆₇	#9718	1	1	yes	characteristic, complemented, Sylow
141	#533	—	67	2	no	—
142	#500	—	67	1	no	—
143	#494	—	67	1	no	—
144	#487	—	67	2	no	—
145	#467	—	134	1	no	—
146	#467	—	134	1	no	—
147	C₄×Q₁₆	—	134	1	no	—
148	C₄×Q₁₆	—	134	1	no	—
149	#431	C₂×D₆₇	1	1	yes	characteristic
150	C₂×#107	—	134	1	no	—
151	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
153	C₄×C₄⋊C₄	—	67	2	no	—
154	#340	—	134	1	no	—
155	#339	—	134	4	no	—
156	#328	—	134	4	no	—
157	#322	—	2	1	no	—
158	C₈²	—	2	1	no	—
159	C₂×Q₁₆	—	134	1	no	—
160	C₂×Q₁₆	—	134	1	no	—
161	C₂×M₁₆	D₂₆₈	1	2	yes	—
162	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
163	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
164	C₄⋊Q₈	—	67	2	no	—
165	C₄⋊Q₈	—	67	2	no	—
166	#125	—	134	2	no	—
167	C₂²⋊Q₈	—	134	2	no	—
168	C₂²⋊Q₈	—	134	2	no	—
169	C₄×Q₈	—	134	2	no	—
170	C₄×Q₈	—	134	2	no	—
171	#117	—	134	2	no	—
172	C₂×C₄⋊C₄	—	134	2	no	—
173	C₂×C₄²	—	134	2	no	—
174	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
175	#107	—	134	4	no	—
176	#107	—	134	1	no	—
177	#107	—	134	1	no	—
178	C₄⋊C₈	—	2	1	no	—
179	#103	—	134	2	no	—
180	#103	—	134	2	no	—
181	C₂²⋊C₈	D₂₆₈	1	2	yes	—
182	#97	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
186	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
189	#95	—	134	2	no	—
190	C₂×Q₈	—	134	2	no	—
191	C₂×Q₈	—	134	2	no	—
192	C₂²×C₄	—	268	2	no	—
193	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
196	Q₁₆	—	268	2	no	—
197	Q₁₆	—	268	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	Dic₂₆₈	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
208	C₄⋊C₄	—	268	2	no	—
209	C₄⋊C₄	—	268	2	no	—
210	C₄⋊C₄	—	268	2	no	—
211	C₄⋊C₄	—	134	4	no	—
212	C₄⋊C₄	—	134	2	no	—
213	C₄⋊C₄	—	134	2	no	—
214	C₄⋊C₄	—	134	2	no	—
215	C₄⋊C₄	—	134	2	no	—
216	C₂²⋊C₄	—	268	2	no	—
217	C₄²	—	268	2	no	—
218	C₄²	—	268	4	no	—
219	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
221	C₄²	#11761628	1	1	yes	characteristic
222	C₄²	#11761628	1	1	yes	characteristic
223	C₂³	#14524040	1	1	yes	characteristic
224	Q₈	—	268	2	no	—
225	Q₈	—	268	2	no	—
226	Q₈	—	536	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	268	4	no	—
231	C₂×C₄	—	268	2	no	—
232	C₂×C₄	—	268	2	no	—
233	C₂×C₄	—	536	2	no	—
234	C₂×C₄	—	536	2	no	—
235	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
238	C₂×C₄	#14524017	1	1	yes	characteristic
239	C₂×C₄	#14524017	1	1	yes	characteristic
240	C₂×C₄	#14524031	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
242	C₂×C₄	#14524065	1	2	yes	—
243	C₈	—	2	4	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	4	2	no	—
248	C₈	#14523969	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	#16863298	1	1	yes	characteristic
252	C₂²	#16863776	1	1	yes	characteristic
253	C₂²	#16863780	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	536	4	no	—
258	C₄	—	536	2	no	—
259	C₄	—	536	2	no	—
260	C₄	C₂×#14523969	1	1	yes	characteristic
261	C₄	#16863412	1	1	yes	characteristic
262	C₄	C₄×Dic₂₆₈	1	2	yes	—
263	C₄	#16863637	1	2	yes	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#22561070	1	1	yes	characteristic
267	C₂	#22561107	1	1	yes	characteristic
268	C₂	#22561128	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₈×M₁₆	D₆₇
hypocenter	C₆₇	#9718
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
7. #16863801	C₄	semidirect
8. #16863776	C₄	semidirect
9. #16863747	C₄	semidirect
75. M₁₆×C₆₇	Q₁₆	semidirect
140. C₆₇	#9718	semidirect
199. M₁₆	Dic₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
#22570795	2
#22561153	2
C₄×#14523977	2
#22558709	2
C₈×M₁₆×C₆₇	2
#9718	67