Subgroups — #30874840

Order 17152.

622 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	M₁₆×Dic₁₃₄	C₂	1	2	yes	maximal
2	C₂×#16863526	C₂	1	1	yes	maximal, characteristic
3	#22560984	C₂	1	2	yes	maximal
4	#22560981	C₂	1	4	yes	maximal
5	#22560348	C₂	1	2	yes	maximal
6	#22560347	C₂	1	1	yes	maximal, characteristic
7	#22560114	C₂	1	2	yes	maximal
8	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	#16863526	—	2	2	no	—
11	#16863526	C₄	1	4	yes	—
12	#16863526	C₄	1	4	yes	—
13	#16863524	—	2	4	no	—
14	C₈×Dic₁₃₄	—	2	2	no	—
15	C₂×#14523978	C₂²	1	4	yes	—
16	#16863420	C₂²	1	2	yes	—
17	#16863419	C₂²	1	4	yes	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×Dic₆₇⋊C₈	C₂²	1	2	yes	—
20	M₁₆×Dic₆₇	C₂²	1	4	yes	—
21	C₂×#14523953	C₂²	1	2	yes	—
22	#16863161	C₂²	1	2	yes	—
23	#16863161	C₂²	1	2	yes	—
24	#16863160	C₂²	1	4	yes	—
25	C₂×#108×C₆₇	C₂²	1	2	yes	—
26	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	#341×C₆₇	C₂²	1	1	yes	characteristic
29	#341×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
31	C₂×#11761612	C₂³	1	4	yes	—
32	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
33	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₃₄	D₄	1	2	yes	—
36	C₄×Dic₁₃₄	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₆₇	Q₈	1	1	yes	characteristic
38	C₂×Dic₆₇⋊C₄	Q₈	1	2	yes	—
39	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
40	#14523978	—	2	8	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
43	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
44	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
45	Dic₆₇⋊C₈	—	2	4	no	—
46	Dic₆₇⋊C₈	—	2	4	no	—
47	Dic₆₇⋊C₈	C₂×C₄	1	8	yes	—
48	#14523953	—	2	4	no	—
49	#14523953	C₂×C₄	1	8	yes	—
50	C₈×Dic₆₇	—	2	4	no	—
51	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
52	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	#108×C₆₇	—	2	4	no	—
57	#97×C₆₇	—	2	2	no	—
58	C₄×C₈×C₆₇	—	2	2	no	—
59	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×Dic₁₃₄	—	4	2	no	—
62	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
63	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₆₇	C₂×Q₈	1	2	yes	—
65	C₂²×Dic₆₇	C₂×Q₈	1	2	yes	—
66	#11761612	—	2	4	no	—
67	#11761612	—	2	4	no	—
68	C₂×C₆₇⋊C₈	—	2	4	no	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
71	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
72	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
73	Dic₆₇⋊C₄	C₄⋊C₄	1	4	yes	—
74	Dic₆₇⋊C₄	C₄⋊C₄	1	4	yes	—
75	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
76	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₆₇	—	2	2	no	—
81	M₁₆×C₆₇	—	2	2	no	—
82	M₁₆×C₆₇	—	2	2	no	—
83	M₁₆×C₆₇	#41	1	4	yes	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	4	1	no	—
97	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
104	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
105	C₂×Dic₆₇	—	4	2	no	—
106	C₂×Dic₆₇	—	4	2	no	—
107	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
109	Dic₁₃₄	—	2	4	no	—
110	Dic₁₃₄	—	4	2	no	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	—	2	2	no	—
113	C₈×C₆₇	—	2	4	no	—
114	C₈×C₆₇	—	4	2	no	—
115	C₆₇⋊C₈	—	2	4	no	—
116	C₆₇⋊C₈	—	4	4	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₆₇	#557	1	2	yes	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	#430	1	2	yes	—
125	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
126	C₄×C₆₇	#519	1	1	yes	characteristic
127	C₄×C₆₇	#548	1	2	yes	—
128	Dic₆₇	—	2	4	no	—
129	Dic₆₇	—	2	4	no	—
130	#22747	—	67	1	no	maximal, Sylow
131	C₂×C₆₇	—	2	2	no	—
132	C₂×C₆₇	#2351	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
134	C₂×C₆₇	#2916	1	1	yes	characteristic
135	Q₈×M₁₆	—	67	2	no	—
136	C₂×#446	—	67	1	no	—
137	#1950	—	67	4	no	—
138	#1950	—	67	2	no	—
139	#1838	—	67	2	no	—
140	#1837	—	67	1	no	—
141	#1778	—	67	2	no	—
142	C₄×#108	D₆₇	1	1	yes	characteristic
143	C₆₇	#22747	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄×Q₈	—	67	1	no	—
145	#446	—	67	4	no	—
146	#446	—	67	4	no	—
147	#446	—	134	2	no	—
148	#446	—	134	4	no	—
149	C₈×Q₈	—	134	2	no	—
150	C₂×#108	—	67	4	no	—
151	C₂×#108	C₂×D₆₇	1	2	yes	—
152	#423	—	67	4	no	—
153	#423	—	67	2	no	—
154	C₂×C₄⋊C₈	—	67	2	no	—
155	C₂×C₄⋊C₈	—	67	1	no	—
156	C₄×M₁₆	—	67	4	no	—
157	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
158	C₂×#97	—	67	2	no	—
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	#341	—	67	4	no	—
161	#341	—	67	2	no	—
162	#341	—	67	2	no	—
163	#341	C₂×D₆₇	1	1	yes	characteristic
164	#341	C₂×D₆₇	1	1	yes	characteristic
165	C₂²×Q₈	—	67	1	no	—
166	C₂×M₁₆	—	67	4	no	—
167	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
168	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
169	C₂²×C₈	—	67	2	no	—
170	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
172	C₄×Q₈	—	67	2	no	—
173	C₄×Q₈	—	67	2	no	—
174	C₄×Q₈	—	67	2	no	—
175	C₄×Q₈	—	67	2	no	—
176	C₂×C₄⋊C₄	—	67	2	no	—
177	C₂×C₄⋊C₄	—	67	1	no	—
178	C₂×C₄²	—	67	2	no	—
179	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
180	#108	—	2	4	no	—
181	#108	—	134	8	no	—
182	C₄⋊C₈	—	67	8	no	—
183	C₄⋊C₈	—	67	2	no	—
184	C₄⋊C₈	—	67	2	no	—
185	C₄⋊C₈	—	134	4	no	—
186	C₄⋊C₈	—	134	4	no	—
187	C₄⋊C₈	—	134	2	no	—
188	C₄⋊C₈	—	134	2	no	—
189	#97	—	2	2	no	—
190	#97	—	67	8	no	—
191	#97	—	134	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	134	4	no	—
194	C₄×C₈	C₄×D₆₇	1	2	yes	—
195	C₄×C₈	C₄×D₆₇	1	2	yes	—
196	C₂×Q₈	—	268	2	no	—
197	C₂×Q₈	—	67	2	no	—
198	C₂×Q₈	—	67	2	no	—
199	C₂²×C₄	—	67	2	no	—
200	C₂²×C₄	—	67	2	no	—
201	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
202	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	134	4	no	—
208	M₁₆	—	134	4	no	—
209	M₁₆	#11761628	1	4	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	67	4	no	—
215	C₂×C₈	—	134	4	no	—
216	C₂×C₈	—	134	4	no	—
217	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
219	C₄⋊C₄	—	67	4	no	—
220	C₄⋊C₄	—	67	4	no	—
221	C₄⋊C₄	—	67	2	no	—
222	C₄⋊C₄	—	67	2	no	—
223	C₄²	—	67	4	no	—
224	C₄²	—	67	4	no	—
225	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
226	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
227	C₄²	D₄×D₆₇	1	1	yes	characteristic
228	C₄²	D₄×D₆₇	1	1	yes	characteristic
229	C₂³	C₂×Q₈×D₆₇	1	1	yes	characteristic
230	Q₈	—	268	2	no	—
231	Q₈	—	134	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	268	2	no	—
235	C₂×C₄	—	268	2	no	—
236	C₂×C₄	—	67	4	no	—
237	C₂×C₄	—	67	4	no	—
238	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
244	C₂×C₄	C₂×#11761628	1	2	yes	—
245	C₂×C₄	#14524090	1	2	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	268	4	no	—
251	C₈	—	134	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
255	C₂²	#16864124	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	134	4	no	—
260	C₄	—	134	4	no	—
261	C₄	#16863443	1	2	yes	—
262	C₄	C₄×#11761628	1	1	yes	characteristic
263	C₄	#16863873	1	1	yes	characteristic
264	C₄	#16864055	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#22563129	1	1	yes	characteristic
267	C₂	C₂×#16863443	1	1	yes	characteristic
268	C₂	#22568817	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₆₇	C₂²×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#108	D₆₇
hypocenter	C₆₇	#22747
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
143. C₆₇	#22747	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₃₄	2
C₂×#16863526	2
#22560984	2
#22560981	2
#22560348	2
#22560347	2
#22560114	2
C₄×#108×C₆₇	2
#22747	67