Subgroups — #31011356

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863801	C₂	1	1	yes	maximal, characteristic
2	#22562496	C₂	1	4	yes	maximal, complemented
3	#22562489	C₂	1	2	yes	maximal, complemented
4	#22562482	C₂	1	2	yes	maximal, complemented
5	#22561998	C₂	1	4	yes	maximal
6	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524043	—	2	1	no	—
9	C₂×#14523983	C₂²	1	2	yes	—
10	#16863801	—	2	1	no	—
11	#16863801	—	2	1	no	—
12	#16863801	C₂²	1	2	yes	—
13	#16863801	C₂²	1	2	yes	—
14	C₂×#14523977	—	2	1	no	—
15	C₂×#14523977	C₂²	1	2	yes	—
16	#16863223	C₂²	1	8	yes	—
17	#16863221	C₂²	1	8	yes	—
18	#16863172	C₂²	1	2	yes	—
19	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
20	#533×C₆₇	—	2	1	no	—
21	#500×C₆₇	C₂²	1	1	yes	characteristic
22	#500×C₆₇	C₂²	1	1	yes	characteristic
23	#499×C₆₇	—	2	1	no	—
24	#499×C₆₇	C₂²	1	2	yes	—
25	#498×C₆₇	—	2	1	no	—
26	#427×C₆₇	—	2	1	no	—
27	#427×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₃₄	—	4	1	no	—
30	#14524043	—	2	2	no	—
31	#14524043	—	2	2	no	—
32	#14524043	—	4	1	no	—
33	#14524043	—	4	1	no	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄⋊Dic₆₇	—	4	1	no	—
36	#14523983	Q₈	1	8	yes	complemented
37	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
38	#14523977	—	2	2	no	—
39	#14523977	—	2	2	no	—
40	#14523977	—	2	2	no	—
41	#14523977	C₂³	1	4	yes	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₆₇	—	2	1	no	—
45	C₄⋊Q₈×C₆₇	—	2	1	no	—
46	#125×C₆₇	—	2	2	no	—
47	C₂²⋊Q₈×C₆₇	—	4	1	no	—
48	C₂²⋊Q₈×C₆₇	—	4	1	no	—
49	C₄×Q₈×C₆₇	—	4	1	no	—
50	C₄×Q₈×C₆₇	—	4	1	no	—
51	#117×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	#107×C₆₇	—	2	1	no	—
54	#107×C₆₇	—	2	1	no	—
55	#107×C₆₇	C₂³	1	2	yes	—
56	#107×C₆₇	C₂³	1	2	yes	—
57	#106×C₆₇	—	2	2	no	—
58	#106×C₆₇	—	2	2	no	—
59	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₃₄	—	4	2	no	—
64	C₂×Dic₁₃₄	—	4	2	no	—
65	C₂×Dic₁₃₄	—	8	4	no	—
66	C₂²×Dic₆₇	—	4	2	no	—
67	C₄⋊Dic₆₇	—	2	4	no	—
68	C₄⋊Dic₆₇	—	4	2	no	—
69	C₄⋊Dic₆₇	—	4	2	no	—
70	C₄⋊Dic₆₇	—	4	2	no	—
71	C₆₇⋊C₁₆	C₂×Q₈	1	8	yes	complemented
72	C₂×Q₈×C₆₇	—	4	1	no	—
73	C₂×Q₈×C₆₇	—	4	1	no	—
74	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₄⋊C₄×C₆₇	—	4	2	no	—
86	C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₄⋊C₄×C₆₇	—	4	1	no	—
88	C₄⋊C₄×C₆₇	—	4	1	no	—
89	C₂²⋊C₄×C₆₇	—	4	2	no	—
90	C₄²×C₆₇	—	4	2	no	—
91	C₄²×C₆₇	D₈	1	1	yes	characteristic
92	C₄²×C₆₇	D₈	1	1	yes	characteristic
93	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
96	Q₈×C₆₇	—	8	2	no	—
97	Q₈×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	4	2	no	—
102	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₆₇	—	4	4	no	—
111	C₂×Dic₆₇	—	8	4	no	—
112	Dic₁₃₄	—	8	4	no	—
113	Dic₁₃₄	—	8	4	no	—
114	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
115	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
116	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
117	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	—	2	1	no	—
120	C₂²×C₆₇	#500	1	2	yes	—
121	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	8	2	no	—
125	C₄×C₆₇	—	8	2	no	—
126	C₄×C₆₇	#500	1	2	yes	—
127	C₄×C₆₇	#509	1	2	yes	—
128	C₄×C₆₇	#510	1	2	yes	—
129	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
131	Dic₆₇	—	8	4	no	—
132	#33731	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	—	2	2	no	—
134	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
136	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
137	#3160	D₆₇	1	1	yes	characteristic
138	C₂×#500	—	67	1	no	—
139	#2255	—	67	4	no	—
140	#2254	—	67	2	no	—
141	#2252	—	67	2	no	—
142	#2159	—	67	4	no	—
143	C₄×M₃₂	—	67	1	no	—
144	C₆₇	#33731	1	1	yes	characteristic, complemented, Sylow
145	#533	—	2	1	no	—
146	C₂×C₄⋊Q₈	—	134	1	no	—
147	C₂×M₃₂	—	67	2	no	—
148	#500	—	67	2	no	—
149	#500	—	67	2	no	—
150	#500	—	134	1	no	—
151	#500	—	134	1	no	—
152	#500	C₂×D₆₇	1	1	yes	characteristic
153	#500	C₂×D₆₇	1	1	yes	characteristic
154	#499	—	2	1	no	—
155	#499	C₂×D₆₇	1	2	yes	—
156	#498	—	2	1	no	—
157	#427	—	2	1	no	—
158	#427	C₂×D₆₇	1	2	yes	—
159	C₂×#107	—	67	2	no	—
160	C₂×#107	—	134	1	no	—
161	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
162	#367	—	67	8	no	—
163	#366	—	67	8	no	—
164	#346	—	67	2	no	—
165	C₄×C₁₆	—	67	2	no	—
166	C₂²×Q₈	—	268	1	no	—
167	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	2	1	no	—
171	C₄⋊Q₈	—	268	1	no	—
172	C₄⋊Q₈	—	268	1	no	—
173	C₄⋊Q₈	—	134	2	no	—
174	C₄⋊Q₈	—	134	2	no	—
175	#125	—	2	2	no	—
176	C₂²⋊Q₈	—	4	1	no	—
177	C₂²⋊Q₈	—	4	1	no	—
178	C₄×Q₈	—	4	1	no	—
179	C₄×Q₈	—	4	1	no	—
180	#117	—	2	2	no	—
181	C₂×C₄⋊C₄	—	268	1	no	—
182	C₂×C₄⋊C₄	—	134	2	no	—
183	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
184	M₃₂	—	67	8	no	—
185	C₂×C₁₆	—	67	4	no	—
186	#107	—	2	1	no	—
187	#107	—	2	1	no	—
188	#107	—	67	4	no	—
189	#107	—	134	2	no	—
190	#107	—	134	2	no	—
191	#107	—	134	2	no	—
192	#107	C₂²×D₆₇	1	2	yes	—
193	#107	C₂²×D₆₇	1	2	yes	—
194	#106	—	2	2	no	—
195	#106	—	2	2	no	—
196	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
197	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	268	2	no	—
203	C₂×Q₈	—	268	2	no	—
204	C₂×Q₈	—	536	4	no	—
205	C₂²×C₄	—	268	2	no	—
206	C₂²×C₄	#11761613	1	1	yes	characteristic
207	C₂²×C₄	#11761613	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
209	C₂×C₈	#11761618	1	4	yes	—
210	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
212	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂×C₈	Q₈×D₆₇	1	4	yes	—
214	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	268	2	no	—
222	C₄⋊C₄	—	268	2	no	—
223	C₄⋊C₄	—	268	2	no	—
224	C₄⋊C₄	—	134	4	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₄²	—	4	2	no	—
227	C₄²	#11761613	1	1	yes	characteristic
228	C₄²	#11761613	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
230	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
231	C₁₆	—	67	8	no	—
232	C₂³	C₂×#11761613	1	1	yes	characteristic
233	Q₈	—	8	2	no	—
234	Q₈	—	8	2	no	—
235	Q₈	—	536	4	no	—
236	Q₈	—	536	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	268	4	no	—
242	C₂×C₄	—	536	4	no	—
243	C₂×C₄	#14524041	1	2	yes	—
244	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
245	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
246	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
247	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
248	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
249	C₂×C₄	C₂×#11761618	1	2	yes	—
250	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
251	C₈	#14524041	1	2	yes	—
252	C₈	C₂×#11761618	1	2	yes	—
253	C₈	C₂×#11761618	1	2	yes	—
254	C₈	C₂×Q₈×D₆₇	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	#16863799	1	2	yes	—
258	C₂²	C₂²×#11761613	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	8	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	536	4	no	—
264	C₄	#16863799	1	2	yes	—
265	C₄	#16863831	1	2	yes	—
266	C₄	#16863836	1	2	yes	—
267	C₄	C₂×#14524041	1	1	yes	characteristic
268	C₄	C₂²×#11761618	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	C₂×#16863799	1	1	yes	characteristic
271	C₂	C₂×#16863831	1	1	yes	characteristic
272	C₂	C₂×#16863836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#33731
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #22562496	C₂	semidirect
3. #22562489	C₂	semidirect
4. #22562482	C₂	semidirect
36. #14523983	Q₈	semidirect
71. C₆₇⋊C₁₆	C₂×Q₈	semidirect
144. C₆₇	#33731	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863801	2
#22562496	2
#22562489	2
#22562482	2
#22561998	2
C₄×#14523983	2
C₆₇×#3160	2
#33731	67