Subgroups — #31725511

Order 17152.

1834 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574028	C₂	1	2	yes	maximal
2	#22570755	C₂	1	6	yes	maximal, complemented
3	#22570740	C₂	1	12	yes	maximal, complemented
4	#22570723	C₂	1	3	yes	maximal, complemented
5	C₂×#16863794	C₂	1	3	yes	maximal, complemented
6	#22568804	C₂	1	4	yes	maximal
7	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₃₄	—	2	2	no	—
9	#16864125	—	2	2	no	—
10	#16864103	—	2	6	no	—
11	D₄×Dic₁₃₄	—	2	6	no	—
12	#16864024	C₂²	1	12	yes	—
13	#16863953	C₂²	1	6	yes	—
14	C₂×#14524043	C₂²	1	6	yes	—
15	#16863879	C₂²	1	4	yes	—
16	#16863805	C₂²	1	24	yes	—
17	#16863801	C₂²	1	6	yes	—
18	#16863798	C₂²	1	12	yes	—
19	#16863794	C₂²	1	6	yes	—
20	#16863794	C₂²	1	3	yes	—
21	#16863794	C₂²	1	12	yes	—
22	#16863794	C₂²	1	1	yes	characteristic
23	#16863438	C₂²	1	24	yes	—
24	#16863434	C₂²	1	12	yes	—
25	C₂×#14523975	C₂²	1	12	yes	—
26	C₂×#131×C₆₇	C₂²	1	2	yes	—
27	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
28	#405×C₆₇	C₂²	1	6	yes	—
29	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
31	C₂²×Dic₁₃₄	—	2	6	no	—
32	#14524043	—	2	6	no	—
33	#14524043	—	2	6	no	—
34	#14524043	C₂³	1	12	yes	—
35	#14524043	C₂³	1	6	yes	—
36	#14524043	C₂³	1	2	yes	—
37	#14524042	—	2	12	no	—
38	#14524035	C₂³	1	12	yes	—
39	#14524034	—	2	12	no	—
40	#14524022	—	2	6	no	—
41	#14524022	—	2	6	no	—
42	#14524020	—	2	24	no	—
43	Q₈×Dic₆₇	—	2	4	no	—
44	C₄×Dic₁₃₄	—	2	6	no	—
45	C₄×Dic₁₃₄	—	2	6	no	—
46	D₄×Dic₆₇	—	2	12	no	—
47	#14524002	C₂³	1	12	yes	—
48	C₂×C₄⋊Dic₆₇	—	2	6	no	—
49	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
50	#14523977	C₂³	1	24	yes	—
51	#14523975	C₂³	1	12	yes	—
52	#14523975	C₂³	1	24	yes	—
53	#14523975	C₂³	1	4	yes	—
54	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
55	#131×C₆₇	Q₈	1	16	yes	complemented
56	C₂×M₁₆×C₆₇	C₂³	1	6	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
58	C₄×Q₈×C₆₇	D₄	1	2	yes	—
59	C₄×D₄×C₆₇	D₄	1	6	yes	—
60	#117×C₆₇	C₂³	1	3	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
62	#97×C₆₇	C₂³	1	6	yes	—
63	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂³	1	6	yes	—
65	C₄×C₈×C₆₇	C₂³	1	3	yes	—
66	C₂×Dic₁₃₄	—	2	2	no	—
67	C₂×Dic₁₃₄	—	2	6	no	—
68	C₂×Dic₁₃₄	—	2	12	no	—
69	C₂×Dic₁₃₄	—	8	12	no	—
70	C₂²×Dic₆₇	—	2	12	no	—
71	C₄⋊Dic₆₇	—	2	6	no	—
72	C₄⋊Dic₆₇	—	2	12	no	—
73	C₄⋊Dic₆₇	—	2	12	no	—
74	C₄⋊Dic₆₇	—	2	2	no	—
75	C₄⋊Dic₆₇	C₂⁴	1	12	yes	—
76	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
77	Dic₆₇⋊C₄	—	2	12	no	—
78	Dic₆₇⋊C₄	—	2	12	no	—
79	C₂²⋊Dic₆₇	—	2	12	no	—
80	C₄×Dic₆₇	—	2	12	no	—
81	#41×C₆₇	C₂×Q₈	1	8	yes	—
82	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
85	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
86	M₁₆×C₆₇	C₂×Q₈	1	24	yes	complemented
87	C₂×C₈×C₆₇	C₂×Q₈	1	24	yes	complemented
88	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
89	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
90	C₄⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
91	C₂²⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
92	C₄²×C₆₇	C₂×D₄	1	6	yes	—
93	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₄²×C₆₇	C₂⁴	1	3	yes	—
95	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
96	Q₈×C₆₇	C₄⋊Q₈	1	4	yes	—
97	D₄×C₆₇	C₄⋊Q₈	1	12	yes	—
98	C₂×C₄×C₆₇	—	2	12	no	—
99	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
100	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
101	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
103	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
104	C₂×Dic₆₇	—	2	4	no	—
105	C₂×Dic₆₇	—	2	12	no	—
106	C₂×Dic₆₇	—	8	12	no	—
107	Dic₁₃₄	—	4	4	no	—
108	Dic₁₃₄	—	4	12	no	—
109	Dic₁₃₄	—	8	12	no	—
110	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
111	C₈×C₆₇	C₂²×Q₈	1	12	yes	—
112	C₂²×C₆₇	—	2	6	no	—
113	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
114	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
115	C₄×C₆₇	—	2	6	no	—
116	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
117	C₄×C₆₇	#577	1	4	yes	—
118	C₄×C₆₇	C₂³×Q₈	1	2	yes	—
119	Dic₆₇	—	4	12	no	—
120	Dic₆₇	—	4	4	no	—
121	#60665	—	67	1	no	maximal, Sylow
122	C₂×C₆₇	—	2	6	no	—
123	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₆₇	C₂×#577	1	2	yes	—
125	#3503	—	67	2	no	—
126	#3165	—	67	6	no	—
127	#3163	—	67	12	no	—
128	#3160	—	67	3	no	—
129	C₂×#498	—	67	3	no	—
130	#2912	—	67	4	no	—
131	C₄×#131	D₆₇	1	1	yes	characteristic
132	C₆₇	#60665	1	1	yes	characteristic, complemented, Sylow
133	C₂×#131	C₂×D₆₇	1	2	yes	—
134	Q₈²	—	134	2	no	—
135	#557	—	134	2	no	—
136	#554	—	134	6	no	—
137	D₄×Q₈	—	134	6	no	—
138	#544	—	67	12	no	—
139	#533	—	67	6	no	—
140	C₂×C₄⋊Q₈	—	67	6	no	—
141	#520	—	67	4	no	—
142	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
143	#501	—	67	24	no	—
144	#500	—	67	6	no	—
145	#499	—	67	12	no	—
146	#498	—	67	3	no	—
147	#498	—	67	12	no	—
148	#498	—	67	6	no	—
149	#498	—	67	1	no	—
150	#428	—	67	24	no	—
151	#427	—	67	12	no	—
152	C₂×#106	—	67	12	no	—
153	#405	C₂×D₆₇	1	6	yes	—
154	C₄×M₁₆	C₂×D₆₇	1	3	yes	—
155	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
156	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
157	C₂²×Q₈	—	134	6	no	—
158	#131	Dic₁₃₄	1	16	yes	complemented
159	C₂×M₁₆	C₂²×D₆₇	1	6	yes	—
160	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
161	C₄⋊Q₈	—	67	6	no	—
162	C₄⋊Q₈	—	67	12	no	—
163	C₄⋊Q₈	—	67	2	no	—
164	C₄⋊Q₈	—	134	12	no	—
165	C₄⋊Q₈	—	134	6	no	—
166	C₄⋊Q₈	—	134	6	no	—
167	#125	—	67	12	no	—
168	#125	—	134	12	no	—
169	C₂²⋊Q₈	—	134	6	no	—
170	C₂²⋊Q₈	—	134	6	no	—
171	C₂²⋊Q₈	—	134	24	no	—
172	C₄×Q₈	—	134	4	no	—
173	C₄×Q₈	—	134	6	no	—
174	C₄×Q₈	—	134	6	no	—
175	C₄×Q₈	D₂₆₈	1	2	yes	—
176	C₄×D₄	—	134	12	no	—
177	C₄×D₄	D₂₆₈	1	6	yes	—
178	#117	—	67	12	no	—
179	#117	C₂²×D₆₇	1	3	yes	—
180	C₂×C₄⋊C₄	—	67	12	no	—
181	C₂×C₄⋊C₄	—	134	6	no	—
182	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
183	#107	—	67	24	no	—
184	#106	—	67	12	no	—
185	#106	—	67	24	no	—
186	#106	—	67	4	no	—
187	#97	C₂²×D₆₇	1	6	yes	—
188	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₆₇	1	3	yes	—
190	C₄×C₈	C₂²×D₆₇	1	6	yes	—
191	#41	C₂×Dic₁₃₄	1	8	yes	—
192	C₂×Q₈	—	536	12	no	—
193	C₂×Q₈	—	134	2	no	—
194	C₂×Q₈	—	134	6	no	—
195	C₂×Q₈	—	134	12	no	—
196	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
197	C₂×D₄	C₂×D₂₆₈	1	3	yes	—
198	C₂²×C₄	—	134	12	no	—
199	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
200	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
201	M₁₆	C₂×Dic₁₃₄	1	24	yes	complemented
202	C₂×C₈	C₂×Dic₁₃₄	1	24	yes	complemented
203	C₂×C₈	C₂³×D₆₇	1	2	yes	—
204	C₂×C₈	C₂³×D₆₇	1	6	yes	—
205	C₄⋊C₄	—	67	12	no	—
206	C₄⋊C₄	—	67	4	no	—
207	C₄⋊C₄	—	134	12	no	—
208	C₄⋊C₄	—	134	6	no	—
209	C₄⋊C₄	—	134	12	no	—
210	C₄⋊C₄	—	134	2	no	—
211	C₄⋊C₄	—	134	12	no	—
212	C₄⋊C₄	—	134	12	no	—
213	C₄⋊C₄	C₂×D₂₆₈	1	6	yes	—
214	C₂²⋊C₄	—	134	12	no	—
215	C₂²⋊C₄	C₂×D₂₆₈	1	6	yes	—
216	C₄²	—	134	12	no	—
217	C₄²	C₂×D₂₆₈	1	6	yes	—
218	C₄²	C₂³×D₆₇	1	1	yes	characteristic
219	C₄²	C₂³×D₆₇	1	3	yes	—
220	C₂³	C₂²×D₂₆₈	1	3	yes	—
221	Q₈	—	268	12	no	—
222	Q₈	—	268	4	no	—
223	Q₈	—	536	12	no	—
224	Q₈	#14524043	1	4	yes	—
225	D₄	#14524043	1	12	yes	—
226	C₂×C₄	—	2	12	no	—
227	C₂×C₄	—	536	12	no	—
228	C₂×C₄	—	134	4	no	—
229	C₂×C₄	—	134	12	no	—
230	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
231	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
232	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
233	C₂×C₄	C₂²×Dic₁₃₄	1	12	yes	—
234	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
235	C₈	C₂²×Dic₁₃₄	1	4	yes	—
236	C₈	C₂²×Dic₁₃₄	1	12	yes	—
237	C₂²	—	2	6	no	—
238	C₂²	C₂×#14524043	1	6	yes	—
239	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	268	12	no	—
242	C₄	—	268	4	no	—
243	C₄	C₂×#14524043	1	6	yes	—
244	C₄	#16864208	1	4	yes	—
245	C₄	C₂³×Dic₁₃₄	1	2	yes	—
246	C₂	—	2	6	no	—
247	C₂	C₂²×#14524043	1	1	yes	characteristic
248	C₂	C₂×#16864208	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₄×#131×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂×#41×C₆₇	C₂³
hypercenter	C₄×#131	D₆₇
hypocenter	C₆₇	#60665
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #22570755	C₂	semidirect
3. #22570740	C₂	semidirect
4. #22570723	C₂	semidirect
5. C₂×#16863794	C₂	semidirect
55. #131×C₆₇	Q₈	semidirect
86. M₁₆×C₆₇	C₂×Q₈	semidirect
87. C₂×C₈×C₆₇	C₂×Q₈	semidirect
132. C₆₇	#60665	semidirect
158. #131	Dic₁₃₄	semidirect
201. M₁₆	C₂×Dic₁₃₄	semidirect
202. C₂×C₈	C₂×Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22574028	2
#22570755	2
#22570740	2
#22570723	2
C₂×#16863794	2
#22568804	2
C₄×#131×C₆₇	2
#60665	67