Subgroups — #31725520

Order 17152.

1762 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574035	C₂	1	2	yes	maximal
2	#22570761	C₂	1	6	yes	maximal, complemented
3	#22570748	C₂	1	6	yes	maximal, complemented
4	#22570740	C₂	1	6	yes	maximal, complemented
5	#22570723	C₂	1	3	yes	maximal, complemented
6	C₂×#16863798	C₂	1	3	yes	maximal, complemented
7	#22568809	C₂	1	2	yes	maximal
8	#22568804	C₂	1	2	yes	maximal
9	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#16864122	—	2	4	no	—
11	#16864105	—	2	12	no	—
12	#16864024	C₂²	1	12	yes	—
13	#16863953	C₂²	1	6	yes	—
14	C₂×#14524035	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	6	yes	—
19	#16863798	C₂²	1	6	yes	—
20	#16863798	C₂²	1	3	yes	—
21	#16863798	C₂²	1	6	yes	—
22	#16863798	C₂²	1	6	yes	—
23	#16863798	C₂²	1	1	yes	characteristic
24	#16863794	C₂²	1	6	yes	—
25	#16863438	C₂²	1	12	yes	—
26	#16863438	C₂²	1	12	yes	—
27	#16863434	C₂²	1	6	yes	—
28	#16863434	C₂²	1	6	yes	—
29	C₂×#14523977	C₂²	1	6	yes	—
30	C₂×#14523975	C₂²	1	6	yes	—
31	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
34	#405×C₆₇	C₂²	1	3	yes	—
35	#405×C₆₇	C₂²	1	3	yes	—
36	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
37	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
38	#14524043	C₂³	1	12	yes	—
39	#14524042	—	2	12	no	—
40	#14524035	—	2	12	no	—
41	#14524035	C₂³	1	12	yes	—
42	#14524035	C₂³	1	6	yes	—
43	#14524035	C₂³	1	2	yes	—
44	#14524034	—	2	12	no	—
45	#14524022	—	2	12	no	—
46	#14524020	—	2	24	no	—
47	Q₈×Dic₆₇	—	2	4	no	—
48	C₄×Dic₁₃₄	—	2	12	no	—
49	D₄×Dic₆₇	—	2	12	no	—
50	#14524002	C₂³	1	12	yes	—
51	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
52	C₂×Dic₆₇⋊C₄	—	2	12	no	—
53	#14523977	C₂³	1	6	yes	—
54	#14523977	C₂³	1	12	yes	—
55	#14523977	C₂³	1	2	yes	—
56	#14523977	C₂³	1	12	yes	—
57	#14523975	C₂³	1	2	yes	—
58	#14523975	C₂³	1	6	yes	—
59	#14523975	C₂³	1	12	yes	—
60	#14523975	C₂³	1	12	yes	—
61	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
62	#131×C₆₇	Q₈	1	8	yes	complemented
63	#131×C₆₇	Q₈	1	8	yes	complemented
64	C₂×M₁₆×C₆₇	C₂³	1	3	yes	—
65	C₂×M₁₆×C₆₇	C₂³	1	3	yes	—
66	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
68	C₄×Q₈×C₆₇	D₄	1	2	yes	—
69	C₄×D₄×C₆₇	D₄	1	6	yes	—
70	#117×C₆₇	C₂³	1	3	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
72	#97×C₆₇	C₂³	1	6	yes	—
73	C₄×C₈×C₆₇	C₂³	1	3	yes	—
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂³	1	3	yes	—
76	C₄×C₈×C₆₇	C₂³	1	3	yes	—
77	C₂×Dic₁₃₄	—	2	12	no	—
78	C₂²×Dic₆₇	—	2	12	no	—
79	C₄⋊Dic₆₇	—	2	12	no	—
80	C₄⋊Dic₆₇	—	2	12	no	—
81	C₄⋊Dic₆₇	C₂⁴	1	12	yes	—
82	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
83	Dic₆₇⋊C₄	—	2	12	no	—
84	Dic₆₇⋊C₄	—	2	24	no	—
85	Dic₆₇⋊C₄	—	2	4	no	—
86	C₂²⋊Dic₆₇	—	2	12	no	—
87	C₄×Dic₆₇	—	2	12	no	—
88	#41×C₆₇	C₂×Q₈	1	4	yes	—
89	#41×C₆₇	C₂×Q₈	1	4	yes	—
90	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
93	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
94	M₁₆×C₆₇	C₂×Q₈	1	12	yes	—
95	M₁₆×C₆₇	C₂×Q₈	1	12	yes	—
96	C₂×C₈×C₆₇	C₂×Q₈	1	12	yes	—
97	C₂×C₈×C₆₇	C₂×Q₈	1	12	yes	—
98	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂⁴	1	3	yes	—
101	C₂×C₈×C₆₇	C₂⁴	1	3	yes	—
102	C₄⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
103	C₂²⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
104	C₄²×C₆₇	C₂×D₄	1	6	yes	—
105	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
106	C₄²×C₆₇	C₂⁴	1	3	yes	—
107	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
108	Q₈×C₆₇	C₄⋊Q₈	1	4	yes	—
109	D₄×C₆₇	C₄⋊Q₈	1	12	yes	—
110	C₂×C₄×C₆₇	—	4	6	no	—
111	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
115	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
116	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
117	C₂×Dic₆₇	—	2	4	no	—
118	C₂×Dic₆₇	—	2	12	no	—
119	C₂×Dic₆₇	—	8	12	no	—
120	Dic₁₃₄	—	8	12	no	—
121	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
122	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
123	C₈×C₆₇	C₂²×Q₈	1	6	yes	—
124	C₈×C₆₇	C₂²×Q₈	1	6	yes	—
125	C₂²×C₆₇	—	2	6	no	—
126	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
127	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
128	C₄×C₆₇	—	2	6	no	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
131	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
132	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
133	Dic₆₇	—	4	4	no	—
134	Dic₆₇	—	4	12	no	—
135	#60666	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	—	2	6	no	—
137	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
138	C₂×C₆₇	#3584	1	2	yes	—
139	#3504	—	67	2	no	—
140	#3166	—	67	6	no	—
141	#3164	—	67	6	no	—
142	#3163	—	67	6	no	—
143	#3160	—	67	3	no	—
144	C₂×#499	—	67	3	no	—
145	#2913	—	67	2	no	—
146	#2912	—	67	2	no	—
147	C₄×#131	D₆₇	1	1	yes	characteristic
148	C₆₇	#60666	1	1	yes	characteristic, complemented, Sylow
149	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
150	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
151	#557	—	134	4	no	—
152	#554	—	134	12	no	—
153	#544	—	67	12	no	—
154	#533	—	67	6	no	—
155	C₂×#125	—	67	6	no	—
156	#520	—	67	4	no	—
157	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
158	#501	—	67	24	no	—
159	#500	—	67	6	no	—
160	#499	—	67	3	no	—
161	#499	—	67	6	no	—
162	#499	—	67	6	no	—
163	#499	—	67	6	no	—
164	#499	—	67	1	no	—
165	#499	—	67	6	no	—
166	#498	—	67	6	no	—
167	#428	—	67	12	no	—
168	#428	—	67	12	no	—
169	#427	—	67	6	no	—
170	#427	—	67	6	no	—
171	C₂×#107	—	67	6	no	—
172	C₂×#106	—	67	6	no	—
173	#405	C₂×D₆₇	1	3	yes	—
174	#405	C₂×D₆₇	1	3	yes	—
175	C₄×M₁₆	C₂×D₆₇	1	3	yes	—
176	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
177	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
178	#131	Dic₁₃₄	1	8	yes	complemented
179	#131	Dic₁₃₄	1	8	yes	complemented
180	C₂×M₁₆	C₂²×D₆₇	1	3	yes	—
181	C₂×M₁₆	C₂²×D₆₇	1	3	yes	—
182	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
183	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
184	C₄⋊Q₈	—	67	12	no	—
185	C₄⋊Q₈	—	134	12	no	—
186	#125	—	67	6	no	—
187	#125	—	67	12	no	—
188	#125	—	67	2	no	—
189	#125	—	134	12	no	—
190	#125	—	134	12	no	—
191	C₂²⋊Q₈	—	134	12	no	—
192	C₂²⋊Q₈	—	134	24	no	—
193	C₄×Q₈	—	134	4	no	—
194	C₄×Q₈	—	134	12	no	—
195	C₄×Q₈	D₂₆₈	1	2	yes	—
196	C₄×D₄	—	134	12	no	—
197	C₄×D₄	D₂₆₈	1	6	yes	—
198	#117	—	67	12	no	—
199	#117	C₂²×D₆₇	1	3	yes	—
200	C₂×C₄⋊C₄	—	67	12	no	—
201	C₂×C₄⋊C₄	—	134	12	no	—
202	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
203	#107	—	67	6	no	—
204	#107	—	67	12	no	—
205	#107	—	67	2	no	—
206	#107	—	67	12	no	—
207	#106	—	67	6	no	—
208	#106	—	67	12	no	—
209	#106	—	67	2	no	—
210	#106	—	67	12	no	—
211	#97	C₂²×D₆₇	1	6	yes	—
212	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₆₇	1	3	yes	—
214	C₄×C₈	C₂²×D₆₇	1	3	yes	—
215	C₄×C₈	C₂²×D₆₇	1	3	yes	—
216	#41	C₂×Dic₁₃₄	1	4	yes	—
217	#41	C₂×Dic₁₃₄	1	4	yes	—
218	C₂×Q₈	—	134	12	no	—
219	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
220	C₂×D₄	C₂×D₂₆₈	1	3	yes	—
221	C₂²×C₄	—	134	12	no	—
222	C₂²×C₄	C₂×D₂₆₈	1	6	yes	—
223	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
224	M₁₆	C₂×Dic₁₃₄	1	12	yes	—
225	M₁₆	C₂×Dic₁₃₄	1	12	yes	—
226	C₂×C₈	C₂×Dic₁₃₄	1	12	yes	—
227	C₂×C₈	C₂×Dic₁₃₄	1	12	yes	—
228	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂³×D₆₇	1	3	yes	—
231	C₂×C₈	C₂³×D₆₇	1	3	yes	—
232	C₄⋊C₄	—	67	12	no	—
233	C₄⋊C₄	—	67	4	no	—
234	C₄⋊C₄	—	134	12	no	—
235	C₄⋊C₄	—	134	12	no	—
236	C₄⋊C₄	—	134	24	no	—
237	C₄⋊C₄	—	134	4	no	—
238	C₄⋊C₄	—	134	12	no	—
239	C₄⋊C₄	C₂×D₂₆₈	1	6	yes	—
240	C₂²⋊C₄	—	134	12	no	—
241	C₂²⋊C₄	C₂×D₂₆₈	1	6	yes	—
242	C₄²	—	134	12	no	—
243	C₄²	C₂×D₂₆₈	1	6	yes	—
244	C₄²	C₂³×D₆₇	1	1	yes	characteristic
245	C₄²	C₂³×D₆₇	1	3	yes	—
246	C₂³	C₂²×D₂₆₈	1	3	yes	—
247	Q₈	—	536	12	no	—
248	Q₈	#14524043	1	4	yes	—
249	D₄	#14524043	1	12	yes	—
250	C₂×C₄	—	4	6	no	—
251	C₂×C₄	—	536	12	no	—
252	C₂×C₄	—	134	4	no	—
253	C₂×C₄	—	134	12	no	—
254	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
255	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
256	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
257	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
258	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
259	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
260	C₈	C₂²×Dic₁₃₄	1	2	yes	—
261	C₈	C₂²×Dic₁₃₄	1	2	yes	—
262	C₈	C₂²×Dic₁₃₄	1	6	yes	—
263	C₈	C₂²×Dic₁₃₄	1	6	yes	—
264	C₂²	—	2	6	no	—
265	C₂²	C₂×#14524043	1	6	yes	—
266	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	6	no	—
269	C₄	—	268	12	no	—
270	C₄	—	268	4	no	—
271	C₄	C₂×#14524043	1	6	yes	—
272	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
273	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
274	C₂	—	2	6	no	—
275	C₂	C₂²×#14524043	1	1	yes	characteristic
276	C₂	#22574983	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	#60666
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. #22570761	C₂	semidirect
3. #22570748	C₂	semidirect
4. #22570740	C₂	semidirect
5. #22570723	C₂	semidirect
6. C₂×#16863798	C₂	semidirect
62. #131×C₆₇	Q₈	semidirect
63. #131×C₆₇	Q₈	semidirect
148. C₆₇	#60666	semidirect
178. #131	Dic₁₃₄	semidirect
179. #131	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22574035	2
#22570761	2
#22570748	2
#22570740	2
#22570723	2
C₂×#16863798	2
#22568809	2
#22568804	2
C₄×#131×C₆₇	2
#60666	67