Subgroups — #31725527

Order 17152.

1834 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	#22574028	C₂	1	2	yes	maximal
2	#22570755	C₂	1	6	yes	maximal, complemented
3	#22570748	C₂	1	12	yes	maximal, complemented
4	#22570723	C₂	1	3	yes	maximal, complemented
5	C₂×#16863801	C₂	1	3	yes	maximal, complemented
6	#22568809	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	#16863801	C₂²	1	3	yes	—
19	#16863801	C₂²	1	12	yes	—
20	#16863801	C₂²	1	1	yes	characteristic
21	#16863798	C₂²	1	12	yes	—
22	#16863794	C₂²	1	6	yes	—
23	#16863438	C₂²	1	24	yes	—
24	#16863434	C₂²	1	12	yes	—
25	C₂×#14523977	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	2	yes	—
36	#14524043	C₂³	1	6	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	4	yes	—
51	#14523977	C₂³	1	12	yes	—
52	#14523977	C₂³	1	24	yes	—
53	#14523975	C₂³	1	24	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	1	yes	characteristic
59	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄×D₄×C₆₇	D₄	1	3	yes	—
61	C₄×D₄×C₆₇	D₄	1	3	yes	—
62	#117×C₆₇	C₂³	1	3	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
64	#97×C₆₇	C₂³	1	6	yes	—
65	C₄×C₈×C₆₇	C₂³	1	3	yes	—
66	C₄×C₈×C₆₇	C₂³	1	6	yes	—
67	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₃₄	—	2	2	no	—
69	C₂×Dic₁₃₄	—	2	6	no	—
70	C₂×Dic₁₃₄	—	2	12	no	—
71	C₂×Dic₁₃₄	—	8	12	no	—
72	C₂²×Dic₆₇	—	2	12	no	—
73	C₄⋊Dic₆₇	—	2	6	no	—
74	C₄⋊Dic₆₇	—	2	12	no	—
75	C₄⋊Dic₆₇	—	2	12	no	—
76	C₄⋊Dic₆₇	—	2	2	no	—
77	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
78	C₄⋊Dic₆₇	C₂⁴	1	12	yes	—
79	Dic₆₇⋊C₄	—	2	12	no	—
80	Dic₆₇⋊C₄	—	2	12	no	—
81	C₂²⋊Dic₆₇	—	2	12	no	—
82	C₄×Dic₆₇	—	2	12	no	—
83	#41×C₆₇	C₂×Q₈	1	8	yes	—
84	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
88	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
89	M₁₆×C₆₇	C₂×Q₈	1	24	yes	complemented
90	C₂×C₈×C₆₇	C₂×Q₈	1	24	yes	complemented
91	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
92	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
93	C₄⋊C₄×C₆₇	C₂×D₄	1	3	yes	—
94	C₄⋊C₄×C₆₇	C₂×D₄	1	3	yes	—
95	C₂²⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
96	C₄²×C₆₇	C₂×D₄	1	3	yes	—
97	C₄²×C₆₇	C₂×D₄	1	3	yes	—
98	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₄²×C₆₇	C₂⁴	1	3	yes	—
100	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
101	Q₈×C₆₇	C₄⋊Q₈	1	4	yes	—
102	D₄×C₆₇	C₄⋊Q₈	1	12	yes	—
103	C₂×C₄×C₆₇	—	2	6	no	—
104	C₂×C₄×C₆₇	—	2	6	no	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
107	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
108	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
111	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
112	C₂×Dic₆₇	—	2	4	no	—
113	C₂×Dic₆₇	—	2	12	no	—
114	C₂×Dic₆₇	—	8	12	no	—
115	Dic₁₃₄	—	4	12	no	—
116	Dic₁₃₄	—	4	4	no	—
117	Dic₁₃₄	—	8	12	no	—
118	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
119	C₈×C₆₇	C₂²×Q₈	1	12	yes	—
120	C₂²×C₆₇	—	2	3	no	—
121	C₂²×C₆₇	—	2	3	no	—
122	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
123	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
124	C₄×C₆₇	—	2	3	no	—
125	C₄×C₆₇	—	2	3	no	—
126	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
127	C₄×C₆₇	#576	1	2	yes	—
128	C₄×C₆₇	#578	1	2	yes	—
129	C₄×C₆₇	C₂³×Q₈	1	2	yes	—
130	Dic₆₇	—	4	12	no	—
131	Dic₆₇	—	4	4	no	—
132	#60667	—	67	1	no	maximal, Sylow
133	C₂×C₆₇	—	2	6	no	—
134	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
136	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
137	#3503	—	67	2	no	—
138	#3165	—	67	6	no	—
139	#3164	—	67	12	no	—
140	#3160	—	67	3	no	—
141	C₂×#500	—	67	3	no	—
142	#2913	—	67	4	no	—
143	C₄×#131	D₆₇	1	1	yes	characteristic
144	C₆₇	#60667	1	1	yes	characteristic, complemented, Sylow
145	C₂×#131	C₂×D₆₇	1	2	yes	—
146	Q₈²	—	134	2	no	—
147	#557	—	134	2	no	—
148	#554	—	134	6	no	—
149	D₄×Q₈	—	134	6	no	—
150	#544	—	67	12	no	—
151	#533	—	67	6	no	—
152	C₂×C₄⋊Q₈	—	67	6	no	—
153	#520	—	67	4	no	—
154	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
155	#501	—	67	24	no	—
156	#500	—	67	3	no	—
157	#500	—	67	12	no	—
158	#500	—	67	6	no	—
159	#500	—	67	1	no	—
160	#499	—	67	12	no	—
161	#498	—	67	6	no	—
162	#428	—	67	24	no	—
163	#427	—	67	12	no	—
164	C₂×#107	—	67	12	no	—
165	#405	C₂×D₆₇	1	6	yes	—
166	C₄×M₁₆	C₂×D₆₇	1	3	yes	—
167	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
168	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
169	C₂²×Q₈	—	134	6	no	—
170	#131	Dic₁₃₄	1	16	yes	complemented
171	C₂×M₁₆	C₂²×D₆₇	1	6	yes	—
172	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
173	C₄⋊Q₈	—	67	6	no	—
174	C₄⋊Q₈	—	67	12	no	—
175	C₄⋊Q₈	—	67	2	no	—
176	C₄⋊Q₈	—	134	12	no	—
177	C₄⋊Q₈	—	134	6	no	—
178	C₄⋊Q₈	—	134	6	no	—
179	#125	—	67	12	no	—
180	#125	—	134	12	no	—
181	C₂²⋊Q₈	—	134	6	no	—
182	C₂²⋊Q₈	—	134	6	no	—
183	C₂²⋊Q₈	—	134	24	no	—
184	C₄×Q₈	—	134	4	no	—
185	C₄×Q₈	—	134	6	no	—
186	C₄×Q₈	—	134	6	no	—
187	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
188	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
189	C₄×D₄	—	134	12	no	—
190	C₄×D₄	D₂₆₈	1	3	yes	—
191	C₄×D₄	D₂₆₈	1	3	yes	—
192	#117	—	67	12	no	—
193	#117	C₂²×D₆₇	1	3	yes	—
194	C₂×C₄⋊C₄	—	67	12	no	—
195	C₂×C₄⋊C₄	—	134	6	no	—
196	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
197	#107	—	67	12	no	—
198	#107	—	67	24	no	—
199	#107	—	67	4	no	—
200	#106	—	67	24	no	—
201	#97	C₂²×D₆₇	1	6	yes	—
202	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₆₇	1	3	yes	—
204	C₄×C₈	C₂²×D₆₇	1	6	yes	—
205	#41	C₂×Dic₁₃₄	1	8	yes	—
206	C₂×Q₈	—	536	12	no	—
207	C₂×Q₈	—	134	2	no	—
208	C₂×Q₈	—	134	6	no	—
209	C₂×Q₈	—	134	12	no	—
210	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
211	C₂×D₄	C₂×D₂₆₈	1	3	yes	—
212	C₂²×C₄	—	134	12	no	—
213	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
214	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
215	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
216	M₁₆	C₂×Dic₁₃₄	1	24	yes	complemented
217	C₂×C₈	C₂×Dic₁₃₄	1	24	yes	complemented
218	C₂×C₈	C₂³×D₆₇	1	2	yes	—
219	C₂×C₈	C₂³×D₆₇	1	6	yes	—
220	C₄⋊C₄	—	67	4	no	—
221	C₄⋊C₄	—	67	12	no	—
222	C₄⋊C₄	—	134	12	no	—
223	C₄⋊C₄	—	134	6	no	—
224	C₄⋊C₄	—	134	12	no	—
225	C₄⋊C₄	—	134	2	no	—
226	C₄⋊C₄	—	134	12	no	—
227	C₄⋊C₄	—	134	12	no	—
228	C₄⋊C₄	C₂×D₂₆₈	1	3	yes	—
229	C₄⋊C₄	C₂×D₂₆₈	1	3	yes	—
230	C₂²⋊C₄	—	134	12	no	—
231	C₂²⋊C₄	C₂×D₂₆₈	1	6	yes	—
232	C₄²	—	134	12	no	—
233	C₄²	C₂×D₂₆₈	1	3	yes	—
234	C₄²	C₂×D₂₆₈	1	3	yes	—
235	C₄²	C₂³×D₆₇	1	1	yes	characteristic
236	C₄²	C₂³×D₆₇	1	3	yes	—
237	C₂³	C₂²×D₂₆₈	1	3	yes	—
238	Q₈	—	268	12	no	—
239	Q₈	—	268	4	no	—
240	Q₈	—	536	12	no	—
241	Q₈	#14524043	1	4	yes	—
242	D₄	#14524043	1	12	yes	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	536	12	no	—
246	C₂×C₄	—	134	4	no	—
247	C₂×C₄	—	134	12	no	—
248	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
249	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
250	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
251	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₁₃₄	1	12	yes	—
254	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
255	C₈	C₂²×Dic₁₃₄	1	4	yes	—
256	C₈	C₂²×Dic₁₃₄	1	12	yes	—
257	C₂²	—	2	3	no	—
258	C₂²	—	2	3	no	—
259	C₂²	C₂×#14524043	1	6	yes	—
260	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
261	C₄	—	2	3	no	—
262	C₄	—	2	3	no	—
263	C₄	—	268	12	no	—
264	C₄	—	268	4	no	—
265	C₄	C₂×#14524043	1	6	yes	—
266	C₄	#16864201	1	2	yes	—
267	C₄	#16864213	1	2	yes	—
268	C₄	C₂³×Dic₁₃₄	1	2	yes	—
269	C₂	—	2	6	no	—
270	C₂	C₂²×#14524043	1	1	yes	characteristic
271	C₂	C₂×#16864201	1	1	yes	characteristic
272	C₂	C₂×#16864213	1	1	yes	characteristic

	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	#60667
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. #22570748	C₂	semidirect
4. #22570723	C₂	semidirect
5. C₂×#16863801	C₂	semidirect
55. #131×C₆₇	Q₈	semidirect
89. M₁₆×C₆₇	C₂×Q₈	semidirect
90. C₂×C₈×C₆₇	C₂×Q₈	semidirect
144. C₆₇	#60667	semidirect
170. #131	Dic₁₃₄	semidirect
216. M₁₆	C₂×Dic₁₃₄	semidirect
217. 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
#22570748	2
#22570723	2
C₂×#16863801	2
#22568809	2
C₄×#131×C₆₇	2
#60667	67