Subgroups — #31010351

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863828	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569879	C₂	1	2	yes	maximal, complemented
3	#22562322	C₂	1	4	yes	maximal, complemented
4	#22562300	C₂	1	2	yes	maximal, complemented
5	#22562295	C₂	1	2	yes	maximal, complemented
6	C₂×#16863222	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561922	C₂	1	2	yes	maximal, complemented
8	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	2	yes	—
10	#16863974	—	2	2	no	—
11	#16863828	—	2	4	no	—
12	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
13	C₂×#14523988	C₂²	1	2	yes	—
14	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
15	#16863642	—	2	4	no	—
16	#16863637	C₂²	1	4	yes	—
17	#16863632	C₂²	1	4	yes	—
18	C₂×#14523977	C₂²	1	2	yes	—
19	#16863398	—	2	2	no	—
20	#16863222	C₂²	1	2	yes	—
21	#16863222	C₂²	1	2	yes	—
22	#16863201	C₂²	1	4	yes	—
23	#16863198	C₂²	1	4	yes	—
24	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
25	#504×C₆₇	—	2	4	no	—
26	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
27	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#11761628	—	2	2	no	—
31	#14524065	—	2	4	no	—
32	#14524065	—	2	4	no	—
33	C₂×Dic₂₆₈	C₂³	1	2	yes	—
34	C₂×#11761617	—	2	2	no	—
35	C₂×D₅₃₆	C₂³	1	2	yes	—
36	#14524022	—	2	4	no	—
37	#14524019	—	4	2	no	—
38	#14524017	—	2	4	no	—
39	#14524015	—	4	4	no	—
40	#14524014	—	4	2	no	—
41	C₂×C₄⋊Dic₆₇	—	2	2	no	—
42	Dic₅₃₆	—	2	2	no	—
43	#14523988	—	2	4	no	—
44	D₁₀₇₂	—	2	2	no	—
45	#14523977	—	2	2	no	—
46	#14523977	C₂³	1	4	yes	—
47	#14523968	—	2	4	no	—
48	#14523965	—	2	4	no	—
49	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
50	#131×C₆₇	—	2	2	no	—
51	#131×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
53	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	M₃₂×C₆₇	—	2	2	no	—
57	M₃₂×C₆₇	—	2	4	no	—
58	C₂×C₁₆×C₆₇	—	2	4	no	—
59	C₂×C₁₆×C₆₇	D₄	1	4	yes	complemented
60	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
63	#11761628	—	4	4	no	—
64	#11761628	—	4	4	no	—
65	C₂×Dic₁₃₄	—	2	2	no	—
66	C₂×D₂₆₈	—	2	2	no	—
67	C₂×C₂²⋊D₆₇	—	4	2	no	—
68	C₂×C₂²⋊D₆₇	—	4	4	no	—
69	C₂²×Dic₆₇	—	4	2	no	—
70	C₂×C₄×D₆₇	—	4	2	no	—
71	Dic₂₆₈	—	2	2	no	—
72	Dic₂₆₈	—	2	2	no	—
73	#11761617	—	4	4	no	—
74	D₅₃₆	—	2	2	no	—
75	D₅₃₆	—	2	2	no	—
76	C₄⋊Dic₆₇	—	2	4	no	—
77	C₄⋊Dic₆₇	—	4	2	no	—
78	Dic₆₇⋊C₄	—	4	4	no	—
79	C₂²⋊Dic₆₇	—	4	4	no	—
80	(C₂×D₆₇)⋊C₄	—	4	4	no	—
81	#41×C₆₇	—	2	2	no	—
82	#41×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
84	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
85	C₂×D₄×C₆₇	D₈	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	M₁₆×C₆₇	—	2	2	no	—
89	M₁₆×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	#41	1	2	yes	—
96	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₁₆×C₆₇	—	2	2	no	—
98	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
99	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
100	C₂³×C₆₇	C₂×D₈	1	2	yes	—
101	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₆₇	—	4	2	no	—
103	Q₈×C₆₇	—	2	2	no	—
104	D₄×C₆₇	—	2	4	no	—
105	D₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	#121	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#135	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₆₇	—	8	4	no	—
114	C₂²⋊D₆₇	—	16	4	no	—
115	C₂×Dic₆₇	—	4	4	no	—
116	C₂×Dic₆₇	—	4	2	no	—
117	C₂×Dic₆₇	—	16	2	no	—
118	D₂₆₈	—	4	2	no	—
119	D₂₆₈	—	4	2	no	—
120	C₄×D₆₇	—	8	4	no	—
121	Dic₁₃₄	—	4	2	no	—
122	Dic₁₃₄	—	4	2	no	—
123	C₈×C₆₇	—	2	2	no	—
124	C₈×C₆₇	#121	1	2	yes	—
125	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₂²×C₆₇	—	2	1	no	—
128	C₂²×C₆₇	—	2	1	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	#466	1	2	yes	—
133	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
134	C₂×D₆₇	—	8	2	no	—
135	C₂×D₆₇	—	8	2	no	—
136	C₂×D₆₇	—	8	2	no	—
137	C₄×C₆₇	—	2	2	no	—
138	C₄×C₆₇	#466	1	2	yes	—
139	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
140	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
141	Dic₆₇	—	8	4	no	—
142	Dic₆₇	—	8	2	no	—
143	#33610	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	—	2	2	no	—
145	C₂×C₆₇	—	4	2	no	—
146	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
147	C₂×C₆₇	#3415	1	1	yes	characteristic
148	C₂×C₆₇	#3417	1	1	yes	characteristic
149	D₆₇	—	16	2	no	—
150	C₂×#508	—	67	1	no	—
151	C₂×#504	D₆₇	1	1	yes	characteristic, complemented
152	#3061	—	67	2	no	—
153	#2220	—	67	4	no	—
154	#2216	—	67	2	no	—
155	#2215	—	67	2	no	—
156	C₂×#366	—	67	1	no	—
157	#2139	—	67	2	no	—
158	C₆₇	#33610	1	1	yes	characteristic, complemented, Sylow
159	C₂×#135	—	67	2	no	—
160	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
161	#537	—	134	2	no	—
162	#508	—	134	4	no	—
163	C₂×Q₃₂	—	67	1	no	—
164	C₂×SD₃₂	—	67	2	no	—
165	C₂×D₁₆	—	67	1	no	—
166	#504	—	2	4	no	—
167	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
168	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
169	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
170	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
171	#468	—	134	4	no	—
172	#467	—	67	4	no	—
173	#466	—	67	4	no	—
174	C₂×#107	—	67	2	no	—
175	#417	—	134	2	no	—
176	#366	—	67	2	no	—
177	#366	—	67	2	no	—
178	#358	—	67	4	no	—
179	#357	—	67	4	no	—
180	C₂×#41	—	134	2	no	—
181	C₂×#41	D₂₆₈	1	1	yes	characteristic
182	#135	—	134	4	no	—
183	#135	—	134	4	no	—
184	C₂×Q₁₆	—	67	2	no	—
185	C₂×SD₁₆	—	134	2	no	—
186	C₂×D₈	—	67	2	no	—
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
190	C₂×M₁₆	D₂₆₈	1	2	yes	—
191	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
192	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
193	C₂²⋊Q₈	—	268	2	no	—
194	C₂²⋊Q₈	—	134	4	no	—
195	#121	—	268	2	no	—
196	#121	—	268	4	no	—
197	#121	—	134	4	no	—
198	C₂×C₄⋊C₄	—	134	2	no	—
199	Q₃₂	—	134	2	no	—
200	SD₃₂	—	134	4	no	—
201	D₁₆	—	134	2	no	—
202	M₃₂	—	2	2	no	—
203	M₃₂	—	2	4	no	—
204	C₂×C₁₆	—	2	4	no	—
205	C₂×C₁₆	C₂²⋊D₆₇	1	4	yes	complemented
206	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
207	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
208	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
209	#107	—	67	4	no	—
210	#107	—	134	2	no	—
211	#103	—	134	4	no	—
212	#102	—	134	4	no	—
213	#41	—	2	2	no	—
214	#41	—	2	2	no	—
215	#41	—	268	4	no	—
216	#41	—	268	4	no	—
217	C₂×Q₈	—	134	2	no	—
218	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
219	C₂×D₄	—	268	2	no	—
220	C₂×D₄	—	268	4	no	—
221	C₂×D₄	—	134	2	no	—
222	C₂×D₄	D₅₃₆	1	2	yes	—
223	C₂×D₄	D₅₃₆	1	1	yes	characteristic
224	C₂²×C₄	—	268	2	no	—
225	C₂²×C₄	—	268	2	no	—
226	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
227	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
228	Q₁₆	—	134	2	no	—
229	Q₁₆	—	134	2	no	—
230	SD₁₆	—	268	4	no	—
231	D₈	—	134	2	no	—
232	D₈	—	134	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
238	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
239	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
240	C₂×C₈	#11761628	1	2	yes	—
241	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
242	C₄⋊C₄	—	268	2	no	—
243	C₄⋊C₄	—	268	4	no	—
244	C₄⋊C₄	—	134	4	no	—
245	C₂²⋊C₄	—	268	4	no	—
246	C₂²⋊C₄	—	268	4	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
249	C₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
250	C₂³	—	268	2	no	—
251	C₂³	C₂×D₅₃₆	1	2	yes	—
252	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	268	2	no	—
255	Q₈	—	268	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	D₄	—	268	2	no	—
259	D₄	—	268	2	no	—
260	D₄	—	536	4	no	—
261	D₄	—	1072	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	268	4	no	—
265	C₂×C₄	—	268	2	no	—
266	C₂×C₄	—	536	4	no	—
267	C₂×C₄	—	1072	2	no	—
268	C₂×C₄	#14524017	1	2	yes	—
269	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
270	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
271	C₂×C₄	#14524065	1	2	yes	—
272	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	#14524017	1	2	yes	—
275	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
276	C₈	C₂×#11761628	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	536	2	no	—
283	C₂²	—	536	2	no	—
284	C₂²	—	536	2	no	—
285	C₂²	#16863632	1	2	yes	—
286	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	536	4	no	—
289	C₄	—	536	2	no	—
290	C₄	#16863632	1	2	yes	—
291	C₄	C₂×#14524017	1	1	yes	characteristic
292	C₄	C₂×#14524065	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	1072	2	no	—
296	C₂	C₂×#16863632	1	1	yes	characteristic
297	C₂	#22573350	1	1	yes	characteristic
298	C₂	#22573362	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₃₆
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#33610
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16863828	C₂	semidirect
2. #22569879	C₂	semidirect
3. #22562322	C₂	semidirect
4. #22562300	C₂	semidirect
5. #22562295	C₂	semidirect
6. C₂×#16863222	C₂	semidirect
7. #22561922	C₂	semidirect
8. C₂×#504×C₆₇	C₂	semidirect
59. C₂×C₁₆×C₆₇	D₄	semidirect
151. C₂×#504	D₆₇	semidirect
158. C₆₇	#33610	semidirect
205. C₂×C₁₆	C₂²⋊D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863828	2
#22569879	2
#22562322	2
#22562300	2
#22562295	2
C₂×#16863222	2
#22561922	2
C₂×#504×C₆₇	2
#33610	67