Subgroups — #30761559

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863159	C₂	1	2	yes	maximal
2	C₂×#16863159	C₂	1	2	yes	maximal
3	#22558439	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	—
6	C₂×C₆₇⋊#95	—	2	2	no	—
7	C₂×C₆₇⋊#95	—	2	2	no	—
8	#16863159	—	2	4	no	—
9	#16863159	C₄	1	8	yes	complemented
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₆₇	—	2	4	no	—
14	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	complemented
15	C₂³×Dic₆₇	—	2	4	no	—
16	C₂×C₄⋊Dic₆₇	—	2	4	no	—
17	C₂×C₄⋊Dic₆₇	—	2	4	no	—
18	C₂×C₄⋊Dic₆₇	—	2	4	no	—
19	C₂×C₄⋊Dic₆₇	—	2	4	no	—
20	C₂×C₄⋊Dic₆₇	—	2	4	no	—
21	C₂×C₄⋊Dic₆₇	C₂×C₄	1	8	yes	—
22	C₆₇⋊#95	—	4	4	no	—
23	C₆₇⋊#95	—	4	4	no	—
24	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	D₄	1	4	yes	—
35	C₂×C₄²×C₆₇	—	2	2	no	—
36	C₂×C₄²×C₆₇	—	2	2	no	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
39	C₄×C₈×C₆₇	—	2	8	no	—
40	C₂²×Dic₆₇	—	2	8	no	—
41	C₂²×Dic₆₇	—	4	4	no	—
42	C₂²×Dic₆₇	—	4	4	no	—
43	C₂²×Dic₆₇	—	4	4	no	—
44	C₂²×Dic₆₇	—	4	4	no	—
45	C₂²×Dic₆₇	—	4	4	no	—
46	C₂²×Dic₆₇	—	4	4	no	—
47	C₄⋊Dic₆₇	—	2	8	no	—
48	C₄⋊Dic₆₇	—	2	8	no	—
49	C₄⋊Dic₆₇	—	4	4	no	—
50	C₄⋊Dic₆₇	—	4	4	no	—
51	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	—	2	2	no	—
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	D₈	1	2	yes	—
59	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
60	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	#41	1	2	yes	—
66	C₂²×C₄×C₆₇	#41	1	2	yes	—
67	C₂²×C₄×C₆₇	#41	1	2	yes	—
68	C₂²×C₄×C₆₇	#41	1	2	yes	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	8	no	—
74	C₂×C₈×C₆₇	—	2	8	no	—
75	C₂×C₈×C₆₇	C₄²	1	4	yes	—
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	—	2	2	no	—
79	C₄²×C₆₇	—	2	4	no	—
80	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₆₇	C₄×D₄	1	2	yes	—
82	C₂³×C₆₇	#121	1	2	yes	—
83	C₂³×C₆₇	#124	1	2	yes	—
84	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
86	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂³×C₆₇	#135	1	2	yes	—
88	C₂³×C₆₇	#135	1	2	yes	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	8	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	#106	1	4	yes	—
97	C₂×C₄×C₆₇	#107	1	4	yes	—
98	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#117	1	4	yes	—
101	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
106	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	#123	1	4	yes	—
109	C₂×C₄×C₆₇	#125	1	2	yes	—
110	C₂×C₄×C₆₇	#126	1	4	yes	—
111	C₂×Dic₆₇	—	4	8	no	—
112	C₂×Dic₆₇	—	4	8	no	—
113	C₂×Dic₆₇	—	4	8	no	—
114	C₂×Dic₆₇	—	8	4	no	—
115	C₂×Dic₆₇	—	8	4	no	—
116	C₂×Dic₆₇	—	8	4	no	—
117	C₂×Dic₆₇	—	8	4	no	—
118	C₈×C₆₇	—	2	4	no	—
119	C₈×C₆₇	—	2	4	no	—
120	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₆₇	#380	1	1	yes	characteristic
122	C₂²×C₆₇	#382	1	1	yes	characteristic
123	C₂²×C₆₇	#388	1	4	yes	—
124	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
126	C₂²×C₆₇	#427	1	1	yes	characteristic
127	C₂²×C₆₇	#427	1	1	yes	characteristic
128	C₂²×C₆₇	C₄×D₈	1	2	yes	—
129	C₂²×C₆₇	C₄×SD₁₆	1	4	yes	—
130	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
131	C₂²×C₆₇	#465	1	4	yes	—
132	C₂²×C₆₇	#466	1	2	yes	—
133	C₂²×C₆₇	#467	1	2	yes	—
134	C₂²×C₆₇	#486	1	2	yes	—
135	C₂²×C₆₇	#487	1	2	yes	—
136	C₂²×C₆₇	#488	1	4	yes	—
137	C₄×C₆₇	—	2	4	no	—
138	C₄×C₆₇	—	2	4	no	—
139	C₄×C₆₇	#376	1	1	yes	characteristic
140	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₆₇	#381	1	1	yes	characteristic
142	C₄×C₆₇	#383	1	1	yes	characteristic
143	C₄×C₆₇	#387	1	4	yes	—
144	Dic₆₇	—	8	8	no	—
145	#10820	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	#1434	1	1	yes	characteristic
147	C₂×C₆₇	C₄×#106	1	1	yes	characteristic
148	C₂×C₆₇	C₄×#107	1	1	yes	characteristic
149	C₂×C₆₇	#1807	1	1	yes	characteristic
150	C₂×C₆₇	#1808	1	1	yes	characteristic
151	C₂×C₆₇	#1834	1	1	yes	characteristic
152	C₂×C₆₇	#1835	1	1	yes	characteristic
153	C₂×C₆₇	#1933	1	4	yes	—
154	C₂×C₆₇	#1934	1	4	yes	—
155	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
156	C₂×#340	—	67	2	no	—
157	C₂×#340	—	67	2	no	—
158	C₄⋊#95	—	67	2	no	—
159	C₆₇	#10820	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
161	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	—	67	4	no	—
163	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
164	C₂×C₄×C₈	—	2	4	no	—
165	C₂×C₄×C₈	Dic₆₇	1	4	yes	complemented
166	C₂×#95	—	134	2	no	—
167	C₂×#95	—	134	2	no	—
168	#340	—	67	8	no	—
169	#340	—	134	4	no	—
170	C₂³×C₄	—	134	4	no	—
171	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
172	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
173	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	C₄×D₆₇	1	4	yes	—
179	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
180	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
182	C₂×C₄⋊C₄	—	67	8	no	—
183	C₂×C₄⋊C₄	—	134	4	no	—
184	C₂×C₄⋊C₄	—	134	4	no	—
185	C₂×C₄⋊C₄	—	134	4	no	—
186	C₂×C₄⋊C₄	—	134	4	no	—
187	C₂×C₄⋊C₄	—	134	4	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	Dic₁₃₄	1	2	yes	—
191	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
192	C₄×C₈	—	2	8	no	—
193	#95	—	268	4	no	—
194	#95	—	268	4	no	—
195	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	268	4	no	—
201	C₂²×C₄	—	268	4	no	—
202	C₂²×C₄	—	268	4	no	—
203	C₂²×C₄	—	268	4	no	—
204	C₂²×C₄	—	268	4	no	—
205	C₂²×C₄	—	268	4	no	—
206	C₂²×C₄	—	134	8	no	—
207	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
208	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
209	C₂²×C₄	D₅₃₆	1	2	yes	—
210	C₂²×C₄	#11761617	1	4	yes	—
211	C₂²×C₄	Dic₂₆₈	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
213	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
215	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
216	C₂²×C₄	#11761628	1	2	yes	—
217	C₂²×C₄	#11761628	1	2	yes	—
218	C₂²×C₄	#11761628	1	2	yes	—
219	C₂²×C₄	#11761628	1	2	yes	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
227	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
228	C₄⋊C₄	—	268	4	no	—
229	C₄⋊C₄	—	268	4	no	—
230	C₄⋊C₄	—	134	8	no	—
231	C₄⋊C₄	—	134	8	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	4	no	—
235	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
236	C₂³	C₄×D₂₆₈	1	2	yes	—
237	C₂³	#14524017	1	2	yes	—
238	C₂³	#14524031	1	2	yes	—
239	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
240	C₂³	C₂×#11761617	1	2	yes	—
241	C₂³	C₂×Dic₂₆₈	1	1	yes	characteristic
242	C₂³	#14524065	1	2	yes	—
243	C₂³	#14524065	1	2	yes	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	268	8	no	—
252	C₂×C₄	—	268	8	no	—
253	C₂×C₄	—	268	8	no	—
254	C₂×C₄	—	536	4	no	—
255	C₂×C₄	—	536	4	no	—
256	C₂×C₄	—	536	4	no	—
257	C₂×C₄	—	536	4	no	—
258	C₂×C₄	#14523975	1	4	yes	—
259	C₂×C₄	#14523977	1	4	yes	—
260	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
262	C₂×C₄	#14524000	1	4	yes	—
263	C₂×C₄	#14524002	1	1	yes	characteristic
264	C₂×C₄	#14524002	1	1	yes	characteristic
265	C₂×C₄	#14524002	1	1	yes	characteristic
266	C₂×C₄	#14524002	1	1	yes	characteristic
267	C₂×C₄	C₄×C₂²⋊D₆₇	1	4	yes	—
268	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
269	C₂×C₄	#14524022	1	2	yes	—
270	C₂×C₄	#14524026	1	4	yes	—
271	C₂×C₄	#14524035	1	2	yes	—
272	C₂×C₄	#14524037	1	4	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	2	4	no	—
275	C₂²	C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
276	C₂²	#16863256	1	1	yes	characteristic
277	C₂²	#16863262	1	1	yes	characteristic
278	C₂²	#16863287	1	4	yes	—
279	C₂²	C₂×#14523975	1	1	yes	characteristic
280	C₂²	C₂×#14523977	1	1	yes	characteristic
281	C₂²	#16863434	1	1	yes	characteristic
282	C₂²	#16863434	1	1	yes	characteristic
283	C₂²	C₄×D₅₃₆	1	2	yes	—
284	C₂²	C₄×#11761617	1	4	yes	—
285	C₂²	C₄×Dic₂₆₈	1	2	yes	—
286	C₂²	#16863625	1	4	yes	—
287	C₂²	#16863632	1	2	yes	—
288	C₂²	#16863637	1	2	yes	—
289	C₂²	#16863743	1	2	yes	—
290	C₂²	#16863747	1	2	yes	—
291	C₂²	#16863751	1	4	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	4	no	—
294	C₄	—	536	8	no	—
295	C₄	#16863242	1	1	yes	characteristic
296	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
297	C₄	#16863259	1	1	yes	characteristic
298	C₄	#16863264	1	1	yes	characteristic
299	C₄	#16863280	1	4	yes	—
300	C₂	#22558436	1	1	yes	characteristic
301	C₂	C₄×#14523975	1	1	yes	characteristic
302	C₂	C₄×#14523977	1	1	yes	characteristic
303	C₂	#22560234	1	1	yes	characteristic
304	C₂	#22560238	1	1	yes	characteristic
305	C₂	#22560337	1	1	yes	characteristic
306	C₂	#22560340	1	1	yes	characteristic
307	C₂	#22560885	1	4	yes	—
308	C₂	#22560892	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	#10820
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. #16863159	C₄	semidirect
14. C₂×C₄×C₈×C₆₇	C₄	semidirect
159. C₆₇	#10820	semidirect
165. C₂×C₄×C₈	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863159	2
#22558439	2
C₂²×C₄×C₈×C₆₇	2
#10820	67