Subgroups — #30770749

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	#22558314	C₂	1	4	yes	maximal, complemented
4	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	—	2	1	no	—
6	C₂×#14523983	—	2	2	no	—
7	C₂×#14523983	—	2	1	no	—
8	C₂×#14523983	C₄	1	4	yes	complemented
9	C₂×#14523983	C₂²	1	2	yes	—
10	C₂²×C₆₇⋊C₁₆	—	2	1	no	—
11	#16863434	—	2	2	no	—
12	#16863434	C₄	1	4	yes	complemented
13	C₂×#14523977	—	2	1	no	—
14	C₂×#14523975	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	—	2	1	no	—
17	#427×C₆₇	—	2	1	no	—
18	#427×C₆₇	—	2	1	no	—
19	#427×C₆₇	C₄	1	2	yes	complemented
20	#427×C₆₇	C₂²	1	2	yes	—
21	C₂×#107×C₆₇	C₄	1	2	yes	complemented
22	C₂×#106×C₆₇	—	2	1	no	—
23	#14524002	—	2	4	no	—
24	#14524002	—	4	2	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
27	C₂×C₄×Dic₆₇	—	4	1	no	—
28	#14523983	—	2	4	no	—
29	#14523983	—	2	2	no	—
30	#14523983	C₂×C₄	1	4	yes	complemented
31	C₂×C₆₇⋊C₁₆	—	2	4	no	—
32	C₂×C₆₇⋊C₁₆	—	2	2	no	—
33	#14523977	—	2	4	no	—
34	#14523975	—	2	2	no	—
35	#14523975	C₂×C₄	1	4	yes	complemented
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	2	yes	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	#117×C₆₇	—	2	2	no	—
48	#117×C₆₇	—	2	2	no	—
49	#117×C₆₇	—	4	1	no	—
50	#117×C₆₇	—	4	1	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
53	C₂×C₄²×C₆₇	—	4	1	no	—
54	#107×C₆₇	—	2	2	no	—
55	#107×C₆₇	C₂×C₄	1	4	yes	—
56	#106×C₆₇	—	2	2	no	—
57	#106×C₆₇	—	2	2	no	—
58	C₂²×Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₂²⋊Dic₆₇	—	4	4	no	—
62	C₄×Dic₆₇	—	4	4	no	—
63	C₆₇⋊C₁₆	—	2	4	no	—
64	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₄²	1	2	yes	—
77	C₂×C₈×C₆₇	C₄²	1	2	yes	—
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	4	2	no	—
96	C₂²⋊C₄×C₆₇	—	4	2	no	—
97	C₂²⋊C₄×C₆₇	—	4	2	no	—
98	C₄²×C₆₇	—	4	2	no	—
99	C₄²×C₆₇	—	4	2	no	—
100	C₂³×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	#102	1	2	yes	—
102	C₂³×C₆₇	#102	1	1	yes	characteristic
103	C₂³×C₆₇	#102	1	1	yes	characteristic
104	C₂³×C₆₇	#107	1	2	yes	—
105	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	4	4	no	—
107	C₂×C₄×C₆₇	—	8	4	no	—
108	C₂×C₄×C₆₇	#95	1	2	yes	—
109	C₂×C₄×C₆₇	#95	1	2	yes	—
110	C₂×C₄×C₆₇	#102	1	2	yes	—
111	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#103	1	2	yes	—
114	C₂×C₄×C₆₇	#103	1	2	yes	—
115	C₂×C₄×C₆₇	#103	1	2	yes	—
116	C₂×C₄×C₆₇	#103	1	2	yes	—
117	C₂×C₄×C₆₇	#106	1	2	yes	—
118	C₂×C₄×C₆₇	#106	1	2	yes	—
119	C₂×C₄×C₆₇	#107	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₆₇	—	4	4	no	—
127	C₂×Dic₆₇	—	8	4	no	—
128	C₈×C₆₇	#95	1	2	yes	—
129	C₈×C₆₇	#95	1	2	yes	—
130	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
131	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	4	2	no	—
138	C₂²×C₆₇	—	4	2	no	—
139	C₂²×C₆₇	#340	1	2	yes	—
140	C₂²×C₆₇	#340	1	2	yes	—
141	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
142	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
144	C₄×C₆₇	—	8	4	no	—
145	C₄×C₆₇	#340	1	2	yes	—
146	C₄×C₆₇	#340	1	2	yes	—
147	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
148	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
149	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
150	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
151	Dic₆₇	—	8	4	no	—
152	#12567	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	—	2	2	no	—
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	—	2	1	no	—
157	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
158	C₂×C₆₇	#2145	1	2	yes	—
159	C₂²×M₃₂	—	67	1	no	—
160	C₂×#427	—	67	1	no	—
161	C₂×#427	D₆₇	1	1	yes	characteristic
162	#1386	—	67	4	no	—
163	C₆₇	#12567	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
165	C₂×#117	—	2	1	no	—
166	C₂×#117	—	134	1	no	—
167	C₂×M₃₂	—	67	4	no	—
168	C₂×M₃₂	—	67	2	no	—
169	C₂×M₃₂	—	134	2	no	—
170	C₂×M₃₂	—	134	1	no	—
171	C₂²×C₁₆	—	134	1	no	—
172	#427	—	2	1	no	—
173	#427	—	2	1	no	—
174	#427	—	67	4	no	—
175	#427	—	134	2	no	—
176	#427	Dic₆₇	1	2	yes	complemented
177	#427	C₂×D₆₇	1	2	yes	—
178	C₂×#107	—	134	1	no	—
179	C₂×#107	Dic₆₇	1	2	yes	complemented
180	C₂×#106	—	2	1	no	—
181	C₂×#106	—	67	2	no	—
182	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
183	C₂²×C₈	Dic₁₃₄	1	2	yes	—
184	C₂²×C₈	C₄×D₆₇	1	2	yes	—
185	C₂²×C₈	C₄×D₆₇	1	2	yes	—
186	C₂²×C₈	D₂₆₈	1	2	yes	—
187	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	4	1	no	—
196	#117	—	4	1	no	—
197	#117	—	268	2	no	—
198	#117	—	134	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	134	2	no	—
201	C₂×C₂²⋊C₄	—	4	1	no	—
202	C₂×C₂²⋊C₄	—	268	1	no	—
203	C₂×C₄²	—	4	1	no	—
204	C₂×C₄²	—	268	1	no	—
205	M₃₂	—	67	4	no	—
206	M₃₂	—	134	4	no	—
207	M₃₂	—	134	2	no	—
208	C₂×C₁₆	—	134	4	no	—
209	C₂×C₁₆	—	134	2	no	—
210	#107	—	2	2	no	—
211	#107	—	134	4	no	—
212	#107	C₂×Dic₆₇	1	4	yes	—
213	#106	—	2	2	no	—
214	#106	—	2	2	no	—
215	#106	—	67	4	no	—
216	#106	—	134	2	no	—
217	C₂⁴	#11761613	1	1	yes	characteristic
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	268	2	no	—
220	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
221	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
223	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
224	C₂²×C₄	#11761613	1	1	yes	characteristic
225	C₂²×C₄	#11761615	1	2	yes	—
226	C₂²×C₄	#11761616	1	1	yes	characteristic
227	C₂²×C₄	#11761616	1	1	yes	characteristic
228	C₂²×C₄	#11761618	1	2	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
230	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
231	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
232	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
233	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
234	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
237	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
238	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
239	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
240	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
241	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
244	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
246	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
247	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	268	2	no	—
251	C₄⋊C₄	—	134	4	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	268	4	no	—
255	C₄²	—	4	2	no	—
256	C₄²	—	4	2	no	—
257	C₄²	—	268	4	no	—
258	C₁₆	—	134	4	no	—
259	C₂³	—	4	2	no	—
260	C₂³	#14523963	1	2	yes	—
261	C₂³	#14523964	1	1	yes	characteristic
262	C₂³	#14523964	1	1	yes	characteristic
263	C₂³	#14523976	1	2	yes	—
264	C₂³	C₂×#11761613	1	1	yes	characteristic
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	268	4	no	—
268	C₂×C₄	—	536	4	no	—
269	C₂×C₄	C₆₇⋊#95	1	2	yes	—
270	C₂×C₄	C₆₇⋊#95	1	2	yes	—
271	C₂×C₄	#14523963	1	2	yes	—
272	C₂×C₄	#14523964	1	1	yes	characteristic
273	C₂×C₄	#14523964	1	1	yes	characteristic
274	C₂×C₄	#14523966	1	2	yes	—
275	C₂×C₄	#14523966	1	2	yes	—
276	C₂×C₄	#14523967	1	2	yes	—
277	C₂×C₄	#14523967	1	2	yes	—
278	C₂×C₄	#14523974	1	2	yes	—
279	C₂×C₄	#14523974	1	2	yes	—
280	C₂×C₄	#14523976	1	2	yes	—
281	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
283	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
285	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
286	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
287	C₈	C₆₇⋊#95	1	2	yes	—
288	C₈	C₆₇⋊#95	1	2	yes	—
289	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
290	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
291	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
292	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	#16863158	1	2	yes	—
299	C₂²	#16863158	1	2	yes	—
300	C₂²	C₂×#14523963	1	1	yes	characteristic
301	C₂²	C₂×#14523964	1	1	yes	characteristic
302	C₂²	C₂×#14523976	1	1	yes	characteristic
303	C₄	—	8	4	no	—
304	C₄	—	536	4	no	—
305	C₄	#16863158	1	2	yes	—
306	C₄	#16863158	1	2	yes	—
307	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
308	C₄	C₂×#14523966	1	1	yes	characteristic
309	C₄	C₂×#14523967	1	1	yes	characteristic
310	C₄	C₂×#14523974	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	C₂×#16863158	1	1	yes	characteristic
316	C₂	#22561945	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14523964
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#12567
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #22558314	C₂	semidirect
8. C₂×#14523983	C₄	semidirect
12. #16863434	C₄	semidirect
19. #427×C₆₇	C₄	semidirect
21. C₂×#107×C₆₇	C₄	semidirect
30. #14523983	C₂×C₄	semidirect
35. #14523975	C₂×C₄	semidirect
163. C₆₇	#12567	semidirect
176. #427	Dic₆₇	semidirect
179. C₂×#107	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂×#16863434	2
#22558314	2
C₂×#427×C₆₇	2
#12567	67