Subgroups — #30770729

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 314 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	#22558306	C₂	1	4	yes	maximal, complemented
4	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524002	—	2	1	no	—
6	C₂×#14523983	—	2	1	no	—
7	C₂×#14523983	—	2	1	no	—
8	C₂×#14523983	—	2	2	no	—
9	C₂×#14523983	C₄	1	4	yes	complemented
10	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#16863434	—	2	1	no	—
12	#16863434	—	2	1	no	—
13	#16863434	C₄	1	2	yes	—
14	#16863434	C₂²	1	2	yes	—
15	C₂×#14523977	C₄	1	2	yes	—
16	C₂×#14523975	—	2	1	no	—
17	C₂²×M₁₆×C₆₇	—	2	1	no	—
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#108×C₆₇	—	2	1	no	—
20	C₂×#108×C₆₇	—	2	2	no	—
21	C₂×#108×C₆₇	C₄	1	4	yes	complemented
22	C₂×#108×C₆₇	C₂²	1	2	yes	—
23	#14524002	—	2	2	no	—
24	#14524002	—	2	2	no	—
25	#14524002	—	4	1	no	—
26	#14524002	—	4	1	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
29	C₂×C₄×Dic₆₇	—	4	1	no	—
30	#14523983	—	2	4	no	—
31	#14523983	—	2	4	no	—
32	C₂×C₆₇⋊C₁₆	—	2	2	no	—
33	C₂×C₆₇⋊C₁₆	—	2	2	no	—
34	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
35	#14523977	—	2	2	no	—
36	#14523977	C₂×C₄	1	4	yes	—
37	#14523975	—	2	2	no	—
38	#14523975	—	2	2	no	—
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	4	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	4	1	no	—
43	C₂×M₁₆×C₆₇	—	4	2	no	—
44	C₂²×C₈×C₆₇	—	4	1	no	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	D₄	1	2	yes	—
53	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	#108×C₆₇	—	2	4	no	—
56	#108×C₆₇	—	2	2	no	—
57	#108×C₆₇	C₂×C₄	1	4	yes	complemented
58	C₂²×Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₂²⋊Dic₆₇	—	4	2	no	—
62	C₂²⋊Dic₆₇	—	4	2	no	—
63	C₄×Dic₆₇	—	4	2	no	—
64	C₄×Dic₆₇	—	4	2	no	—
65	C₆₇⋊C₁₆	—	2	4	no	—
66	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	M₁₆×C₆₇	—	2	4	no	—
78	M₁₆×C₆₇	—	4	4	no	—
79	M₁₆×C₆₇	—	4	2	no	—
80	C₂×C₈×C₆₇	—	4	4	no	—
81	C₂×C₈×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	C₄²	1	2	yes	—
83	C₂×C₈×C₆₇	C₄²	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
100	C₂³×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	#102	1	1	yes	characteristic
102	C₂³×C₆₇	#102	1	1	yes	characteristic
103	C₂³×C₆₇	#103	1	2	yes	—
104	C₂³×C₆₇	#106	1	2	yes	—
105	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#95	1	2	yes	—
107	C₂×C₄×C₆₇	#95	1	2	yes	—
108	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#102	1	2	yes	—
111	C₂×C₄×C₆₇	#102	1	2	yes	—
112	C₂×C₄×C₆₇	#103	1	2	yes	—
113	C₂×C₄×C₆₇	#103	1	2	yes	—
114	C₂×C₄×C₆₇	#103	1	2	yes	—
115	C₂×C₄×C₆₇	#106	1	2	yes	—
116	C₂×C₄×C₆₇	#107	1	2	yes	—
117	C₂×C₄×C₆₇	#107	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×Dic₆₇	—	4	4	no	—
125	C₂×Dic₆₇	—	8	4	no	—
126	C₈×C₆₇	—	4	4	no	—
127	C₈×C₆₇	#95	1	2	yes	—
128	C₈×C₆₇	#95	1	2	yes	—
129	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
130	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	2	2	no	—
136	C₂²×C₆₇	—	4	2	no	—
137	C₂²×C₆₇	—	4	2	no	—
138	C₂²×C₆₇	#340	1	2	yes	—
139	C₂²×C₆₇	#340	1	2	yes	—
140	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
142	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
143	C₄×C₆₇	#340	1	2	yes	—
144	C₄×C₆₇	#340	1	2	yes	—
145	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
146	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
147	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
148	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
149	Dic₆₇	—	8	4	no	—
150	#12561	—	67	1	no	maximal, Sylow
151	C₂×C₆₇	—	2	2	no	—
152	C₂×C₆₇	—	2	2	no	—
153	C₂×C₆₇	—	2	1	no	—
154	C₂×C₆₇	—	2	1	no	—
155	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
156	C₂×C₆₇	#2143	1	2	yes	—
157	C₂²×M₃₂	—	67	1	no	—
158	C₂²×#108	D₆₇	1	1	yes	characteristic
159	C₂×#427	—	67	1	no	—
160	#1382	—	67	4	no	—
161	C₆₇	#12561	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	2	1	no	—
163	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#117	—	134	1	no	—
165	C₂×M₃₂	—	67	4	no	—
166	C₂×M₃₂	—	134	1	no	—
167	C₂×M₃₂	—	134	1	no	—
168	C₂×M₃₂	—	134	2	no	—
169	C₂²×C₁₆	—	67	2	no	—
170	C₂×#108	—	2	1	no	—
171	C₂×#108	—	2	2	no	—
172	C₂×#108	Dic₆₇	1	4	yes	complemented
173	C₂×#108	C₂×D₆₇	1	2	yes	—
174	#427	—	67	2	no	—
175	#427	—	67	2	no	—
176	#427	—	134	1	no	—
177	#427	—	134	1	no	—
178	C₂×#107	—	67	2	no	—
179	C₂×#106	—	134	1	no	—
180	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	4	1	no	—
184	C₂×M₁₆	—	4	2	no	—
185	C₂²×C₈	—	4	1	no	—
186	C₂²×C₈	Dic₁₃₄	1	2	yes	—
187	C₂²×C₈	C₄×D₆₇	1	2	yes	—
188	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
191	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
192	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
193	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
195	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
196	#117	—	268	1	no	—
197	#117	—	268	1	no	—
198	#117	—	134	2	no	—
199	#117	—	134	2	no	—
200	C₂×C₄⋊C₄	—	134	2	no	—
201	C₂×C₂²⋊C₄	—	268	1	no	—
202	C₂×C₄²	—	268	1	no	—
203	M₃₂	—	134	4	no	—
204	M₃₂	—	134	4	no	—
205	C₂×C₁₆	—	67	4	no	—
206	C₂×C₁₆	—	134	2	no	—
207	C₂×C₁₆	—	134	2	no	—
208	#108	—	2	4	no	—
209	#108	—	2	2	no	—
210	#108	C₂×Dic₆₇	1	4	yes	complemented
211	#107	—	67	4	no	—
212	#107	—	134	2	no	—
213	#106	—	134	2	no	—
214	#106	—	134	2	no	—
215	C₂⁴	#11761615	1	1	yes	characteristic
216	C₂²×C₄	—	268	2	no	—
217	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
220	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
221	C₂²×C₄	#11761613	1	1	yes	characteristic
222	C₂²×C₄	#11761613	1	1	yes	characteristic
223	C₂²×C₄	#11761615	1	1	yes	characteristic
224	C₂²×C₄	#11761616	1	2	yes	—
225	C₂²×C₄	#11761618	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	4	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
233	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
234	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
235	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
237	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
238	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
239	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
240	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
241	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
242	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
243	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
244	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
249	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
250	C₄⋊C₄	—	268	2	no	—
251	C₄⋊C₄	—	134	4	no	—
252	C₂²⋊C₄	—	268	2	no	—
253	C₂²⋊C₄	—	268	2	no	—
254	C₄²	—	268	2	no	—
255	C₄²	—	268	2	no	—
256	C₁₆	—	134	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	#14523963	1	1	yes	characteristic
259	C₂³	#14523963	1	1	yes	characteristic
260	C₂³	#14523967	1	2	yes	—
261	C₂³	#14523974	1	2	yes	—
262	C₂³	C₂×#11761615	1	1	yes	characteristic
263	C₂×C₄	—	268	4	no	—
264	C₂×C₄	—	536	4	no	—
265	C₂×C₄	C₆₇⋊#95	1	2	yes	—
266	C₂×C₄	C₆₇⋊#95	1	2	yes	—
267	C₂×C₄	#14523963	1	1	yes	characteristic
268	C₂×C₄	#14523963	1	1	yes	characteristic
269	C₂×C₄	#14523964	1	2	yes	—
270	C₂×C₄	#14523964	1	2	yes	—
271	C₂×C₄	#14523966	1	2	yes	—
272	C₂×C₄	#14523966	1	2	yes	—
273	C₂×C₄	#14523967	1	2	yes	—
274	C₂×C₄	#14523974	1	2	yes	—
275	C₂×C₄	#14523976	1	2	yes	—
276	C₂×C₄	#14523976	1	2	yes	—
277	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
281	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
282	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
283	C₈	—	4	4	no	—
284	C₈	C₆₇⋊#95	1	2	yes	—
285	C₈	C₆₇⋊#95	1	2	yes	—
286	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
287	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
288	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
289	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#16863158	1	2	yes	—
296	C₂²	#16863158	1	2	yes	—
297	C₂²	C₂×#14523963	1	1	yes	characteristic
298	C₂²	C₂×#14523967	1	1	yes	characteristic
299	C₂²	C₂×#14523974	1	1	yes	characteristic
300	C₄	—	536	4	no	—
301	C₄	#16863158	1	2	yes	—
302	C₄	#16863158	1	2	yes	—
303	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
304	C₄	C₂×#14523964	1	1	yes	characteristic
305	C₄	C₂×#14523966	1	1	yes	characteristic
306	C₄	C₂×#14523976	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	C₂×#16863158	1	1	yes	characteristic
312	C₂	#22561936	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	#12561
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #22558306	C₂	semidirect
9. C₂×#14523983	C₄	semidirect
21. C₂×#108×C₆₇	C₄	semidirect
34. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
57. #108×C₆₇	C₂×C₄	semidirect
161. C₆₇	#12561	semidirect
172. C₂×#108	Dic₆₇	semidirect
210. #108	C₂×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
#22558306	2
C₂²×#108×C₆₇	2
#12561	67