Subgroups — C₄×#16863825 #31009108

Order 17152.

690 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863825	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524062	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562067	C₂	1	1	yes	maximal, characteristic, complemented
5	#22562064	C₂	1	2	yes	maximal, complemented
6	C₄×#14523987	C₂	1	2	yes	maximal, complemented
7	D₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#22561990	C₂	1	1	yes	maximal, characteristic, complemented
9	#22561913	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524062	C₂²	1	2	yes	—
13	C₄×#11761627	—	2	2	no	—
14	#16863825	C₄	1	8	yes	direct product factor
15	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic
16	C₂×#14523987	C₂²	1	2	yes	—
17	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₆₇	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₆₈	C₂²	1	2	yes	—
20	#16863490	C₂²	1	2	yes	—
21	#16863487	—	2	2	no	—
22	C₄×#11761616	—	2	2	no	—
23	D₈×Dic₆₇	C₂²	1	2	yes	—
24	#16863433	C₂²	1	2	yes	—
25	#16863389	—	2	2	no	—
26	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
27	#16863223	C₂²	1	2	yes	—
28	#16863222	C₂²	1	1	yes	characteristic
29	#16863201	C₂²	1	2	yes	—
30	#16863199	C₂²	1	2	yes	—
31	#16863197	C₂²	1	2	yes	—
32	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
35	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₆₇	C₂²	1	2	yes	—
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#357×C₆₇	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#11761627	—	2	2	no	—
41	#14524062	—	2	4	no	—
42	#14524062	C₂×C₄	1	8	yes	complemented
43	C₂×Dic₂₆₈	C₂³	1	2	yes	—
44	C₂×#11761616	—	2	2	no	—
45	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
46	C₂×C₈×D₆₇	C₂³	1	1	yes	characteristic
47	C₄×Dic₁₃₄	—	2	2	no	—
48	#14524008	—	2	2	no	—
49	D₄×Dic₆₇	—	2	2	no	—
50	C₄×C₂²⋊D₆₇	—	4	2	no	—
51	(C₂²⋊D₆₇)⋊C₄	—	4	2	no	—
52	#14524001	—	2	2	no	—
53	#14523999	—	4	2	no	—
54	C₂×C₄×Dic₆₇	—	4	2	no	—
55	C₄²×D₆₇	D₄	1	1	yes	characteristic
56	Dic₅₃₆	C₂×C₄	1	4	yes	complemented
57	#14523987	C₂×C₄	1	8	yes	complemented
58	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₁₆×D₆₇	—	2	2	no	—
61	C₁₆×D₆₇	C₂×C₄	1	4	yes	complemented
62	#14523977	C₂³	1	2	yes	—
63	#14523974	—	2	2	no	—
64	#14523968	—	2	2	no	—
65	#14523966	—	2	2	no	—
66	#14523964	—	2	2	no	—
67	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
68	C₈×Dic₆₇	D₄	1	1	yes	characteristic
69	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₆₇	C₂³	1	2	yes	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	D₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	C₂³	1	2	yes	—
76	#102×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	#11761627	—	2	8	no	—
79	#11761627	—	4	4	no	—
80	C₂×Dic₁₃₄	—	2	2	no	—
81	C₂×C₂²⋊D₆₇	—	4	2	no	—
82	C₂²×Dic₆₇	—	4	2	no	—
83	C₂²×Dic₆₇	—	4	2	no	—
84	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
85	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
86	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
87	Dic₂₆₈	—	2	2	no	—
88	Dic₂₆₈	C₂²×C₄	1	4	yes	—
89	#11761616	—	2	8	no	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₈×D₆₇	—	2	2	no	—
93	C₈×D₆₇	C₂²×C₄	1	2	yes	—
94	C₈×D₆₇	#41	1	2	yes	—
95	C₄⋊Dic₆₇	—	2	2	no	—
96	Dic₆₇⋊C₄	—	4	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	C₂²⋊Dic₆₇	—	4	2	no	—
99	(C₂×D₆₇)⋊C₄	—	4	2	no	—
100	C₄×Dic₆₇	—	4	2	no	—
101	C₄×Dic₆₇	—	4	2	no	—
102	C₄×Dic₆₇	—	4	2	no	—
103	C₄×Dic₆₇	D₈	1	1	yes	characteristic
104	C₄×Dic₆₇	D₈	1	1	yes	characteristic
105	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
106	C₆₇⋊C₁₆	—	2	1	no	—
107	C₆₇⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	D₈×C₆₇	—	2	2	no	—
111	D₈×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	2	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₇	—	2	1	no	—
118	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂³×C₆₇	—	4	2	no	—
120	C₂²×D₆₇	C₂×D₈	1	1	yes	characteristic
121	D₄×C₆₇	—	2	4	no	—
122	D₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	8	4	no	—
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₆₇	—	4	8	no	—
129	C₂×Dic₆₇	—	4	2	no	—
130	C₂×Dic₆₇	—	4	8	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	C₂×Dic₆₇	—	8	4	no	—
133	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
134	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×Dic₆₇	C₂×D₈	1	1	yes	characteristic
136	C₄×D₆₇	—	2	2	no	—
137	C₄×D₆₇	—	2	2	no	—
138	C₄×D₆₇	C₄×D₄	1	2	yes	—
139	C₄×D₆₇	#135	1	2	yes	—
140	Dic₁₃₄	—	2	4	no	—
141	Dic₁₃₄	—	4	2	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
144	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₆₇⋊C₈	—	2	1	no	—
146	C₆₇⋊C₈	C₄×D₄	1	2	yes	—
147	C₂²×C₆₇	—	4	4	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₆₇	—	2	1	no	—
152	C₂×D₆₇	—	2	1	no	—
153	C₂×D₆₇	C₄×D₈	1	2	yes	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	#508	1	2	yes	complemented
157	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
159	Dic₆₇	—	2	1	no	—
160	Dic₆₇	—	4	4	no	—
161	Dic₆₇	—	8	2	no	—
162	Dic₆₇	C₄×D₈	1	2	yes	—
163	Dic₆₇	#508	1	2	yes	complemented
164	C₄×#508	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	—	8	4	no	—
166	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
169	D₆₇	—	2	2	no	—
170	C₂×#508	—	67	1	no	—
171	C₄×#135	—	67	2	no	—
172	C₄×Q₃₂	—	67	1	no	—
173	C₄×Q₃₂	—	67	1	no	—
174	C₄×SD₃₂	—	67	2	no	—
175	C₄×SD₃₂	—	67	2	no	—
176	C₄×D₁₆	—	67	1	no	—
177	C₄×D₁₆	D₆₇	1	1	yes	characteristic, complemented
178	#2158	—	67	1	no	—
179	#2138	—	67	2	no	—
180	C₂×C₄×C₁₆	—	67	1	no	—
181	C₆₇	C₄×#508	1	1	yes	characteristic, complemented, Sylow
182	C₂×#135	—	67	2	no	—
183	C₄×#41	—	134	2	no	—
184	#508	—	67	8	no	—
185	C₂×Q₃₂	—	67	1	no	—
186	C₂×SD₃₂	—	67	2	no	—
187	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic
188	C₂²×C₁₆	—	67	1	no	—
189	C₂²×C₁₆	—	67	1	no	—
190	C₄×Q₁₆	—	67	2	no	—
191	C₄×Q₁₆	—	67	2	no	—
192	C₄×SD₁₆	—	134	2	no	—
193	C₄×SD₁₆	—	134	2	no	—
194	C₄×D₈	—	67	2	no	—
195	C₄×D₈	C₂×D₆₇	1	2	yes	—
196	#427	—	67	2	no	—
197	#416	—	134	2	no	—
198	C₂×C₄×C₈	—	67	1	no	—
199	#367	—	67	2	no	—
200	#366	—	67	1	no	—
201	#366	C₂×D₆₇	1	1	yes	characteristic
202	#358	—	67	2	no	—
203	#358	—	67	2	no	—
204	#357	—	67	2	no	—
205	#357	C₂×D₆₇	1	2	yes	—
206	C₄×C₁₆	—	67	1	no	—
207	C₄×C₁₆	—	67	1	no	—
208	C₄×C₁₆	—	67	1	no	—
209	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
210	C₂×#41	—	134	2	no	—
211	#135	—	67	8	no	—
212	#135	—	134	4	no	—
213	C₂×Q₁₆	—	67	2	no	—
214	C₂×SD₁₆	—	134	2	no	—
215	C₂×D₈	C₂²×D₆₇	1	2	yes	—
216	C₂²×C₈	—	67	1	no	—
217	C₂²×C₈	—	67	1	no	—
218	C₄×Q₈	—	134	2	no	—
219	C₄×Q₈	—	134	2	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	268	2	no	—
222	C₄×D₄	—	268	2	no	—
223	C₄×D₄	—	134	2	no	—
224	#117	—	268	2	no	—
225	#117	—	134	2	no	—
226	C₂×C₄²	—	268	2	no	—
227	C₂×C₄²	—	67	1	no	—
228	Q₃₂	—	67	4	no	—
229	SD₃₂	—	67	8	no	—
230	D₁₆	C₄×D₆₇	1	4	yes	complemented
231	C₂×C₁₆	—	67	1	no	—
232	C₂×C₁₆	—	67	4	no	—
233	C₂×C₁₆	—	67	1	no	—
234	C₂×C₁₆	—	134	2	no	—
235	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
237	#107	—	67	2	no	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	#106	—	134	2	no	—
240	#103	—	134	2	no	—
241	#103	—	134	2	no	—
242	#102	—	2	2	no	—
243	#102	—	134	2	no	—
244	C₄×C₈	—	67	1	no	—
245	C₄×C₈	—	67	1	no	—
246	C₄×C₈	—	67	1	no	—
247	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
248	#41	—	268	4	no	—
249	#41	—	134	8	no	—
250	C₂×Q₈	—	134	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	268	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	268	2	no	—
255	C₂²×C₄	—	268	2	no	—
256	C₂²×C₄	—	67	1	no	—
257	C₂²×C₄	—	67	1	no	—
258	C₂²×C₄	—	67	1	no	—
259	Q₁₆	—	67	4	no	—
260	Q₁₆	—	134	2	no	—
261	SD₁₆	—	134	8	no	—
262	D₈	—	2	2	no	—
263	D₈	C₂×C₄×D₆₇	1	4	yes	—
264	C₂×C₈	—	67	1	no	—
265	C₂×C₈	—	67	2	no	—
266	C₂×C₈	—	67	2	no	—
267	C₂×C₈	—	67	1	no	—
268	C₂×C₈	—	134	2	no	—
269	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
270	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	268	2	no	—
273	C₄⋊C₄	—	268	2	no	—
274	C₄⋊C₄	—	134	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	268	2	no	—
277	C₂²⋊C₄	—	268	2	no	—
278	C₄²	—	268	2	no	—
279	C₄²	—	268	2	no	—
280	C₄²	—	268	2	no	—
281	C₄²	—	67	1	no	—
282	C₄²	—	67	1	no	—
283	C₄²	—	67	1	no	—
284	C₄²	D₄×D₆₇	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	67	2	no	—
287	C₁₆	—	134	1	no	—
288	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
289	C₂³	—	4	2	no	—
290	C₂³	—	67	1	no	—
291	Q₈	—	268	2	no	—
292	Q₈	—	134	4	no	—
293	D₄	—	2	4	no	—
294	D₄	—	4	2	no	—
295	D₄	—	268	8	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	—	268	2	no	—
299	C₂×C₄	—	268	8	no	—
300	C₂×C₄	—	268	2	no	—
301	C₂×C₄	—	536	4	no	—
302	C₂×C₄	—	67	1	no	—
303	C₂×C₄	—	67	1	no	—
304	C₂×C₄	—	67	2	no	—
305	C₂×C₄	—	67	2	no	—
306	C₂×C₄	—	67	1	no	—
307	C₂×C₄	—	134	2	no	—
308	C₂×C₄	—	134	2	no	—
309	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
310	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	67	2	no	—
314	C₈	—	134	1	no	—
315	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
316	C₈	#41×D₆₇	1	1	yes	characteristic
317	C₂²	—	4	4	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	67	2	no	—
321	C₂²	—	134	1	no	—
322	C₂²	—	134	1	no	—
323	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	268	4	no	—
327	C₄	—	536	2	no	—
328	C₄	—	67	2	no	—
329	C₄	—	67	2	no	—
330	C₄	—	134	1	no	—
331	C₄	#16863825	1	2	yes	direct product factor
332	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
333	C₄	#135×D₆₇	1	1	yes	characteristic
334	C₂	—	8	4	no	—
335	C₂	—	134	2	no	—
336	C₂	C₄×D₈×D₆₇	1	1	yes	characteristic
337	C₂	#508×D₆₇	1	1	yes	characteristic
338	C₂	C₂×#16863825	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×D₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#16863825	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×D₁₆	D₆₇
hypocenter	C₆₇	C₄×#508
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#14524062	C₂	semidirect
3. C₄×Dic₅₃₆	C₂	semidirect
4. #22562067	C₂	semidirect
5. #22562064	C₂	semidirect
6. C₄×#14523987	C₂	semidirect
7. D₁₆×Dic₆₇	C₂	semidirect
8. #22561990	C₂	semidirect
9. #22561913	C₂	semidirect
10. C₄×C₁₆×D₆₇	C₂	semidirect
11. C₄×D₁₆×C₆₇	C₂	semidirect
14. #16863825	C₄	direct (N × H)
42. #14524062	C₂×C₄	semidirect
56. Dic₅₃₆	C₂×C₄	semidirect
57. #14523987	C₂×C₄	semidirect
61. C₁₆×D₆₇	C₂×C₄	semidirect
72. D₁₆×C₆₇	C₂×C₄	semidirect
156. C₄×C₆₇	#508	semidirect
163. Dic₆₇	#508	semidirect
177. C₄×D₁₆	D₆₇	semidirect
181. C₆₇	C₄×#508	semidirect
230. D₁₆	C₄×D₆₇	semidirect
331. C₄	#16863825	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863825	2
C₄×#14524062	2
C₄×Dic₅₃₆	2
#22562067	2
#22562064	2
C₄×#14523987	2
D₁₆×Dic₆₇	2
#22561990	2
#22561913	2
C₄×C₁₆×D₆₇	2
C₄×D₁₆×C₆₇	2
C₄×#508	67

Subgroups — C4×#16863825 #31009108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863825 #31009108