Subgroups — C₄×#10587604 #14537604

Order 2176.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587604	C₂	1	1	yes	maximal, characteristic
2	C₄×#63256	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762916	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762913	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762912	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762908	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#63231	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#63219	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#63230	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762816	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762809	C₂	1	1	yes	maximal, characteristic
14	C₄×#63218	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63256	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
23	C₂×#63231	C₂²	1	1	yes	characteristic
24	C₂×#63230	C₂²	1	1	yes	characteristic
25	#10587604	C₄	1	8	yes	direct product factor
26	C₂×#63219	C₂²	1	1	yes	characteristic
27	C₂×#63218	C₂²	1	1	yes	characteristic
28	C₂×#63218	C₂²	1	1	yes	characteristic
29	#10587579	—	2	1	no	—
30	#10587579	C₂²	1	1	yes	characteristic
31	#10587578	—	2	1	no	—
32	#10587578	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	—	2	1	no	—
34	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
35	#10587573	—	2	1	no	—
36	#10587573	C₂²	1	1	yes	characteristic
37	C₄×D₆₈	—	2	1	no	—
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	C₄²×D₁₇	—	2	1	no	—
40	C₄²×D₁₇	—	2	1	no	—
41	C₄²×D₁₇	C₂²	1	1	yes	characteristic
42	C₄×Dic₃₄	—	2	1	no	—
43	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
44	C₁₇×#107	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	C₁₇×#103	C₂²	1	1	yes	characteristic
47	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
48	#10587506	C₂²	1	1	yes	characteristic
49	#10587506	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587486	C₂²	1	1	yes	characteristic
54	#10587486	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₄×#3622	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	#63256	—	2	2	no	—
63	#63256	C₂×C₄	1	4	yes	—
64	Q₈×D₁₇	—	2	2	no	—
65	Q₈×D₁₇	C₂×C₄	1	4	yes	complemented
66	C₂×D₆₈	—	2	1	no	—
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
76	C₂×Dic₃₄	—	2	1	no	—
77	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
78	Q₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	—	2	1	no	—
82	C₄⋊C₄×C₁₇	—	2	1	no	—
83	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
85	C₄²×C₁₇	—	2	1	no	—
86	C₄²×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	C₂³	1	1	yes	characteristic
88	#63231	C₂×C₄	1	4	yes	complemented
89	#63230	C₂×C₄	1	4	yes	complemented
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	C₄⋊Dic₁₇	—	2	1	no	—
94	C₄⋊Dic₁₇	—	2	1	no	—
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₄×Dic₁₇	—	2	1	no	—
105	C₄×Dic₁₇	D₄	1	1	yes	characteristic
106	C₄×Dic₁₇	D₄	1	1	yes	characteristic
107	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	C₂×#3622	C₂³	1	1	yes	characteristic
110	#63219	C₂×C₄	1	4	yes	complemented
111	#63218	—	2	2	no	—
112	#63218	C₂×C₄	1	4	yes	complemented
113	C₂²×D₁₇	—	2	1	no	—
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
118	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
132	D₆₈	—	2	4	no	—
133	D₆₈	—	2	1	no	—
134	D₆₈	C₂²×C₄	1	2	yes	—
135	C₄×D₁₇	—	2	4	no	—
136	C₄×D₁₇	—	2	2	no	—
137	C₄×D₁₇	—	2	4	no	—
138	C₄×D₁₇	—	4	1	no	—
139	C₄×D₁₇	—	4	2	no	—
140	C₄×D₁₇	—	4	2	no	—
141	C₄×D₁₇	—	4	2	no	—
142	C₄×D₁₇	C₂²×C₄	1	2	yes	—
143	C₄×D₁₇	#41	1	2	yes	—
144	Dic₃₄	—	2	1	no	—
145	Dic₃₄	—	2	4	no	—
146	Dic₃₄	C₂²×C₄	1	2	yes	—
147	C₈×C₁₇	—	2	1	no	—
148	C₈×C₁₇	C₂²×C₄	1	2	yes	—
149	#3622	—	2	1	no	—
150	#3622	C₂²×C₄	1	2	yes	—
151	C₄×#137	—	17	1	no	maximal, Sylow
152	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₇	—	2	1	no	—
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	—	2	2	no	—
156	C₂×D₁₇	—	4	1	no	—
157	C₂×D₁₇	—	4	1	no	—
158	C₂×D₁₇	C₄×D₄	1	2	yes	—
159	C₄×C₁₇	—	2	2	no	—
160	C₄×C₁₇	—	2	2	no	—
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	4	1	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	#137	1	2	yes	complemented
165	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
166	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
167	Dic₁₇	—	2	2	no	—
168	Dic₁₇	—	2	1	no	—
169	Dic₁₇	—	2	1	no	—
170	Dic₁₇	—	4	1	no	—
171	Dic₁₇	—	4	1	no	—
172	Dic₁₇	C₄×D₄	1	2	yes	—
173	C₂×#137	—	17	1	no	—
174	C₄×#41	—	17	1	no	—
175	C₂×C₄×Q₈	—	17	1	no	—
176	#441	—	17	1	no	—
177	#441	—	17	1	no	—
178	#440	—	17	1	no	—
179	#440	—	17	1	no	—
180	C₄×Q₁₆	—	17	1	no	—
181	C₄×Q₁₆	D₁₇	1	1	yes	characteristic, complemented
182	C₄×SD₁₆	—	17	1	no	—
183	C₄×SD₁₆	—	17	1	no	—
184	#428	—	17	1	no	—
185	#419	—	17	1	no	—
186	#417	—	17	1	no	—
187	C₄×M₁₆	—	17	1	no	—
188	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
190	C₂×C₁₇	#575	1	1	yes	characteristic
191	D₁₇	—	2	2	no	—
192	D₁₇	—	4	2	no	—
193	C₂×#41	—	17	1	no	—
194	C₂²×Q₈	—	17	1	no	—
195	#137	—	17	8	no	—
196	C₂×Q₁₆	—	17	1	no	—
197	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
198	C₂×SD₁₆	—	17	1	no	—
199	C₂×SD₁₆	—	17	1	no	—
200	C₂×M₁₆	—	17	1	no	—
201	C₂×M₁₆	—	17	1	no	—
202	C₄×Q₈	—	17	1	no	—
203	C₄×Q₈	—	17	1	no	—
204	C₄×Q₈	—	17	1	no	—
205	C₄×Q₈	—	17	1	no	—
206	C₄×Q₈	—	34	1	no	—
207	C₄×Q₈	—	34	1	no	—
208	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
209	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
210	C₄×D₄	—	17	1	no	—
211	C₄×D₄	—	17	1	no	—
212	C₄×D₄	—	34	1	no	—
213	C₄×D₄	—	34	1	no	—
214	#117	—	17	1	no	—
215	#117	—	34	1	no	—
216	C₂×C₄⋊C₄	—	17	1	no	—
217	C₂×C₄⋊C₄	—	34	1	no	—
218	C₂×C₄²	—	17	1	no	—
219	C₂×C₄²	—	34	1	no	—
220	C₂×C₄²	—	34	1	no	—
221	#107	—	17	1	no	—
222	#107	C₂×D₁₇	1	1	yes	characteristic
223	#106	—	17	1	no	—
224	#106	—	17	1	no	—
225	#103	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	C₂×D₁₇	1	1	yes	characteristic
230	#103	C₂×D₁₇	1	1	yes	characteristic
231	#102	—	17	1	no	—
232	#102	—	17	1	no	—
233	#97	—	17	1	no	—
234	#97	—	17	1	no	—
235	C₄×C₈	—	17	1	no	—
236	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
237	C₁₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
238	#41	—	17	4	no	—
239	#41	—	34	2	no	—
240	C₂×Q₈	—	17	4	no	—
241	C₂×Q₈	—	17	1	no	—
242	C₂×Q₈	—	34	1	no	—
243	C₂×Q₈	—	34	2	no	—
244	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₂×D₄	—	17	1	no	—
247	C₂×D₄	—	34	1	no	—
248	C₂²×C₄	—	17	1	no	—
249	C₂²×C₄	—	17	1	no	—
250	C₂²×C₄	—	17	1	no	—
251	C₂²×C₄	—	34	1	no	—
252	C₂²×C₄	—	34	1	no	—
253	C₂²×C₄	—	34	1	no	—
254	C₂²×C₄	—	34	1	no	—
255	C₂²×C₄	—	34	1	no	—
256	Q₁₆	—	17	4	no	—
257	Q₁₆	C₄×D₁₇	1	4	yes	complemented
258	SD₁₆	—	17	4	no	—
259	SD₁₆	—	17	4	no	—
260	M₁₆	—	17	4	no	—
261	M₁₆	—	34	2	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	17	1	no	—
264	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	17	1	no	—
269	C₄⋊C₄	—	17	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	—	34	1	no	—
274	C₄⋊C₄	—	34	1	no	—
275	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
277	C₂²⋊C₄	—	34	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	17	1	no	—
283	C₄²	—	17	1	no	—
284	C₄²	—	17	1	no	—
285	C₄²	—	34	1	no	—
286	C₄²	—	34	1	no	—
287	C₄²	—	34	1	no	—
288	C₄²	—	34	1	no	—
289	C₄²	—	34	1	no	—
290	C₄²	C₂²×D₁₇	1	1	yes	characteristic
291	C₂³	—	17	1	no	—
292	C₂³	—	34	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	17	2	no	—
296	Q₈	—	34	4	no	—
297	Q₈	—	34	1	no	—
298	Q₈	C₂×C₄×D₁₇	1	2	yes	—
299	Q₈	C₂×C₄×D₁₇	1	2	yes	—
300	D₄	—	17	2	no	—
301	D₄	—	34	4	no	—
302	D₄	—	34	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	17	1	no	—
308	C₂×C₄	—	17	2	no	—
309	C₂×C₄	—	17	2	no	—
310	C₂×C₄	—	17	1	no	—
311	C₂×C₄	—	17	1	no	—
312	C₂×C₄	—	34	4	no	—
313	C₂×C₄	—	34	4	no	—
314	C₂×C₄	—	34	1	no	—
315	C₂×C₄	—	34	2	no	—
316	C₂×C₄	—	34	1	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	68	1	no	—
319	C₂×C₄	—	68	2	no	—
320	C₂×C₄	—	68	2	no	—
321	C₂×C₄	—	68	2	no	—
322	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
323	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	17	2	no	—
327	C₈	—	34	1	no	—
328	C₈	C₂×C₄×D₁₇	1	2	yes	—
329	C₂²	—	17	2	no	—
330	C₂²	—	34	1	no	—
331	C₂²	—	34	1	no	—
332	C₂²	—	34	2	no	—
333	C₂²	—	68	1	no	—
334	C₂²	—	68	1	no	—
335	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	17	2	no	—
342	C₄	—	34	2	no	—
343	C₄	—	34	1	no	—
344	C₄	—	34	1	no	—
345	C₄	—	68	1	no	—
346	C₄	—	68	1	no	—
347	C₄	#10587604	1	2	yes	direct product factor
348	C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
349	C₄	#41×D₁₇	1	1	yes	characteristic
350	C₂	—	34	2	no	—
351	C₂	—	68	2	no	—
352	C₂	C₄×D₄×D₁₇	1	1	yes	characteristic
353	C₂	C₂×#10587604	1	1	yes	characteristic
354	C₂	#11763606	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₄×Q₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63256	C₂²
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₄×Q₁₆	D₁₇
hypocenter	C₁₇	C₄×#137
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Q₈×D₁₇	C₂	semidirect
4. #11762916	C₂	semidirect
5. #11762913	C₂	semidirect
6. #11762912	C₂	semidirect
7. #11762908	C₂	semidirect
8. C₄×#63231	C₂	semidirect
9. C₄×#63219	C₂	semidirect
10. C₄×#63230	C₂	semidirect
11. #11762850	C₂	semidirect
12. #11762816	C₂	semidirect
14. C₄×#63218	C₂	semidirect
15. C₄×Q₁₆×C₁₇	C₂	semidirect
25. #10587604	C₄	direct (N × H)
65. Q₈×D₁₇	C₂×C₄	semidirect
78. Q₁₆×C₁₇	C₂×C₄	semidirect
88. #63231	C₂×C₄	semidirect
89. #63230	C₂×C₄	semidirect
110. #63219	C₂×C₄	semidirect
112. #63218	C₂×C₄	semidirect
164. C₄×C₁₇	#137	semidirect
181. C₄×Q₁₆	D₁₇	semidirect
237. C₁₇	C₄×#137	semidirect
257. Q₁₆	C₄×D₁₇	semidirect
347. C₄	#10587604	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×#10587604	2
C₄×#63256	2
C₄×Q₈×D₁₇	2
#11762916	2
#11762913	2
#11762912	2
#11762908	2
C₄×#63231	2
C₄×#63219	2
C₄×#63230	2
#11762850	2
#11762816	2
#11762809	2
C₄×#63218	2
C₄×Q₁₆×C₁₇	2
C₄×#137	17

Subgroups — C4×#10587604 #14537604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#10587604 #14537604