Subgroups — #14538813

Order 2176.

428 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763356	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763129	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763126	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763093	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763008	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763007	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762949	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762946	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#63219	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#63230	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#63229	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762860	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63252	—	2	1	no	—
18	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₂×#63230	C₂²	1	1	yes	characteristic
23	#10587626	C₂²	1	1	yes	characteristic
24	#10587624	C₂²	1	1	yes	characteristic
25	C₂×#63229	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊D₁₇	—	2	1	no	—
27	C₂×#63219	C₂²	1	1	yes	characteristic
28	#10587581	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₁₇	—	2	1	no	—
30	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587573	—	2	1	no	—
33	#10587573	C₂²	1	1	yes	characteristic
34	#10587571	—	2	1	no	—
35	#10587570	—	2	1	no	—
36	C₄×D₆₈	C₂²	1	1	yes	characteristic
37	C₄²×D₁₇	—	2	1	no	—
38	C₄²×D₁₇	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	—	2	1	no	—
40	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#102	C₂²	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587491	C₂²	1	1	yes	characteristic
50	#10587488	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587480	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	Q₈×D₁₇	—	2	2	no	—
66	Q₈×D₁₇	—	2	2	no	—
67	#63252	—	4	4	no	—
68	C₂×D₆₈	—	2	1	no	—
69	C₂×D₆₈	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
78	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
79	SD₁₆×C₁₇	—	2	2	no	—
80	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₇	—	2	1	no	—
86	C₄²×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₁₇	—	2	1	no	—
89	#63230	—	2	2	no	—
90	#63229	—	2	2	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	C₄⋊Dic₁₇	—	2	1	no	—
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
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	D₄	1	1	yes	characteristic
104	C₄×Dic₁₇	D₄	1	1	yes	characteristic
105	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	#63219	—	2	2	no	—
109	C₂³×C₁₇	—	2	1	no	—
110	C₂²×D₁₇	—	2	1	no	—
111	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₁₇	—	2	1	no	—
113	Q₈×C₁₇	#41	1	2	yes	complemented
114	D₄×C₁₇	—	2	1	no	—
115	D₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	4	2	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₁₇	—	4	2	no	—
124	C₂²⋊D₁₇	—	4	2	no	—
125	C₂²⋊D₁₇	—	4	2	no	—
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	1	no	—
133	D₆₈	—	2	1	no	—
134	D₆₈	—	4	2	no	—
135	C₄×D₁₇	—	2	4	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	2	1	no	—
138	C₄×D₁₇	—	4	1	no	—
139	C₄×D₁₇	—	4	2	no	—
140	C₄×D₁₇	—	4	2	no	—
141	C₄×D₁₇	SD₁₆	1	4	yes	complemented
142	Dic₃₄	—	2	4	no	—
143	Dic₃₄	—	2	1	no	—
144	Dic₃₄	—	4	2	no	—
145	Dic₃₄	#41	1	2	yes	complemented
146	C₈×C₁₇	—	2	1	no	—
147	C₈×C₁₇	—	2	1	no	—
148	#3622	—	2	1	no	—
149	#3622	—	2	1	no	—
150	#3103	—	17	1	no	maximal, Sylow
151	C₂²×C₁₇	—	4	1	no	—
152	C₂²×C₁₇	—	4	1	no	—
153	C₂²×C₁₇	—	4	1	no	—
154	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	—	2	1	no	—
157	C₂×D₁₇	—	4	1	no	—
158	C₂×D₁₇	—	4	1	no	—
159	C₂×D₁₇	—	4	1	no	—
160	C₂×D₁₇	C₂×SD₁₆	1	2	yes	—
161	C₄×C₁₇	—	2	2	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	—	4	1	no	—
165	C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
166	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
167	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
168	Dic₁₇	—	2	1	no	—
169	Dic₁₇	—	2	2	no	—
170	Dic₁₇	—	2	1	no	—
171	Dic₁₇	—	4	1	no	—
172	Dic₁₇	—	4	1	no	—
173	Dic₁₇	C₂×SD₁₆	1	2	yes	—
174	#532	—	17	1	no	—
175	C₂×C₄×Q₈	—	17	1	no	—
176	#481	—	17	1	no	—
177	#481	—	17	1	no	—
178	#475	—	17	1	no	—
179	#475	—	17	1	no	—
180	#460	—	17	1	no	—
181	#460	—	17	1	no	—
182	#448	—	17	1	no	—
183	#448	—	17	1	no	—
184	C₄×SD₁₆	—	17	1	no	—
185	C₄×SD₁₆	—	17	1	no	—
186	C₄×SD₁₆	—	17	1	no	—
187	C₄×SD₁₆	D₁₇	1	1	yes	characteristic, complemented
188	#431	—	17	1	no	—
189	C₂×C₁₇	—	8	1	no	—
190	C₂×C₁₇	#525	1	1	yes	characteristic
191	C₂×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
192	C₂×C₁₇	#575	1	1	yes	characteristic
193	D₁₇	—	2	2	no	—
194	D₁₇	—	8	1	no	—
195	C₂×#41	—	34	1	no	—
196	C₂²×Q₈	—	17	1	no	—
197	C₂×SD₁₆	—	17	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂×SD₁₆	—	17	1	no	—
200	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
201	C₄⋊Q₈	—	17	1	no	—
202	#127	—	17	1	no	—
203	#124	—	34	1	no	—
204	#121	—	17	1	no	—
205	#121	—	17	1	no	—
206	#121	—	34	1	no	—
207	C₄×Q₈	—	17	1	no	—
208	C₄×Q₈	—	17	1	no	—
209	C₄×Q₈	—	17	1	no	—
210	C₄×Q₈	—	34	1	no	—
211	C₄×Q₈	—	34	1	no	—
212	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
213	C₄×D₄	—	17	1	no	—
214	C₄×D₄	—	34	1	no	—
215	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
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	#106	—	17	1	no	—
221	#106	—	17	1	no	—
222	#106	—	17	1	no	—
223	#106	C₂×D₁₇	1	1	yes	characteristic
224	C₄⋊C₈	—	17	1	no	—
225	C₄⋊C₈	—	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	#102	—	17	1	no	—
231	#102	—	17	1	no	—
232	#102	—	17	1	no	—
233	#102	C₂×D₁₇	1	1	yes	characteristic
234	C₂²⋊C₈	—	17	1	no	—
235	C₂²⋊C₈	—	17	1	no	—
236	C₄×C₈	—	17	1	no	—
237	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
238	C₁₇	#3103	1	1	yes	characteristic, complemented, Sylow
239	#41	—	68	4	no	—
240	C₂×Q₈	—	17	1	no	—
241	C₂×Q₈	—	34	1	no	—
242	C₂×Q₈	—	34	1	no	—
243	C₂×Q₈	—	34	2	no	—
244	C₂×Q₈	—	34	2	no	—
245	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₂×D₄	—	17	1	no	—
247	C₂×D₄	—	34	1	no	—
248	C₂×D₄	—	34	1	no	—
249	C₂×D₄	—	34	1	no	—
250	C₂×D₄	—	34	1	no	—
251	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	17	1	no	—
254	C₂²×C₄	—	17	1	no	—
255	C₂²×C₄	—	17	1	no	—
256	C₂²×C₄	—	34	1	no	—
257	C₂²×C₄	—	34	1	no	—
258	C₂²×C₄	—	34	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	34	2	no	—
261	SD₁₆	—	34	2	no	—
262	SD₁₆	—	34	2	no	—
263	C₂×C₈	—	17	1	no	—
264	C₂×C₈	—	17	1	no	—
265	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
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	—	2	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	17	1	no	—
291	C₂³	—	34	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	17	2	no	—
294	Q₈	—	34	4	no	—
295	Q₈	—	34	1	no	—
296	Q₈	—	68	2	no	—
297	Q₈	#63252	1	2	yes	complemented
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	34	1	no	—
301	D₄	—	34	1	no	—
302	D₄	—	68	2	no	—
303	D₄	—	68	2	no	—
304	D₄	—	68	2	no	—
305	D₄	—	68	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	17	1	no	—
311	C₂×C₄	—	17	4	no	—
312	C₂×C₄	—	17	1	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	34	4	no	—
315	C₂×C₄	—	34	1	no	—
316	C₂×C₄	—	34	1	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	34	1	no	—
319	C₂×C₄	—	34	1	no	—
320	C₂×C₄	—	68	1	no	—
321	C₂×C₄	—	68	2	no	—
322	C₂×C₄	—	68	2	no	—
323	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
324	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	34	1	no	—
329	C₈	—	34	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	17	2	no	—
334	C₂²	—	34	1	no	—
335	C₂²	—	34	1	no	—
336	C₂²	—	68	1	no	—
337	C₂²	—	68	1	no	—
338	C₂²	—	68	1	no	—
339	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	17	2	no	—
345	C₄	—	34	1	no	—
346	C₄	—	34	2	no	—
347	C₄	—	34	1	no	—
348	C₄	—	68	1	no	—
349	C₄	—	68	1	no	—
350	C₄	SD₁₆×D₁₇	1	2	yes	—
351	C₄	C₂×#63252	1	1	yes	characteristic
352	C₄	#41×D₁₇	1	1	yes	characteristic
353	C₂	—	8	1	no	—
354	C₂	—	34	2	no	—
355	C₂	—	136	1	no	—
356	C₂	#11763318	1	1	yes	characteristic
357	C₂	C₂×SD₁₆×D₁₇	1	1	yes	characteristic
358	C₂	#11763606	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₄×SD₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763356	C₂
hypercenter	C₄×SD₁₆	D₁₇
hypocenter	C₁₇	#3103
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₁₇	C₂	semidirect
3. #11763129	C₂	semidirect
4. #11763126	C₂	semidirect
5. #11763093	C₂	semidirect
6. #11763092	C₂	semidirect
7. #11763008	C₂	semidirect
8. #11763007	C₂	semidirect
9. #11762949	C₂	semidirect
10. #11762946	C₂	semidirect
11. C₄×#63219	C₂	semidirect
12. C₄×#63230	C₂	semidirect
13. C₄×#63229	C₂	semidirect
14. #11762860	C₂	semidirect
15. C₄×SD₁₆×C₁₇	C₂	semidirect
113. Q₈×C₁₇	#41	semidirect
141. C₄×D₁₇	SD₁₆	semidirect
145. Dic₃₄	#41	semidirect
187. C₄×SD₁₆	D₁₇	semidirect
238. C₁₇	#3103	semidirect
297. Q₈	#63252	semidirect

Maximal subgroups

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

subgroup	index
#11763356	2
C₄×Q₈×D₁₇	2
#11763129	2
#11763126	2
#11763093	2
#11763092	2
#11763008	2
#11763007	2
#11762949	2
#11762946	2
C₄×#63219	2
C₄×#63230	2
C₄×#63229	2
#11762860	2
C₄×SD₁₆×C₁₇	2
#3103	17