Subgroups — #14537752

Order 2176.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763339	C₂	1	1	yes	maximal, characteristic
2	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762884	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762881	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762877	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762867	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762863	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762764	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762763	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587625	C₄	1	1	yes	characteristic
20	#10587621	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
23	#10587612	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
25	C₂×#63223	C₂²	1	1	yes	characteristic
26	C₂²×#3622	C₂²	1	1	yes	characteristic
27	C₂×#63218	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
29	#10587584	C₄	1	1	yes	characteristic
30	#10587575	C₄	1	1	yes	characteristic
31	#10587572	C₄	1	1	yes	characteristic
32	#10587571	C₄	1	1	yes	characteristic
33	#10587565	C₄	1	1	yes	characteristic
34	#10587564	C₄	1	1	yes	characteristic
35	#10587558	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
37	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
41	C₂²⋊#3622	C₂²	1	1	yes	characteristic
42	C₂²⋊#3622	C₂²	1	1	yes	characteristic
43	#10587491	C₂²	1	1	yes	characteristic
44	#10587491	C₂²	1	1	yes	characteristic
45	#10587486	C₂²	1	1	yes	characteristic
46	#10587486	C₂²	1	1	yes	characteristic
47	#10587486	C₂²	1	1	yes	characteristic
48	#10587485	C₂²	1	1	yes	characteristic
49	#10587485	C₂²	1	1	yes	characteristic
50	#10587485	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
56	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
57	C₄⋊#3622	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
67	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	—	2	2	no	—
69	C₂×C₈×C₁₇	—	2	2	no	—
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
77	C₄²×C₁₇	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
82	#63227	C₂×C₄	1	1	yes	characteristic
83	#63227	C₂×C₄	1	1	yes	characteristic
84	#63227	C₂×C₄	1	1	yes	characteristic
85	#63227	C₂×C₄	1	1	yes	characteristic
86	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
89	#63225	C₂×C₄	1	1	yes	characteristic
90	#63225	C₂×C₄	1	1	yes	characteristic
91	#63225	C₂×C₄	1	1	yes	characteristic
92	#63225	C₂×C₄	1	1	yes	characteristic
93	#63225	C₂×C₄	1	1	yes	characteristic
94	#63225	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
98	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
99	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
100	#63223	—	2	2	no	—
101	C₂×#3622	—	2	2	no	—
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	#63218	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
111	D₄×C₁₇	—	2	2	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₇	—	2	2	no	—
122	C₂²⋊D₁₇	—	2	2	no	—
123	C₂×Dic₁₇	—	2	2	no	—
124	C₂×Dic₁₇	—	2	2	no	—
125	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	Dic₃₄	—	2	2	no	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	C₈×C₁₇	#41	1	2	yes	—
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3622	#41	1	2	yes	—
143	#2966	—	17	1	no	maximal, Sylow
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	—	2	1	no	—
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	#131	1	2	yes	—
150	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₁₇	—	2	1	no	—
152	C₂×D₁₇	—	2	1	no	—
153	C₂×D₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
160	Dic₁₇	—	2	1	no	—
161	Dic₁₇	—	2	1	no	—
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	2	1	no	—
166	Dic₁₇	#131	1	2	yes	—
167	#529	—	17	1	no	—
168	#446	—	17	1	no	—
169	C₈×Q₈	—	17	1	no	—
170	#436	—	17	1	no	—
171	#435	—	17	1	no	—
172	#435	—	17	1	no	—
173	C₈×D₄	—	17	1	no	—
174	C₈×D₄	—	17	1	no	—
175	C₈×D₄	D₁₇	1	1	yes	characteristic, complemented
176	#433	—	17	1	no	—
177	#433	—	17	1	no	—
178	#432	—	17	1	no	—
179	#405	—	17	1	no	—
180	#405	—	17	1	no	—
181	C₂×C₄×C₈	—	17	1	no	—
182	C₂×C₁₇	—	2	2	no	—
183	C₂×C₁₇	—	4	1	no	—
184	C₂×C₁₇	C₄×#41	1	1	yes	characteristic
185	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
186	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
187	D₁₇	—	4	1	no	—
188	C₂×M₁₆	—	17	1	no	—
189	C₂×M₁₆	—	17	1	no	—
190	C₂²×C₈	—	17	1	no	—
191	C₂²×C₈	—	17	1	no	—
192	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
193	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
194	#126	—	17	1	no	—
195	#126	—	17	1	no	—
196	#125	—	17	1	no	—
197	#124	—	17	1	no	—
198	#123	—	17	1	no	—
199	#123	—	17	1	no	—
200	C₂²⋊Q₈	—	17	1	no	—
201	#121	—	17	1	no	—
202	C₄×Q₈	—	17	1	no	—
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
206	#117	—	17	1	no	—
207	#117	—	17	1	no	—
208	C₂×C₄²	—	17	1	no	—
209	C₄⋊C₈	—	17	1	no	—
210	C₄⋊C₈	—	17	1	no	—
211	C₄⋊C₈	—	17	1	no	—
212	C₄⋊C₈	—	17	1	no	—
213	C₄⋊C₈	—	17	1	no	—
214	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
215	C₂²⋊C₈	—	17	1	no	—
216	C₂²⋊C₈	—	17	1	no	—
217	C₂²⋊C₈	—	17	1	no	—
218	C₂²⋊C₈	—	17	1	no	—
219	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
220	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
221	#97	—	17	1	no	—
222	#97	—	17	1	no	—
223	#97	—	17	1	no	—
224	#97	—	17	1	no	—
225	C₄×C₈	—	17	1	no	—
226	C₄×C₈	—	17	1	no	—
227	C₄×C₈	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₄×C₈	—	17	1	no	—
230	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
231	C₁₇	#2966	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	17	1	no	—
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	17	1	no	—
235	C₂×D₄	C₄×D₁₇	1	1	yes	characteristic
236	C₂²×C₄	—	17	1	no	—
237	C₂²×C₄	—	17	1	no	—
238	C₂²×C₄	—	17	1	no	—
239	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
241	M₁₆	—	34	2	no	—
242	M₁₆	—	34	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	17	1	no	—
246	C₂×C₈	—	17	1	no	—
247	C₂×C₈	—	17	1	no	—
248	C₂×C₈	—	17	1	no	—
249	C₂×C₈	—	34	2	no	—
250	C₂×C₈	—	34	2	no	—
251	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	17	1	no	—
262	C₄⋊C₄	—	17	1	no	—
263	C₄⋊C₄	—	17	1	no	—
264	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	17	1	no	—
266	C₂²⋊C₄	—	17	1	no	—
267	C₂²⋊C₄	—	17	1	no	—
268	C₂²⋊C₄	—	17	1	no	—
269	C₂²⋊C₄	—	17	1	no	—
270	C₂²⋊C₄	—	17	1	no	—
271	C₂²⋊C₄	—	17	1	no	—
272	C₂²⋊C₄	—	17	1	no	—
273	C₂²⋊C₄	C₄×D₁₇	1	1	yes	characteristic
274	C₂²⋊C₄	C₄×D₁₇	1	1	yes	characteristic
275	C₄²	—	17	1	no	—
276	C₄²	—	17	1	no	—
277	C₄²	—	17	1	no	—
278	C₄²	—	17	1	no	—
279	C₄²	—	17	1	no	—
280	C₄²	C₂²×D₁₇	1	1	yes	characteristic
281	C₂³	—	17	1	no	—
282	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
283	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
284	Q₈	—	34	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	34	2	no	—
287	D₄	—	34	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	17	1	no	—
293	C₂×C₄	—	17	1	no	—
294	C₂×C₄	—	17	1	no	—
295	C₂×C₄	—	17	1	no	—
296	C₂×C₄	—	17	1	no	—
297	C₂×C₄	—	17	1	no	—
298	C₂×C₄	—	17	1	no	—
299	C₂×C₄	—	34	2	no	—
300	C₂×C₄	—	34	2	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	17	2	no	—
312	C₈	—	34	1	no	—
313	C₈	—	34	1	no	—
314	C₈	—	34	1	no	—
315	C₈	#63254	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	34	1	no	—
322	C₂²	—	34	1	no	—
323	C₂²	—	34	1	no	—
324	C₂²	#10587587	1	2	yes	—
325	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	17	2	no	—
331	C₄	—	34	1	no	—
332	C₄	—	34	1	no	—
333	C₄	—	34	1	no	—
334	C₄	—	34	1	no	—
335	C₄	—	34	1	no	—
336	C₄	—	34	1	no	—
337	C₄	C₂×#63254	1	1	yes	characteristic
338	C₄	#41×D₁₇	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	4	1	no	—
341	C₂	—	68	1	no	—
342	C₂	C₄×#63254	1	1	yes	characteristic
343	C₂	#131×D₁₇	1	1	yes	characteristic
344	C₂	C₂×#10587587	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₈×D₄×C₁₇	C₂
prime radical	#10587625	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₈×D₄	D₁₇
hypocenter	C₁₇	#2966
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #11762938	C₂	semidirect
3. C₈×Dic₃₄	C₂	semidirect
4. #11762884	C₂	semidirect
5. #11762881	C₂	semidirect
6. #11762877	C₂	semidirect
7. C₈×C₂²⋊D₁₇	C₂	semidirect
8. C₈×C₂²⋊D₁₇	C₂	semidirect
9. #11762869	C₂	semidirect
10. #11762867	C₂	semidirect
11. #11762863	C₂	semidirect
12. #11762764	C₂	semidirect
13. #11762763	C₂	semidirect
14. C₂×C₈×Dic₁₇	C₂	semidirect
15. C₈×D₄×C₁₇	C₂	semidirect
175. C₈×D₄	D₁₇	semidirect
231. C₁₇	#2966	semidirect

Maximal subgroups

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

subgroup	index
#11763339	2
#11762938	2
C₈×Dic₃₄	2
#11762884	2
#11762881	2
#11762877	2
C₈×C₂²⋊D₁₇	2
#11762869	2
#11762867	2
#11762863	2
#11762764	2
#11762763	2
C₂×C₈×Dic₁₇	2
C₈×D₄×C₁₇	2
#2966	17