Subgroups — #14537794

Order 2176.

406 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	C₄×#63254	C₂	1	1	yes	maximal, characteristic
2	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762883	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762881	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762878	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762872	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762865	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762863	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762863	C₂	1	1	yes	maximal, characteristic, complemented
12	M₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#10587486	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63254	C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
20	D₄×Dic₁₇	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	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×C₈×D₁₇	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
29	#10587578	C₄	1	1	yes	characteristic
30	#10587573	C₄	1	1	yes	characteristic
31	#10587567	C₄	1	1	yes	characteristic
32	#10587567	C₄	1	1	yes	characteristic
33	#10587563	C₄	1	1	yes	characteristic
34	#10587563	C₄	1	1	yes	characteristic
35	C₄²×D₁₇	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	C₁₇×#97	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	#10587486	C₂²	1	1	yes	characteristic
49	#10587486	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	#10587485	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	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
68	#63254	—	2	4	no	—
69	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
72	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
73	M₁₆×C₁₇	—	2	2	no	—
74	C₂×C₈×C₁₇	—	2	2	no	—
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
85	#63227	C₂×C₄	1	1	yes	characteristic
86	#63227	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
88	#63225	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	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₇	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	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
99	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	#63223	—	2	2	no	—
102	C₂×#3622	—	2	2	no	—
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	C₂×#3622	C₂³	1	1	yes	characteristic
107	C₈×D₁₇	—	2	2	no	—
108	C₈×D₁₇	—	2	2	no	—
109	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
112	D₄×C₁₇	—	2	2	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	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₇	—	2	2	no	—
123	C₂²⋊D₁₇	M₁₆	1	4	yes	complemented
124	C₂×Dic₁₇	—	2	2	no	—
125	C₂×Dic₁₇	—	2	2	no	—
126	C₂×Dic₁₇	—	4	1	no	—
127	C₂×Dic₁₇	—	4	1	no	—
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	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₁₇	—	2	2	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	2	1	no	—
138	C₄×D₁₇	—	4	1	no	—
139	Dic₃₄	—	2	2	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	#41	1	2	yes	complemented
144	#3622	—	2	1	no	—
145	#3622	—	2	1	no	—
146	#3622	—	2	1	no	—
147	#3622	#41	1	2	yes	complemented
148	#2970	—	17	1	no	maximal, Sylow
149	C₂²×C₁₇	—	2	1	no	—
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	C₂×M₁₆	1	2	yes	—
155	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×D₁₇	—	2	1	no	—
157	C₂×D₁₇	—	2	1	no	—
158	C₂×D₁₇	C₂×M₁₆	1	2	yes	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
164	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
165	Dic₁₇	—	2	1	no	—
166	Dic₁₇	—	2	1	no	—
167	Dic₁₇	—	2	1	no	—
168	Dic₁₇	—	2	1	no	—
169	Dic₁₇	—	2	1	no	—
170	Dic₁₇	—	2	1	no	—
171	Dic₁₇	C₂×M₁₆	1	2	yes	—
172	C₄×#41	—	17	1	no	—
173	#446	—	17	1	no	—
174	#446	—	17	1	no	—
175	#435	—	17	1	no	—
176	#435	—	17	1	no	—
177	#435	—	17	1	no	—
178	#435	D₁₇	1	1	yes	characteristic, complemented
179	C₈×D₄	—	17	1	no	—
180	C₈×D₄	—	17	1	no	—
181	#432	—	17	1	no	—
182	#432	—	17	1	no	—
183	#432	—	17	1	no	—
184	C₄×M₁₆	—	17	1	no	—
185	C₂×#97	—	17	1	no	—
186	C₂×#97	—	17	1	no	—
187	C₂×C₁₇	—	2	2	no	—
188	C₂×C₁₇	—	4	1	no	—
189	C₂×C₁₇	C₄×#41	1	1	yes	characteristic
190	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
191	C₂×C₁₇	#568	1	1	yes	characteristic
192	D₁₇	—	2	2	no	—
193	C₂×#41	—	17	1	no	—
194	C₂×M₁₆	—	17	1	no	—
195	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
196	C₂²×C₈	—	17	1	no	—
197	C₂²×C₈	—	17	1	no	—
198	C₂²×C₈	—	17	1	no	—
199	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
200	C₄×Q₈	—	17	1	no	—
201	C₄×Q₈	—	17	1	no	—
202	C₄×D₄	—	17	1	no	—
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	17	1	no	—
206	C₄×D₄	—	17	1	no	—
207	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
208	#117	—	17	1	no	—
209	#117	—	17	1	no	—
210	#117	—	17	1	no	—
211	C₂×C₄²	—	17	1	no	—
212	C₂×C₄²	—	17	1	no	—
213	C₂×C₄²	—	17	1	no	—
214	C₄⋊C₈	—	17	1	no	—
215	C₄⋊C₈	—	17	1	no	—
216	C₄⋊C₈	—	17	1	no	—
217	C₄⋊C₈	—	17	1	no	—
218	C₄⋊C₈	—	17	1	no	—
219	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
220	C₂²⋊C₈	—	17	1	no	—
221	C₂²⋊C₈	—	17	1	no	—
222	C₂²⋊C₈	—	17	1	no	—
223	C₂²⋊C₈	—	17	1	no	—
224	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
226	#97	—	17	1	no	—
227	#97	—	17	1	no	—
228	#97	—	17	1	no	—
229	#97	—	17	1	no	—
230	#97	—	17	1	no	—
231	#97	—	17	1	no	—
232	#97	—	17	1	no	—
233	#97	C₂×D₁₇	1	1	yes	characteristic
234	C₄×C₈	—	17	1	no	—
235	C₄×C₈	—	17	1	no	—
236	C₁₇	#2970	1	1	yes	characteristic, complemented, Sylow
237	#41	—	34	4	no	—
238	C₂×Q₈	—	17	1	no	—
239	C₂×D₄	—	17	1	no	—
240	C₂×D₄	—	17	1	no	—
241	C₂×D₄	C₄×D₁₇	1	1	yes	characteristic
242	C₂²×C₄	—	17	1	no	—
243	C₂²×C₄	—	17	1	no	—
244	C₂²×C₄	—	17	1	no	—
245	C₂²×C₄	—	17	1	no	—
246	C₂²×C₄	—	17	1	no	—
247	C₂²×C₄	—	17	1	no	—
248	C₂²×C₄	—	17	1	no	—
249	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	34	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	17	1	no	—
257	C₂×C₈	—	17	1	no	—
258	C₂×C₈	—	34	2	no	—
259	C₂×C₈	—	34	2	no	—
260	C₂×C₈	—	34	2	no	—
261	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	1	no	—
267	C₄⋊C₄	—	17	1	no	—
268	C₄⋊C₄	—	17	1	no	—
269	C₄⋊C₄	—	17	1	no	—
270	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
271	C₂²⋊C₄	—	17	1	no	—
272	C₂²⋊C₄	—	17	1	no	—
273	C₂²⋊C₄	—	17	1	no	—
274	C₂²⋊C₄	—	17	1	no	—
275	C₂²⋊C₄	C₄×D₁₇	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₁₇	1	1	yes	characteristic
277	C₄²	—	17	1	no	—
278	C₄²	—	17	1	no	—
279	C₄²	—	17	1	no	—
280	C₄²	—	17	1	no	—
281	C₄²	—	17	1	no	—
282	C₄²	—	17	1	no	—
283	C₄²	—	17	1	no	—
284	C₄²	—	17	1	no	—
285	C₄²	—	17	1	no	—
286	C₄²	C₂²×D₁₇	1	1	yes	characteristic
287	C₂³	—	17	1	no	—
288	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
290	Q₈	—	34	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	17	4	no	—
293	D₄	—	34	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	17	1	no	—
299	C₂×C₄	—	17	1	no	—
300	C₂×C₄	—	17	1	no	—
301	C₂×C₄	—	17	1	no	—
302	C₂×C₄	—	17	1	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	17	1	no	—
305	C₂×C₄	—	34	2	no	—
306	C₂×C₄	—	34	2	no	—
307	C₂×C₄	—	34	2	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	34	1	no	—
310	C₂×C₄	—	68	1	no	—
311	C₂×C₄	—	68	1	no	—
312	C₂×C₄	—	68	1	no	—
313	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	17	2	no	—
322	C₈	—	34	1	no	—
323	C₈	—	34	1	no	—
324	C₈	—	34	1	no	—
325	C₈	#63254	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	17	2	no	—
332	C₂²	—	34	1	no	—
333	C₂²	—	34	1	no	—
334	C₂²	M₁₆×D₁₇	1	2	yes	—
335	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	17	2	no	—
341	C₄	—	34	1	no	—
342	C₄	—	34	1	no	—
343	C₄	—	34	1	no	—
344	C₄	—	34	1	no	—
345	C₄	—	34	1	no	—
346	C₄	—	34	1	no	—
347	C₄	C₂×#63254	1	1	yes	characteristic
348	C₄	#41×D₁₇	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	34	2	no	—
352	C₂	C₄×#63254	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₁₇	1	1	yes	characteristic
354	C₂	#11763574	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#435	C₂
prime radical	C₂×#63254	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#435	D₁₇
hypocenter	C₁₇	#2970
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #11762938	C₂	semidirect
3. #11762937	C₂	semidirect
4. #11762883	C₂	semidirect
5. #11762881	C₂	semidirect
6. #11762878	C₂	semidirect
7. C₈×C₂²⋊D₁₇	C₂	semidirect
8. #11762872	C₂	semidirect
9. #11762865	C₂	semidirect
10. #11762863	C₂	semidirect
11. #11762863	C₂	semidirect
12. M₁₆×Dic₁₇	C₂	semidirect
13. C₂×#10587486	C₂	semidirect
14. #97×D₁₇	C₂	semidirect
15. C₁₇×#435	C₂	semidirect
123. C₂²⋊D₁₇	M₁₆	semidirect
143. C₈×C₁₇	#41	semidirect
147. #3622	#41	semidirect
178. #435	D₁₇	semidirect
236. C₁₇	#2970	semidirect
325. C₈	#63254	semidirect

Maximal subgroups

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

subgroup	index
C₄×#63254	2
#11762938	2
#11762937	2
#11762883	2
#11762881	2
#11762878	2
C₈×C₂²⋊D₁₇	2
#11762872	2
#11762865	2
#11762863	2
M₁₆×Dic₁₇	2
C₂×#10587486	2
#97×D₁₇	2
C₁₇×#435	2
#2970	17