Subgroups — #14537447

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763277	C₂	1	1	yes	maximal, characteristic
2	#11762870	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762864	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762863	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762860	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762859	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762831	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762774	C₂	1	1	yes	maximal, characteristic
12	#11762774	C₂	1	1	yes	maximal, characteristic
13	#11762766	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762763	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63252	C₂²	1	1	yes	characteristic
17	C₁₇×#117	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂×#63218	C₂²	1	1	yes	characteristic
20	C₂×#63218	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
23	#10587579	C₄	1	1	yes	characteristic
24	#10587579	C₄	1	1	yes	characteristic
25	#10587573	C₄	1	1	yes	characteristic
26	#10587573	C₄	1	1	yes	characteristic
27	#10587567	C₄	1	1	yes	characteristic
28	#10587567	C₄	1	1	yes	characteristic
29	#10587567	C₄	1	1	yes	characteristic
30	#10587567	C₄	1	1	yes	characteristic
31	#10587558	C₂²	1	1	yes	characteristic
32	#10587558	C₂²	1	1	yes	characteristic
33	C₄²×D₁₇	C₂²	1	1	yes	characteristic
34	C₄²×D₁₇	C₂²	1	1	yes	characteristic
35	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₇×#97	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	#10587491	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587486	C₂²	1	1	yes	characteristic
48	#10587486	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	#10587485	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	C₄⋊#3622	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
64	#63252	—	2	2	no	—
65	#63252	—	2	2	no	—
66	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
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×C₄×D₁₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₁₇	C₂×C₄	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	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	#63227	C₂×C₄	1	1	yes	characteristic
85	#63227	C₂×C₄	1	1	yes	characteristic
86	#63227	C₂×C₄	1	1	yes	characteristic
87	#63227	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	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	#63218	—	2	2	no	—
105	#63218	—	2	2	no	—
106	C₈×D₁₇	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₇	—	2	2	no	—
120	C₂²⋊D₁₇	—	2	2	no	—
121	C₂×Dic₁₇	—	2	2	no	—
122	C₂×Dic₁₇	—	4	1	no	—
123	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
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	D₆₈	—	2	2	no	—
130	C₄×D₁₇	—	2	1	no	—
131	C₄×D₁₇	—	2	1	no	—
132	C₄×D₁₇	—	2	2	no	—
133	C₄×D₁₇	—	2	2	no	—
134	C₄×D₁₇	—	4	1	no	—
135	C₄×D₁₇	—	4	1	no	—
136	C₄×D₁₇	#41	1	2	yes	—
137	C₄×D₁₇	#41	1	2	yes	—
138	Dic₃₄	—	2	2	no	—
139	C₈×C₁₇	—	2	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3622	—	2	1	no	—
146	#3622	—	2	1	no	—
147	#2925	—	17	1	no	maximal, Sylow
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	—	2	1	no	—
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
152	C₂×D₁₇	—	2	1	no	—
153	C₂×D₁₇	—	2	1	no	—
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	—	2	1	no	—
157	C₂×D₁₇	#117	1	2	yes	—
158	C₄×C₁₇	—	2	1	no	—
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	#117	1	2	yes	—
170	Dic₁₇	#131	1	2	yes	—
171	C₄×#41	—	17	1	no	—
172	#433	—	17	1	no	—
173	#433	—	17	1	no	—
174	#433	—	17	1	no	—
175	#433	D₁₇	1	1	yes	characteristic, complemented
176	#432	—	17	1	no	—
177	#432	—	17	1	no	—
178	#431	—	17	1	no	—
179	#431	—	17	1	no	—
180	#423	—	17	1	no	—
181	C₂×C₄⋊C₈	—	17	1	no	—
182	#408	—	17	1	no	—
183	#408	—	17	1	no	—
184	#405	—	17	1	no	—
185	#405	—	17	1	no	—
186	C₂×C₁₇	—	4	1	no	—
187	C₂×C₁₇	C₂×#117	1	1	yes	characteristic
188	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
189	C₂×C₁₇	#568	1	1	yes	characteristic
190	D₁₇	—	2	2	no	—
191	D₁₇	—	4	1	no	—
192	C₂×#41	—	17	1	no	—
193	C₂×M₁₆	—	17	1	no	—
194	C₂×M₁₆	—	17	1	no	—
195	C₂²×C₈	—	17	1	no	—
196	C₂²×C₈	—	17	1	no	—
197	C₄×Q₈	—	17	1	no	—
198	C₄×Q₈	—	17	1	no	—
199	C₄×D₄	—	17	1	no	—
200	C₄×D₄	—	17	1	no	—
201	C₄×D₄	—	17	1	no	—
202	C₄×D₄	—	17	1	no	—
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	#117	—	17	1	no	—
206	#117	—	17	1	no	—
207	#117	C₂×D₁₇	1	1	yes	characteristic
208	C₂×C₄²	—	17	1	no	—
209	C₂×C₄²	—	17	1	no	—
210	C₂×C₄²	—	17	1	no	—
211	C₄⋊C₈	—	17	1	no	—
212	C₄⋊C₈	—	17	1	no	—
213	C₄⋊C₈	—	17	1	no	—
214	C₄⋊C₈	—	17	1	no	—
215	C₄⋊C₈	—	17	1	no	—
216	C₄⋊C₈	—	17	1	no	—
217	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
218	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
219	C₂²⋊C₈	—	17	1	no	—
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	—	17	1	no	—
225	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
226	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
227	#97	—	17	1	no	—
228	#97	—	17	1	no	—
229	#97	—	17	1	no	—
230	#97	C₂×D₁₇	1	1	yes	characteristic
231	C₄×C₈	—	17	1	no	—
232	C₄×C₈	—	17	1	no	—
233	C₄×C₈	—	17	1	no	—
234	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
235	C₁₇	#2925	1	1	yes	characteristic, complemented, Sylow
236	#41	—	34	2	no	—
237	#41	—	34	2	no	—
238	C₂×Q₈	—	17	1	no	—
239	C₂×D₄	—	17	1	no	—
240	C₂×D₄	—	17	1	no	—
241	C₂×D₄	—	17	1	no	—
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	—	17	1	no	—
250	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
251	M₁₆	—	34	2	no	—
252	M₁₆	—	34	2	no	—
253	C₂×C₈	—	17	1	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	17	1	no	—
257	C₂×C₈	—	34	2	no	—
258	C₂×C₈	—	34	2	no	—
259	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
263	C₄⋊C₄	—	17	1	no	—
264	C₄⋊C₄	—	17	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	1	no	—
267	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
268	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
281	C₄²	—	17	1	no	—
282	C₄²	—	17	1	no	—
283	C₄²	C₂²×D₁₇	1	1	yes	characteristic
284	C₄²	C₂²×D₁₇	1	1	yes	characteristic
285	C₂³	—	17	1	no	—
286	C₂³	—	17	1	no	—
287	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
288	Q₈	—	34	2	no	—
289	D₄	—	34	2	no	—
290	D₄	—	34	2	no	—
291	D₄	—	34	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	17	1	no	—
295	C₂×C₄	—	17	1	no	—
296	C₂×C₄	—	17	2	no	—
297	C₂×C₄	—	17	2	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	—	34	2	no	—
303	C₂×C₄	—	34	2	no	—
304	C₂×C₄	—	34	2	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	68	1	no	—
308	C₂×C₄	—	68	1	no	—
309	C₂×C₄	—	68	1	no	—
310	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	34	1	no	—
321	C₈	—	34	1	no	—
322	C₈	—	34	1	no	—
323	C₈	—	34	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	17	2	no	—
328	C₂²	—	34	1	no	—
329	C₂²	—	34	1	no	—
330	C₂²	—	34	1	no	—
331	C₂²	—	34	1	no	—
332	C₂²	—	34	1	no	—
333	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	17	2	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	#41×D₁₇	1	1	yes	characteristic
346	C₄	#41×D₁₇	1	1	yes	characteristic
347	C₂	—	4	1	no	—
348	C₂	—	34	2	no	—
349	C₂	—	68	1	no	—
350	C₂	#117×D₁₇	1	1	yes	characteristic
351	C₂	#131×D₁₇	1	1	yes	characteristic
352	C₂	#11763574	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#433	C₂
prime radical	C₂×#63252	C₂²
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#433	D₁₇
hypocenter	C₁₇	#2925
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #11762870	C₂	semidirect
3. #11762869	C₂	semidirect
4. #11762869	C₂	semidirect
5. #11762864	C₂	semidirect
6. #11762863	C₂	semidirect
7. #11762860	C₂	semidirect
8. #11762859	C₂	semidirect
9. #11762831	C₂	semidirect
10. C₄⋊C₈×D₁₇	C₂	semidirect
13. #11762766	C₂	semidirect
14. #11762763	C₂	semidirect
15. C₁₇×#433	C₂	semidirect
175. #433	D₁₇	semidirect
235. C₁₇	#2925	semidirect

Maximal subgroups

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

subgroup	index
#11763277	2
#11762870	2
#11762869	2
#11762864	2
#11762863	2
#11762860	2
#11762859	2
#11762831	2
C₄⋊C₈×D₁₇	2
#11762774	2
#11762766	2
#11762763	2
C₁₇×#433	2
#2925	17