Subgroups — #17186973

Order 4352.

818 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763227	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763225	C₂	1	1	yes	maximal, characteristic
3	#14537206	C₂	1	1	yes	maximal, characteristic
4	#14530382	C₂	1	2	yes	maximal, complemented
5	#14530381	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530379	C₂	1	2	yes	maximal, complemented
7	M₃₂×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10587469	C₂	1	2	yes	maximal, complemented
9	C₂×#11762584	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₇×#346	C₂	1	2	yes	maximal, complemented
11	C₁₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10587593	C₄	1	1	yes	characteristic
13	C₂×#10587587	C₂²	1	1	yes	characteristic
14	#11763277	C₄	1	1	yes	characteristic
15	#11763227	C₄	1	8	yes	complemented
16	#11763225	C₄	1	8	yes	complemented
17	C₂×#10587483	C₂²	1	1	yes	characteristic
18	C₂×#10587469	C₂²	1	2	yes	—
19	C₂×#10587469	C₂²	1	2	yes	—
20	C₂²×#63214	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
22	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
23	#11762766	C₄	1	2	yes	—
24	#11762765	C₄	1	1	yes	characteristic
25	#11762763	C₂²	1	2	yes	—
26	#11762761	C₄	1	2	yes	—
27	M₁₆×Dic₁₇	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₁₇	C₂²	1	2	yes	—
30	#11762585	C₂²	1	2	yes	—
31	#11762584	C₂²	1	1	yes	characteristic
32	#11762584	C₂²	1	2	yes	—
33	#11762584	C₂²	1	1	yes	characteristic
34	#11762584	C₂²	1	1	yes	characteristic
35	#11762584	C₂²	1	1	yes	characteristic
36	C₄×#63214	C₂²	1	2	yes	—
37	C₁₆×Dic₁₇	C₂²	1	2	yes	—
38	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
39	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
40	C₁₇×#405	C₂²	1	1	yes	characteristic
41	C₁₇×#346	C₂²	1	2	yes	—
42	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
43	C₂×#63252	C₂×C₄	1	1	yes	characteristic
44	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
46	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×#63223	C₂×C₄	1	1	yes	characteristic
49	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
50	#10587593	C₈	1	8	yes	—
51	#10587587	C₂×C₄	1	4	yes	—
52	#10587587	C₂×C₄	1	4	yes	—
53	C₂×#63218	C₂×C₄	1	2	yes	—
54	C₂×#63218	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₁₇	C₂³	1	2	yes	—
57	#10587579	C₈	1	2	yes	—
58	#10587573	C₈	1	2	yes	—
59	#10587567	C₈	1	4	yes	—
60	#10587558	C₂×C₄	1	2	yes	—
61	C₄²×D₁₇	C₂×C₄	1	2	yes	—
62	C₁₇×M₃₂	C₂×C₄	1	4	yes	complemented
63	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
64	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
67	C₁₇×#97	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₇	C₂³	1	2	yes	—
69	#10587486	C₂×C₄	1	2	yes	—
70	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
71	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
74	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
75	#10587483	C₂×C₄	1	4	yes	complemented
76	C₂×#63214	C₂×C₄	1	4	yes	complemented
77	C₂×#63214	C₂³	1	1	yes	characteristic
78	C₂×#63214	C₂³	1	2	yes	—
79	C₂×#63214	C₂³	1	1	yes	characteristic
80	#10587474	C₂×C₄	1	2	yes	—
81	C₄×#3622	C₂×C₄	1	2	yes	—
82	#10587469	C₂×C₄	1	8	yes	complemented
83	#10587469	C₂×C₄	1	8	yes	complemented
84	C₁₆×D₁₇	C₂×C₄	1	8	yes	complemented
85	C₁₆×D₁₇	C₂×C₄	1	8	yes	complemented
86	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₁₇	C₂×C₈	1	2	yes	—
88	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
89	#63252	C₄²	1	4	yes	—
90	#63252	C₂×C₈	1	4	yes	—
91	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
93	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
94	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
95	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
96	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
102	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
103	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
104	#63227	C₂×C₈	1	4	yes	—
105	#63225	C₂×C₈	1	4	yes	—
106	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
107	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
108	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
110	#63223	C₄²	1	4	yes	—
111	C₂×#3622	C₂×C₈	1	4	yes	—
112	C₂×#3622	C₂²×C₄	1	2	yes	—
113	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
114	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
115	#63218	C₄²	1	8	yes	—
116	#63218	C₂×C₈	1	8	yes	—
117	C₈×D₁₇	C₂×C₈	1	8	yes	—
118	C₈×D₁₇	C₂²×C₄	1	4	yes	—
119	C₈×D₁₇	C₂²×C₄	1	4	yes	—
120	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
121	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
122	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
123	#63214	C₂²×C₄	1	2	yes	—
124	#63214	C₂²×C₄	1	4	yes	—
125	#63214	C₂²×C₄	1	2	yes	—
126	#33326	—	17	1	no	maximal, Sylow
127	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
128	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
131	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
134	C₂²⋊D₁₇	C₄×C₈	1	8	yes	—
135	C₂×Dic₁₇	—	4	2	no	—
136	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
137	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
138	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
139	D₆₈	C₄×C₈	1	4	yes	—
140	C₄×D₁₇	—	4	4	no	—
141	C₄×D₁₇	C₂×C₄²	1	4	yes	—
142	C₄×D₁₇	C₂²×C₈	1	4	yes	—
143	Dic₃₄	C₄×C₈	1	4	yes	—
144	C₈×C₁₇	C₂×C₄²	1	2	yes	—
145	C₈×C₁₇	C₂²×C₈	1	4	yes	—
146	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
147	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
148	#3622	C₂×C₄²	1	4	yes	—
149	#3622	C₂²×C₈	1	2	yes	—
150	#3622	C₂²×C₈	1	2	yes	—
151	C₂×#504	—	17	1	no	—
152	C₂×#504	—	17	1	no	—
153	C₄×#131	—	17	1	no	—
154	#2108	—	17	1	no	—
155	#2108	—	17	2	no	—
156	#2108	—	17	2	no	—
157	#2108	D₁₇	1	1	yes	characteristic, complemented
158	C₄×M₃₂	—	17	1	no	—
159	C₄×M₃₂	—	17	2	no	—
160	C₂×#346	—	17	1	no	—
161	C₂×#346	—	17	2	no	—
162	C₂²×C₁₇	—	2	2	no	—
163	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
164	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
165	C₂×D₁₇	—	2	4	no	—
166	C₂×D₁₇	C₂×C₄×C₈	1	4	yes	—
167	C₄×C₁₇	—	2	4	no	—
168	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
169	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
170	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
171	Dic₁₇	—	2	2	no	—
172	Dic₁₇	—	2	2	no	—
173	Dic₁₇	C₂×C₄×C₈	1	4	yes	—
174	C₂×#131	—	17	1	no	—
175	C₂×#131	—	17	1	no	—
176	C₄×#41	—	17	1	no	—
177	#504	—	17	8	no	—
178	#504	—	17	8	no	—
179	C₂×M₃₂	—	17	1	no	—
180	C₂×M₃₂	—	17	2	no	—
181	C₂×M₃₂	—	17	2	no	—
182	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
183	C₂²×C₁₆	—	17	1	no	—
184	C₂²×C₁₆	—	17	2	no	—
185	C₂²×C₁₆	—	17	2	no	—
186	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
187	#405	—	17	1	no	—
188	#405	—	17	2	no	—
189	#405	—	17	2	no	—
190	#405	C₂×D₁₇	1	1	yes	characteristic
191	C₄×M₁₆	—	17	1	no	—
192	C₄×M₁₆	—	17	2	no	—
193	C₂×C₄×C₈	—	17	1	no	—
194	C₂×C₄×C₈	—	17	2	no	—
195	#346	—	17	1	no	—
196	#346	—	17	2	no	—
197	#346	—	17	1	no	—
198	#346	—	17	1	no	—
199	#346	—	17	1	no	—
200	#346	—	17	2	no	—
201	#346	C₂×D₁₇	1	2	yes	—
202	C₄×C₁₆	—	17	2	no	—
203	C₄×C₁₆	—	17	2	no	—
204	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
205	C₂×C₁₇	—	2	2	no	—
206	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
207	C₂×C₁₇	#3407	1	2	yes	—
208	D₁₇	—	2	4	no	—
209	C₂×#41	—	17	1	no	—
210	#131	—	17	8	no	—
211	#131	—	17	4	no	—
212	#131	—	17	4	no	—
213	C₂×M₁₆	—	17	1	no	—
214	C₂×M₁₆	—	17	2	no	—
215	C₂×M₁₆	—	17	2	no	—
216	C₂×M₁₆	C₄×D₁₇	1	1	yes	characteristic
217	C₂²×C₈	—	17	1	no	—
218	C₂²×C₈	—	17	2	no	—
219	C₂²×C₈	—	17	2	no	—
220	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
221	C₄×Q₈	—	17	2	no	—
222	C₄×D₄	—	17	2	no	—
223	C₄×D₄	—	17	4	no	—
224	#117	—	17	2	no	—
225	#117	C₄×D₁₇	1	1	yes	characteristic
226	C₂×C₄²	—	17	1	no	—
227	C₂×C₄²	—	17	2	no	—
228	M₃₂	—	17	4	no	—
229	M₃₂	—	17	8	no	—
230	M₃₂	—	17	8	no	—
231	M₃₂	C₄×D₁₇	1	4	yes	complemented
232	C₂×C₁₆	—	17	4	no	—
233	C₂×C₁₆	—	17	2	no	—
234	C₂×C₁₆	—	17	1	no	—
235	C₂×C₁₆	—	17	1	no	—
236	C₂×C₁₆	—	17	8	no	—
237	C₂×C₁₆	—	17	8	no	—
238	C₂×C₁₆	C₄×D₁₇	1	4	yes	complemented
239	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
240	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
241	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
242	#97	—	17	2	no	—
243	#97	—	17	2	no	—
244	#97	C₄×D₁₇	1	2	yes	—
245	C₄×C₈	—	17	1	no	—
246	C₄×C₈	—	17	2	no	—
247	C₄×C₈	—	17	1	no	—
248	C₄×C₈	—	17	1	no	—
249	C₄×C₈	—	17	1	no	—
250	C₄×C₈	—	17	2	no	—
251	C₄×C₈	C₂²×D₁₇	1	2	yes	—
252	C₁₇	#33326	1	1	yes	characteristic, complemented, Sylow
253	#41	—	17	4	no	—
254	#41	—	17	4	no	—
255	C₂×Q₈	—	17	1	no	—
256	C₂×D₄	—	17	2	no	—
257	C₂×D₄	—	17	1	no	—
258	C₂²×C₄	—	17	2	no	—
259	C₂²×C₄	—	17	2	no	—
260	C₂²×C₄	—	17	4	no	—
261	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
262	M₁₆	—	17	4	no	—
263	M₁₆	—	17	8	no	—
264	M₁₆	—	17	8	no	—
265	M₁₆	C₈×D₁₇	1	4	yes	—
266	C₂×C₈	—	17	1	no	—
267	C₂×C₈	—	17	2	no	—
268	C₂×C₈	—	17	1	no	—
269	C₂×C₈	—	17	4	no	—
270	C₂×C₈	—	17	4	no	—
271	C₂×C₈	—	17	4	no	—
272	C₂×C₈	—	17	8	no	—
273	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
274	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
275	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
276	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
277	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
278	C₄⋊C₄	—	17	4	no	—
279	C₄⋊C₄	C₈×D₁₇	1	2	yes	—
280	C₂²⋊C₄	—	17	4	no	—
281	C₂²⋊C₄	C₈×D₁₇	1	2	yes	—
282	C₄²	—	17	1	no	—
283	C₄²	—	17	1	no	—
284	C₄²	—	17	4	no	—
285	C₄²	—	17	2	no	—
286	C₄²	C₂×C₄×D₁₇	1	2	yes	—
287	C₁₆	—	17	2	no	—
288	C₁₆	—	17	2	no	—
289	C₁₆	—	17	4	no	—
290	C₁₆	C₂×C₄×D₁₇	1	2	yes	—
291	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
292	C₁₆	C₂×C₄×D₁₇	1	2	yes	—
293	C₂³	—	17	2	no	—
294	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
295	Q₈	—	17	4	no	—
296	D₄	—	17	8	no	—
297	D₄	—	17	4	no	—
298	C₂×C₄	—	17	2	no	—
299	C₂×C₄	—	17	4	no	—
300	C₂×C₄	—	17	4	no	—
301	C₂×C₄	—	17	2	no	—
302	C₂×C₄	—	17	2	no	—
303	C₂×C₄	—	68	4	no	—
304	C₂×C₄	—	68	2	no	—
305	C₂×C₄	C₄²×D₁₇	1	2	yes	—
306	C₂×C₄	C₂×C₈×D₁₇	1	4	yes	—
307	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
310	C₈	—	17	4	no	—
311	C₈	—	17	2	no	—
312	C₈	—	17	2	no	—
313	C₈	C₄²×D₁₇	1	2	yes	—
314	C₈	C₂×C₈×D₁₇	1	4	yes	—
315	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
316	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	17	4	no	—
319	C₂²	—	34	4	no	—
320	C₂²	C₄×C₈×D₁₇	1	2	yes	—
321	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
322	C₄	—	2	4	no	—
323	C₄	—	17	4	no	—
324	C₄	—	34	2	no	—
325	C₄	—	34	2	no	—
326	C₄	C₄×C₈×D₁₇	1	2	yes	—
327	C₄	C₂×C₄²×D₁₇	1	1	yes	characteristic
328	C₄	C₂²×C₈×D₁₇	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	34	4	no	—
331	C₂	C₂×C₄×C₈×D₁₇	1	1	yes	characteristic
332	C₂	#14541885	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₇
commutator	C₂×C₁₇	C₂²×C₄×C₈
Fitting	C₁₇×#2108	C₂
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2108	D₁₇
hypocenter	C₁₇	#33326
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
4. #14530382	C₂	semidirect
5. #14530381	C₂	semidirect
6. #14530379	C₂	semidirect
7. M₃₂×Dic₁₇	C₂	semidirect
8. C₄×#10587469	C₂	semidirect
9. C₂×#11762584	C₂	semidirect
10. D₁₇×#346	C₂	semidirect
11. C₁₇×#2108	C₂	semidirect
15. #11763227	C₄	semidirect
16. #11763225	C₄	semidirect
62. C₁₇×M₃₂	C₂×C₄	semidirect
63. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
75. #10587483	C₂×C₄	semidirect
76. C₂×#63214	C₂×C₄	semidirect
82. #10587469	C₂×C₄	semidirect
83. #10587469	C₂×C₄	semidirect
84. C₁₆×D₁₇	C₂×C₄	semidirect
85. C₁₆×D₁₇	C₂×C₄	semidirect
157. #2108	D₁₇	semidirect
231. M₃₂	C₄×D₁₇	semidirect
238. C₂×C₁₆	C₄×D₁₇	semidirect
252. C₁₇	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763227	2
C₂×#11763225	2
#14537206	2
#14530382	2
#14530381	2
#14530379	2
M₃₂×Dic₁₇	2
C₄×#10587469	2
C₂×#11762584	2
D₁₇×#346	2
C₁₇×#2108	2
#33326	17