Subgroups — #17978827

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#63216	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545170	C₂	1	1	yes	maximal, characteristic, complemented
4	#14545167	C₂	1	2	yes	maximal, complemented
5	#14545163	C₂	1	1	yes	maximal, characteristic, complemented
6	#14545163	C₂	1	2	yes	maximal, complemented
7	#14545151	C₂	1	2	yes	maximal, complemented
8	#14545143	C₂	1	1	yes	maximal, characteristic, complemented
9	#14545120	C₂	1	2	yes	maximal, complemented
10	#14545114	C₂	1	1	yes	maximal, characteristic, complemented
11	#14537758	C₂	1	1	yes	maximal, characteristic
12	C₂×#10587530	C₂²	1	2	yes	—
13	C₂×#10587529	C₂²	1	2	yes	—
14	C₂×#10587529	C₂²	1	2	yes	—
15	#11763730	C₂²	1	2	yes	—
16	#11763730	C₂²	1	1	yes	characteristic
17	#11763730	C₂²	1	1	yes	characteristic
18	C₄⋊#63216	C₂²	1	1	yes	characteristic
19	C₄⋊#63216	C₂²	1	2	yes	—
20	C₄⋊#63216	C₂²	1	2	yes	—
21	C₄⋊#63216	C₂²	1	1	yes	characteristic
22	#11763704	C₂²	1	1	yes	characteristic
23	#11763704	C₂²	1	2	yes	—
24	#11763704	C₂²	1	2	yes	—
25	#11763704	C₂²	1	1	yes	characteristic
26	#11763697	C₂²	1	2	yes	—
27	C₄×#63216	C₂²	1	2	yes	—
28	C₄×#63216	C₂²	1	1	yes	characteristic
29	C₄×#63216	C₂²	1	1	yes	characteristic
30	#11763346	C₄	1	1	yes	characteristic
31	Q₈×#3622	C₂²	1	1	yes	characteristic
32	#11762888	C₄	1	1	yes	characteristic
33	#11762882	C₄	1	2	yes	—
34	#11762874	C₂²	1	2	yes	—
35	#11762874	C₂²	1	1	yes	characteristic
36	#11762869	C₄	1	2	yes	—
37	#11762865	C₄	1	1	yes	characteristic
38	#11762766	C₂²	1	2	yes	—
39	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
40	C₁₇×#446	C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	#10587633	C₈	1	1	yes	characteristic
43	#10587632	C₈	1	1	yes	characteristic
44	C₂×#63218	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₁₇	C₂³	1	2	yes	—
48	#10587584	C₈	1	2	yes	—
49	#10587581	C₈	1	1	yes	characteristic
50	#10587580	C₈	1	2	yes	—
51	#10587576	C₈	1	1	yes	characteristic
52	C₄×D₆₈	C₂×C₄	1	2	yes	—
53	C₄×D₆₈	C₂×C₄	1	1	yes	characteristic
54	#10587558	C₂×C₄	1	2	yes	—
55	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
56	C₂×#63216	C₂³	1	2	yes	—
57	C₂×#63216	C₂³	1	1	yes	characteristic
58	C₂×#63216	C₂³	1	2	yes	—
59	C₂×#63216	C₂³	1	1	yes	characteristic
60	C₂×#63216	C₂³	1	1	yes	characteristic
61	C₂×#63216	C₂³	1	1	yes	characteristic
62	#10587530	—	4	2	no	—
63	#10587529	—	4	2	no	—
64	#10587529	—	4	2	no	—
65	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
66	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
67	C₁₇×#97	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
69	#10587491	C₂×C₄	1	2	yes	—
70	#10587491	C₂×C₄	1	2	yes	—
71	#10587491	C₂×C₄	1	2	yes	—
72	#10587486	C₂×C₄	1	2	yes	—
73	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
76	C₄⋊#3622	C₂×C₄	1	2	yes	—
77	C₄×#3622	C₂³	1	1	yes	characteristic
78	C₄×#3622	C₂³	1	2	yes	—
79	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
80	C₂×D₆₈	C₂×C₈	1	2	yes	—
81	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
89	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
90	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
91	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
93	#63227	C₂×C₈	1	2	yes	—
94	#63227	C₂×C₈	1	2	yes	—
95	#63227	C₂×C₈	1	2	yes	—
96	#63227	C₂×C₈	1	2	yes	—
97	#63227	C₂×C₈	1	2	yes	—
98	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
100	#63225	C₂×C₈	1	2	yes	—
101	#63225	C₂×C₈	1	2	yes	—
102	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
106	C₂×#3622	C₂²×C₄	1	2	yes	—
107	C₂×#3622	C₂⁴	1	1	yes	characteristic
108	#63218	—	4	2	no	—
109	C₈×D₁₇	—	2	1	no	—
110	C₈×D₁₇	—	2	2	no	—
111	C₈×D₁₇	—	2	1	no	—
112	C₈×D₁₇	—	2	2	no	—
113	C₈×D₁₇	—	4	1	no	—
114	#63216	—	2	1	no	—
115	#63216	—	2	2	no	—
116	#63216	—	2	2	no	—
117	#63216	—	2	1	no	—
118	#63216	—	2	1	no	—
119	#63216	—	2	1	no	—
120	#33508	—	17	1	no	maximal, Sylow
121	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
122	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₁₇	—	2	2	no	—
124	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
133	D₆₈	—	2	4	no	—
134	D₆₈	—	2	2	no	—
135	C₄×D₁₇	—	2	2	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	2	2	no	—
138	C₄×D₁₇	—	2	1	no	—
139	C₄×D₁₇	—	2	2	no	—
140	C₄×D₁₇	—	2	2	no	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	2	no	—
143	C₈×C₁₇	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3622	—	2	2	no	—
146	#3622	H(5,𝔽₂)	1	1	yes	characteristic
147	#3622	2₋¹⁺⁴	1	1	yes	characteristic
148	#2966	—	17	1	no	—
149	#2183	—	17	1	no	—
150	Q₈×C₁₆	—	17	1	no	—
151	#2169	—	17	1	no	—
152	#2168	—	17	2	no	—
153	D₄×C₁₆	—	17	1	no	—
154	D₄×C₁₆	—	17	2	no	—
155	#2164	—	17	2	no	—
156	#2162	—	17	1	no	—
157	#2151	—	17	2	no	—
158	C₂×C₄⋊C₁₆	—	17	1	no	—
159	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
160	C₂×D₁₇	—	2	1	no	—
161	C₂×D₁₇	—	2	1	no	—
162	C₂×D₁₇	—	2	2	no	—
163	C₂×D₁₇	—	2	2	no	—
164	C₂×D₁₇	—	2	2	no	—
165	C₂×D₁₇	#504	1	2	yes	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	2	2	no	—
168	C₄×C₁₇	—	2	2	no	—
169	C₄×C₁₇	#504	1	2	yes	—
170	C₄×C₁₇	#520	1	1	yes	characteristic
171	C₄×C₁₇	#568	1	1	yes	characteristic
172	Dic₁₇	—	2	1	no	—
173	Dic₁₇	—	2	1	no	—
174	Dic₁₇	—	2	1	no	—
175	Dic₁₇	—	2	2	no	—
176	#529	—	17	1	no	—
177	C₂×M₃₂	—	17	2	no	—
178	C₂²×C₁₆	—	17	2	no	—
179	C₂²×C₁₆	—	17	2	no	—
180	#446	#595	1	1	yes	characteristic
181	C₈×Q₈	—	17	1	no	—
182	#436	—	17	1	no	—
183	#435	—	17	2	no	—
184	C₈×D₄	—	17	2	no	—
185	C₈×D₄	—	17	1	no	—
186	#433	—	17	2	no	—
187	#432	—	17	1	no	—
188	#405	—	17	2	no	—
189	C₂×C₄×C₈	—	17	1	no	—
190	C₄⋊C₁₆	—	17	1	no	—
191	C₄⋊C₁₆	—	17	1	no	—
192	C₄⋊C₁₆	—	17	2	no	—
193	C₄⋊C₁₆	—	17	2	no	—
194	C₄⋊C₁₆	—	17	2	no	—
195	C₄⋊C₁₆	—	17	1	no	—
196	C₄⋊C₁₆	—	17	1	no	—
197	C₂²⋊C₁₆	—	17	1	no	—
198	C₂²⋊C₁₆	—	17	1	no	—
199	C₂²⋊C₁₆	—	17	2	no	—
200	C₂²⋊C₁₆	—	17	2	no	—
201	#346	—	17	2	no	—
202	C₄×C₁₆	—	17	1	no	—
203	C₄×C₁₆	—	17	1	no	—
204	C₄×C₁₆	—	17	2	no	—
205	C₂×C₁₇	#2988	1	1	yes	characteristic
206	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
207	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
208	D₁₇	—	2	2	no	—
209	D₁₇	—	4	2	no	—
210	C₂×M₁₆	—	17	2	no	—
211	C₂²×C₈	—	17	1	no	—
212	C₂²×C₈	—	17	2	no	—
213	C₂²×C₈	—	17	1	no	—
214	#126	—	17	2	no	—
215	#125	—	17	1	no	—
216	#124	—	17	1	no	—
217	#123	—	17	2	no	—
218	C₂²⋊Q₈	—	17	1	no	—
219	#121	—	17	1	no	—
220	C₄×Q₈	C₂×#595	1	1	yes	characteristic
221	C₄×D₄	—	17	2	no	—
222	C₄×D₄	—	17	1	no	—
223	#117	—	17	2	no	—
224	C₂×C₄²	—	17	1	no	—
225	M₃₂	—	68	2	no	—
226	C₂×C₁₆	—	17	1	no	—
227	C₂×C₁₆	—	17	1	no	—
228	C₂×C₁₆	—	17	1	no	—
229	C₂×C₁₆	—	17	2	no	—
230	C₂×C₁₆	—	17	1	no	—
231	C₂×C₁₆	—	17	2	no	—
232	C₂×C₁₆	—	68	2	no	—
233	C₂×C₁₆	—	68	2	no	—
234	C₄⋊C₈	—	17	1	no	—
235	C₄⋊C₈	—	17	2	no	—
236	C₄⋊C₈	C₂×#595	1	1	yes	characteristic
237	C₄⋊C₈	C₂×#595	1	2	yes	—
238	C₂²⋊C₈	—	17	2	no	—
239	C₂²⋊C₈	—	17	2	no	—
240	C₂²⋊C₈	—	17	2	no	—
241	#97	—	17	2	no	—
242	#97	C₂×#595	1	2	yes	—
243	C₄×C₈	—	17	1	no	—
244	C₄×C₈	—	17	1	no	—
245	C₄×C₈	—	17	1	no	—
246	C₄×C₈	—	17	2	no	—
247	C₄×C₈	C₂×#595	1	1	yes	characteristic
248	C₁₇	#33508	1	1	yes	characteristic, complemented, Sylow
249	C₂×Q₈	#63242	1	1	yes	characteristic
250	C₂×D₄	—	17	2	no	—
251	C₂×D₄	—	17	1	no	—
252	C₂²×C₄	—	17	1	no	—
253	C₂²×C₄	—	17	1	no	—
254	C₂²×C₄	—	17	1	no	—
255	C₂²×C₄	—	17	2	no	—
256	M₁₆	—	68	2	no	—
257	C₂×C₈	—	17	1	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	2	no	—
260	C₂×C₈	—	34	1	no	—
261	C₂×C₈	—	34	2	no	—
262	C₂×C₈	—	34	1	no	—
263	C₂×C₈	—	34	2	no	—
264	C₂×C₈	—	68	1	no	—
265	C₂×C₈	C₂²×#595	1	1	yes	characteristic
266	C₂×C₈	C₂²×#595	1	1	yes	characteristic
267	C₂×C₈	C₂²×#595	1	2	yes	—
268	C₄⋊C₄	—	17	2	no	—
269	C₄⋊C₄	—	17	2	no	—
270	C₄⋊C₄	—	17	1	no	—
271	C₄⋊C₄	—	17	2	no	—
272	C₄⋊C₄	#63242	1	1	yes	characteristic
273	C₄⋊C₄	#63242	1	2	yes	—
274	C₂²⋊C₄	—	17	2	no	—
275	C₂²⋊C₄	—	17	2	no	—
276	C₂²⋊C₄	—	17	2	no	—
277	C₂²⋊C₄	—	17	2	no	—
278	C₂²⋊C₄	—	17	2	no	—
279	C₄²	—	17	1	no	—
280	C₄²	—	17	1	no	—
281	C₄²	—	17	1	no	—
282	C₄²	C₂²×#595	1	1	yes	characteristic
283	C₄²	C₂²×#595	1	2	yes	—
284	C₁₆	—	34	1	no	—
285	C₁₆	—	34	2	no	—
286	C₁₆	—	34	2	no	—
287	C₁₆	—	34	1	no	—
288	C₁₆	—	34	1	no	—
289	C₁₆	—	34	1	no	—
290	C₂³	—	17	1	no	—
291	C₂³	—	17	2	no	—
292	Q₈	—	2	2	no	—
293	D₄	—	34	4	no	—
294	D₄	—	34	2	no	—
295	C₂×C₄	—	17	1	no	—
296	C₂×C₄	—	17	1	no	—
297	C₂×C₄	—	17	1	no	—
298	C₂×C₄	—	17	2	no	—
299	C₂×C₄	—	34	2	no	—
300	C₂×C₄	—	34	2	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	2	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	34	2	no	—
305	C₂×C₄	C₂×#63242	1	1	yes	characteristic
306	C₂×C₄	C₂×#63242	1	2	yes	—
307	C₂×C₄	C₂×#63242	1	2	yes	—
308	C₂×C₄	C₂×#63242	1	1	yes	characteristic
309	C₂×C₄	C₂³×#595	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	—	17	1	no	—
314	C₈	—	17	1	no	—
315	C₈	—	34	1	no	—
316	C₈	—	34	2	no	—
317	C₂²	—	17	2	no	—
318	C₂²	—	34	1	no	—
319	C₂²	—	34	1	no	—
320	C₂²	—	34	2	no	—
321	C₂²	—	34	2	no	—
322	C₂²	—	34	2	no	—
323	C₂²	C₂²×#63242	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	34	1	no	—
328	C₄	—	34	2	no	—
329	C₄	—	34	1	no	—
330	C₄	—	34	1	no	—
331	C₄	#11763868	1	2	yes	—
332	C₄	#11763878	1	1	yes	characteristic
333	C₄	#11763889	1	1	yes	characteristic
334	C₂	—	34	2	no	—
335	C₂	—	68	2	no	—
336	C₂	#14545702	1	1	yes	characteristic
337	C₂	#14545749	1	1	yes	characteristic
338	C₂	C₂×#11763868	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#446	C₄
prime radical	#10587581	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#446	#595
hypocenter	C₁₇	#33508
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
2. Q₈×#63216	C₂	semidirect
3. #14545170	C₂	semidirect
4. #14545167	C₂	semidirect
5. #14545163	C₂	semidirect
6. #14545163	C₂	semidirect
7. #14545151	C₂	semidirect
8. #14545143	C₂	semidirect
9. #14545120	C₂	semidirect
10. #14545114	C₂	semidirect
248. C₁₇	#33508	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
Q₈×#63216	2
#14545170	2
#14545167	2
#14545163	2
#14545151	2
#14545143	2
#14545120	2
#14545114	2
#14537758	2
#33508	17