Subgroups — #17978828

Order 4352.

484 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	2	yes	maximal, complemented
2	#14545168	C₂	1	2	yes	maximal, complemented
3	#14545164	C₂	1	2	yes	maximal, complemented
4	#14545164	C₂	1	2	yes	maximal, complemented
5	#14545149	C₂	1	2	yes	maximal, complemented
6	#14545145	C₂	1	1	yes	maximal, characteristic, complemented
7	#14545119	C₂	1	2	yes	maximal, complemented
8	#14545116	C₂	1	1	yes	maximal, characteristic, complemented
9	#14537760	C₂	1	1	yes	maximal, characteristic
10	C₂×#10587538	C₂²	1	1	yes	characteristic
11	C₂×#10587538	C₂²	1	2	yes	—
12	C₂×#10587538	C₂²	1	1	yes	characteristic
13	C₂²×#63216	C₂²	1	2	yes	—
14	C₄⋊#63216	C₂²	1	2	yes	—
15	C₄⋊#63216	C₂²	1	2	yes	—
16	C₄⋊#63216	C₂²	1	2	yes	—
17	C₄⋊#63216	C₂²	1	2	yes	—
18	C₄⋊#63216	C₂²	1	1	yes	characteristic
19	C₄⋊#63216	C₂²	1	1	yes	characteristic
20	C₂²⋊#63216	C₂²	1	2	yes	—
21	C₂²⋊#63216	C₂²	1	2	yes	—
22	C₂²⋊#63216	C₂²	1	2	yes	—
23	#11763697	C₂²	1	2	yes	—
24	#11763697	C₂²	1	2	yes	—
25	C₄×#63216	C₂²	1	2	yes	—
26	#11762940	C₄	1	1	yes	characteristic
27	Q₈×#3622	C₂²	1	1	yes	characteristic
28	#11762887	C₄	1	1	yes	characteristic
29	#11762881	C₄	1	2	yes	—
30	D₄×#3622	C₂²	1	1	yes	characteristic
31	D₄×#3622	C₂²	1	2	yes	—
32	#11762870	C₄	1	2	yes	—
33	#11762866	C₄	1	1	yes	characteristic
34	#11762765	C₂²	1	2	yes	—
35	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
36	C₁₇×#529	C₄	1	1	yes	characteristic
37	C₁₇×#126	C₈	1	2	yes	—
38	C₁₇×#125	C₈	1	1	yes	characteristic
39	C₁₇×#124	C₈	1	1	yes	characteristic
40	C₁₇×#123	C₈	1	2	yes	—
41	C₁₇×C₂²⋊Q₈	C₈	1	1	yes	characteristic
42	C₁₇×#121	C₈	1	1	yes	characteristic
43	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×D₄×C₁₇	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₁₇×#117	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×#63223	C₂×C₄	1	2	yes	—
49	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
50	C₂²×#3622	C₂³	1	1	yes	characteristic
51	C₂²×#3622	C₂³	1	2	yes	—
52	#10587538	—	4	1	no	—
53	#10587538	—	4	2	no	—
54	#10587538	—	4	1	no	—
55	C₂×#63216	—	4	2	no	—
56	C₂×#63216	C₂³	1	2	yes	—
57	C₂×#63216	C₂³	1	2	yes	—
58	C₂×#63216	C₂³	1	2	yes	—
59	C₂×#63216	C₂³	1	2	yes	—
60	C₂²⋊#3622	C₂×C₄	1	2	yes	—
61	C₂²⋊#3622	C₂×C₄	1	2	yes	—
62	C₂²⋊#3622	C₂×C₄	1	2	yes	—
63	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
64	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
65	C₄⋊#3622	C₂×C₄	1	2	yes	—
66	C₄⋊#3622	C₂×C₄	1	2	yes	—
67	#10587474	C₂×C₄	1	2	yes	—
68	#10587474	C₂×C₄	1	2	yes	—
69	C₄×#3622	C₂×C₄	1	1	yes	characteristic
70	C₄×#3622	C₂×C₄	1	1	yes	characteristic
71	C₄×#3622	C₂³	1	1	yes	characteristic
72	C₄×#3622	C₂³	1	1	yes	characteristic
73	C₄×#3622	C₂³	1	2	yes	—
74	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
76	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
81	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
84	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
85	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
86	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
88	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
89	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
92	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
97	#63223	—	4	2	no	—
98	C₂×#3622	—	2	2	no	—
99	C₂×#3622	—	2	1	no	—
100	C₂×#3622	—	2	1	no	—
101	C₂×#3622	—	2	2	no	—
102	C₂×#3622	—	4	1	no	—
103	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
104	C₂×#3622	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	C₂²×C₄	1	2	yes	—
108	C₂×#3622	C₂⁴	1	1	yes	characteristic
109	#63216	—	2	2	no	—
110	#63216	—	2	2	no	—
111	#63216	—	2	2	no	—
112	#63216	—	2	2	no	—
113	#33509	—	17	1	no	maximal, Sylow
114	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
115	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
116	Q₈×C₁₇	—	4	1	no	—
117	D₄×C₁₇	—	4	1	no	—
118	D₄×C₁₇	—	4	2	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	2	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	2	no	—
123	C₂×C₄×C₁₇	—	4	1	no	—
124	C₂×C₄×C₁₇	—	4	1	no	—
125	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
133	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
134	#3622	—	2	2	no	—
135	#3622	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3622	—	2	2	no	—
139	#3622	H(5,𝔽₂)	1	1	yes	characteristic
140	#3622	2₋¹⁺⁴	1	1	yes	characteristic
141	#2966	—	17	1	no	—
142	#2183	—	17	2	no	—
143	#2169	—	17	2	no	—
144	#2168	—	17	2	no	—
145	#2168	—	17	2	no	—
146	#2164	—	17	2	no	—
147	#2163	—	17	1	no	—
148	#2151	—	17	2	no	—
149	#2150	—	17	1	no	—
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	2	2	no	—
153	C₂²×C₁₇	—	2	2	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	—	2	2	no	—
156	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	2	2	no	—
162	C₄×C₁₇	—	2	2	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	2	2	no	—
165	C₄×C₁₇	#520	1	1	yes	characteristic
166	C₄×C₁₇	#568	1	1	yes	characteristic
167	#529	#595	1	1	yes	characteristic
168	C₂×M₃₂	—	17	1	no	—
169	C₂×M₃₂	—	17	1	no	—
170	C₂×M₃₂	—	17	2	no	—
171	C₂²×C₁₆	—	17	2	no	—
172	#446	—	17	1	no	—
173	C₈×Q₈	—	17	1	no	—
174	#436	—	17	1	no	—
175	#435	—	17	2	no	—
176	C₈×D₄	—	17	1	no	—
177	C₈×D₄	—	17	2	no	—
178	#433	—	17	2	no	—
179	#432	—	17	1	no	—
180	#405	—	17	2	no	—
181	C₂×C₄×C₈	—	17	1	no	—
182	C₄⋊C₁₆	—	17	2	no	—
183	C₄⋊C₁₆	—	17	2	no	—
184	C₄⋊C₁₆	—	17	1	no	—
185	C₄⋊C₁₆	—	17	2	no	—
186	C₄⋊C₁₆	—	17	2	no	—
187	C₄⋊C₁₆	—	17	1	no	—
188	C₂²⋊C₁₆	—	17	2	no	—
189	C₂²⋊C₁₆	—	17	2	no	—
190	C₂²⋊C₁₆	—	17	2	no	—
191	#346	—	17	2	no	—
192	#346	—	17	2	no	—
193	C₄×C₁₆	—	17	2	no	—
194	C₂×C₁₇	—	4	1	no	—
195	C₂×C₁₇	—	4	2	no	—
196	C₂×C₁₇	#2988	1	1	yes	characteristic
197	C₂×C₁₇	#3407	1	1	yes	characteristic
198	C₂×C₁₇	#3407	1	1	yes	characteristic
199	C₂×M₁₆	—	17	2	no	—
200	C₂²×C₈	—	17	1	no	—
201	C₂²×C₈	—	17	2	no	—
202	C₂²×C₈	—	17	1	no	—
203	#126	#3624	1	2	yes	—
204	#125	#3624	1	1	yes	characteristic
205	#124	#3624	1	1	yes	characteristic
206	#123	#3624	1	2	yes	—
207	C₂²⋊Q₈	#3624	1	1	yes	characteristic
208	#121	#3624	1	1	yes	characteristic
209	C₄×Q₈	C₂×#595	1	1	yes	characteristic
210	C₄×D₄	C₂×#595	1	2	yes	—
211	C₄×D₄	C₂×#595	1	1	yes	characteristic
212	#117	C₂×#595	1	2	yes	—
213	C₂×C₄²	C₂×#595	1	1	yes	characteristic
214	M₃₂	—	68	1	no	—
215	M₃₂	—	68	2	no	—
216	M₃₂	—	68	1	no	—
217	C₂×C₁₆	—	17	2	no	—
218	C₂×C₁₆	—	17	2	no	—
219	C₂×C₁₆	—	17	2	no	—
220	C₂×C₁₆	—	17	2	no	—
221	C₂×C₁₆	—	68	2	no	—
222	C₄⋊C₈	—	17	1	no	—
223	C₄⋊C₈	—	17	1	no	—
224	C₄⋊C₈	—	17	2	no	—
225	C₄⋊C₈	—	17	2	no	—
226	C₂²⋊C₈	—	17	2	no	—
227	C₂²⋊C₈	—	17	2	no	—
228	C₂²⋊C₈	—	17	2	no	—
229	#97	—	17	2	no	—
230	#97	—	17	2	no	—
231	C₄×C₈	—	17	1	no	—
232	C₄×C₈	—	17	1	no	—
233	C₄×C₈	—	17	2	no	—
234	C₄×C₈	—	17	1	no	—
235	C₄×C₈	—	17	1	no	—
236	C₁₇	#33509	1	1	yes	characteristic, complemented, Sylow
237	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
238	C₂×D₄	C₂×#3624	1	2	yes	—
239	C₂×D₄	C₂×#3624	1	1	yes	characteristic
240	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
241	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
242	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
243	C₂²×C₄	C₂²×#595	1	2	yes	—
244	M₁₆	—	68	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	—	17	2	no	—
250	C₂×C₈	—	17	2	no	—
251	C₂×C₈	—	34	2	no	—
252	C₂×C₈	—	34	1	no	—
253	C₂×C₈	—	34	1	no	—
254	C₂×C₈	—	34	2	no	—
255	C₂×C₈	—	68	1	no	—
256	C₄⋊C₄	C₂×#3624	1	2	yes	—
257	C₄⋊C₄	C₂×#3624	1	2	yes	—
258	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
259	C₄⋊C₄	C₂×#3624	1	2	yes	—
260	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
261	C₄⋊C₄	C₂×#3624	1	2	yes	—
262	C₂²⋊C₄	C₂×#3624	1	2	yes	—
263	C₂²⋊C₄	C₂×#3624	1	2	yes	—
264	C₂²⋊C₄	C₂×#3624	1	2	yes	—
265	C₂²⋊C₄	C₂×#3624	1	2	yes	—
266	C₂²⋊C₄	C₂×#3624	1	2	yes	—
267	C₄²	C₂×#3624	1	1	yes	characteristic
268	C₄²	C₂×#3624	1	1	yes	characteristic
269	C₄²	C₂²×#595	1	1	yes	characteristic
270	C₄²	C₂²×#595	1	1	yes	characteristic
271	C₄²	C₂²×#595	1	2	yes	—
272	C₁₆	—	34	2	no	—
273	C₁₆	—	34	2	no	—
274	C₁₆	—	34	2	no	—
275	C₁₆	—	34	2	no	—
276	C₂³	C₂²×#3624	1	2	yes	—
277	C₂³	C₂²×#3624	1	1	yes	characteristic
278	Q₈	—	4	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
288	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
289	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
290	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
291	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
292	C₂×C₄	C₂²×#3624	1	2	yes	—
293	C₂×C₄	C₂²×#3624	1	2	yes	—
294	C₂×C₄	C₂²×#3624	1	2	yes	—
295	C₂×C₄	C₂³×#595	1	1	yes	characteristic
296	C₈	—	17	1	no	—
297	C₈	—	17	1	no	—
298	C₈	—	34	1	no	—
299	C₈	—	34	2	no	—
300	C₈	—	34	1	no	—
301	C₈	—	34	1	no	—
302	C₈	—	34	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂³×#3624	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	#11763878	1	1	yes	characteristic
319	C₄	#11763888	1	1	yes	characteristic
320	C₂	—	4	1	no	—
321	C₂	—	4	2	no	—
322	C₂	#14545699	1	1	yes	characteristic
323	C₂	#14545751	1	1	yes	characteristic
324	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#529	C₄
prime radical	C₁₇×#121	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#529	#595
hypocenter	C₁₇	#33509
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
2. #14545168	C₂	semidirect
3. #14545164	C₂	semidirect
4. #14545164	C₂	semidirect
5. #14545149	C₂	semidirect
6. #14545145	C₂	semidirect
7. #14545119	C₂	semidirect
8. #14545116	C₂	semidirect
236. C₁₇	#33509	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
#14545168	2
#14545164	2
#14545149	2
#14545145	2
#14545119	2
#14545116	2
#14537760	2
#33509	17