Subgroups — #17978489

Order 4352.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545149	C₂	1	4	yes	maximal, complemented
2	#14545145	C₂	1	2	yes	maximal, complemented
3	#14545141	C₂	1	2	yes	maximal, complemented
4	#14545116	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic, complemented
6	#14545014	C₂	1	2	yes	maximal
7	#14544993	C₂	1	2	yes	maximal, complemented
8	C₄×#10587634	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587538	C₂²	1	2	yes	—
10	C₂²×#63216	C₂²	1	2	yes	—
11	C₄⋊#63216	C₂²	1	1	yes	characteristic
12	C₄⋊#63216	C₂²	1	1	yes	characteristic
13	C₄⋊#63216	C₂²	1	2	yes	—
14	C₄⋊#63216	C₂²	1	2	yes	—
15	C₄⋊#63216	C₂²	1	2	yes	—
16	C₂²⋊#63216	C₂²	1	4	yes	—
17	C₂²⋊#63216	C₂²	1	4	yes	—
18	#11763697	C₂²	1	4	yes	—
19	C₄×#63216	C₂²	1	4	yes	—
20	C₂×#10587634	C₄	1	1	yes	characteristic
21	C₂×#10587634	C₂²	1	1	yes	characteristic
22	#11762765	C₄	1	1	yes	characteristic
23	#11762765	C₄	1	2	yes	—
24	#11762765	C₂²	1	1	yes	characteristic
25	#11762765	C₂²	1	2	yes	—
26	C₄×#63223	C₄	1	2	yes	—
27	C₄×#63223	C₄	1	1	yes	characteristic
28	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
29	C₂×C₄×#3622	C₂²	1	2	yes	—
30	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
31	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
34	C₄×D₄×C₁₇	C₈	1	2	yes	—
35	C₄×D₄×C₁₇	C₈	1	1	yes	characteristic
36	C₄×D₄×C₁₇	C₈	1	2	yes	—
37	C₄×D₄×C₁₇	C₈	1	1	yes	characteristic
38	C₁₇×#117	C₂×C₄	1	2	yes	—
39	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
42	#10587634	—	2	4	no	—
43	#10587634	—	2	2	no	—
44	#10587634	—	2	2	no	—
45	C₂×#63223	C₂×C₄	1	2	yes	—
46	C₂×#63223	C₂×C₄	1	1	yes	characteristic
47	C₂×#63223	C₂×C₄	1	2	yes	—
48	C₂×#63223	C₂×C₄	1	1	yes	characteristic
49	C₂²×#3622	C₂×C₄	1	2	yes	—
50	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
51	C₂²×#3622	C₂³	1	2	yes	—
52	C₂²×#3622	C₂³	1	1	yes	characteristic
53	#10587538	—	2	4	no	—
54	C₂×#63216	—	2	4	no	—
55	C₂×#63216	C₂³	1	2	yes	—
56	C₂×#63216	C₂³	1	2	yes	—
57	C₂×#63216	C₂³	1	4	yes	—
58	#10587474	C₂×C₄	1	2	yes	—
59	#10587474	C₂×C₄	1	2	yes	—
60	#10587474	C₂×C₄	1	2	yes	—
61	C₄×#3622	C₂×C₄	1	2	yes	—
62	C₄×#3622	C₂×C₄	1	1	yes	characteristic
63	C₄×#3622	C₂×C₄	1	2	yes	—
64	C₄×#3622	C₂×C₄	1	1	yes	characteristic
65	C₄×#3622	C₂³	1	1	yes	characteristic
66	C₄×#3622	C₂³	1	1	yes	characteristic
67	C₄×#3622	C₂³	1	2	yes	—
68	#41×C₁₇	—	2	2	no	—
69	#41×C₁₇	—	2	2	no	—
70	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
71	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
72	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
80	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
81	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
84	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
86	C₄²×C₁₇	C₂×C₈	1	2	yes	—
87	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₁₇	C₂×C₈	1	2	yes	—
89	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
93	#63223	—	2	2	no	—
94	#63223	—	2	4	no	—
95	#63223	—	2	4	no	—
96	#63223	M₁₆	1	4	yes	—
97	C₂×#3622	—	2	2	no	—
98	C₂×#3622	—	2	4	no	—
99	C₂×#3622	—	2	2	no	—
100	C₂×#3622	—	2	2	no	—
101	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
102	C₂×#3622	C₂²×C₄	1	2	yes	—
103	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
104	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
105	C₂×#3622	C₂²×C₄	1	2	yes	—
106	C₂×#3622	#41	1	2	yes	—
107	C₂×#3622	#41	1	2	yes	—
108	C₂×#3622	C₂⁴	1	1	yes	characteristic
109	#63216	—	2	2	no	—
110	#63216	—	2	2	no	—
111	#63216	—	2	4	no	—
112	#33423	—	17	1	no	maximal, Sylow
113	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
114	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	2	no	—
116	D₄×C₁₇	—	2	4	no	—
117	D₄×C₁₇	—	2	2	no	—
118	C₂×C₄×C₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	2	no	—
120	C₂×C₄×C₁₇	—	2	2	no	—
121	C₂×C₄×C₁₇	—	2	4	no	—
122	C₂×C₄×C₁₇	—	4	1	no	—
123	C₂×C₄×C₁₇	—	4	2	no	—
124	C₂×C₄×C₁₇	#117	1	2	yes	—
125	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
134	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
135	#3622	—	2	2	no	—
136	#3622	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3622	#117	1	2	yes	—
139	#3622	C₂×M₁₆	1	4	yes	—
140	#3622	C₂×#41	1	1	yes	characteristic
141	#3622	C₂×#41	1	1	yes	characteristic
142	C₄×#131	—	17	1	no	—
143	#2164	—	17	4	no	—
144	#2163	—	17	2	no	—
145	#2162	—	17	2	no	—
146	#2150	—	17	1	no	—
147	C₂×C₄⋊C₁₆	—	17	1	no	—
148	#2113	—	17	2	no	—
149	#2108	—	17	2	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	2	no	—
155	C₂²×C₁₇	#431	1	2	yes	—
156	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	2	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	#431	1	2	yes	—
164	C₄×C₁₇	#504	1	2	yes	—
165	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
166	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
167	C₂×#131	—	17	1	no	—
168	C₂×#131	—	17	1	no	—
169	C₄×#41	#595	1	1	yes	characteristic
170	C₂×M₃₂	—	17	2	no	—
171	C₂²×C₁₆	—	17	2	no	—
172	#405	—	17	1	no	—
173	#405	—	17	2	no	—
174	#405	—	17	1	no	—
175	#405	—	17	2	no	—
176	C₄×M₁₆	—	17	1	no	—
177	C₄×M₁₆	—	17	2	no	—
178	C₂×C₄×C₈	—	17	1	no	—
179	C₂×C₄×C₈	—	17	2	no	—
180	C₄⋊C₁₆	—	17	1	no	—
181	C₄⋊C₁₆	—	17	1	no	—
182	C₄⋊C₁₆	—	17	2	no	—
183	C₄⋊C₁₆	—	17	2	no	—
184	C₄⋊C₁₆	—	17	2	no	—
185	C₂²⋊C₁₆	—	17	4	no	—
186	C₂²⋊C₁₆	—	17	4	no	—
187	#346	—	17	4	no	—
188	C₄×C₁₆	—	17	4	no	—
189	C₂×C₁₇	—	2	2	no	—
190	C₂×C₁₇	—	4	2	no	—
191	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
192	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
193	C₂×C₁₇	#3407	1	1	yes	characteristic
194	C₂×#41	C₂×#595	1	1	yes	characteristic
195	#131	—	34	2	no	—
196	#131	—	34	2	no	—
197	#131	—	34	4	no	—
198	C₂×M₁₆	—	17	1	no	—
199	C₂×M₁₆	—	17	2	no	—
200	C₂×M₁₆	—	17	2	no	—
201	C₂×M₁₆	—	17	1	no	—
202	C₂²×C₈	—	17	1	no	—
203	C₂²×C₈	—	17	2	no	—
204	C₂²×C₈	—	17	2	no	—
205	C₂²×C₈	—	17	1	no	—
206	C₄×Q₈	#3624	1	1	yes	characteristic
207	C₄×Q₈	#3624	1	1	yes	characteristic
208	C₄×D₄	#3624	1	2	yes	—
209	C₄×D₄	#3624	1	1	yes	characteristic
210	C₄×D₄	#3624	1	2	yes	—
211	C₄×D₄	#3624	1	1	yes	characteristic
212	#117	C₂×#595	1	2	yes	—
213	#117	C₂×#595	1	1	yes	characteristic
214	C₂×C₄²	C₂×#595	1	2	yes	—
215	C₂×C₄²	C₂×#595	1	1	yes	characteristic
216	M₃₂	—	34	4	no	—
217	C₂×C₁₆	—	17	2	no	—
218	C₂×C₁₆	—	17	4	no	—
219	C₂×C₁₆	—	17	2	no	—
220	C₂×C₁₆	—	34	4	no	—
221	#97	—	17	2	no	—
222	#97	—	17	2	no	—
223	#97	—	17	2	no	—
224	C₄×C₈	—	17	1	no	—
225	C₄×C₈	—	17	2	no	—
226	C₄×C₈	—	17	1	no	—
227	C₄×C₈	—	17	2	no	—
228	C₄×C₈	—	17	2	no	—
229	C₄×C₈	—	17	1	no	—
230	C₄×C₈	—	17	1	no	—
231	C₁₇	#33423	1	1	yes	characteristic, complemented, Sylow
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
235	C₂×D₄	C₂×#3624	1	2	yes	—
236	C₂×D₄	C₂×#3624	1	1	yes	characteristic
237	C₂²×C₄	C₂×#3624	1	2	yes	—
238	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
239	C₂²×C₄	C₂×#3624	1	2	yes	—
240	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
241	C₂²×C₄	C₂²×#595	1	2	yes	—
242	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
243	M₁₆	—	17	4	no	—
244	M₁₆	—	34	4	no	—
245	M₁₆	—	34	4	no	—
246	M₁₆	—	34	2	no	—
247	C₂×C₈	—	17	2	no	—
248	C₂×C₈	—	17	2	no	—
249	C₂×C₈	—	17	1	no	—
250	C₂×C₈	—	17	1	no	—
251	C₂×C₈	—	17	2	no	—
252	C₂×C₈	—	17	1	no	—
253	C₂×C₈	—	17	2	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	34	2	no	—
256	C₂×C₈	—	34	2	no	—
257	C₂×C₈	—	34	4	no	—
258	C₂×C₈	—	34	2	no	—
259	C₄⋊C₄	C₂×#3624	1	2	yes	—
260	C₄⋊C₄	C₂×#3624	1	2	yes	—
261	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
262	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
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	C₄²	C₂×#3624	1	2	yes	—
267	C₄²	C₂×#3624	1	1	yes	characteristic
268	C₄²	C₂×#3624	1	2	yes	—
269	C₄²	C₂×#3624	1	1	yes	characteristic
270	C₄²	C₂²×#595	1	1	yes	characteristic
271	C₄²	C₂²×#595	1	2	yes	—
272	C₄²	C₂²×#595	1	1	yes	characteristic
273	C₁₆	—	34	2	no	—
274	C₁₆	—	34	2	no	—
275	C₁₆	—	34	4	no	—
276	C₂³	C₂²×#3624	1	2	yes	—
277	C₂³	C₂²×#3624	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
288	C₂×C₄	#10587677	1	2	yes	—
289	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
290	C₂×C₄	C₂²×#3624	1	2	yes	—
291	C₂×C₄	C₂²×#3624	1	2	yes	—
292	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
293	C₂×C₄	C₂²×#3624	1	2	yes	—
294	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
295	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
296	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
297	C₂×C₄	C₂³×#595	1	1	yes	characteristic
298	C₈	—	17	1	no	—
299	C₈	—	17	1	no	—
300	C₈	—	17	2	no	—
301	C₈	—	17	4	no	—
302	C₈	—	34	2	no	—
303	C₈	—	34	1	no	—
304	C₈	—	34	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#11763825	1	2	yes	—
311	C₂²	C₂³×#3624	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	#11763825	1	2	yes	—
319	C₄	#11763867	1	2	yes	—
320	C₄	C₂×#10587677	1	1	yes	characteristic
321	C₄	C₂×D₁₇⋊M₁₆	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#11763825	1	1	yes	characteristic
325	C₂	C₂×#11763867	1	1	yes	characteristic
326	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₄×#41×C₁₇	C₄
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂×#3622	C₂⁴
hypercenter	C₄×#41	#595
hypocenter	C₁₇	#33423
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #14545149	C₂	semidirect
2. #14545145	C₂	semidirect
3. #14545141	C₂	semidirect
4. #14545116	C₂	semidirect
5. C₂×C₄⋊#63216	C₂	semidirect
7. #14544993	C₂	semidirect
231. C₁₇	#33423	semidirect

Maximal subgroups

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

subgroup	index
#14545149	2
#14545145	2
#14545141	2
#14545116	2
C₂×C₄⋊#63216	2
#14545014	2
#14544993	2
C₄×#10587634	2
#33423	17