Subgroups — #17978800

Order 4352.

536 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#14545164	C₂	1	4	yes	maximal, complemented
3	D₄×#63216	C₂	1	2	yes	maximal, complemented
4	#14545145	C₂	1	2	yes	maximal, complemented
5	#14545141	C₂	1	1	yes	maximal, characteristic, complemented
6	#14545116	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊#63216	C₂	1	2	yes	maximal, complemented
8	#41×#3622	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587538	C₂²	1	2	yes	—
10	C₂²×#63216	C₂²	1	4	yes	—
11	C₄⋊#63216	C₂²	1	2	yes	—
12	C₄⋊#63216	C₂²	1	4	yes	—
13	C₄⋊#63216	C₂²	1	1	yes	characteristic
14	C₄⋊#63216	C₂²	1	2	yes	—
15	C₄⋊#63216	C₂²	1	1	yes	characteristic
16	C₂²⋊#63216	C₂²	1	4	yes	—
17	C₂²⋊#63216	C₂²	1	2	yes	—
18	#11763697	C₂²	1	4	yes	—
19	C₄×#63216	C₂²	1	2	yes	—
20	Q₈×#3622	C₄	1	1	yes	characteristic
21	Q₈×#3622	C₂²	1	1	yes	characteristic
22	D₄×#3622	C₄	1	2	yes	—
23	D₄×#3622	C₄	1	1	yes	characteristic
24	D₄×#3622	C₂²	1	1	yes	characteristic
25	D₄×#3622	C₂²	1	2	yes	—
26	#11762862	C₄	1	1	yes	characteristic
27	#11762862	C₄	1	2	yes	—
28	C₂×C₄×#3622	C₂²	1	2	yes	—
29	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
30	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
31	C₂×#41×C₁₇	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₁₇	C₂×C₄	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	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₁₇×#117	C₈	1	1	yes	characteristic
39	C₁₇×#117	C₈	1	2	yes	—
40	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×#3622	C₂×C₄	1	2	yes	—
43	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
44	C₂²×#3622	C₂³	1	2	yes	—
45	C₂²×#3622	C₂³	1	1	yes	characteristic
46	#10587538	—	4	2	no	—
47	C₂×#63216	—	4	4	no	—
48	C₂×#63216	C₂³	1	2	yes	—
49	C₂×#63216	C₂³	1	4	yes	—
50	C₂×#63216	C₂³	1	2	yes	—
51	C₂²⋊#3622	C₂×C₄	1	2	yes	—
52	C₂²⋊#3622	C₂×C₄	1	2	yes	—
53	C₂²⋊#3622	C₂×C₄	1	2	yes	—
54	C₄⋊#3622	C₂×C₄	1	2	yes	—
55	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#3622	C₂×C₄	1	2	yes	—
58	C₄×#3622	C₂×C₄	1	2	yes	—
59	C₄×#3622	C₂×C₄	1	1	yes	characteristic
60	C₄×#3622	C₂×C₄	1	2	yes	—
61	C₄×#3622	C₂×C₄	1	1	yes	characteristic
62	C₄×#3622	C₂³	1	1	yes	characteristic
63	C₄×#3622	C₂³	1	1	yes	characteristic
64	C₄×#3622	C₂³	1	2	yes	—
65	#41×C₁₇	—	2	4	no	—
66	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
75	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
76	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
78	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
80	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
81	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
82	C₄²×C₁₇	C₂×C₈	1	2	yes	—
83	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₇	C₂×C₈	1	2	yes	—
85	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
86	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×#3622	—	2	2	no	—
90	C₂×#3622	—	2	1	no	—
91	C₂×#3622	—	2	2	no	—
92	C₂×#3622	—	2	1	no	—
93	C₂×#3622	—	4	2	no	—
94	C₂×#3622	—	4	1	no	—
95	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
96	C₂×#3622	C₂²×C₄	1	2	yes	—
97	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
98	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
99	C₂×#3622	C₂²×C₄	1	2	yes	—
100	C₂×#3622	C₂⁴	1	1	yes	characteristic
101	#63216	—	2	2	no	—
102	#63216	—	2	4	no	—
103	#63216	—	2	2	no	—
104	#33504	—	17	1	no	maximal, Sylow
105	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
106	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
107	Q₈×C₁₇	—	2	2	no	—
108	D₄×C₁₇	—	2	4	no	—
109	D₄×C₁₇	M₃₂	1	4	yes	complemented
110	C₂×C₄×C₁₇	—	2	4	no	—
111	C₂×C₄×C₁₇	—	2	2	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	2	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	2	no	—
116	C₂×C₄×C₁₇	—	2	4	no	—
117	C₂×C₄×C₁₇	—	2	2	no	—
118	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
127	#3622	—	2	2	no	—
128	#3622	—	2	1	no	—
129	#3622	—	2	1	no	—
130	#3622	—	2	1	no	—
131	#3622	—	2	2	no	—
132	#3622	H(5,𝔽₂)	1	1	yes	characteristic
133	#3622	2₋¹⁺⁴	1	1	yes	characteristic
134	C₈×#41	—	17	1	no	—
135	#2183	—	17	2	no	—
136	#2168	—	17	4	no	—
137	D₄×C₁₆	—	17	2	no	—
138	#2163	—	17	2	no	—
139	#2162	—	17	1	no	—
140	#2150	—	17	1	no	—
141	C₂×C₄⋊C₁₆	—	17	2	no	—
142	C₂²×C₁₇	—	2	2	no	—
143	C₂²×C₁₇	—	2	2	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	C₂×M₃₂	1	4	yes	—
148	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	2	no	—
151	C₄×C₁₇	—	2	2	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	2	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
156	C₄×C₁₇	#504	1	2	yes	—
157	C₄×C₁₇	#520	1	1	yes	characteristic
158	C₄×C₁₇	#568	1	1	yes	characteristic
159	C₄×#41	#595	1	1	yes	characteristic
160	C₂×M₃₂	—	17	2	no	—
161	C₂²×C₁₆	—	17	4	no	—
162	C₈×Q₈	—	17	1	no	—
163	C₈×Q₈	—	17	1	no	—
164	C₈×D₄	—	17	1	no	—
165	C₈×D₄	—	17	2	no	—
166	C₈×D₄	—	17	2	no	—
167	C₈×D₄	—	17	1	no	—
168	#431	—	17	1	no	—
169	#431	—	17	2	no	—
170	C₂×C₄×C₈	—	17	2	no	—
171	C₂×C₄×C₈	—	17	1	no	—
172	C₄⋊C₁₆	—	17	1	no	—
173	C₄⋊C₁₆	—	17	2	no	—
174	C₄⋊C₁₆	—	17	4	no	—
175	C₄⋊C₁₆	—	17	2	no	—
176	C₄⋊C₁₆	—	17	1	no	—
177	C₂²⋊C₁₆	—	17	2	no	—
178	C₂²⋊C₁₆	—	17	4	no	—
179	#346	—	17	4	no	—
180	C₄×C₁₆	—	17	2	no	—
181	C₂×C₁₇	—	2	4	no	—
182	C₂×C₁₇	—	4	1	no	—
183	C₂×C₁₇	#2988	1	1	yes	characteristic
184	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
185	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
186	C₂×#41	#3624	1	1	yes	characteristic
187	C₂²×C₈	—	17	2	no	—
188	C₂²×C₈	—	17	1	no	—
189	C₂²×C₈	—	17	2	no	—
190	C₂²×C₈	—	17	1	no	—
191	C₄×Q₈	#3624	1	1	yes	characteristic
192	C₄×Q₈	C₂×#595	1	1	yes	characteristic
193	C₄×D₄	#3624	1	2	yes	—
194	C₄×D₄	#3624	1	1	yes	characteristic
195	C₄×D₄	C₂×#595	1	2	yes	—
196	C₄×D₄	C₂×#595	1	1	yes	characteristic
197	#117	#3624	1	1	yes	characteristic
198	#117	#3624	1	2	yes	—
199	C₂×C₄²	C₂×#595	1	2	yes	—
200	C₂×C₄²	C₂×#595	1	1	yes	characteristic
201	M₃₂	—	68	2	no	—
202	C₂×C₁₆	—	17	2	no	—
203	C₂×C₁₆	—	17	2	no	—
204	C₂×C₁₆	—	17	4	no	—
205	C₂×C₁₆	—	68	4	no	—
206	C₄⋊C₈	—	17	2	no	—
207	C₄⋊C₈	—	17	1	no	—
208	C₄⋊C₈	—	17	1	no	—
209	C₄⋊C₈	—	17	2	no	—
210	C₂²⋊C₈	—	17	2	no	—
211	C₂²⋊C₈	—	17	2	no	—
212	C₂²⋊C₈	—	17	2	no	—
213	C₄×C₈	—	17	2	no	—
214	C₄×C₈	—	17	1	no	—
215	C₄×C₈	—	17	2	no	—
216	C₄×C₈	—	17	1	no	—
217	C₄×C₈	—	17	1	no	—
218	C₄×C₈	—	17	2	no	—
219	C₄×C₈	—	17	1	no	—
220	C₁₇	#33504	1	1	yes	characteristic, complemented, Sylow
221	#41	—	2	4	no	—
222	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
223	C₂×D₄	C₂×#3624	1	2	yes	—
224	C₂×D₄	C₂×#3624	1	1	yes	characteristic
225	C₂²×C₄	C₂×#3624	1	2	yes	—
226	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
227	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
228	C₂²×C₄	C₂×#3624	1	2	yes	—
229	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
230	C₂²×C₄	C₂²×#595	1	2	yes	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	—	17	1	no	—
233	C₂×C₈	—	17	2	no	—
234	C₂×C₈	—	17	1	no	—
235	C₂×C₈	—	17	1	no	—
236	C₂×C₈	—	17	2	no	—
237	C₂×C₈	—	34	2	no	—
238	C₂×C₈	—	34	1	no	—
239	C₂×C₈	—	34	2	no	—
240	C₂×C₈	—	34	1	no	—
241	C₂×C₈	—	68	2	no	—
242	C₂×C₈	—	68	1	no	—
243	C₄⋊C₄	C₂×#3624	1	2	yes	—
244	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
245	C₄⋊C₄	C₂×#3624	1	2	yes	—
246	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
247	C₂²⋊C₄	C₂×#3624	1	2	yes	—
248	C₂²⋊C₄	C₂×#3624	1	2	yes	—
249	C₂²⋊C₄	C₂×#3624	1	2	yes	—
250	C₄²	C₂×#3624	1	2	yes	—
251	C₄²	C₂×#3624	1	1	yes	characteristic
252	C₄²	C₂×#3624	1	2	yes	—
253	C₄²	C₂×#3624	1	1	yes	characteristic
254	C₄²	C₂²×#595	1	1	yes	characteristic
255	C₄²	C₂²×#595	1	2	yes	—
256	C₄²	C₂²×#595	1	1	yes	characteristic
257	C₁₆	—	34	2	no	—
258	C₁₆	—	34	4	no	—
259	C₁₆	—	34	2	no	—
260	C₂³	C₂²×#3624	1	1	yes	characteristic
261	C₂³	C₂²×#3624	1	2	yes	—
262	Q₈	—	2	2	no	—
263	D₄	—	2	4	no	—
264	D₄	#10587538	1	4	yes	complemented
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
274	C₂×C₄	C₂²×#3624	1	2	yes	—
275	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
276	C₂×C₄	C₂²×#3624	1	2	yes	—
277	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
278	C₂×C₄	C₂²×#3624	1	2	yes	—
279	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
280	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
281	C₂×C₄	C₂³×#595	1	1	yes	characteristic
282	C₈	—	17	1	no	—
283	C₈	—	17	1	no	—
284	C₈	—	34	2	no	—
285	C₈	—	34	1	no	—
286	C₈	—	34	1	no	—
287	C₈	—	34	1	no	—
288	C₈	—	34	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	C₂×#10587538	1	4	yes	—
295	C₂²	C₂³×#3624	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	C₂×#10587538	1	2	yes	—
303	C₄	#11763867	1	2	yes	—
304	C₄	#11763878	1	1	yes	characteristic
305	C₄	#11763888	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	—	4	1	no	—
308	C₂	#14545699	1	1	yes	characteristic
309	C₂	C₂²×#10587538	1	1	yes	characteristic
310	C₂	C₂×#11763867	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	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₄×#41	#595
hypocenter	C₁₇	#33504
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. #14545199	C₂	semidirect
2. #14545164	C₂	semidirect
3. D₄×#63216	C₂	semidirect
4. #14545145	C₂	semidirect
5. #14545141	C₂	semidirect
6. #14545116	C₂	semidirect
7. C₂×C₄⋊#63216	C₂	semidirect
109. D₄×C₁₇	M₃₂	semidirect
220. C₁₇	#33504	semidirect
264. D₄	#10587538	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
#14545164	2
D₄×#63216	2
#14545145	2
#14545141	2
#14545116	2
C₂×C₄⋊#63216	2
#41×#3622	2
#33504	17