Subgroups — C₂³×#10587570 #17940960

Order 4352.

19998 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#10587570	C₂	1	56	yes	maximal, direct product factor
6	C₂³×#63225	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#63227	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
10	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
11	C₂³×D₆₈	—	2	16	no	—
12	C₂³×D₆₈	C₂²	1	28	yes	complemented
13	C₂³×C₂²⋊D₁₇	—	2	16	no	—
14	C₂³×C₂²⋊D₁₇	—	2	16	no	—
15	C₂³×C₂²⋊D₁₇	C₂²	1	28	yes	complemented
16	C₂³×C₂²⋊D₁₇	C₂²	1	28	yes	complemented
17	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂³×C₄×D₁₇	—	2	16	no	—
20	C₂³×C₄×D₁₇	C₂²	1	28	yes	complemented
21	C₂×#10587570	C₂²	1	448	yes	direct product factor
22	C₂²×#63225	C₂²	1	28	yes	complemented
23	C₂²×#63227	C₂²	1	28	yes	complemented
24	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic
25	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
26	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	28	yes	complemented
28	C₂⁵×C₁₇	—	2	8	no	—
29	C₂⁵×C₁₇	—	2	8	no	—
30	C₂⁵×C₁₇	—	2	8	no	—
31	C₂⁵×C₁₇	C₂³	1	14	yes	—
32	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
33	C₂⁴×D₁₇	—	2	8	no	—
34	C₂⁴×D₁₇	—	2	8	no	—
35	C₂⁴×D₁₇	—	2	8	no	—
36	C₂⁴×D₁₇	—	2	8	no	—
37	C₂⁴×D₁₇	—	2	8	no	—
38	C₂⁴×D₁₇	D₄	1	16	yes	complemented
39	C₂⁴×D₁₇	C₂³	1	14	yes	—
40	C₂⁴×D₁₇	C₂³	1	14	yes	—
41	C₂³×C₄×C₁₇	—	2	8	no	—
42	C₂³×C₄×C₁₇	—	2	8	no	—
43	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
44	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
45	C₂²×C₂²⋊D₁₇	—	2	224	no	—
46	C₂²×C₂²⋊D₁₇	—	2	224	no	—
47	C₂²×C₂²⋊D₁₇	C₂³	1	112	yes	complemented
48	C₂²×C₂²⋊D₁₇	C₂³	1	112	yes	complemented
49	C₂³×Dic₁₇	—	2	8	no	—
50	C₂³×Dic₁₇	D₄	1	16	yes	complemented
51	C₂³×Dic₁₇	C₂³	1	14	yes	—
52	C₂³×Dic₁₇	C₂³	1	14	yes	—
53	C₂²×D₆₈	—	2	224	no	—
54	C₂²×D₆₈	C₂³	1	112	yes	complemented
55	C₂²×C₄×D₁₇	—	2	224	no	—
56	C₂²×C₄×D₁₇	C₂³	1	112	yes	complemented
57	C₂×C₂²⋊C₄×C₁₇	C₂³	1	112	yes	complemented
58	C₂×#63227	C₂³	1	112	yes	complemented
59	C₂×#63225	C₂³	1	112	yes	complemented
60	#10587570	C₂³	1	512	yes	direct product factor
61	C₂⁴×C₁₇	—	2	56	no	—
62	C₂⁴×C₁₇	—	2	56	no	—
63	C₂⁴×C₁₇	—	2	56	no	—
64	C₂⁴×C₁₇	—	4	64	no	—
65	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
66	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
67	C₂⁴×C₁₇	#41	1	8	yes	—
68	C₂⁴×C₁₇	C₂⁴	1	28	yes	—
69	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
70	C₂³×D₁₇	—	2	128	no	—
71	C₂³×D₁₇	—	2	56	no	—
72	C₂³×D₁₇	—	2	56	no	—
73	C₂³×D₁₇	—	2	56	no	—
74	C₂³×D₁₇	—	2	56	no	—
75	C₂³×D₁₇	—	2	56	no	—
76	C₂³×D₁₇	—	4	64	no	—
77	C₂³×D₁₇	C₂×D₄	1	112	yes	complemented
78	C₂³×D₁₇	C₂⁴	1	28	yes	—
79	C₂³×D₁₇	C₂⁴	1	28	yes	—
80	C₂²×C₄×C₁₇	—	2	56	no	—
81	C₂²×C₄×C₁₇	—	2	56	no	—
82	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
83	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
84	C₂×C₂²⋊D₁₇	—	2	448	no	—
85	C₂×C₂²⋊D₁₇	—	2	448	no	—
86	C₂×C₂²⋊D₁₇	C₂⁴	1	64	yes	complemented
87	C₂×C₂²⋊D₁₇	C₂⁴	1	64	yes	complemented
88	C₂²×Dic₁₇	—	2	56	no	—
89	C₂²×Dic₁₇	C₂×D₄	1	112	yes	complemented
90	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
91	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
92	C₂×D₆₈	—	2	448	no	—
93	C₂×D₆₈	C₂⁴	1	64	yes	complemented
94	C₂×C₄×D₁₇	—	2	448	no	—
95	C₂×C₄×D₁₇	C₂⁴	1	64	yes	complemented
96	C₂²⋊C₄×C₁₇	C₂⁴	1	64	yes	complemented
97	#63227	C₂⁴	1	64	yes	complemented
98	#63225	C₂⁴	1	64	yes	complemented
99	C₂³×#121	—	17	1	no	maximal, Sylow
100	C₂³×C₁₇	—	2	56	no	—
101	C₂³×C₁₇	—	2	56	no	—
102	C₂³×C₁₇	—	2	56	no	—
103	C₂³×C₁₇	—	4	224	no	—
104	C₂³×C₁₇	#121	1	64	yes	complemented
105	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
106	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
107	C₂³×C₁₇	C₂×#41	1	28	yes	—
108	C₂³×C₁₇	C₂⁵	1	8	yes	—
109	C₂³×C₁₇	C₂⁵	1	7	yes	—
110	C₂²×D₁₇	—	2	448	no	—
111	C₂²×D₁₇	—	2	56	no	—
112	C₂²×D₁₇	—	2	56	no	—
113	C₂²×D₁₇	—	2	56	no	—
114	C₂²×D₁₇	—	2	56	no	—
115	C₂²×D₁₇	—	2	56	no	—
116	C₂²×D₁₇	—	4	224	no	—
117	C₂²×D₁₇	C₂²×D₄	1	112	yes	complemented
118	C₂²×D₁₇	C₂⁵	1	8	yes	—
119	C₂²×D₁₇	C₂⁵	1	8	yes	—
120	C₂×C₄×C₁₇	—	2	56	no	—
121	C₂×C₄×C₁₇	—	2	56	no	—
122	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
123	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
124	C₂²⋊D₁₇	—	2	128	no	—
125	C₂²⋊D₁₇	—	2	128	no	—
126	C₂×Dic₁₇	—	2	56	no	—
127	C₂×Dic₁₇	C₂²×D₄	1	112	yes	complemented
128	C₂×Dic₁₇	C₂⁵	1	8	yes	—
129	C₂×Dic₁₇	C₂⁵	1	8	yes	—
130	D₆₈	—	2	128	no	—
131	C₄×D₁₇	—	2	128	no	—
132	C₂⁴×D₄	—	17	1	no	—
133	C₂⁴×D₄	—	17	1	no	—
134	C₂⁴×D₄	—	17	1	no	—
135	C₂⁵×C₄	—	17	1	no	—
136	C₂²×#121	—	17	56	no	—
137	C₂³×C₄⋊C₄	—	17	1	no	—
138	C₂³×C₂²⋊C₄	—	17	1	no	—
139	C₂³×C₂²⋊C₄	D₁₇	1	1	yes	characteristic, complemented
140	C₂²×C₁₇	—	2	8	no	—
141	C₂²×C₁₇	—	2	8	no	—
142	C₂²×C₁₇	—	2	8	no	—
143	C₂²×C₁₇	—	4	112	no	—
144	C₂²×C₁₇	C₂×#121	1	112	yes	complemented
145	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
146	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
147	C₂²×C₁₇	C₂²×#41	1	14	yes	—
148	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
149	C₂×D₁₇	—	2	8	no	—
150	C₂×D₁₇	—	2	8	no	—
151	C₂×D₁₇	—	2	224	no	—
152	C₂×D₁₇	—	2	8	no	—
153	C₂×D₁₇	—	2	8	no	—
154	C₂×D₁₇	—	2	8	no	—
155	C₂×D₁₇	—	4	112	no	—
156	C₂×D₁₇	C₂³×D₄	1	16	yes	complemented
157	C₄×C₁₇	—	2	8	no	—
158	C₄×C₁₇	—	2	8	no	—
159	Dic₁₇	—	2	8	no	—
160	Dic₁₇	C₂³×D₄	1	16	yes	complemented
161	C₂⁶	—	17	1	no	—
162	C₂⁶	—	17	1	no	—
163	C₂⁶	C₂×D₁₇	1	1	yes	characteristic
164	C₂³×D₄	—	17	28	no	—
165	C₂³×D₄	—	17	28	no	—
166	C₂³×D₄	—	17	28	no	—
167	C₂³×D₄	—	34	16	no	—
168	C₂³×D₄	—	34	16	no	—
169	C₂³×D₄	—	34	16	no	—
170	C₂⁴×C₄	—	17	1	no	—
171	C₂⁴×C₄	—	17	1	no	—
172	C₂⁴×C₄	—	17	28	no	—
173	C₂⁴×C₄	—	34	16	no	—
174	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
175	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
176	C₂×#121	—	17	448	no	—
177	C₂²×C₄⋊C₄	—	17	28	no	—
178	C₂²×C₂²⋊C₄	—	17	28	no	—
179	C₂²×C₂²⋊C₄	C₂×D₁₇	1	28	yes	complemented
180	C₂×C₁₇	—	4	8	no	—
181	C₂×C₁₇	C₂²×#121	1	28	yes	complemented
182	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
183	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
184	C₂×C₁₇	C₂³×#41	1	1	yes	characteristic
185	D₁₇	—	2	16	no	—
186	D₁₇	—	4	8	no	—
187	C₂⁵	—	2	8	no	—
188	C₂⁵	—	2	8	no	—
189	C₂⁵	—	2	8	no	—
190	C₂⁵	—	17	16	no	—
191	C₂⁵	—	17	14	no	—
192	C₂⁵	—	17	14	no	—
193	C₂⁵	—	34	8	no	—
194	C₂⁵	—	34	8	no	—
195	C₂⁵	—	34	8	no	—
196	C₂⁵	—	34	8	no	—
197	C₂⁵	—	34	8	no	—
198	C₂⁵	C₂²×D₁₇	1	14	yes	—
199	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
200	C₂²×D₄	—	17	112	no	—
201	C₂²×D₄	—	17	112	no	—
202	C₂²×D₄	—	17	112	no	—
203	C₂²×D₄	—	34	224	no	—
204	C₂²×D₄	—	34	224	no	—
205	C₂²×D₄	—	34	224	no	—
206	C₂³×C₄	—	2	8	no	—
207	C₂³×C₄	—	2	8	no	—
208	C₂³×C₄	—	17	16	no	—
209	C₂³×C₄	—	17	14	no	—
210	C₂³×C₄	—	17	14	no	—
211	C₂³×C₄	—	17	112	no	—
212	C₂³×C₄	—	34	224	no	—
213	C₂³×C₄	—	34	8	no	—
214	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
215	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
216	#121	—	17	512	no	—
217	C₂×C₄⋊C₄	—	17	112	no	—
218	C₂×C₂²⋊C₄	—	17	112	no	—
219	C₂×C₂²⋊C₄	C₂²×D₁₇	1	112	yes	complemented
220	C₁₇	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
221	C₂⁴	—	2	56	no	—
222	C₂⁴	—	2	56	no	—
223	C₂⁴	—	2	56	no	—
224	C₂⁴	—	4	64	no	—
225	C₂⁴	—	17	112	no	—
226	C₂⁴	—	17	28	no	—
227	C₂⁴	—	17	28	no	—
228	C₂⁴	—	34	56	no	—
229	C₂⁴	—	34	56	no	—
230	C₂⁴	—	34	128	no	—
231	C₂⁴	—	34	56	no	—
232	C₂⁴	—	34	56	no	—
233	C₂⁴	—	34	56	no	—
234	C₂⁴	—	68	64	no	—
235	C₂⁴	#63252	1	8	yes	—
236	C₂⁴	D₄×D₁₇	1	8	yes	—
237	C₂⁴	D₄×D₁₇	1	8	yes	—
238	C₂⁴	C₂³×D₁₇	1	28	yes	—
239	C₂⁴	C₂³×D₁₇	1	7	yes	—
240	C₂×D₄	—	17	64	no	—
241	C₂×D₄	—	17	64	no	—
242	C₂×D₄	—	17	64	no	—
243	C₂×D₄	—	34	448	no	—
244	C₂×D₄	—	34	448	no	—
245	C₂×D₄	—	34	448	no	—
246	C₂²×C₄	—	2	56	no	—
247	C₂²×C₄	—	2	56	no	—
248	C₂²×C₄	—	17	112	no	—
249	C₂²×C₄	—	17	28	no	—
250	C₂²×C₄	—	17	28	no	—
251	C₂²×C₄	—	17	64	no	—
252	C₂²×C₄	—	34	448	no	—
253	C₂²×C₄	—	34	56	no	—
254	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
255	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
256	C₄⋊C₄	—	17	64	no	—
257	C₂²⋊C₄	—	17	64	no	—
258	C₂²⋊C₄	C₂³×D₁₇	1	64	yes	complemented
259	C₂³	—	2	56	no	—
260	C₂³	—	2	56	no	—
261	C₂³	—	2	56	no	—
262	C₂³	—	4	224	no	—
263	C₂³	—	17	112	no	—
264	C₂³	—	17	8	no	—
265	C₂³	—	17	8	no	—
266	C₂³	—	34	448	no	—
267	C₂³	—	34	56	no	—
268	C₂³	—	34	56	no	—
269	C₂³	—	34	56	no	—
270	C₂³	—	34	56	no	—
271	C₂³	—	34	56	no	—
272	C₂³	—	68	224	no	—
273	C₂³	#10587570	1	64	yes	direct product factor
274	C₂³	C₂×#63252	1	28	yes	—
275	C₂³	C₂×D₄×D₁₇	1	28	yes	—
276	C₂³	C₂×D₄×D₁₇	1	28	yes	—
277	C₂³	C₂⁴×D₁₇	1	8	yes	—
278	C₂³	C₂⁴×D₁₇	1	7	yes	—
279	D₄	—	34	128	no	—
280	D₄	—	34	128	no	—
281	D₄	—	34	128	no	—
282	C₂×C₄	—	2	56	no	—
283	C₂×C₄	—	2	56	no	—
284	C₂×C₄	—	17	112	no	—
285	C₂×C₄	—	17	8	no	—
286	C₂×C₄	—	17	8	no	—
287	C₂×C₄	—	34	128	no	—
288	C₂×C₄	—	34	56	no	—
289	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
290	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	8	no	—
294	C₂²	—	4	112	no	—
295	C₂²	—	17	16	no	—
296	C₂²	—	34	224	no	—
297	C₂²	—	34	8	no	—
298	C₂²	—	34	8	no	—
299	C₂²	—	34	8	no	—
300	C₂²	—	34	8	no	—
301	C₂²	—	34	8	no	—
302	C₂²	—	68	112	no	—
303	C₂²	C₂×#10587570	1	112	yes	direct product factor
304	C₂²	C₂²×D₄×D₁₇	1	14	yes	—
305	C₂²	C₂²×D₄×D₁₇	1	14	yes	—
306	C₂²	C₂²×#63252	1	14	yes	—
307	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
308	C₄	—	2	8	no	—
309	C₄	—	2	8	no	—
310	C₄	—	17	16	no	—
311	C₄	—	34	8	no	—
312	C₂	—	4	8	no	—
313	C₂	—	34	16	no	—
314	C₂	—	68	8	no	—
315	C₂	C₂²×#10587570	1	28	yes	direct product factor
316	C₂	C₂³×D₄×D₁₇	1	1	yes	characteristic
317	C₂	C₂³×D₄×D₁₇	1	1	yes	characteristic
318	C₂	C₂³×#63252	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₇
commutator, cosocle	C₂²×C₁₇	C₂⁶
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂³×C₂²⋊C₄×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
hypercenter	C₂³×C₂²⋊C₄	D₁₇
hypocenter	C₁₇	C₂³×#121
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₆₈	C₂	semidirect
2. C₂⁴×C₂²⋊D₁₇	C₂	semidirect
3. C₂⁴×C₂²⋊D₁₇	C₂	semidirect
4. C₂⁴×C₄×D₁₇	C₂	semidirect
5. C₂²×#10587570	C₂	direct (N × H)
6. C₂³×#63225	C₂	semidirect
7. C₂³×#63227	C₂	semidirect
8. C₂³×C₂²⋊C₄×C₁₇	C₂	semidirect
12. C₂³×D₆₈	C₂²	semidirect
15. C₂³×C₂²⋊D₁₇	C₂²	semidirect
16. C₂³×C₂²⋊D₁₇	C₂²	semidirect
20. C₂³×C₄×D₁₇	C₂²	semidirect
21. C₂×#10587570	C₂²	direct (N × H)
22. C₂²×#63225	C₂²	semidirect
23. C₂²×#63227	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₁₇	C₂²	semidirect
38. C₂⁴×D₁₇	D₄	semidirect
47. C₂²×C₂²⋊D₁₇	C₂³	semidirect
48. C₂²×C₂²⋊D₁₇	C₂³	semidirect
50. C₂³×Dic₁₇	D₄	semidirect
54. C₂²×D₆₈	C₂³	semidirect
56. C₂²×C₄×D₁₇	C₂³	semidirect
57. C₂×C₂²⋊C₄×C₁₇	C₂³	semidirect
58. C₂×#63227	C₂³	semidirect
59. C₂×#63225	C₂³	semidirect
60. #10587570	C₂³	direct (N × H)
77. C₂³×D₁₇	C₂×D₄	semidirect
86. C₂×C₂²⋊D₁₇	C₂⁴	semidirect
87. C₂×C₂²⋊D₁₇	C₂⁴	semidirect
89. C₂²×Dic₁₇	C₂×D₄	semidirect
93. C₂×D₆₈	C₂⁴	semidirect
95. C₂×C₄×D₁₇	C₂⁴	semidirect
96. C₂²⋊C₄×C₁₇	C₂⁴	semidirect
97. #63227	C₂⁴	semidirect
98. #63225	C₂⁴	semidirect
104. C₂³×C₁₇	#121	semidirect
117. C₂²×D₁₇	C₂²×D₄	semidirect
127. C₂×Dic₁₇	C₂²×D₄	semidirect
139. C₂³×C₂²⋊C₄	D₁₇	semidirect
144. C₂²×C₁₇	C₂×#121	semidirect
156. C₂×D₁₇	C₂³×D₄	semidirect
160. Dic₁₇	C₂³×D₄	semidirect
179. C₂²×C₂²⋊C₄	C₂×D₁₇	semidirect
181. C₂×C₁₇	C₂²×#121	semidirect
219. C₂×C₂²⋊C₄	C₂²×D₁₇	semidirect
220. C₁₇	C₂³×#121	semidirect
258. C₂²⋊C₄	C₂³×D₁₇	semidirect
273. C₂³	#10587570	direct (N × H)
303. C₂²	C₂×#10587570	direct (N × H)
315. C₂	C₂²×#10587570	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₆₈	2
C₂⁴×C₂²⋊D₁₇	2
C₂⁴×C₄×D₁₇	2
C₂²×#10587570	2
C₂³×#63225	2
C₂³×#63227	2
C₂³×C₂²⋊C₄×C₁₇	2
C₂³×#121	17

Subgroups — C23×#10587570 #17940960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#10587570 #17940960