Subgroups — C₂²×#11763283 #17940904

Order 4352.

14590 conjugacy classes of subgroups.

The subgroup lattice has 282 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
2	C₂×#11763283	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×D₆₈	C₂	1	4	yes	maximal, complemented
4	C₂²×#10587579	C₂	1	4	yes	maximal, complemented
5	C₂²×#10587612	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊C₄×D₁₇	C₂	1	4	yes	maximal, complemented
7	C₂²×C₁₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁵×D₁₇	C₂²	1	2	yes	—
9	C₂³×D₆₈	C₄	1	32	yes	complemented
10	C₂³×D₆₈	C₂²	1	2	yes	—
11	C₂³×D₆₈	C₂²	1	2	yes	—
12	C₂³×D₆₈	C₂²	1	24	yes	—
13	C₂³×C₄×D₁₇	C₂²	1	8	yes	complemented
14	#11763283	C₂²	1	256	yes	direct product factor
15	C₂×C₄×D₆₈	C₂²	1	96	yes	complemented
16	C₂×#10587579	C₂²	1	96	yes	complemented
17	C₂×#10587612	C₂²	1	24	yes	complemented
18	C₂²×C₄⋊Dic₁₇	C₂²	1	2	yes	complemented
19	C₂²×C₂²⋊Dic₁₇	C₂²	1	2	yes	—
20	C₂²×#63227	C₂²	1	8	yes	complemented
21	C₂×C₂²⋊C₄×D₁₇	C₂²	1	96	yes	complemented
22	C₂²×C₄×Dic₁₇	C₂²	1	2	yes	complemented
23	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×C₁₇×#117	C₂²	1	24	yes	complemented
25	C₂²×C₄⋊C₄×C₁₇	C₂²	1	2	yes	complemented
26	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
27	C₂²×C₄²×C₁₇	C₂²	1	2	yes	complemented
28	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
29	C₂⁴×D₁₇	—	4	16	no	—
30	C₂⁴×D₁₇	C₂×C₄	1	32	yes	complemented
31	C₂⁴×D₁₇	C₂³	1	4	yes	—
32	C₂⁴×D₁₇	C₂³	1	24	yes	—
33	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	complemented
34	C₂³×C₄×C₁₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₁₇	C₂³	1	12	yes	—
38	C₂³×Dic₁₇	C₂³	1	4	yes	—
39	C₂²×D₆₈	C₂×C₄	1	32	yes	complemented
40	C₂²×D₆₈	C₂×C₄	1	32	yes	complemented
41	C₂²×D₆₈	C₂×C₄	1	384	yes	complemented
42	C₂²×D₆₈	C₂³	1	24	yes	—
43	C₂²×D₆₈	C₂³	1	24	yes	—
44	C₂²×D₆₈	C₂³	1	64	yes	—
45	C₂²×C₄×D₁₇	—	4	32	no	—
46	C₂²×C₄×D₁₇	C₂³	1	96	yes	complemented
47	C₁₇×#117	C₂³	1	64	yes	complemented
48	C₂×C₄⋊C₄×C₁₇	C₂³	1	24	yes	complemented
49	C₂×C₂²⋊C₄×C₁₇	C₂³	1	24	yes	—
50	C₂×C₄²×C₁₇	C₂³	1	24	yes	complemented
51	C₂×C₂²⋊Dic₁₇	C₂³	1	24	yes	—
52	C₂×#63227	C₂³	1	96	yes	complemented
53	#10587612	C₂³	1	64	yes	complemented
54	C₂×C₄⋊Dic₁₇	C₂³	1	24	yes	complemented
55	C₂×C₄×Dic₁₇	C₂³	1	24	yes	complemented
56	#10587579	C₂³	1	256	yes	complemented
57	C₂²⋊C₄×D₁₇	C₂³	1	256	yes	complemented
58	C₄×D₆₈	C₂³	1	256	yes	complemented
59	C₂⁴×C₁₇	—	2	8	no	—
60	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
61	C₂⁴×C₁₇	C₂⁴	1	6	yes	—
62	C₂⁴×C₁₇	C₂⁴	1	1	yes	characteristic
63	C₂³×D₁₇	—	2	32	no	—
64	C₂³×D₁₇	—	4	128	no	—
65	C₂³×D₁₇	—	4	96	no	—
66	C₂³×D₁₇	C₂²×C₄	1	32	yes	—
67	C₂³×D₁₇	C₂²×C₄	1	192	yes	complemented
68	C₂³×D₁₇	C₂⁴	1	24	yes	—
69	C₂³×D₁₇	C₂⁴	1	32	yes	—
70	C₂²×C₄×C₁₇	—	2	16	no	—
71	C₂²×C₄×C₁₇	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₁₇	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₁₇	C₂²×C₄	1	96	yes	complemented
74	C₂²×C₄×C₁₇	C₂⁴	1	24	yes	—
75	C₂²×C₄×C₁₇	C₂⁴	1	6	yes	—
76	C₂²×C₄×C₁₇	C₂⁴	1	6	yes	—
77	C₂²×C₄×C₁₇	C₂⁴	1	16	yes	—
78	C₂²×Dic₁₇	—	2	16	no	—
79	C₂²×Dic₁₇	C₂⁴	1	24	yes	—
80	C₂×D₆₈	C₂²×C₄	1	192	yes	complemented
81	C₂×D₆₈	C₂²×C₄	1	192	yes	complemented
82	C₂×D₆₈	C₂²×C₄	1	512	yes	complemented
83	C₂×D₆₈	C₂⁴	1	32	yes	—
84	C₂×D₆₈	C₂⁴	1	32	yes	—
85	C₂×C₄×D₁₇	—	4	192	no	—
86	C₂×C₄×D₁₇	C₂⁴	1	128	yes	complemented
87	C₄⋊C₄×C₁₇	C₂⁴	1	32	yes	complemented
88	C₂²⋊C₄×C₁₇	C₂⁴	1	32	yes	—
89	C₄²×C₁₇	C₂⁴	1	32	yes	complemented
90	C₂²⋊Dic₁₇	C₂⁴	1	32	yes	—
91	#63227	C₂⁴	1	128	yes	complemented
92	C₄⋊Dic₁₇	C₂⁴	1	32	yes	complemented
93	C₄×Dic₁₇	C₂⁴	1	32	yes	complemented
94	C₂²×#518	—	17	1	no	maximal, Sylow
95	C₂³×C₁₇	—	2	32	no	—
96	C₂³×C₁₇	—	2	24	no	—
97	C₂³×C₁₇	C₂³×C₄	1	24	yes	—
98	C₂³×C₁₇	C₂³×C₄	1	4	yes	—
99	C₂³×C₁₇	H(5,𝔽₂)	1	8	yes	—
100	C₂³×C₁₇	C₂⁵	1	4	yes	—
101	C₂³×C₁₇	C₂⁵	1	3	yes	—
102	C₂²×D₁₇	—	2	128	no	—
103	C₂²×D₁₇	—	2	96	no	—
104	C₂²×D₁₇	—	4	384	no	—
105	C₂²×D₁₇	—	4	64	no	—
106	C₂²×D₁₇	C₂³×C₄	1	96	yes	—
107	C₂²×D₁₇	C₂³×C₄	1	128	yes	complemented
108	C₂²×D₁₇	C₂⁵	1	16	yes	—
109	C₂×C₄×C₁₇	—	2	48	no	—
110	C₂×C₄×C₁₇	C₂³×C₄	1	24	yes	—
111	C₂×C₄×C₁₇	C₂³×C₄	1	24	yes	—
112	C₂×C₄×C₁₇	C₂³×C₄	1	64	yes	complemented
113	C₂×C₄×C₁₇	C₂⁵	1	16	yes	—
114	C₂×C₄×C₁₇	C₂⁵	1	4	yes	—
115	C₂×C₄×C₁₇	C₂⁵	1	4	yes	—
116	C₂×Dic₁₇	—	2	48	no	—
117	C₂×Dic₁₇	C₂⁵	1	16	yes	—
118	D₆₈	C₂³×C₄	1	128	yes	complemented
119	D₆₈	C₂³×C₄	1	128	yes	complemented
120	C₄×D₁₇	—	4	128	no	—
121	C₂⁴×D₄	—	17	1	no	—
122	C₂×#518	—	17	48	no	—
123	C₂²×C₄×D₄	—	17	4	no	—
124	C₂²×C₄×D₄	—	17	4	no	—
125	C₂²×#117	—	17	1	no	—
126	C₂²×#117	D₁₇	1	1	yes	characteristic, complemented
127	C₂³×C₂²⋊C₄	—	17	4	no	—
128	C₂²×C₁₇	—	2	8	no	—
129	C₂²×C₁₇	—	2	48	no	—
130	C₂²×C₁₇	#518	1	16	yes	complemented
131	C₂²×C₁₇	C₂⁴×C₄	1	8	yes	—
132	C₂²×C₁₇	C₂⁴×C₄	1	6	yes	—
133	C₂²×C₁₇	C₂×H(5,𝔽₂)	1	12	yes	—
134	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
135	C₂×D₁₇	—	2	32	no	—
136	C₂×D₁₇	—	2	192	no	—
137	C₂×D₁₇	—	4	128	no	—
138	C₂×D₁₇	C₂⁴×C₄	1	32	yes	—
139	C₄×C₁₇	—	2	16	no	—
140	C₄×C₁₇	C₂⁴×C₄	1	8	yes	—
141	C₄×C₁₇	C₂⁴×C₄	1	8	yes	—
142	Dic₁₇	—	2	16	no	—
143	C₂⁶	—	17	2	no	—
144	C₂³×D₄	—	17	32	no	—
145	C₂³×D₄	—	17	2	no	—
146	C₂³×D₄	—	17	2	no	—
147	C₂³×D₄	—	17	24	no	—
148	C₂⁴×C₄	—	17	8	no	—
149	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
150	#518	—	17	256	no	—
151	C₂×C₄×D₄	—	17	96	no	—
152	C₂×C₄×D₄	—	17	96	no	—
153	C₂×#117	—	17	24	no	—
154	C₂×#117	C₂×D₁₇	1	24	yes	complemented
155	C₂²×C₄⋊C₄	—	17	2	no	—
156	C₂²×C₄⋊C₄	C₂×D₁₇	1	2	yes	complemented
157	C₂²×C₂²⋊C₄	—	17	2	no	—
158	C₂²×C₂²⋊C₄	—	17	8	no	—
159	C₂²×C₂²⋊C₄	—	17	96	no	—
160	C₂²×C₂²⋊C₄	C₂×D₁₇	1	2	yes	—
161	C₂²×C₄²	—	17	2	no	—
162	C₂²×C₄²	C₂×D₁₇	1	2	yes	complemented
163	C₂×C₁₇	—	2	8	no	—
164	C₂×C₁₇	C₂×#518	1	12	yes	complemented
165	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
166	C₂×C₁₇	C₂²×H(5,𝔽₂)	1	2	yes	—
167	D₁₇	—	2	32	no	—
168	C₂⁵	—	17	4	no	—
169	C₂⁵	—	17	32	no	—
170	C₂⁵	—	17	24	no	—
171	C₂⁵	—	68	16	no	—
172	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
173	C₂²×D₄	—	17	32	no	—
174	C₂²×D₄	—	17	32	no	—
175	C₂²×D₄	—	17	384	no	—
176	C₂²×D₄	—	17	24	no	—
177	C₂²×D₄	—	17	24	no	—
178	C₂²×D₄	—	17	64	no	—
179	C₂³×C₄	—	17	4	no	—
180	C₂³×C₄	—	17	96	no	—
181	C₂³×C₄	—	68	32	no	—
182	C₂³×C₄	C₄×D₁₇	1	16	yes	complemented
183	C₂³×C₄	C₂²×D₁₇	1	4	yes	—
184	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₁₇	1	12	yes	—
187	C₄×D₄	—	17	256	no	—
188	C₄×D₄	—	17	256	no	—
189	#117	—	17	64	no	—
190	#117	C₂²×D₁₇	1	64	yes	complemented
191	C₂×C₄⋊C₄	—	17	24	no	—
192	C₂×C₄⋊C₄	C₂²×D₁₇	1	24	yes	complemented
193	C₂×C₂²⋊C₄	—	17	24	no	—
194	C₂×C₂²⋊C₄	—	17	96	no	—
195	C₂×C₂²⋊C₄	—	17	256	no	—
196	C₂×C₂²⋊C₄	C₂²×D₁₇	1	24	yes	—
197	C₂×C₄²	—	17	24	no	—
198	C₂×C₄²	C₂²×D₁₇	1	24	yes	complemented
199	C₁₇	C₂²×#518	1	1	yes	characteristic, complemented, Sylow
200	C₂⁴	—	2	8	no	—
201	C₂⁴	—	17	32	no	—
202	C₂⁴	—	17	24	no	—
203	C₂⁴	—	17	192	no	—
204	C₂⁴	—	17	32	no	—
205	C₂⁴	—	34	32	no	—
206	C₂⁴	—	68	128	no	—
207	C₂⁴	—	68	96	no	—
208	C₂⁴	C₂×C₄×D₁₇	1	8	yes	—
209	C₂⁴	C₂³×D₁₇	1	6	yes	—
210	C₂⁴	C₂³×D₁₇	1	1	yes	characteristic
211	C₂×D₄	—	17	192	no	—
212	C₂×D₄	—	17	192	no	—
213	C₂×D₄	—	17	512	no	—
214	C₂×D₄	—	17	32	no	—
215	C₂×D₄	—	17	32	no	—
216	C₂²×C₄	—	2	16	no	—
217	C₂²×C₄	—	17	24	no	—
218	C₂²×C₄	—	17	128	no	—
219	C₂²×C₄	—	34	16	no	—
220	C₂²×C₄	—	68	192	no	—
221	C₂²×C₄	C₂×C₄×D₁₇	1	8	yes	—
222	C₂²×C₄	C₂×C₄×D₁₇	1	8	yes	—
223	C₂²×C₄	C₂×C₄×D₁₇	1	96	yes	complemented
224	C₂²×C₄	C₂³×D₁₇	1	24	yes	—
225	C₂²×C₄	C₂³×D₁₇	1	6	yes	—
226	C₂²×C₄	C₂³×D₁₇	1	6	yes	—
227	C₂²×C₄	C₂³×D₁₇	1	16	yes	—
228	C₄⋊C₄	—	17	32	no	—
229	C₄⋊C₄	C₂³×D₁₇	1	32	yes	complemented
230	C₂²⋊C₄	—	17	32	no	—
231	C₂²⋊C₄	—	17	128	no	—
232	C₂²⋊C₄	C₂³×D₁₇	1	32	yes	—
233	C₄²	—	17	32	no	—
234	C₄²	C₂³×D₁₇	1	32	yes	complemented
235	C₂³	—	2	32	no	—
236	C₂³	—	2	24	no	—
237	C₂³	—	17	96	no	—
238	C₂³	—	17	16	no	—
239	C₂³	—	17	128	no	—
240	C₂³	—	34	128	no	—
241	C₂³	—	34	96	no	—
242	C₂³	—	68	384	no	—
243	C₂³	—	68	64	no	—
244	C₂³	C₂²×C₄×D₁₇	1	24	yes	—
245	C₂³	C₂²×C₄×D₁₇	1	4	yes	—
246	C₂³	#10587696	1	8	yes	—
247	C₂³	C₂⁴×D₁₇	1	4	yes	—
248	C₂³	C₂⁴×D₁₇	1	3	yes	—
249	D₄	—	17	128	no	—
250	D₄	—	17	128	no	—
251	C₂×C₄	—	2	48	no	—
252	C₂×C₄	—	17	16	no	—
253	C₂×C₄	—	34	48	no	—
254	C₂×C₄	—	68	128	no	—
255	C₂×C₄	C₂²×C₄×D₁₇	1	24	yes	—
256	C₂×C₄	C₂²×C₄×D₁₇	1	24	yes	—
257	C₂×C₄	C₂²×C₄×D₁₇	1	64	yes	complemented
258	C₂×C₄	C₂⁴×D₁₇	1	16	yes	—
259	C₂×C₄	C₂⁴×D₁₇	1	4	yes	—
260	C₂×C₄	C₂⁴×D₁₇	1	4	yes	—
261	C₂²	—	2	48	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	17	32	no	—
264	C₂²	—	34	192	no	—
265	C₂²	—	34	32	no	—
266	C₂²	—	68	128	no	—
267	C₂²	#11763283	1	16	yes	direct product factor
268	C₂²	C₂³×C₄×D₁₇	1	8	yes	—
269	C₂²	C₂³×C₄×D₁₇	1	6	yes	—
270	C₂²	C₂×#10587696	1	12	yes	—
271	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
272	C₄	—	2	16	no	—
273	C₄	—	34	16	no	—
274	C₄	C₂³×C₄×D₁₇	1	8	yes	—
275	C₄	C₂³×C₄×D₁₇	1	8	yes	—
276	C₂	—	2	8	no	—
277	C₂	—	34	32	no	—
278	C₂	C₂×#11763283	1	12	yes	direct product factor
279	C₂	C₂⁴×C₄×D₁₇	1	1	yes	characteristic
280	C₂	C₂²×#10587696	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂²×C₁₇×#117	C₂
socle	C₂⁴×C₁₇	C₂⁴
prime radical	C₂⁴×D₆₈	C₂
cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂²×#117	D₁₇
hypocenter	C₁₇	C₂²×#518
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#11763283	C₂	direct (N × H)
3. C₂²×C₄×D₆₈	C₂	semidirect
4. C₂²×#10587579	C₂	semidirect
5. C₂²×#10587612	C₂	semidirect
6. C₂²×C₂²⋊C₄×D₁₇	C₂	semidirect
7. C₂²×C₁₇×#117	C₂	semidirect
9. C₂³×D₆₈	C₄	semidirect
13. C₂³×C₄×D₁₇	C₂²	semidirect
14. #11763283	C₂²	direct (N × H)
15. C₂×C₄×D₆₈	C₂²	semidirect
16. C₂×#10587579	C₂²	semidirect
17. C₂×#10587612	C₂²	semidirect
18. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
20. C₂²×#63227	C₂²	semidirect
21. C₂×C₂²⋊C₄×D₁₇	C₂²	semidirect
22. C₂²×C₄×Dic₁₇	C₂²	semidirect
24. C₂×C₁₇×#117	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₁₇	C₂²	semidirect
27. C₂²×C₄²×C₁₇	C₂²	semidirect
30. C₂⁴×D₁₇	C₂×C₄	semidirect
33. C₂³×C₄×C₁₇	C₂×C₄	semidirect
39. C₂²×D₆₈	C₂×C₄	semidirect
40. C₂²×D₆₈	C₂×C₄	semidirect
41. C₂²×D₆₈	C₂×C₄	semidirect
46. C₂²×C₄×D₁₇	C₂³	semidirect
47. C₁₇×#117	C₂³	semidirect
48. C₂×C₄⋊C₄×C₁₇	C₂³	semidirect
50. C₂×C₄²×C₁₇	C₂³	semidirect
52. C₂×#63227	C₂³	semidirect
53. #10587612	C₂³	semidirect
54. C₂×C₄⋊Dic₁₇	C₂³	semidirect
55. C₂×C₄×Dic₁₇	C₂³	semidirect
56. #10587579	C₂³	semidirect
57. C₂²⋊C₄×D₁₇	C₂³	semidirect
58. C₄×D₆₈	C₂³	semidirect
67. C₂³×D₁₇	C₂²×C₄	semidirect
73. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
80. C₂×D₆₈	C₂²×C₄	semidirect
81. C₂×D₆₈	C₂²×C₄	semidirect
82. C₂×D₆₈	C₂²×C₄	semidirect
86. C₂×C₄×D₁₇	C₂⁴	semidirect
87. C₄⋊C₄×C₁₇	C₂⁴	semidirect
89. C₄²×C₁₇	C₂⁴	semidirect
91. #63227	C₂⁴	semidirect
92. C₄⋊Dic₁₇	C₂⁴	semidirect
93. C₄×Dic₁₇	C₂⁴	semidirect
107. C₂²×D₁₇	C₂³×C₄	semidirect
112. C₂×C₄×C₁₇	C₂³×C₄	semidirect
118. D₆₈	C₂³×C₄	semidirect
119. D₆₈	C₂³×C₄	semidirect
126. C₂²×#117	D₁₇	semidirect
130. C₂²×C₁₇	#518	semidirect
154. C₂×#117	C₂×D₁₇	semidirect
156. C₂²×C₄⋊C₄	C₂×D₁₇	semidirect
162. C₂²×C₄²	C₂×D₁₇	semidirect
164. C₂×C₁₇	C₂×#518	semidirect
182. C₂³×C₄	C₄×D₁₇	semidirect
190. #117	C₂²×D₁₇	semidirect
192. C₂×C₄⋊C₄	C₂²×D₁₇	semidirect
198. C₂×C₄²	C₂²×D₁₇	semidirect
199. C₁₇	C₂²×#518	semidirect
223. C₂²×C₄	C₂×C₄×D₁₇	semidirect
229. C₄⋊C₄	C₂³×D₁₇	semidirect
234. C₄²	C₂³×D₁₇	semidirect
257. C₂×C₄	C₂²×C₄×D₁₇	semidirect
267. C₂²	#11763283	direct (N × H)
278. C₂	C₂×#11763283	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
C₂×#11763283	2
C₂²×C₄×D₆₈	2
C₂²×#10587579	2
C₂²×#10587612	2
C₂²×C₂²⋊C₄×D₁₇	2
C₂²×C₁₇×#117	2
C₂²×#518	17

Subgroups — C22×#11763283 #17940904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11763283 #17940904