Subgroups — C₂×#14530577 #17187378

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587469	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₁₇	C₂	1	2	yes	maximal
3	#14530577	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762639	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11762638	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×D₁₇	—	2	2	no	—
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂×#10587469	C₄	1	4	yes	—
10	C₂×#10587469	C₄	1	4	yes	—
11	C₂×#10587469	C₂²	1	4	yes	—
12	C₂²×#63214	C₂²	1	1	yes	characteristic
13	C₂×#10587490	C₂²	1	2	yes	—
14	C₂×#10587493	—	2	2	no	—
15	#108×D₁₇	—	2	4	no	—
16	#108×D₁₇	C₂²	1	8	yes	—
17	#11762639	C₂²	1	4	yes	complemented
18	#11762638	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₁₇×#108	C₂²	1	2	yes	—
21	C₁₇×#368	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
23	C₂×M₁₆×C₁₇	—	2	2	no	—
24	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×#63223	—	2	2	no	—
26	C₂²×#3622	D₄	1	1	yes	characteristic
27	M₁₆×D₁₇	—	2	8	no	—
28	M₁₆×D₁₇	—	4	4	no	—
29	C₂×#63218	—	4	2	no	—
30	C₂×C₈×D₁₇	—	4	2	no	—
31	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₁₇	D₄	1	2	yes	—
34	C₂×C₈×D₁₇	Q₈	1	2	yes	—
35	C₂×C₈×D₁₇	Q₈	1	2	yes	—
36	C₂×C₈×D₁₇	C₂³	1	2	yes	—
37	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
40	C₁₇×#108	—	2	2	no	—
41	C₁₇×#108	C₂³	1	4	yes	—
42	#10587493	—	2	8	no	—
43	#10587490	—	2	2	no	—
44	#10587490	C₂³	1	4	yes	—
45	C₂×#63214	C₂×C₄	1	2	yes	—
46	C₂×#63214	C₂×C₄	1	2	yes	—
47	C₂×#63214	C₂³	1	2	yes	—
48	#10587469	C₂×C₄	1	16	yes	—
49	C₂³×D₁₇	Q₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
52	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×D₁₇	D₈	1	2	yes	—
55	C₂×C₄×D₁₇	D₈	1	2	yes	—
56	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
57	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
58	M₁₆×C₁₇	—	2	4	no	—
59	M₁₆×C₁₇	—	4	2	no	—
60	C₂×C₈×C₁₇	—	4	2	no	—
61	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
67	#63223	—	2	4	no	—
68	#63223	—	4	2	no	—
69	C₂×#3622	—	4	2	no	—
70	C₂×#3622	C₄⋊C₄	1	2	yes	—
71	C₂×#3622	C₄⋊C₄	1	2	yes	—
72	C₂×#3622	C₂×D₄	1	2	yes	—
73	#63218	—	4	8	no	—
74	C₈×D₁₇	—	4	8	no	—
75	C₈×D₁₇	C₄⋊C₄	1	4	yes	—
76	C₈×D₁₇	C₄⋊C₄	1	4	yes	—
77	C₈×D₁₇	C₂²×C₄	1	4	yes	—
78	C₈×D₁₇	C₂×Q₈	1	4	yes	—
79	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
80	#63214	C₂²×C₄	1	4	yes	—
81	C₂×#2161	—	17	1	no	maximal, Sylow
82	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂²×D₁₇	—	4	2	no	—
84	C₂²×D₁₇	#107	1	2	yes	—
85	C₂²×D₁₇	#107	1	2	yes	—
86	C₂²×D₁₇	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×Dic₁₇	#107	1	2	yes	—
94	C₂×Dic₁₇	#107	1	2	yes	—
95	C₂×Dic₁₇	C₂×Q₁₆	1	2	yes	—
96	C₄×D₁₇	#107	1	4	yes	—
97	C₄×D₁₇	#107	1	4	yes	—
98	C₄×D₁₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₄×D₁₇	C₂×D₈	1	4	yes	—
100	C₈×C₁₇	—	4	4	no	—
101	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
105	#3622	—	4	4	no	—
106	#3622	C₂×C₄⋊C₄	1	4	yes	—
107	C₂²×M₃₂	—	17	1	no	—
108	C₂²×#108	—	17	2	no	—
109	#2161	—	17	8	no	—
110	C₂×#368	—	17	1	no	—
111	C₂×#368	—	17	2	no	—
112	C₂×#368	D₁₇	1	1	yes	characteristic, complemented
113	C₂²×C₁₇	—	2	2	no	—
114	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
115	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
116	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
117	C₂×D₁₇	—	2	4	no	—
118	C₂×D₁₇	—	4	2	no	—
119	C₂×D₁₇	—	4	2	no	—
120	C₂×D₁₇	C₂×#107	1	4	yes	—
121	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
122	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
123	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
125	Dic₁₇	C₂×#107	1	4	yes	—
126	C₂²×M₁₆	—	34	2	no	—
127	C₂³×C₈	—	17	1	no	—
128	C₂×M₃₂	—	17	4	no	—
129	C₂×M₃₂	—	17	4	no	—
130	C₂×M₃₂	—	17	4	no	—
131	C₂²×C₁₆	—	17	1	no	—
132	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
133	C₂×#108	—	17	2	no	—
134	C₂×#108	—	17	8	no	—
135	C₂×#108	—	34	4	no	—
136	C₂×#108	—	34	2	no	—
137	C₂×#108	C₂×D₁₇	1	2	yes	—
138	#368	—	17	4	no	—
139	#368	—	17	8	no	—
140	#368	C₂×D₁₇	1	4	yes	complemented
141	C₂×C₁₇	—	2	1	no	—
142	C₂×C₁₇	—	2	1	no	—
143	C₂×C₁₇	#2161	1	2	yes	complemented
144	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
145	D₁₇	—	2	4	no	—
146	C₂³×C₄	—	17	1	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	—	34	8	no	—
149	C₂×M₁₆	—	34	2	no	—
150	C₂×M₁₆	—	68	2	no	—
151	C₂×M₁₆	—	68	4	no	—
152	C₂²×C₈	—	17	1	no	—
153	C₂²×C₈	—	17	2	no	—
154	C₂²×C₈	—	17	2	no	—
155	C₂²×C₈	—	17	2	no	—
156	C₂²×C₈	—	17	2	no	—
157	C₂²×C₈	—	17	2	no	—
158	C₂²×C₈	—	17	2	no	—
159	C₂²×C₈	—	68	2	no	—
160	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
161	M₃₂	—	17	16	no	—
162	C₂×C₁₆	—	17	2	no	—
163	C₂×C₁₆	—	17	2	no	—
164	C₂×C₁₆	—	17	2	no	—
165	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
166	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
167	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
168	#108	—	2	2	no	—
169	#108	—	17	4	no	—
170	#108	—	34	8	no	—
171	#108	—	34	2	no	—
172	#108	C₂²×D₁₇	1	4	yes	—
173	C₁₇	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
174	C₂⁴	—	17	1	no	—
175	C₂²×C₄	—	17	1	no	—
176	C₂²×C₄	—	17	2	no	—
177	C₂²×C₄	—	17	2	no	—
178	C₂²×C₄	—	17	2	no	—
179	C₂²×C₄	—	17	2	no	—
180	C₂²×C₄	—	17	2	no	—
181	C₂²×C₄	—	17	2	no	—
182	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
183	M₁₆	—	2	4	no	—
184	M₁₆	—	4	2	no	—
185	M₁₆	—	34	4	no	—
186	M₁₆	—	68	8	no	—
187	M₁₆	—	68	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	17	2	no	—
190	C₂×C₈	—	17	2	no	—
191	C₂×C₈	—	17	2	no	—
192	C₂×C₈	—	17	4	no	—
193	C₂×C₈	—	17	4	no	—
194	C₂×C₈	—	17	4	no	—
195	C₂×C₈	—	17	4	no	—
196	C₂×C₈	—	68	8	no	—
197	C₂×C₈	—	68	2	no	—
198	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
200	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
202	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
204	C₁₆	—	17	4	no	—
205	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
206	C₂³	—	17	2	no	—
207	C₂³	—	17	2	no	—
208	C₂³	—	17	2	no	—
209	C₂³	—	68	2	no	—
210	C₂³	Q₁₆×D₁₇	1	1	yes	characteristic
211	C₂×C₄	—	17	2	no	—
212	C₂×C₄	—	17	2	no	—
213	C₂×C₄	—	17	2	no	—
214	C₂×C₄	—	17	4	no	—
215	C₂×C₄	—	17	4	no	—
216	C₂×C₄	—	17	4	no	—
217	C₂×C₄	—	17	4	no	—
218	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
219	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
220	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
221	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
222	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
223	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
224	C₈	—	4	4	no	—
225	C₈	—	17	4	no	—
226	C₈	—	68	4	no	—
227	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
228	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
229	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
230	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	17	4	no	—
233	C₂²	—	34	4	no	—
234	C₂²	—	68	2	no	—
235	C₂²	—	68	2	no	—
236	C₂²	#107×D₁₇	1	1	yes	characteristic
237	C₂²	#107×D₁₇	1	1	yes	characteristic
238	C₂²	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
239	C₄	—	17	4	no	—
240	C₄	#107×D₁₇	1	1	yes	characteristic
241	C₄	#107×D₁₇	1	1	yes	characteristic
242	C₄	C₂×C₄⋊C₄×D₁₇	1	1	yes	characteristic
243	C₄	C₂×D₈×D₁₇	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	—	34	4	no	—
247	C₂	#14530577	1	2	yes	direct product factor
248	C₂	C₂×#107×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₇
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#368	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×D₁₇	Q₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#368	D₁₇
hypocenter	C₁₇	C₂×#2161
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
3. #14530577	C₂	direct (N × H)
4. C₂×#11762639	C₂	semidirect
5. C₂×#11762638	C₂	semidirect
6. C₂×C₁₇×#368	C₂	semidirect
17. #11762639	C₂²	semidirect
18. #11762638	C₂²	semidirect
21. C₁₇×#368	C₂²	semidirect
112. C₂×#368	D₁₇	semidirect
140. #368	C₂×D₁₇	semidirect
143. C₂×C₁₇	#2161	semidirect
173. C₁₇	C₂×#2161	semidirect
247. C₂	#14530577	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#10587469	2
C₂×#108×D₁₇	2
#14530577	2
C₂×#11762639	2
C₂×#11762638	2
C₂×C₁₇×#368	2
C₂×#2161	17

Subgroups — C2×#14530577 #17187378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14530577 #17187378