Subgroups — #16952913

Order 4352.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	#14530540	C₂	1	1	yes	maximal, characteristic
3	#14526894	C₂	1	4	yes	maximal
4	C₄×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587555	—	2	1	no	—
6	C₂×#10587483	C₂²	1	2	yes	—
7	#11763214	—	2	1	no	—
8	#11763214	—	2	1	no	—
9	#11763214	C₄	1	2	yes	—
10	#11763214	C₄	1	2	yes	—
11	C₂×#10587489	—	2	1	no	—
12	C₂×#10587489	C₂²	1	2	yes	—
13	C₄⋊#63214	—	2	1	no	—
14	C₄⋊#63214	—	2	1	no	—
15	C₂×C₁₇×#108	C₂²	1	2	yes	—
16	C₄×M₁₆×C₁₇	—	2	1	no	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#341	—	2	1	no	—
19	C₂²×Dic₃₄	—	4	1	no	—
20	C₂×M₁₆×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	—	4	1	no	—
26	#10587555	—	2	2	no	—
27	#10587555	—	2	2	no	—
28	#10587555	—	4	1	no	—
29	#10587555	—	4	1	no	—
30	C₁₇×#108	—	2	2	no	—
31	C₁₇×#108	Q₈	1	4	yes	complemented
32	C₁₇×#97	—	4	1	no	—
33	C₄×C₈×C₁₇	—	4	1	no	—
34	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
38	#10587489	—	2	2	no	—
39	#10587489	—	2	2	no	—
40	#10587489	—	2	2	no	—
41	#10587489	C₂×C₄	1	4	yes	—
42	#10587483	—	2	4	no	—
43	C₂×#63214	—	2	2	no	—
44	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	—	4	2	no	—
48	C₂×Dic₃₄	—	4	2	no	—
49	C₂×Dic₃₄	—	4	2	no	—
50	C₂×Dic₃₄	—	8	4	no	—
51	M₁₆×C₁₇	—	2	4	no	—
52	M₁₆×C₁₇	—	4	2	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	4	2	no	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₇	#41	1	2	yes	—
61	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₇	D₈	1	1	yes	characteristic
64	C₄²×C₁₇	D₈	1	1	yes	characteristic
65	C₄⋊Dic₁₇	—	2	4	no	—
66	C₄⋊Dic₁₇	—	4	2	no	—
67	C₄⋊Dic₁₇	—	4	2	no	—
68	C₄⋊Dic₁₇	—	4	2	no	—
69	#63214	—	4	2	no	—
70	#13279	—	17	1	no	maximal, Sylow
71	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#137	1	2	yes	—
82	C₂×Dic₁₇	—	4	4	no	—
83	C₂×Dic₁₇	—	8	4	no	—
84	Dic₃₄	—	8	4	no	—
85	Dic₃₄	—	8	4	no	—
86	C₈×C₁₇	—	2	1	no	—
87	C₈×C₁₇	—	2	1	no	—
88	C₈×C₁₇	—	4	4	no	—
89	C₈×C₁₇	C₄×Q₈	1	2	yes	—
90	C₈×C₁₇	#124	1	1	yes	characteristic
91	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×#500	—	17	1	no	—
93	#2150	—	17	1	no	—
94	C₄×#108	D₁₇	1	1	yes	characteristic
95	#1387	—	17	4	no	—
96	C₂²×C₁₇	—	2	1	no	—
97	C₂²×C₁₇	—	2	1	no	—
98	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
99	C₂²×C₁₇	#475	1	2	yes	—
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	—	2	1	no	—
102	C₄×C₁₇	#361	1	2	yes	—
103	C₄×C₁₇	#362	1	2	yes	—
104	C₄×C₁₇	#391	1	1	yes	characteristic
105	C₄×C₁₇	#419	1	1	yes	characteristic
106	C₄×C₁₇	#474	1	2	yes	—
107	Dic₁₇	—	8	4	no	—
108	C₂×C₄⋊Q₈	—	34	1	no	—
109	C₂×M₃₂	—	17	2	no	—
110	#500	—	17	2	no	—
111	#500	—	17	2	no	—
112	#500	—	34	1	no	—
113	#500	—	34	1	no	—
114	C₂×#108	C₂×D₁₇	1	2	yes	—
115	C₂×#107	—	17	2	no	—
116	C₂×#107	—	34	1	no	—
117	C₄×M₁₆	—	2	1	no	—
118	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
119	C₄⋊C₁₆	—	34	1	no	—
120	C₄⋊C₁₆	—	34	1	no	—
121	#341	—	2	1	no	—
122	C₂×C₁₇	—	2	2	no	—
123	C₂×C₁₇	#1988	1	1	yes	characteristic
124	C₂×C₁₇	C₂×#361	1	1	yes	characteristic
125	C₂×C₁₇	C₂×#362	1	1	yes	characteristic
126	C₂²×Q₈	—	68	1	no	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
129	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
130	C₄⋊Q₈	—	34	2	no	—
131	C₄⋊Q₈	—	34	2	no	—
132	C₄⋊Q₈	—	68	1	no	—
133	C₄⋊Q₈	—	68	1	no	—
134	C₂×C₄⋊C₄	—	34	2	no	—
135	C₂×C₄⋊C₄	—	68	1	no	—
136	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
137	M₃₂	—	34	4	no	—
138	C₂×C₁₆	—	34	2	no	—
139	#108	—	2	2	no	—
140	#108	Dic₃₄	1	4	yes	complemented
141	#107	—	17	4	no	—
142	#107	—	34	2	no	—
143	#107	—	34	2	no	—
144	#107	—	34	2	no	—
145	#97	—	4	1	no	—
146	C₄×C₈	—	4	1	no	—
147	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
148	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
149	C₄×C₈	D₆₈	1	1	yes	characteristic
150	C₄×C₈	D₆₈	1	1	yes	characteristic
151	C₁₇	#13279	1	1	yes	characteristic, complemented, Sylow
152	C₂×Q₈	—	68	2	no	—
153	C₂×Q₈	—	68	2	no	—
154	C₂×Q₈	—	136	4	no	—
155	C₂²×C₄	—	68	2	no	—
156	C₂²×C₄	#63230	1	1	yes	characteristic
157	C₂²×C₄	#63230	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
159	M₁₆	—	2	4	no	—
160	M₁₆	—	4	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	#63227	1	1	yes	characteristic
164	C₂×C₈	#63227	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
166	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
167	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
168	C₂×C₈	#63252	1	2	yes	—
169	C₄⋊C₄	—	34	4	no	—
170	C₄⋊C₄	—	68	2	no	—
171	C₄⋊C₄	—	68	2	no	—
172	C₄⋊C₄	—	68	2	no	—
173	C₄²	#63227	1	1	yes	characteristic
174	C₄²	#63227	1	1	yes	characteristic
175	C₄²	#63228	1	1	yes	characteristic
176	C₄²	#63228	1	1	yes	characteristic
177	C₁₆	—	68	2	no	—
178	C₂³	C₂×#63230	1	1	yes	characteristic
179	Q₈	—	136	4	no	—
180	Q₈	—	136	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	68	4	no	—
184	C₂×C₄	—	136	4	no	—
185	C₂×C₄	#10587480	1	1	yes	characteristic
186	C₂×C₄	#10587480	1	1	yes	characteristic
187	C₂×C₄	#10587480	1	1	yes	characteristic
188	C₂×C₄	#10587480	1	1	yes	characteristic
189	C₂×C₄	#10587610	1	2	yes	—
190	C₂×C₄	C₂×#63227	1	1	yes	characteristic
191	C₂×C₄	C₂×#63228	1	1	yes	characteristic
192	C₂×C₄	#10587637	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	4	no	—
196	C₈	C₄×Dic₃₄	1	2	yes	—
197	C₈	#10587555	1	1	yes	characteristic
198	C₈	#10587561	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#10587480	1	1	yes	characteristic
202	C₂²	#11763094	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	136	4	no	—
206	C₄	#11762621	1	2	yes	—
207	C₄	#11762624	1	2	yes	—
208	C₄	#11762715	1	1	yes	characteristic
209	C₄	#11762817	1	1	yes	characteristic
210	C₄	#11763089	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	#14529736	1	1	yes	characteristic
213	C₂	C₂×#11762621	1	1	yes	characteristic
214	C₂	C₂×#11762624	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587480
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×#108	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×#108	D₁₇
hypocenter	C₁₇	#13279
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
31. C₁₇×#108	Q₈	semidirect
140. #108	Dic₃₄	semidirect
151. C₁₇	#13279	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763214	2
#14530540	2
#14526894	2
C₄×C₁₇×#108	2
#13279	17