Subgroups — #14537758

Order 2176.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763346	C₂	1	1	yes	maximal, characteristic
2	Q₈×#3622	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762888	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762882	C₂	1	2	yes	maximal, complemented
5	#11762874	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762874	C₂	1	2	yes	maximal, complemented
7	#11762869	C₂	1	2	yes	maximal, complemented
8	#11762865	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762766	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
13	#10587633	C₄	1	1	yes	characteristic
14	#10587632	C₄	1	1	yes	characteristic
15	C₂×#63218	C₂²	1	2	yes	—
16	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₇	C₂²	1	2	yes	—
19	#10587584	C₄	1	2	yes	—
20	#10587581	C₄	1	1	yes	characteristic
21	#10587580	C₄	1	2	yes	—
22	#10587576	C₄	1	1	yes	characteristic
23	C₄×D₆₈	C₂²	1	2	yes	—
24	C₄×D₆₈	C₂²	1	1	yes	characteristic
25	#10587558	C₂²	1	2	yes	—
26	C₄²×D₁₇	C₂²	1	1	yes	characteristic
27	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
29	C₁₇×#97	C₂²	1	2	yes	—
30	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
31	#10587491	C₂²	1	2	yes	—
32	#10587491	C₂²	1	2	yes	—
33	#10587491	C₂²	1	2	yes	—
34	#10587486	C₂²	1	2	yes	—
35	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
37	C₄⋊#3622	C₂²	1	1	yes	characteristic
38	C₄⋊#3622	C₂²	1	2	yes	—
39	C₄×#3622	C₂²	1	1	yes	characteristic
40	C₄×#3622	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×D₆₈	C₂×C₄	1	2	yes	—
43	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂³	1	2	yes	—
48	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂³	1	2	yes	—
50	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
53	C₄²×C₁₇	C₂³	1	1	yes	characteristic
54	C₄²×C₁₇	C₂³	1	2	yes	—
55	#63227	C₂×C₄	1	2	yes	—
56	#63227	C₂×C₄	1	2	yes	—
57	#63227	C₂×C₄	1	2	yes	—
58	#63227	C₂×C₄	1	2	yes	—
59	#63227	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
62	#63225	C₂×C₄	1	2	yes	—
63	#63225	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
67	C₂×#3622	C₂³	1	1	yes	characteristic
68	C₂×#3622	C₂³	1	2	yes	—
69	C₂×#3622	C₂³	1	1	yes	characteristic
70	#63218	—	2	4	no	—
71	C₈×D₁₇	—	2	2	no	—
72	C₈×D₁₇	—	2	2	no	—
73	C₈×D₁₇	—	2	4	no	—
74	C₂²×D₁₇	C₂²×C₄	1	2	yes	—
75	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
86	D₆₈	—	2	4	no	—
87	D₆₈	—	2	2	no	—
88	C₄×D₁₇	—	2	2	no	—
89	C₄×D₁₇	—	2	2	no	—
90	C₄×D₁₇	—	2	1	no	—
91	C₄×D₁₇	—	2	1	no	—
92	C₄×D₁₇	—	2	2	no	—
93	C₄×D₁₇	—	2	2	no	—
94	C₈×C₁₇	—	2	1	no	—
95	C₈×C₁₇	—	2	2	no	—
96	C₈×C₁₇	#41	1	2	yes	—
97	#3622	—	2	1	no	—
98	#3622	—	2	2	no	—
99	#3622	#41	1	2	yes	—
100	#2966	—	17	1	no	maximal, Sylow
101	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₁₇	—	2	1	no	—
103	C₂×D₁₇	—	2	1	no	—
104	C₂×D₁₇	—	2	2	no	—
105	C₂×D₁₇	—	2	2	no	—
106	C₂×D₁₇	—	2	2	no	—
107	C₂×D₁₇	#131	1	2	yes	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	—	2	2	no	—
111	C₄×C₁₇	#131	1	2	yes	—
112	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
113	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
114	Dic₁₇	—	2	1	no	—
115	Dic₁₇	—	2	2	no	—
116	Dic₁₇	—	2	1	no	—
117	Dic₁₇	—	2	1	no	—
118	#529	—	17	1	no	—
119	#446	D₁₇	1	1	yes	characteristic, complemented
120	C₈×Q₈	—	17	1	no	—
121	#436	—	17	1	no	—
122	#435	—	17	2	no	—
123	C₈×D₄	—	17	1	no	—
124	C₈×D₄	—	17	2	no	—
125	#433	—	17	2	no	—
126	#432	—	17	1	no	—
127	#405	—	17	2	no	—
128	C₂×C₄×C₈	—	17	1	no	—
129	C₂×C₁₇	C₄×#41	1	1	yes	characteristic
130	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
131	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
132	D₁₇	—	2	2	no	—
133	D₁₇	—	4	2	no	—
134	C₂×M₁₆	—	17	2	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	17	2	no	—
138	#126	—	17	2	no	—
139	#125	—	17	1	no	—
140	#124	—	17	1	no	—
141	#123	—	17	2	no	—
142	C₂²⋊Q₈	—	17	1	no	—
143	#121	—	17	1	no	—
144	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
145	C₄×D₄	—	17	2	no	—
146	C₄×D₄	—	17	1	no	—
147	#117	—	17	2	no	—
148	C₂×C₄²	—	17	1	no	—
149	C₄⋊C₈	—	17	1	no	—
150	C₄⋊C₈	—	17	2	no	—
151	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
152	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
153	C₂²⋊C₈	—	17	2	no	—
154	C₂²⋊C₈	—	17	2	no	—
155	C₂²⋊C₈	—	17	2	no	—
156	#97	—	17	2	no	—
157	#97	C₂×D₁₇	1	2	yes	—
158	C₄×C₈	—	17	1	no	—
159	C₄×C₈	—	17	1	no	—
160	C₄×C₈	—	17	1	no	—
161	C₄×C₈	—	17	2	no	—
162	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
163	C₁₇	#2966	1	1	yes	characteristic, complemented, Sylow
164	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
165	C₂×D₄	—	17	2	no	—
166	C₂×D₄	—	17	1	no	—
167	C₂²×C₄	—	17	1	no	—
168	C₂²×C₄	—	17	1	no	—
169	C₂²×C₄	—	17	1	no	—
170	C₂²×C₄	—	17	2	no	—
171	M₁₆	—	34	4	no	—
172	C₂×C₈	—	17	1	no	—
173	C₂×C₈	—	17	1	no	—
174	C₂×C₈	—	17	2	no	—
175	C₂×C₈	—	34	2	no	—
176	C₂×C₈	—	34	4	no	—
177	C₂×C₈	—	34	2	no	—
178	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₁₇	1	2	yes	—
180	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
181	C₄⋊C₄	—	17	2	no	—
182	C₄⋊C₄	—	17	2	no	—
183	C₄⋊C₄	—	17	1	no	—
184	C₄⋊C₄	—	17	2	no	—
185	C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
186	C₄⋊C₄	C₄×D₁₇	1	2	yes	—
187	C₂²⋊C₄	—	17	2	no	—
188	C₂²⋊C₄	—	17	2	no	—
189	C₂²⋊C₄	—	17	2	no	—
190	C₂²⋊C₄	—	17	2	no	—
191	C₂²⋊C₄	—	17	2	no	—
192	C₄²	—	17	1	no	—
193	C₄²	—	17	1	no	—
194	C₄²	—	17	1	no	—
195	C₄²	C₂²×D₁₇	1	1	yes	characteristic
196	C₄²	C₂²×D₁₇	1	2	yes	—
197	C₂³	—	17	1	no	—
198	C₂³	—	17	2	no	—
199	Q₈	—	2	2	no	—
200	D₄	—	34	4	no	—
201	D₄	—	34	2	no	—
202	C₂×C₄	—	17	1	no	—
203	C₂×C₄	—	17	1	no	—
204	C₂×C₄	—	17	1	no	—
205	C₂×C₄	—	17	2	no	—
206	C₂×C₄	—	34	2	no	—
207	C₂×C₄	—	34	1	no	—
208	C₂×C₄	—	34	2	no	—
209	C₂×C₄	—	34	1	no	—
210	C₂×C₄	—	34	2	no	—
211	C₂×C₄	—	34	2	no	—
212	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₁₇	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	17	2	no	—
220	C₈	—	34	1	no	—
221	C₈	—	34	2	no	—
222	C₈	#63256	1	2	yes	—
223	C₂²	—	17	2	no	—
224	C₂²	—	34	1	no	—
225	C₂²	—	34	1	no	—
226	C₂²	—	34	2	no	—
227	C₂²	—	34	2	no	—
228	C₂²	—	34	2	no	—
229	C₂²	C₂²×C₄×D₁₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	34	1	no	—
234	C₄	—	34	2	no	—
235	C₄	—	34	1	no	—
236	C₄	—	34	1	no	—
237	C₄	#10587593	1	2	yes	—
238	C₄	C₂×#63256	1	1	yes	characteristic
239	C₄	#41×D₁₇	1	1	yes	characteristic
240	C₂	—	34	2	no	—
241	C₂	—	68	2	no	—
242	C₂	C₄×#63256	1	1	yes	characteristic
243	C₂	#131×D₁₇	1	1	yes	characteristic
244	C₂	C₂×#10587593	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#446	C₂
prime radical	#10587581	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#446	D₁₇
hypocenter	C₁₇	#2966
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. Q₈×#3622	C₂	semidirect
3. #11762888	C₂	semidirect
4. #11762882	C₂	semidirect
5. #11762874	C₂	semidirect
6. #11762874	C₂	semidirect
7. #11762869	C₂	semidirect
8. #11762865	C₂	semidirect
9. #11762766	C₂	semidirect
10. C₄×C₈×D₁₇	C₂	semidirect
11. C₁₇×#446	C₂	semidirect
119. #446	D₁₇	semidirect
163. C₁₇	#2966	semidirect

Maximal subgroups

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

subgroup	index
#11763346	2
Q₈×#3622	2
#11762888	2
#11762882	2
#11762874	2
#11762869	2
#11762865	2
#11762766	2
C₄×C₈×D₁₇	2
C₁₇×#446	2
#2966	17