Subgroups — #14537788

Order 2176.

370 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	#11763343	C₂	1	1	yes	maximal, characteristic
2	#11762939	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762888	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762885	C₂	1	2	yes	maximal, complemented
6	#11762878	C₂	1	2	yes	maximal, complemented
7	C₈×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762871	C₂	1	2	yes	maximal, complemented
9	#11762862	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762766	C₂	1	2	yes	maximal, complemented
11	C₄×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
15	#10587616	C₄	1	1	yes	characteristic
16	#10587615	C₄	1	2	yes	—
17	C₄×C₂²⋊D₁₇	C₂²	1	2	yes	—
18	#10587610	C₄	1	1	yes	characteristic
19	C₂×#63218	C₂²	1	2	yes	—
20	C₂×C₈×D₁₇	C₂²	1	2	yes	—
21	#10587562	C₄	1	2	yes	—
22	#10587561	C₄	1	1	yes	characteristic
23	C₄×D₆₈	C₂²	1	1	yes	characteristic
24	#10587558	C₂²	1	2	yes	—
25	#10587556	C₄	1	1	yes	characteristic
26	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
27	C₁₇×#97	C₂²	1	1	yes	characteristic
28	C₁₇×#97	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
31	C₂²⋊#3622	C₂²	1	2	yes	—
32	#10587491	C₂²	1	2	yes	—
33	#10587491	C₂²	1	2	yes	—
34	#10587486	C₂²	1	2	yes	—
35	#10587485	C₂²	1	2	yes	—
36	#10587485	C₂²	1	2	yes	—
37	C₈×Dic₁₇	C₂²	1	2	yes	—
38	C₄⋊#3622	C₂²	1	1	yes	characteristic
39	C₄⋊#3622	C₂²	1	1	yes	characteristic
40	C₄×#3622	C₂²	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₇	C₂×C₄	1	2	yes	—
45	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₇	C₂³	1	2	yes	—
47	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
48	M₁₆×C₁₇	—	2	2	no	—
49	M₁₆×C₁₇	—	2	2	no	—
50	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
54	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂³	1	1	yes	characteristic
57	C₄²×C₁₇	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₁₇	C₂×C₄	1	2	yes	—
59	#63227	C₂×C₄	1	2	yes	—
60	#63227	C₂×C₄	1	2	yes	—
61	#63227	C₂×C₄	1	2	yes	—
62	#63227	C₂×C₄	1	2	yes	—
63	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
64	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
65	#63225	C₂×C₄	1	2	yes	—
66	#63225	C₂×C₄	1	2	yes	—
67	#63225	C₂×C₄	1	2	yes	—
68	#63225	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₇	C₂³	1	2	yes	—
70	C₂×#3622	C₂³	1	2	yes	—
71	C₂×#3622	C₂³	1	2	yes	—
72	#63218	—	2	4	no	—
73	C₈×D₁₇	—	2	4	no	—
74	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₁₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₇	—	2	2	no	—
77	C₂×C₄×C₁₇	—	2	1	no	—
78	C₂×C₄×C₁₇	—	2	1	no	—
79	C₂×C₄×C₁₇	—	4	1	no	—
80	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₁₇	—	4	2	no	—
87	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₇	C₂²×C₄	1	2	yes	—
90	D₆₈	—	2	2	no	—
91	C₄×D₁₇	—	2	2	no	—
92	C₄×D₁₇	—	2	2	no	—
93	Dic₃₄	—	2	2	no	—
94	C₈×C₁₇	—	2	1	no	—
95	C₈×C₁₇	—	2	1	no	—
96	C₈×C₁₇	#41	1	2	yes	—
97	C₈×C₁₇	#41	1	2	yes	—
98	#3622	—	2	2	no	—
99	#3622	—	2	2	no	—
100	#2969	—	17	1	no	maximal, Sylow
101	C₂²×C₁₇	—	2	1	no	—
102	C₂²×C₁₇	—	2	1	no	—
103	C₂²×C₁₇	—	2	1	no	—
104	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×D₁₇	—	2	2	no	—
106	C₂×D₁₇	—	2	2	no	—
107	C₂×D₁₇	—	2	2	no	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	—	2	1	no	—
110	C₄×C₁₇	—	2	1	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	#131	1	2	yes	—
113	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
115	Dic₁₇	—	2	2	no	—
116	Dic₁₇	—	2	2	no	—
117	Dic₁₇	—	2	2	no	—
118	#529	—	17	1	no	—
119	#446	—	17	1	no	—
120	C₈×Q₈	—	17	1	no	—
121	#436	—	17	2	no	—
122	#436	—	17	1	no	—
123	#435	—	17	2	no	—
124	C₈×D₄	—	17	1	no	—
125	#433	—	17	2	no	—
126	#431	—	17	1	no	—
127	#405	—	17	2	no	—
128	C₄×M₁₆	D₁₇	1	1	yes	characteristic, complemented
129	C₂×C₁₇	—	4	1	no	—
130	C₂×C₁₇	C₄×#41	1	1	yes	characteristic
131	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
132	C₂×C₁₇	#568	1	1	yes	characteristic
133	D₁₇	—	4	2	no	—
134	C₂×M₁₆	—	17	2	no	—
135	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
136	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
145	C₄×D₄	—	17	2	no	—
146	C₄×D₄	—	17	1	no	—
147	#117	—	17	2	no	—
148	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
149	C₄⋊C₈	—	17	2	no	—
150	C₄⋊C₈	—	17	2	no	—
151	C₄⋊C₈	—	17	1	no	—
152	C₄⋊C₈	—	17	1	no	—
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	1	yes	characteristic
158	#97	C₂×D₁₇	1	1	yes	characteristic
159	C₄×C₈	—	17	2	no	—
160	C₄×C₈	—	17	2	no	—
161	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
162	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
163	C₁₇	#2969	1	1	yes	characteristic, complemented, Sylow
164	C₂×Q₈	—	17	1	no	—
165	C₂×D₄	—	17	2	no	—
166	C₂×D₄	—	17	1	no	—
167	C₂²×C₄	—	17	2	no	—
168	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	—	34	4	no	—
174	C₂×C₈	—	17	2	no	—
175	C₂×C₈	—	17	2	no	—
176	C₂×C₈	—	34	4	no	—
177	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
186	C₄⋊C₄	—	17	2	no	—
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	2	no	—
193	C₄²	C₄×D₁₇	1	1	yes	characteristic
194	C₄²	C₄×D₁₇	1	1	yes	characteristic
195	C₄²	C₂²×D₁₇	1	1	yes	characteristic
196	C₄²	C₂²×D₁₇	1	1	yes	characteristic
197	C₂³	—	17	2	no	—
198	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
199	Q₈	—	34	2	no	—
200	D₄	—	34	2	no	—
201	D₄	—	68	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	17	2	no	—
207	C₂×C₄	—	17	2	no	—
208	C₂×C₄	—	17	2	no	—
209	C₂×C₄	—	34	2	no	—
210	C₂×C₄	—	34	2	no	—
211	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
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	1	no	—
219	C₈	—	34	2	no	—
220	C₈	—	34	2	no	—
221	C₈	#63252	1	2	yes	—
222	C₈	#63252	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	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	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	34	2	no	—
235	C₄	—	34	2	no	—
236	C₄	—	34	2	no	—
237	C₄	#10587593	1	2	yes	—
238	C₄	C₂×#63252	1	1	yes	characteristic
239	C₄	C₂×#63252	1	1	yes	characteristic
240	C₂	—	4	1	no	—
241	C₂	—	68	2	no	—
242	C₂	C₄×#63252	1	1	yes	characteristic
243	C₂	C₂×#10587593	1	1	yes	characteristic
244	C₂	#11763575	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₄×M₁₆×C₁₇	C₂
prime radical	#10587616	C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₄×M₁₆	D₁₇
hypocenter	C₁₇	#2969
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #11762939	C₂	semidirect
3. C₈×Dic₃₄	C₂	semidirect
4. #11762888	C₂	semidirect
5. #11762885	C₂	semidirect
6. #11762878	C₂	semidirect
7. C₈×D₆₈	C₂	semidirect
8. #11762871	C₂	semidirect
9. #11762862	C₂	semidirect
10. #11762766	C₂	semidirect
11. C₄×M₁₆×C₁₇	C₂	semidirect
128. C₄×M₁₆	D₁₇	semidirect
163. C₁₇	#2969	semidirect

Maximal subgroups

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

subgroup	index
#11763343	2
#11762939	2
C₈×Dic₃₄	2
#11762888	2
#11762885	2
#11762878	2
C₈×D₆₈	2
#11762871	2
#11762862	2
#11762766	2
C₄×M₁₆×C₁₇	2
#2969	17