Subgroups — #14538946

Order 2176.

386 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763366	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763150	C₂	1	2	yes	maximal, complemented
3	#11763104	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763099	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763019	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763014	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762971	C₂	1	2	yes	maximal
8	#11762914	C₂	1	2	yes	maximal, complemented
9	#11762909	C₂	1	2	yes	maximal, complemented
10	#11762866	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₇	—	2	1	no	—
16	C₄×D₄×C₁₇	—	2	1	no	—
17	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#117	—	2	1	no	—
20	C₁₇×#117	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₇	—	2	1	no	—
22	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×#63231	C₂²	1	2	yes	—
24	C₂×#63229	C₂²	1	2	yes	—
25	#10587612	—	2	1	no	—
26	#10587610	—	2	1	no	—
27	#10587610	C₂²	1	2	yes	—
28	#10587556	—	2	1	no	—
29	#10587555	C₂²	1	1	yes	characteristic
30	#10587555	C₂²	1	1	yes	characteristic
31	C₄×Dic₃₄	—	2	1	no	—
32	C₄×Dic₃₄	C₂²	1	2	yes	—
33	#10587506	C₂²	1	2	yes	—
34	#10587503	C₂²	1	2	yes	—
35	C₂²⋊#3622	C₂²	1	2	yes	—
36	#10587481	C₂²	1	2	yes	—
37	#10587481	C₂²	1	2	yes	—
38	#10587479	C₂²	1	2	yes	—
39	#10587478	C₂²	1	2	yes	—
40	C₄⋊#3622	C₂²	1	1	yes	characteristic
41	C₄⋊#3622	C₂²	1	1	yes	characteristic
42	#10587474	C₂²	1	2	yes	—
43	#41×C₁₇	—	2	2	no	—
44	#41×C₁₇	—	2	2	no	—
45	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₇	—	2	1	no	—
47	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₇	—	2	1	no	—
49	C₂²×C₄×C₁₇	—	2	1	no	—
50	C₂²×C₄×C₁₇	—	2	1	no	—
51	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
54	C₂×Dic₃₄	—	2	1	no	—
55	C₂×Dic₃₄	C₂³	1	2	yes	—
56	C₄⋊C₄×C₁₇	—	2	1	no	—
57	C₄⋊C₄×C₁₇	—	2	1	no	—
58	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²⋊C₄×C₁₇	—	2	1	no	—
61	C₂²⋊C₄×C₁₇	—	2	1	no	—
62	C₂²⋊C₄×C₁₇	—	2	1	no	—
63	C₄²×C₁₇	—	2	1	no	—
64	C₄²×C₁₇	—	2	1	no	—
65	C₄²×C₁₇	—	2	1	no	—
66	C₄²×C₁₇	D₄	1	1	yes	characteristic
67	C₄²×C₁₇	D₄	1	1	yes	characteristic
68	C₄²×C₁₇	C₂³	1	1	yes	characteristic
69	C₄²×C₁₇	C₂³	1	1	yes	characteristic
70	C₂²⋊Dic₁₇	—	2	2	no	—
71	#63231	—	2	4	no	—
72	#63229	—	2	4	no	—
73	C₄⋊Dic₁₇	—	2	1	no	—
74	C₄⋊Dic₁₇	—	2	1	no	—
75	C₄⋊Dic₁₇	—	2	1	no	—
76	C₄⋊Dic₁₇	C₂³	1	2	yes	—
77	#63225	—	2	2	no	—
78	#63225	—	2	2	no	—
79	C₄×Dic₁₇	—	2	2	no	—
80	C₂×#3622	C₂³	1	2	yes	—
81	C₂×#3622	C₂³	1	2	yes	—
82	C₂³×C₁₇	—	2	1	no	—
83	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₁₇	—	2	1	no	—
85	Q₈×C₁₇	#41	1	2	yes	—
86	D₄×C₁₇	—	2	4	no	—
87	D₄×C₁₇	—	2	1	no	—
88	D₄×C₁₇	#41	1	2	yes	—
89	C₂×C₄×C₁₇	—	2	4	no	—
90	C₂×C₄×C₁₇	—	2	1	no	—
91	C₂×C₄×C₁₇	—	2	2	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	4	1	no	—
97	C₂×C₄×C₁₇	—	4	2	no	—
98	C₂×C₄×C₁₇	—	4	2	no	—
99	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₁₇	—	2	2	no	—
106	C₂×Dic₁₇	—	2	2	no	—
107	Dic₃₄	—	2	2	no	—
108	Dic₃₄	—	2	2	no	—
109	Dic₃₄	—	4	2	no	—
110	#3622	—	2	2	no	—
111	#3622	—	2	2	no	—
112	#3116	—	17	1	no	maximal, Sylow
113	C₂²×C₁₇	—	2	1	no	—
114	C₂²×C₁₇	—	2	2	no	—
115	C₂²×C₁₇	—	2	1	no	—
116	C₂²×C₁₇	—	2	1	no	—
117	C₂²×C₁₇	—	4	1	no	—
118	C₂²×C₁₇	—	4	1	no	—
119	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₁₇	—	2	1	no	—
121	C₄×C₁₇	—	2	2	no	—
122	C₄×C₁₇	—	2	1	no	—
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	—	2	1	no	—
125	C₄×C₁₇	—	4	1	no	—
126	C₄×C₁₇	—	4	1	no	—
127	C₄×C₁₇	#137	1	2	yes	—
128	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
129	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
130	Dic₁₇	—	4	2	no	—
131	Dic₁₇	—	4	2	no	—
132	#533	—	17	1	no	—
133	C₄×#41	D₁₇	1	1	yes	characteristic
134	#485	—	17	2	no	—
135	#477	—	17	1	no	—
136	#476	—	17	1	no	—
137	#462	—	17	1	no	—
138	#461	—	17	1	no	—
139	#452	—	17	2	no	—
140	#441	—	17	2	no	—
141	#440	—	17	2	no	—
142	#432	—	17	1	no	—
143	C₂×C₁₇	—	4	1	no	—
144	C₂×C₁₇	—	4	2	no	—
145	C₂×C₁₇	#525	1	1	yes	characteristic
146	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
147	C₂×C₁₇	#575	1	1	yes	characteristic
148	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
149	C₂×Q₁₆	—	17	2	no	—
150	C₂×SD₁₆	—	17	2	no	—
151	C₄⋊Q₈	—	17	1	no	—
152	C₄⋊Q₈	—	17	1	no	—
153	#125	—	34	1	no	—
154	C₂²⋊Q₈	—	17	2	no	—
155	C₂²⋊Q₈	—	34	1	no	—
156	C₄×Q₈	—	17	2	no	—
157	C₄×Q₈	—	34	1	no	—
158	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
159	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
163	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
164	#117	—	2	1	no	—
165	#117	—	34	1	no	—
166	#117	C₂×D₁₇	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
169	#107	—	17	2	no	—
170	#106	—	17	2	no	—
171	C₄⋊C₈	—	17	1	no	—
172	C₄⋊C₈	—	17	1	no	—
173	#103	—	17	2	no	—
174	#103	—	17	2	no	—
175	#103	—	17	2	no	—
176	#102	—	17	2	no	—
177	C₂²⋊C₈	—	17	2	no	—
178	#97	—	17	2	no	—
179	C₁₇	#3116	1	1	yes	characteristic, complemented, Sylow
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	17	2	no	—
183	C₂×Q₈	—	34	1	no	—
184	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
193	Q₁₆	—	34	4	no	—
194	SD₁₆	—	34	4	no	—
195	C₂×C₈	—	17	2	no	—
196	C₂×C₈	—	17	2	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	17	2	no	—
200	C₄⋊C₄	—	34	1	no	—
201	C₄⋊C₄	—	34	1	no	—
202	C₄⋊C₄	—	34	2	no	—
203	C₄⋊C₄	—	34	2	no	—
204	C₄⋊C₄	—	34	1	no	—
205	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
206	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	34	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	34	2	no	—
215	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
216	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
217	C₄²	C₂²×D₁₇	1	1	yes	characteristic
218	C₄²	C₂²×D₁₇	1	1	yes	characteristic
219	C₂³	—	2	1	no	—
220	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
221	Q₈	—	2	1	no	—
222	Q₈	—	34	2	no	—
223	Q₈	—	34	2	no	—
224	Q₈	—	68	2	no	—
225	Q₈	#63252	1	2	yes	—
226	D₄	—	2	4	no	—
227	D₄	—	2	1	no	—
228	D₄	#63252	1	2	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	34	2	no	—
240	C₂×C₄	—	34	2	no	—
241	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
244	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
246	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
247	C₈	—	34	2	no	—
248	C₈	—	34	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	68	2	no	—
264	C₄	—	68	2	no	—
265	C₄	#10587637	1	2	yes	—
266	C₄	C₂×#63252	1	1	yes	characteristic
267	C₄	C₂×#63252	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	—	4	2	no	—
270	C₂	#11763320	1	1	yes	characteristic
271	C₂	C₂×#10587637	1	1	yes	characteristic
272	C₂	#11763607	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₄×#41×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#41×C₁₇	C₂²
hypercenter	C₄×#41	D₁₇
hypocenter	C₁₇	#3116
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #11763366	C₂	semidirect
2. #11763150	C₂	semidirect
3. #11763104	C₂	semidirect
4. #11763099	C₂	semidirect
5. #11763019	C₂	semidirect
6. #11763014	C₂	semidirect
8. #11762914	C₂	semidirect
9. #11762909	C₂	semidirect
10. #11762866	C₂	semidirect
179. C₁₇	#3116	semidirect

Maximal subgroups

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

subgroup	index
#11763366	2
#11763150	2
#11763104	2
#11763099	2
#11763019	2
#11763014	2
#11762971	2
#11762914	2
#11762909	2
#11762866	2
C₄×#41×C₁₇	2
#3116	17