Subgroups — #14545713

Order 2176.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763861	C₂	1	2	yes	maximal, complemented
2	#11763839	C₂	1	2	yes	maximal, complemented
3	#11763838	C₂	1	2	yes	maximal, complemented
4	#11763836	C₂	1	2	yes	maximal, complemented
5	#11763832	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763829	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763827	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763814	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10587544	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763810	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763277	C₂	1	1	yes	maximal, characteristic
12	C₂×#63252	C₄	1	1	yes	characteristic
13	C₂²×#3624	C₂²	1	2	yes	—
14	C₂×#63243	C₂²	1	2	yes	—
15	C₂×#63242	C₂²	1	2	yes	—
16	C₁₇×#117	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
18	#10587579	C₄	1	1	yes	characteristic
19	#10587579	C₂²	1	1	yes	characteristic
20	#10587573	C₄	1	1	yes	characteristic
21	#10587573	C₂²	1	1	yes	characteristic
22	#10587567	C₄	1	1	yes	characteristic
23	#10587567	C₄	1	1	yes	characteristic
24	#10587567	C₂²	1	1	yes	characteristic
25	#10587567	C₂²	1	1	yes	characteristic
26	#10587558	C₄	1	1	yes	characteristic
27	#10587558	C₄	1	1	yes	characteristic
28	C₄²×D₁₇	C₂²	1	1	yes	characteristic
29	C₄²×D₁₇	C₂²	1	1	yes	characteristic
30	C₂²⋊#3624	C₂²	1	2	yes	—
31	#10587544	C₂²	1	1	yes	characteristic
32	#10587544	C₂²	1	1	yes	characteristic
33	#10587544	C₂²	1	1	yes	characteristic
34	#10587544	C₂²	1	2	yes	—
35	#10587544	C₂²	1	2	yes	—
36	#10587544	C₂²	1	1	yes	characteristic
37	#10587543	C₂²	1	2	yes	—
38	#10587543	C₂²	1	2	yes	—
39	#10587542	C₂²	1	2	yes	—
40	#10587542	C₂²	1	2	yes	—
41	C₄⋊#3624	C₂²	1	2	yes	—
42	C₄×#3624	C₂²	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	C₂³	1	1	yes	characteristic
48	#63252	—	2	4	no	—
49	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
56	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
57	C₂×#3624	—	4	2	no	—
58	C₂×#3624	C₂³	1	2	yes	—
59	C₂×#3624	C₂³	1	2	yes	—
60	C₂×#3624	C₂³	1	2	yes	—
61	C₂×#3624	C₂³	1	2	yes	—
62	#63243	—	4	2	no	—
63	#63242	—	4	2	no	—
64	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
70	#63227	C₂×C₄	1	1	yes	characteristic
71	#63227	C₂×C₄	1	1	yes	characteristic
72	#63227	C₂×C₄	1	1	yes	characteristic
73	#63227	C₂×C₄	1	1	yes	characteristic
74	#63225	C₂×C₄	1	1	yes	characteristic
75	#63225	C₂×C₄	1	1	yes	characteristic
76	#63225	C₂×C₄	1	1	yes	characteristic
77	#63225	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
83	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
84	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
85	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
86	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²⋊D₁₇	—	2	2	no	—
97	C₂²⋊D₁₇	M₁₆	1	4	yes	complemented
98	C₂×Dic₁₇	—	2	2	no	—
99	C₂×Dic₁₇	—	2	1	no	—
100	C₂×Dic₁₇	—	2	1	no	—
101	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₇	C₂⁴	1	1	yes	characteristic
107	D₆₈	—	2	2	no	—
108	C₄×D₁₇	—	2	2	no	—
109	C₄×D₁₇	—	2	2	no	—
110	C₄×D₁₇	—	2	2	no	—
111	C₄×D₁₇	—	2	2	no	—
112	C₄×D₁₇	—	2	1	no	—
113	C₄×D₁₇	—	2	1	no	—
114	C₄×D₁₇	—	2	1	no	—
115	C₄×D₁₇	—	2	1	no	—
116	Dic₃₄	—	2	2	no	—
117	#3624	—	2	2	no	—
118	#3624	—	2	2	no	—
119	#3624	—	2	2	no	—
120	#3624	—	2	2	no	—
121	#2990	—	17	1	no	maximal, Sylow
122	C₂²×C₁₇	—	2	1	no	—
123	C₂²×C₁₇	—	2	1	no	—
124	C₂²×C₁₇	C₂×M₁₆	1	2	yes	—
125	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×D₁₇	—	2	1	no	—
127	C₂×D₁₇	—	2	1	no	—
128	C₂×D₁₇	—	2	1	no	—
129	C₂×D₁₇	—	2	1	no	—
130	C₂×D₁₇	—	2	1	no	—
131	C₂×D₁₇	C₂×M₁₆	1	2	yes	—
132	C₄×C₁₇	—	2	1	no	—
133	C₄×C₁₇	—	2	1	no	—
134	C₄×C₁₇	—	2	1	no	—
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	Dic₁₇	—	2	1	no	—
139	Dic₁₇	—	2	1	no	—
140	Dic₁₇	—	2	1	no	—
141	Dic₁₇	C₂×M₁₆	1	2	yes	—
142	Dic₁₇	#131	1	2	yes	—
143	Dic₁₇	H(5,𝔽₂)	1	1	yes	characteristic
144	Dic₁₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₄×#41	—	17	1	no	—
146	#446	—	17	2	no	—
147	#435	—	17	2	no	—
148	#435	—	17	2	no	—
149	C₈×D₄	—	17	2	no	—
150	#432	—	17	1	no	—
151	#432	—	17	1	no	—
152	#431	—	17	1	no	—
153	#423	—	17	1	no	—
154	C₂×C₄⋊C₈	—	17	1	no	—
155	C₂×C₄⋊C₈	—	17	1	no	—
156	C₂×C₁₇	—	2	2	no	—
157	C₂×C₁₇	#520	1	1	yes	characteristic
158	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
159	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
160	D₁₇	—	2	2	no	—
161	D₁₇	—	4	1	no	—
162	C₂×#41	—	17	1	no	—
163	C₂×M₁₆	—	17	2	no	—
164	C₂²×C₈	—	17	2	no	—
165	C₂²×C₈	—	17	2	no	—
166	C₄×Q₈	—	17	1	no	—
167	C₄×Q₈	—	17	1	no	—
168	C₄×D₄	—	17	1	no	—
169	C₄×D₄	—	17	1	no	—
170	C₄×D₄	—	17	1	no	—
171	C₄×D₄	—	17	1	no	—
172	C₄×D₄	—	17	1	no	—
173	C₄×D₄	—	17	1	no	—
174	#117	—	17	1	no	—
175	#117	—	17	1	no	—
176	#117	#595	1	1	yes	characteristic
177	C₂×C₄²	—	17	1	no	—
178	C₂×C₄²	—	17	1	no	—
179	C₂×C₄²	—	17	1	no	—
180	C₄⋊C₈	—	17	2	no	—
181	C₄⋊C₈	—	17	1	no	—
182	C₄⋊C₈	—	17	1	no	—
183	C₄⋊C₈	—	17	1	no	—
184	C₄⋊C₈	—	17	2	no	—
185	C₄⋊C₈	—	17	2	no	—
186	C₄⋊C₈	—	17	1	no	—
187	C₂²⋊C₈	—	17	2	no	—
188	C₂²⋊C₈	—	17	2	no	—
189	C₂²⋊C₈	—	17	2	no	—
190	#97	—	17	2	no	—
191	#97	—	17	2	no	—
192	C₄×C₈	—	17	2	no	—
193	C₁₇	#2990	1	1	yes	characteristic, complemented, Sylow
194	#41	—	34	4	no	—
195	C₂×Q₈	—	17	1	no	—
196	C₂×D₄	—	17	1	no	—
197	C₂×D₄	—	17	1	no	—
198	C₂×D₄	—	17	1	no	—
199	C₂²×C₄	—	17	1	no	—
200	C₂²×C₄	—	17	1	no	—
201	C₂²×C₄	—	17	1	no	—
202	C₂²×C₄	—	17	1	no	—
203	C₂²×C₄	—	17	1	no	—
204	C₂²×C₄	—	17	1	no	—
205	C₂²×C₄	—	17	1	no	—
206	C₂²×C₄	—	17	1	no	—
207	C₂²×C₄	C₂×#595	1	1	yes	characteristic
208	M₁₆	—	68	2	no	—
209	C₂×C₈	—	17	2	no	—
210	C₂×C₈	—	17	2	no	—
211	C₂×C₈	—	17	2	no	—
212	C₂×C₈	—	17	2	no	—
213	C₂×C₈	—	68	2	no	—
214	C₂×C₈	—	68	2	no	—
215	C₄⋊C₄	—	17	1	no	—
216	C₄⋊C₄	—	17	1	no	—
217	C₄⋊C₄	—	17	1	no	—
218	C₄⋊C₄	—	17	1	no	—
219	C₄⋊C₄	C₂×#595	1	1	yes	characteristic
220	C₄⋊C₄	C₂×#595	1	1	yes	characteristic
221	C₂²⋊C₄	—	17	1	no	—
222	C₂²⋊C₄	—	17	1	no	—
223	C₂²⋊C₄	—	17	1	no	—
224	C₂²⋊C₄	—	17	1	no	—
225	C₂²⋊C₄	C₂×#595	1	1	yes	characteristic
226	C₂²⋊C₄	C₂×#595	1	1	yes	characteristic
227	C₄²	—	17	1	no	—
228	C₄²	—	17	1	no	—
229	C₄²	—	17	1	no	—
230	C₄²	—	17	1	no	—
231	C₄²	—	17	1	no	—
232	C₄²	—	17	1	no	—
233	C₄²	—	17	1	no	—
234	C₄²	—	17	1	no	—
235	C₄²	C₂×#595	1	1	yes	characteristic
236	C₄²	C₂×#595	1	1	yes	characteristic
237	C₂³	—	17	1	no	—
238	C₂³	—	17	1	no	—
239	C₂³	C₂²×#595	1	1	yes	characteristic
240	Q₈	—	34	2	no	—
241	D₄	—	17	4	no	—
242	D₄	—	34	2	no	—
243	D₄	—	34	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	17	1	no	—
246	C₂×C₄	—	17	1	no	—
247	C₂×C₄	—	17	1	no	—
248	C₂×C₄	—	17	1	no	—
249	C₂×C₄	—	17	1	no	—
250	C₂×C₄	—	17	1	no	—
251	C₂×C₄	—	34	1	no	—
252	C₂×C₄	—	34	1	no	—
253	C₂×C₄	—	34	2	no	—
254	C₂×C₄	—	34	1	no	—
255	C₂×C₄	—	34	1	no	—
256	C₂×C₄	—	34	2	no	—
257	C₂×C₄	—	34	2	no	—
258	C₂×C₄	—	34	2	no	—
259	C₂×C₄	—	34	1	no	—
260	C₂×C₄	—	34	1	no	—
261	C₂×C₄	—	34	2	no	—
262	C₂×C₄	C₂²×#595	1	1	yes	characteristic
263	C₂×C₄	C₂²×#595	1	1	yes	characteristic
264	C₂×C₄	C₂²×#595	1	1	yes	characteristic
265	C₂×C₄	C₂²×#595	1	1	yes	characteristic
266	C₂×C₄	C₂²×#595	1	1	yes	characteristic
267	C₂×C₄	C₂²×#595	1	1	yes	characteristic
268	C₈	—	34	2	no	—
269	C₈	—	34	2	no	—
270	C₈	—	34	2	no	—
271	C₈	—	34	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	17	2	no	—
275	C₂²	—	34	1	no	—
276	C₂²	—	34	1	no	—
277	C₂²	—	34	1	no	—
278	C₂²	—	34	1	no	—
279	C₂²	—	34	1	no	—
280	C₂²	D₁₇⋊M₁₆	1	2	yes	—
281	C₂²	C₂³×#595	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	17	1	no	—
289	C₄	—	17	1	no	—
290	C₄	—	17	2	no	—
291	C₄	—	17	2	no	—
292	C₄	—	34	1	no	—
293	C₄	—	34	1	no	—
294	C₄	—	34	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	34	2	no	—
297	C₂	—	68	1	no	—
298	C₂	#11763878	1	1	yes	characteristic
299	C₂	C₂×D₁₇⋊M₁₆	1	1	yes	characteristic
300	C₂	D₁₇⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#117	C₄
prime radical	C₂×#63252	C₄
cosocle	C₂×Dic₁₇	C₂⁴
hypercenter	#117	#595
hypocenter	C₁₇	#2990
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #11763861	C₂	semidirect
2. #11763839	C₂	semidirect
3. #11763838	C₂	semidirect
4. #11763836	C₂	semidirect
5. #11763832	C₂	semidirect
6. #11763829	C₂	semidirect
7. #11763827	C₂	semidirect
8. #11763814	C₂	semidirect
9. C₂×#10587544	C₂	semidirect
10. #11763810	C₂	semidirect
97. C₂²⋊D₁₇	M₁₆	semidirect
193. C₁₇	#2990	semidirect

Maximal subgroups

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

subgroup	index
#11763861	2
#11763839	2
#11763838	2
#11763836	2
#11763832	2
#11763829	2
#11763827	2
#11763814	2
C₂×#10587544	2
#11763810	2
#11763277	2
#2990	17