Subgroups — #14545725

Order 2176.

354 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763861	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763858	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763844	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763842	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763842	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763841	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763840	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763836	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763834	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763833	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763826	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763816	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763815	C₂	1	1	yes	maximal, characteristic, complemented
14	#11763812	C₂	1	1	yes	maximal, characteristic, complemented
15	#11763336	C₂	1	1	yes	maximal, characteristic
16	C₂×#63247	C₂²	1	1	yes	characteristic
17	C₂×#63247	C₂²	1	1	yes	characteristic
18	C₂×#63247	C₂²	1	1	yes	characteristic
19	C₂²×#3624	C₂²	1	1	yes	characteristic
20	C₂×#63243	C₂²	1	1	yes	characteristic
21	C₂×#63242	C₂²	1	1	yes	characteristic
22	C₁₇×#123	C₄	1	1	yes	characteristic
23	#10587625	C₄	1	1	yes	characteristic
24	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
25	#10587615	C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
27	#10587584	C₄	1	1	yes	characteristic
28	#10587578	C₂²	1	1	yes	characteristic
29	#10587575	C₄	1	1	yes	characteristic
30	#10587573	C₂²	1	1	yes	characteristic
31	#10587571	C₄	1	1	yes	characteristic
32	#10587567	C₂²	1	1	yes	characteristic
33	#10587567	C₂²	1	1	yes	characteristic
34	#10587565	C₄	1	1	yes	characteristic
35	#10587564	C₄	1	1	yes	characteristic
36	#10587563	C₂²	1	1	yes	characteristic
37	C₂²⋊#3624	C₂²	1	1	yes	characteristic
38	C₂²⋊#3624	C₂²	1	1	yes	characteristic
39	C₂²⋊#3624	C₂²	1	1	yes	characteristic
40	C₂²⋊#3624	C₂²	1	1	yes	characteristic
41	#10587544	C₂²	1	1	yes	characteristic
42	#10587544	C₂²	1	1	yes	characteristic
43	#10587544	C₂²	1	1	yes	characteristic
44	#10587544	C₂²	1	1	yes	characteristic
45	#10587544	C₂²	1	1	yes	characteristic
46	#10587544	C₂²	1	1	yes	characteristic
47	#10587543	C₂²	1	1	yes	characteristic
48	#10587543	C₂²	1	1	yes	characteristic
49	#10587542	C₂²	1	1	yes	characteristic
50	#10587542	C₂²	1	1	yes	characteristic
51	C₄⋊#3624	C₂²	1	1	yes	characteristic
52	C₄⋊#3624	C₂²	1	1	yes	characteristic
53	C₄⋊#3624	C₂²	1	1	yes	characteristic
54	C₄⋊#3624	C₂²	1	1	yes	characteristic
55	C₄×#3624	C₂²	1	1	yes	characteristic
56	C₄×#3624	C₂²	1	1	yes	characteristic
57	C₄×#3624	C₂²	1	1	yes	characteristic
58	C₄×#3624	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₇	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
67	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
68	#63247	—	4	1	no	—
69	#63247	—	4	1	no	—
70	#63247	—	4	1	no	—
71	C₂×#3624	—	4	1	no	—
72	C₂×#3624	C₂³	1	1	yes	characteristic
73	C₂×#3624	C₂³	1	1	yes	characteristic
74	C₂×#3624	C₂³	1	1	yes	characteristic
75	C₂×#3624	C₂³	1	1	yes	characteristic
76	C₂×#3624	C₂³	1	1	yes	characteristic
77	C₂×#3624	C₂³	1	1	yes	characteristic
78	C₂×#3624	C₂³	1	1	yes	characteristic
79	C₂×#3624	C₂³	1	1	yes	characteristic
80	#63243	—	4	1	no	—
81	#63242	—	4	1	no	—
82	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
90	#63227	C₂×C₄	1	1	yes	characteristic
91	#63227	C₂×C₄	1	1	yes	characteristic
92	#63227	C₂×C₄	1	1	yes	characteristic
93	#63227	C₂×C₄	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
96	#63225	C₂×C₄	1	1	yes	characteristic
97	#63225	C₂×C₄	1	1	yes	characteristic
98	#63225	C₂×C₄	1	1	yes	characteristic
99	#63225	C₂×C₄	1	1	yes	characteristic
100	#63225	C₂×C₄	1	1	yes	characteristic
101	#63225	C₂×C₄	1	1	yes	characteristic
102	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
103	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
104	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
105	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
106	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
107	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
108	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
111	D₄×C₁₇	—	4	1	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²⋊D₁₇	—	2	2	no	—
119	C₂²⋊D₁₇	—	4	1	no	—
120	C₂×Dic₁₇	—	2	2	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂⁴	1	1	yes	characteristic
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	Dic₃₄	—	2	2	no	—
135	#3624	—	2	1	no	—
136	#3624	—	2	1	no	—
137	#3624	—	2	1	no	—
138	#3624	—	2	1	no	—
139	#3624	—	2	1	no	—
140	#3624	—	2	1	no	—
141	#3624	—	2	1	no	—
142	#3624	—	2	1	no	—
143	#2993	—	17	1	no	maximal, Sylow
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	—	2	1	no	—
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	—	2	1	no	—
150	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×D₁₇	—	2	1	no	—
152	C₂×D₁₇	—	2	1	no	—
153	C₂×D₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	Dic₁₇	—	2	1	no	—
160	Dic₁₇	—	2	1	no	—
161	Dic₁₇	—	2	1	no	—
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	#131	1	2	yes	—
165	Dic₁₇	H(5,𝔽₂)	1	1	yes	characteristic
166	Dic₁₇	2₋¹⁺⁴	1	1	yes	characteristic
167	#529	—	17	1	no	—
168	#446	—	17	1	no	—
169	C₈×Q₈	—	17	1	no	—
170	#436	—	17	1	no	—
171	#436	—	17	1	no	—
172	#436	—	17	1	no	—
173	#435	—	17	1	no	—
174	#435	—	17	1	no	—
175	C₈×D₄	—	17	1	no	—
176	#433	—	17	1	no	—
177	#433	—	17	1	no	—
178	#431	—	17	1	no	—
179	#424	—	17	1	no	—
180	#424	—	17	1	no	—
181	#423	—	17	1	no	—
182	C₂×C₁₇	—	4	1	no	—
183	C₂×C₁₇	—	4	1	no	—
184	C₂×C₁₇	#520	1	1	yes	characteristic
185	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
186	C₂×C₁₇	#568	1	1	yes	characteristic
187	D₁₇	—	4	1	no	—
188	C₂×M₁₆	—	17	1	no	—
189	C₂×M₁₆	—	17	1	no	—
190	C₂×M₁₆	—	17	1	no	—
191	C₂×M₁₆	—	17	1	no	—
192	C₂²×C₈	—	17	1	no	—
193	C₂²×C₈	—	17	1	no	—
194	#126	—	17	1	no	—
195	#126	—	17	1	no	—
196	#125	—	17	1	no	—
197	#124	—	17	1	no	—
198	#123	—	17	1	no	—
199	#123	#595	1	1	yes	characteristic
200	C₂²⋊Q₈	—	17	1	no	—
201	#121	—	17	1	no	—
202	C₄×Q₈	—	17	1	no	—
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	17	1	no	—
206	#117	—	17	1	no	—
207	#117	—	17	1	no	—
208	C₂×C₄²	—	17	1	no	—
209	C₄⋊C₈	—	17	1	no	—
210	C₄⋊C₈	—	17	1	no	—
211	C₄⋊C₈	—	17	1	no	—
212	C₄⋊C₈	—	17	1	no	—
213	C₄⋊C₈	—	17	1	no	—
214	C₄⋊C₈	—	17	1	no	—
215	C₄⋊C₈	—	17	1	no	—
216	C₄⋊C₈	—	17	1	no	—
217	C₄⋊C₈	—	17	1	no	—
218	C₄⋊C₈	—	17	1	no	—
219	C₂²⋊C₈	—	17	1	no	—
220	C₂²⋊C₈	—	17	1	no	—
221	C₂²⋊C₈	—	17	1	no	—
222	C₂²⋊C₈	—	17	1	no	—
223	C₂²⋊C₈	—	17	1	no	—
224	C₂²⋊C₈	—	17	1	no	—
225	#97	—	17	1	no	—
226	#97	—	17	1	no	—
227	C₄×C₈	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₄×C₈	—	17	1	no	—
230	C₄×C₈	—	17	1	no	—
231	C₁₇	#2993	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	17	1	no	—
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	17	1	no	—
235	C₂×D₄	C₂×#595	1	1	yes	characteristic
236	C₂²×C₄	—	17	1	no	—
237	C₂²×C₄	—	17	1	no	—
238	C₂²×C₄	—	17	1	no	—
239	C₂²×C₄	—	17	1	no	—
240	C₂²×C₄	C₂×#595	1	1	yes	characteristic
241	M₁₆	—	68	1	no	—
242	M₁₆	—	68	1	no	—
243	M₁₆	—	68	1	no	—
244	M₁₆	—	68	1	no	—
245	C₂×C₈	—	17	1	no	—
246	C₂×C₈	—	17	1	no	—
247	C₂×C₈	—	17	1	no	—
248	C₂×C₈	—	17	1	no	—
249	C₂×C₈	—	17	1	no	—
250	C₂×C₈	—	17	1	no	—
251	C₂×C₈	—	17	1	no	—
252	C₂×C₈	—	17	1	no	—
253	C₂×C₈	—	68	1	no	—
254	C₂×C₈	—	68	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	17	1	no	—
262	C₄⋊C₄	—	17	1	no	—
263	C₄⋊C₄	C₂×#595	1	1	yes	characteristic
264	C₄⋊C₄	C₂×#595	1	1	yes	characteristic
265	C₂²⋊C₄	—	17	1	no	—
266	C₂²⋊C₄	—	17	1	no	—
267	C₂²⋊C₄	—	17	1	no	—
268	C₂²⋊C₄	—	17	1	no	—
269	C₂²⋊C₄	—	17	1	no	—
270	C₂²⋊C₄	—	17	1	no	—
271	C₂²⋊C₄	—	17	1	no	—
272	C₂²⋊C₄	C₂×#595	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#595	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#595	1	1	yes	characteristic
275	C₄²	—	17	1	no	—
276	C₄²	—	17	1	no	—
277	C₄²	—	17	1	no	—
278	C₄²	—	17	1	no	—
279	C₄²	—	17	1	no	—
280	C₄²	—	17	1	no	—
281	C₂³	—	17	1	no	—
282	C₂³	C₂²×#595	1	1	yes	characteristic
283	C₂³	C₂²×#595	1	1	yes	characteristic
284	Q₈	—	34	2	no	—
285	D₄	—	4	1	no	—
286	D₄	—	34	2	no	—
287	D₄	—	68	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	17	1	no	—
290	C₂×C₄	—	17	1	no	—
291	C₂×C₄	—	17	1	no	—
292	C₂×C₄	—	17	1	no	—
293	C₂×C₄	—	17	1	no	—
294	C₂×C₄	—	17	1	no	—
295	C₂×C₄	—	17	1	no	—
296	C₂×C₄	—	34	1	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	—	34	1	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	2	no	—
303	C₂×C₄	C₂²×#595	1	1	yes	characteristic
304	C₂×C₄	C₂²×#595	1	1	yes	characteristic
305	C₂×C₄	C₂²×#595	1	1	yes	characteristic
306	C₂×C₄	C₂²×#595	1	1	yes	characteristic
307	C₂×C₄	C₂²×#595	1	1	yes	characteristic
308	C₈	—	34	1	no	—
309	C₈	—	34	1	no	—
310	C₈	—	34	1	no	—
311	C₈	—	34	1	no	—
312	C₈	—	34	1	no	—
313	C₈	—	34	1	no	—
314	C₈	—	34	1	no	—
315	C₈	—	34	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	34	1	no	—
323	C₂²	—	34	1	no	—
324	C₂²	—	34	1	no	—
325	C₂²	C₂³×#595	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	17	1	no	—
332	C₄	—	17	1	no	—
333	C₄	—	17	2	no	—
334	C₄	—	34	1	no	—
335	C₄	—	34	1	no	—
336	C₄	—	34	1	no	—
337	C₄	—	34	1	no	—
338	C₄	—	34	1	no	—
339	C₂	—	4	1	no	—
340	C₂	—	4	1	no	—
341	C₂	—	68	1	no	—
342	C₂	#11763878	1	1	yes	characteristic
343	C₂	D₁₇⋊#131	1	1	yes	characteristic
344	C₂	#11763887	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#123	C₄
prime radical	#10587625	C₄
cosocle	C₂×Dic₁₇	C₂⁴
hypercenter	#123	#595
hypocenter	C₁₇	#2993
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. #11763861	C₂	semidirect
2. #11763858	C₂	semidirect
3. #11763844	C₂	semidirect
4. #11763842	C₂	semidirect
5. #11763842	C₂	semidirect
6. #11763841	C₂	semidirect
7. #11763840	C₂	semidirect
8. #11763836	C₂	semidirect
9. #11763834	C₂	semidirect
10. #11763833	C₂	semidirect
11. #11763826	C₂	semidirect
12. #11763816	C₂	semidirect
13. #11763815	C₂	semidirect
14. #11763812	C₂	semidirect
231. C₁₇	#2993	semidirect

Maximal subgroups

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

subgroup	index
#11763861	2
#11763858	2
#11763844	2
#11763842	2
#11763841	2
#11763840	2
#11763836	2
#11763834	2
#11763833	2
#11763826	2
#11763816	2
#11763815	2
#11763812	2
#11763336	2
#2993	17