Subgroups — #14540645

Order 2176.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763460	C₂	1	1	yes	maximal, characteristic
3	#11763158	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763139	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763138	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763041	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763038	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763013	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762949	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762948	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762894	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₁₇×#103	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×#63252	—	2	1	no	—
19	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
21	#10587632	—	2	1	no	—
22	#10587632	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
24	#10587630	—	2	1	no	—
25	#10587630	C₂²	1	1	yes	characteristic
26	C₂×#63230	C₂²	1	1	yes	characteristic
27	C₂×#63230	C₂²	1	1	yes	characteristic
28	#10587616	C₂²	1	1	yes	characteristic
29	#10587615	—	2	1	no	—
30	C₄×C₂²⋊D₁₇	—	2	1	no	—
31	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
32	#10587610	—	2	1	no	—
33	#10587610	C₂²	1	1	yes	characteristic
34	C₂×#63219	C₂²	1	1	yes	characteristic
35	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
36	#10587582	—	2	1	no	—
37	#10587581	C₂²	1	1	yes	characteristic
38	#10587580	—	2	1	no	—
39	#10587579	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
41	#10587574	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	C₂²⋊#3622	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587491	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587479	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	—	2	2	no	—
61	C₂×Q₈×C₁₇	—	4	1	no	—
62	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₇	—	2	1	no	—
68	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
69	Q₈×D₁₇	—	2	2	no	—
70	Q₈×D₁₇	—	4	1	no	—
71	#63252	—	4	4	no	—
72	C₂×D₆₈	—	2	1	no	—
73	C₂×D₆₈	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₇	—	2	1	no	—
75	C₂×C₄×D₁₇	—	2	1	no	—
76	C₂×C₄×D₁₇	—	2	1	no	—
77	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
78	C₂×Dic₃₄	—	2	1	no	—
79	C₂×Dic₃₄	—	2	1	no	—
80	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₁₇	—	4	1	no	—
82	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	—	2	1	no	—
85	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
89	#63230	—	2	2	no	—
90	#63230	—	4	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	#63227	D₄	1	1	yes	characteristic
96	C₄⋊Dic₁₇	—	2	1	no	—
97	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	#63225	—	2	1	no	—
104	#63225	—	2	1	no	—
105	#63225	D₄	1	1	yes	characteristic
106	C₄×Dic₁₇	—	2	1	no	—
107	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	C₂×#3622	C₂³	1	1	yes	characteristic
110	#63219	—	4	1	no	—
111	C₈×D₁₇	—	4	1	no	—
112	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₇	—	2	1	no	—
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	—	2	1	no	—
117	Q₈×C₁₇	—	2	1	no	—
118	Q₈×C₁₇	—	4	2	no	—
119	Q₈×C₁₇	#41	1	2	yes	complemented
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	4	2	no	—
127	C₂×C₄×C₁₇	—	4	2	no	—
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₇	—	4	2	no	—
131	C₂²⋊D₁₇	SD₁₆	1	4	yes	complemented
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
138	D₆₈	—	2	1	no	—
139	D₆₈	—	2	1	no	—
140	D₆₈	—	4	2	no	—
141	C₄×D₁₇	—	2	1	no	—
142	C₄×D₁₇	—	2	1	no	—
143	C₄×D₁₇	—	4	2	no	—
144	C₄×D₁₇	—	4	2	no	—
145	C₄×D₁₇	—	4	2	no	—
146	Dic₃₄	—	2	1	no	—
147	Dic₃₄	—	2	1	no	—
148	Dic₃₄	—	4	2	no	—
149	Dic₃₄	—	4	2	no	—
150	C₈×C₁₇	—	2	1	no	—
151	C₈×C₁₇	—	2	1	no	—
152	#3622	—	2	1	no	—
153	#3622	—	2	1	no	—
154	#3298	—	17	1	no	maximal, Sylow
155	C₂²×C₁₇	—	2	1	no	—
156	C₂²×C₁₇	—	2	1	no	—
157	C₂²×C₁₇	C₂×SD₁₆	1	2	yes	—
158	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₁₇	—	2	1	no	—
160	C₂×D₁₇	—	2	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	4	1	no	—
163	C₂×D₁₇	—	4	1	no	—
164	C₂×D₁₇	C₂×SD₁₆	1	2	yes	—
165	C₄×C₁₇	—	2	2	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	4	1	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₁₇	—	2	1	no	—
173	Dic₁₇	—	2	1	no	—
174	Dic₁₇	—	4	1	no	—
175	Dic₁₇	—	4	1	no	—
176	Dic₁₇	—	4	1	no	—
177	Dic₁₇	C₂×SD₁₆	1	2	yes	—
178	D₄×Q₈	—	17	1	no	—
179	#547	—	17	1	no	—
180	#487	—	17	1	no	—
181	#483	—	17	1	no	—
182	#483	—	17	1	no	—
183	#475	—	17	1	no	—
184	#465	—	17	1	no	—
185	#465	—	17	1	no	—
186	#461	—	17	1	no	—
187	#448	—	17	1	no	—
188	#448	—	17	1	no	—
189	C₄×SD₁₆	—	17	1	no	—
190	C₈×D₄	—	17	1	no	—
191	C₂×#103	—	17	1	no	—
192	C₂×#103	D₁₇	1	1	yes	characteristic, complemented
193	C₂×C₁₇	—	2	2	no	—
194	C₂×C₁₇	#546	1	1	yes	characteristic
195	C₂×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
196	C₂×C₁₇	#578	1	1	yes	characteristic
197	D₁₇	—	2	2	no	—
198	D₁₇	—	8	1	no	—
199	C₂×#41	—	34	1	no	—
200	C₂²×Q₈	—	17	1	no	—
201	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
202	C₂×SD₁₆	—	17	1	no	—
203	C₂×SD₁₆	—	17	1	no	—
204	C₂×SD₁₆	—	17	1	no	—
205	C₂²×C₈	—	17	1	no	—
206	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
207	C₄⋊Q₈	—	17	1	no	—
208	C₄⋊Q₈	—	17	1	no	—
209	C₄⋊Q₈	—	34	1	no	—
210	#123	—	34	1	no	—
211	#123	—	34	1	no	—
212	C₂²⋊Q₈	—	17	1	no	—
213	C₂²⋊Q₈	—	17	1	no	—
214	C₂²⋊Q₈	—	34	1	no	—
215	C₂²⋊Q₈	—	34	1	no	—
216	C₂²⋊Q₈	—	34	1	no	—
217	#121	—	17	1	no	—
218	#121	—	17	1	no	—
219	C₄×Q₈	—	17	1	no	—
220	C₄×D₄	—	17	1	no	—
221	C₄×D₄	—	17	1	no	—
222	C₄×D₄	—	34	1	no	—
223	C₂×C₄⋊C₄	—	17	1	no	—
224	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
225	#106	—	17	1	no	—
226	#106	—	17	1	no	—
227	#106	—	17	1	no	—
228	C₄⋊C₈	—	17	1	no	—
229	#103	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#103	—	17	1	no	—
234	#103	C₂×D₁₇	1	1	yes	characteristic
235	#103	C₂×D₁₇	1	1	yes	characteristic
236	#103	C₂×D₁₇	1	1	yes	characteristic
237	#103	C₂×D₁₇	1	1	yes	characteristic
238	#102	—	17	1	no	—
239	C₂²⋊C₈	—	17	1	no	—
240	C₂²⋊C₈	—	17	1	no	—
241	C₄×C₈	—	17	1	no	—
242	C₁₇	#3298	1	1	yes	characteristic, complemented, Sylow
243	#41	—	68	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	17	1	no	—
248	C₂×Q₈	—	34	1	no	—
249	C₂×Q₈	—	34	1	no	—
250	C₂×Q₈	—	34	2	no	—
251	C₂×Q₈	—	68	1	no	—
252	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
254	C₂×D₄	—	17	1	no	—
255	C₂×D₄	—	17	1	no	—
256	C₂×D₄	—	34	1	no	—
257	C₂×D₄	—	34	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	17	1	no	—
261	C₂²×C₄	—	34	1	no	—
262	C₂²×C₄	—	34	1	no	—
263	C₂²×C₄	—	34	1	no	—
264	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
265	SD₁₆	—	34	2	no	—
266	SD₁₆	—	68	1	no	—
267	SD₁₆	—	68	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	17	1	no	—
270	C₂×C₈	—	17	1	no	—
271	C₂×C₈	—	68	1	no	—
272	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	17	1	no	—
276	C₄⋊C₄	—	17	1	no	—
277	C₄⋊C₄	—	17	1	no	—
278	C₄⋊C₄	—	34	1	no	—
279	C₄⋊C₄	—	34	1	no	—
280	C₄⋊C₄	—	34	1	no	—
281	C₄⋊C₄	—	34	1	no	—
282	C₄⋊C₄	—	34	1	no	—
283	C₄⋊C₄	—	34	1	no	—
284	C₄⋊C₄	—	34	1	no	—
285	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
286	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
287	C₂²⋊C₄	—	17	1	no	—
288	C₂²⋊C₄	—	17	1	no	—
289	C₂²⋊C₄	—	34	1	no	—
290	C₂²⋊C₄	—	34	1	no	—
291	C₂²⋊C₄	—	34	1	no	—
292	C₂²⋊C₄	—	34	1	no	—
293	C₂²⋊C₄	—	34	1	no	—
294	C₄²	—	17	1	no	—
295	C₄²	—	34	1	no	—
296	C₂³	—	17	1	no	—
297	C₂³	—	34	1	no	—
298	C₂³	D₄×D₁₇	1	1	yes	characteristic
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	34	1	no	—
304	Q₈	—	34	1	no	—
305	Q₈	—	68	2	no	—
306	Q₈	—	68	2	no	—
307	Q₈	#63252	1	2	yes	complemented
308	D₄	—	17	4	no	—
309	D₄	—	34	1	no	—
310	D₄	—	34	1	no	—
311	D₄	—	68	2	no	—
312	D₄	—	68	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	17	1	no	—
322	C₂×C₄	—	17	1	no	—
323	C₂×C₄	—	17	1	no	—
324	C₂×C₄	—	34	1	no	—
325	C₂×C₄	—	34	1	no	—
326	C₂×C₄	—	34	1	no	—
327	C₂×C₄	—	34	1	no	—
328	C₂×C₄	—	34	1	no	—
329	C₂×C₄	—	68	2	no	—
330	C₂×C₄	—	68	2	no	—
331	C₂×C₄	—	68	2	no	—
332	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	34	1	no	—
337	C₈	—	34	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	17	2	no	—
341	C₂²	—	34	1	no	—
342	C₂²	—	34	1	no	—
343	C₂²	—	68	1	no	—
344	C₂²	—	68	1	no	—
345	C₂²	—	68	1	no	—
346	C₂²	SD₁₆×D₁₇	1	2	yes	—
347	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	17	2	no	—
354	C₄	—	34	1	no	—
355	C₄	—	34	1	no	—
356	C₄	—	68	1	no	—
357	C₄	—	68	1	no	—
358	C₄	—	68	1	no	—
359	C₄	C₂×#63252	1	1	yes	characteristic
360	C₄	#10587691	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	34	2	no	—
363	C₂	—	136	1	no	—
364	C₂	#11763449	1	1	yes	characteristic
365	C₂	C₂×SD₁₆×D₁₇	1	1	yes	characteristic
366	C₂	#11763622	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂×C₁₇×#103	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763460	C₂
hypercenter	C₂×#103	D₁₇
hypocenter	C₁₇	#3298
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
1. Q₈×C₂²⋊D₁₇	C₂	semidirect
3. #11763158	C₂	semidirect
4. #11763139	C₂	semidirect
5. #11763138	C₂	semidirect
6. #11763092	C₂	semidirect
7. #11763041	C₂	semidirect
8. #11763038	C₂	semidirect
9. #11763013	C₂	semidirect
10. #11762949	C₂	semidirect
11. #11762948	C₂	semidirect
12. #11762894	C₂	semidirect
13. C₈×C₂²⋊D₁₇	C₂	semidirect
14. #103×D₁₇	C₂	semidirect
15. C₂×C₁₇×#103	C₂	semidirect
119. Q₈×C₁₇	#41	semidirect
131. C₂²⋊D₁₇	SD₁₆	semidirect
192. C₂×#103	D₁₇	semidirect
242. C₁₇	#3298	semidirect
307. Q₈	#63252	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₂²⋊D₁₇	2
#11763460	2
#11763158	2
#11763139	2
#11763138	2
#11763092	2
#11763041	2
#11763038	2
#11763013	2
#11762949	2
#11762948	2
#11762894	2
C₈×C₂²⋊D₁₇	2
#103×D₁₇	2
C₂×C₁₇×#103	2
#3298	17