Subgroups — #14539029

Order 2176.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763415	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763281	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763136	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763135	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763005	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763004	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762944	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762944	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762920	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762918	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762891	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762889	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#125	C₂²	1	1	yes	characteristic, complemented
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
20	C₂×#63228	C₂²	1	1	yes	characteristic
21	C₂×#63228	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₇	—	2	1	no	—
23	C₂×#63227	—	2	1	no	—
24	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
25	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
26	#10587581	—	2	1	no	—
27	#10587581	—	2	1	no	—
28	#10587581	C₂²	1	1	yes	characteristic
29	#10587581	C₂²	1	1	yes	characteristic
30	#10587580	—	2	1	no	—
31	#10587580	—	2	1	no	—
32	#10587579	C₂²	1	1	yes	characteristic
33	#10587579	C₂²	1	1	yes	characteristic, complemented
34	#10587579	C₂²	1	1	yes	characteristic, complemented
35	#10587578	C₂²	1	1	yes	characteristic
36	#10587567	—	2	1	no	—
37	C₂²⋊C₄×D₁₇	—	2	1	no	—
38	#10587560	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	#10587558	C₂²	1	1	yes	characteristic
41	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
42	C₁₇×#106	C₂²	1	1	yes	characteristic, complemented
43	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
44	C₁₇×#102	C₂²	1	1	yes	characteristic
45	C₁₇×#102	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic, complemented
47	#10587492	C₂²	1	1	yes	characteristic, complemented
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587491	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587480	C₂²	1	1	yes	characteristic, complemented
55	#10587480	C₂²	1	1	yes	characteristic, complemented
56	#10587479	C₂²	1	1	yes	characteristic, complemented
57	#10587478	C₂²	1	1	yes	characteristic, complemented
58	C₄⋊#3622	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂²×Dic₁₇	—	2	1	no	—
64	D₄×D₁₇	—	2	2	no	—
65	D₄×D₁₇	—	2	2	no	—
66	C₂×D₆₈	—	2	1	no	—
67	C₂×D₆₈	—	2	1	no	—
68	C₂×D₆₈	—	2	1	no	—
69	C₂×D₆₈	—	2	1	no	—
70	C₂×D₆₈	C₂³	1	1	yes	characteristic
71	C₂×D₆₈	C₂³	1	1	yes	characteristic
72	C₂×D₆₈	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₇	—	2	1	no	—
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×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	—	2	1	no	—
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₇	—	2	1	no	—
86	C₄²×C₁₇	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	#63228	—	2	2	no	—
89	#63228	—	2	2	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	#63227	—	2	1	no	—
95	#63227	—	2	1	no	—
96	#63227	—	2	1	no	—
97	#63227	D₄	1	1	yes	characteristic
98	#63227	D₄	1	1	yes	characteristic
99	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
100	#63225	D₄	1	1	yes	characteristic
101	#63225	D₄	1	1	yes	characteristic
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	D₁₃₆	—	2	2	no	—
107	D₁₃₆	—	2	2	no	—
108	C₂³×C₁₇	—	2	1	no	—
109	C₂²×D₁₇	—	2	1	no	—
110	C₂²×D₁₇	—	2	4	no	—
111	C₂²×D₁₇	—	2	1	no	—
112	C₂²×D₁₇	—	2	1	no	—
113	C₂²×D₁₇	—	8	1	no	—
114	C₂²×D₁₇	—	8	1	no	—
115	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
116	D₄×C₁₇	—	2	1	no	—
117	D₄×C₁₇	#41	1	2	yes	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	8	1	no	—
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₇	—	2	4	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	8	1	no	—
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
131	D₆₈	—	2	1	no	—
132	D₆₈	—	2	1	no	—
133	D₆₈	—	2	1	no	—
134	D₆₈	—	2	1	no	—
135	D₆₈	—	2	1	no	—
136	D₆₈	—	8	1	no	—
137	D₆₈	—	8	1	no	—
138	D₆₈	—	8	1	no	—
139	D₆₈	—	8	1	no	—
140	D₆₈	#41	1	2	yes	—
141	C₄×D₁₇	—	2	1	no	—
142	C₄×D₁₇	—	2	1	no	—
143	C₄×D₁₇	—	8	1	no	—
144	C₄×D₁₇	—	8	1	no	—
145	C₄×D₁₇	—	8	1	no	—
146	C₄×D₁₇	—	8	1	no	—
147	C₈×C₁₇	—	2	1	no	—
148	C₈×C₁₇	—	2	1	no	—
149	#3622	—	2	1	no	—
150	#3622	—	2	1	no	—
151	#3122	—	17	1	no	maximal, Sylow
152	C₂²×C₁₇	—	2	2	no	—
153	C₂²×C₁₇	—	4	1	no	—
154	C₂²×C₁₇	—	4	1	no	—
155	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₁₇	—	2	1	no	—
157	C₂×D₁₇	—	2	2	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	2	1	no	—
160	C₂×D₁₇	—	4	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	4	1	no	—
163	C₂×D₁₇	—	4	1	no	—
164	C₂×D₁₇	—	4	1	no	—
165	C₂×D₁₇	—	4	1	no	—
166	C₂×D₁₇	—	4	1	no	—
167	C₂×D₁₇	—	4	1	no	—
168	C₂×D₁₇	—	8	4	no	—
169	C₄×C₁₇	—	2	1	no	—
170	C₄×C₁₇	—	2	1	no	—
171	C₄×C₁₇	—	4	1	no	—
172	C₄×C₁₇	—	4	1	no	—
173	C₄×C₁₇	—	4	1	no	—
174	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
175	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
176	Dic₁₇	—	2	1	no	—
177	Dic₁₇	—	2	1	no	—
178	Dic₁₇	—	2	1	no	—
179	Dic₁₇	—	4	1	no	—
180	#540	—	17	1	no	—
181	#518	—	17	1	no	—
182	#482	—	17	1	no	—
183	#482	—	17	1	no	—
184	#478	—	17	1	no	—
185	#478	D₁₇	1	1	yes	characteristic, complemented
186	#459	—	17	1	no	—
187	#459	—	17	1	no	—
188	#447	—	17	1	no	—
189	#447	—	17	1	no	—
190	#442	—	17	1	no	—
191	#442	—	17	1	no	—
192	C₄×D₈	—	17	1	no	—
193	C₄×D₈	—	17	1	no	—
194	#433	—	17	1	no	—
195	C₂×C₁₇	—	4	2	no	—
196	C₂×C₁₇	#525	1	1	yes	characteristic
197	C₂×C₁₇	#576	1	1	yes	characteristic
198	C₂×C₁₇	#576	1	1	yes	characteristic
199	D₁₇	—	4	1	no	—
200	D₁₇	—	4	2	no	—
201	D₁₇	—	8	1	no	—
202	D₁₇	—	8	1	no	—
203	C₂²×D₄	—	17	1	no	—
204	C₂×D₈	—	17	1	no	—
205	C₂×D₈	—	17	1	no	—
206	C₂×D₈	—	17	1	no	—
207	C₂×D₈	—	17	1	no	—
208	#127	—	17	1	no	—
209	#125	C₂×D₁₇	1	1	yes	characteristic, complemented
210	#123	—	34	1	no	—
211	#123	—	34	1	no	—
212	#121	—	17	1	no	—
213	#121	—	17	1	no	—
214	#121	—	34	1	no	—
215	#121	—	34	1	no	—
216	C₄×D₄	—	17	1	no	—
217	C₄×D₄	—	17	1	no	—
218	C₄×D₄	—	17	1	no	—
219	C₄×D₄	—	17	1	no	—
220	C₄×D₄	—	17	1	no	—
221	C₄×D₄	—	34	1	no	—
222	C₄×D₄	—	34	1	no	—
223	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
224	#117	—	17	1	no	—
225	#117	—	17	1	no	—
226	C₂×C₂²⋊C₄	—	34	1	no	—
227	C₂×C₂²⋊C₄	—	34	1	no	—
228	#107	—	17	1	no	—
229	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
230	#106	—	17	1	no	—
231	#106	C₂×D₁₇	1	1	yes	characteristic, complemented
232	C₄⋊C₈	—	17	1	no	—
233	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
234	#102	—	17	1	no	—
235	#102	—	17	1	no	—
236	#102	—	17	1	no	—
237	#102	—	17	1	no	—
238	#102	—	17	1	no	—
239	#102	—	17	1	no	—
240	#102	C₂×D₁₇	1	1	yes	characteristic
241	#102	C₂×D₁₇	1	1	yes	characteristic
242	C₂²⋊C₈	—	17	1	no	—
243	C₂²⋊C₈	—	17	1	no	—
244	#97	—	17	1	no	—
245	C₄×C₈	—	17	1	no	—
246	C₁₇	#3122	1	1	yes	characteristic, complemented, Sylow
247	C₂⁴	—	34	1	no	—
248	C₂×D₄	—	17	1	no	—
249	C₂×D₄	—	17	1	no	—
250	C₂×D₄	—	17	1	no	—
251	C₂×D₄	—	34	1	no	—
252	C₂×D₄	—	34	1	no	—
253	C₂×D₄	—	34	1	no	—
254	C₂×D₄	—	34	2	no	—
255	C₂×D₄	—	34	2	no	—
256	C₂×D₄	—	34	1	no	—
257	C₂×D₄	—	34	1	no	—
258	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
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	—	34	1	no	—
265	C₂²×C₄	—	34	1	no	—
266	D₈	—	34	2	no	—
267	D₈	—	34	2	no	—
268	D₈	—	34	2	no	—
269	D₈	—	34	2	no	—
270	C₂×C₈	—	17	1	no	—
271	C₂×C₈	—	17	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	—	2	1	no	—
276	C₄⋊C₄	—	17	1	no	—
277	C₄⋊C₄	—	17	1	no	—
278	C₄⋊C₄	—	17	1	no	—
279	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
280	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
281	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	17	1	no	—
284	C₂²⋊C₄	—	17	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₂²⋊C₄	—	34	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	C₄²	—	17	1	no	—
294	C₄²	—	34	1	no	—
295	C₄²	C₂²×D₁₇	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	17	1	no	—
298	C₂³	—	34	4	no	—
299	C₂³	—	34	1	no	—
300	C₂³	—	34	1	no	—
301	C₂³	—	34	1	no	—
302	C₂³	—	136	1	no	—
303	C₂³	—	136	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	17	2	no	—
306	D₄	—	34	4	no	—
307	D₄	—	34	1	no	—
308	D₄	—	34	1	no	—
309	D₄	—	34	1	no	—
310	D₄	—	34	1	no	—
311	D₄	—	34	1	no	—
312	D₄	—	136	1	no	—
313	D₄	—	136	1	no	—
314	D₄	—	136	1	no	—
315	D₄	—	136	1	no	—
316	D₄	#63256	1	2	yes	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	8	1	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	—	136	1	no	—
328	C₂×C₄	—	136	1	no	—
329	C₂×C₄	—	136	1	no	—
330	C₂×C₄	—	136	1	no	—
331	C₂×C₄	—	136	1	no	—
332	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
333	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
334	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	34	1	no	—
338	C₈	—	34	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	34	1	no	—
343	C₂²	—	34	2	no	—
344	C₂²	—	34	1	no	—
345	C₂²	—	34	1	no	—
346	C₂²	—	68	1	no	—
347	C₂²	—	68	1	no	—
348	C₂²	—	68	1	no	—
349	C₂²	—	68	1	no	—
350	C₂²	—	68	1	no	—
351	C₂²	—	68	1	no	—
352	C₂²	—	68	1	no	—
353	C₂²	—	68	1	no	—
354	C₂²	—	136	4	no	—
355	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	34	1	no	—
362	C₄	—	34	1	no	—
363	C₄	—	34	1	no	—
364	C₄	—	68	1	no	—
365	C₄	C₂×#63256	1	1	yes	characteristic
366	C₄	#41×D₁₇	1	1	yes	characteristic
367	C₂	—	4	2	no	—
368	C₂	—	68	1	no	—
369	C₂	—	68	2	no	—
370	C₂	—	136	1	no	—
371	C₂	—	136	1	no	—
372	C₂	#11763322	1	1	yes	characteristic
373	C₂	#11763611	1	1	yes	characteristic
374	C₂	#11763613	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#478	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3122
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #11763415	C₂	semidirect
2. #11763281	C₂	semidirect
3. #11763136	C₂	semidirect
4. #11763135	C₂	semidirect
5. #11763106	C₂	semidirect
6. #11763005	C₂	semidirect
7. #11763004	C₂	semidirect
8. #11762944	C₂	semidirect
9. #11762944	C₂	semidirect
10. #11762920	C₂	semidirect
11. #11762918	C₂	semidirect
12. #11762891	C₂	semidirect
13. #11762889	C₂	semidirect
14. #11762869	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
17. C₁₇×#125	C₂²	semidirect
33. #10587579	C₂²	semidirect
34. #10587579	C₂²	semidirect
38. #10587560	C₂²	semidirect
41. C₁₇×#107	C₂²	semidirect
42. C₁₇×#106	C₂²	semidirect
43. C₁₇×C₄⋊C₈	C₂²	semidirect
46. #10587492	C₂²	semidirect
47. #10587492	C₂²	semidirect
50. #10587486	C₂²	semidirect
51. C₈×Dic₁₇	C₂²	semidirect
54. #10587480	C₂²	semidirect
55. #10587480	C₂²	semidirect
56. #10587479	C₂²	semidirect
57. #10587478	C₂²	semidirect
58. C₄⋊#3622	C₂²	semidirect
185. #478	D₁₇	semidirect
209. #125	C₂×D₁₇	semidirect
229. #107	C₂×D₁₇	semidirect
231. #106	C₂×D₁₇	semidirect
233. C₄⋊C₈	C₂×D₁₇	semidirect
246. C₁₇	#3122	semidirect

Maximal subgroups

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

subgroup	index
#11763415	2
#11763281	2
#11763136	2
#11763135	2
#11763106	2
#11763005	2
#11763004	2
#11762944	2
#11762920	2
#11762918	2
#11762891	2
#11762889	2
#11762869	2
C₁₇×#478	2
#3122	17