Subgroups — #14545652

Order 2176.

412 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763843	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763842	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763841	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763839	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763839	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763838	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763837	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763836	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763833	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763833	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763783	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763782	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763780	C₂	1	1	yes	maximal, characteristic, complemented
14	#11763777	C₂	1	1	yes	maximal, characteristic, complemented
15	#11763317	C₂	1	1	yes	maximal, characteristic
16	C₂×#63254	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×#63247	C₂²	1	1	yes	characteristic, complemented
19	C₂²×#3624	C₂²	1	1	yes	characteristic, complemented
20	C₂×#63243	C₂²	1	1	yes	characteristic
21	C₂×#63243	C₂²	1	1	yes	characteristic
22	C₂×#63243	C₂²	1	1	yes	characteristic, complemented
23	C₂×#63242	C₂²	1	1	yes	characteristic
24	C₂×#63242	C₂²	1	1	yes	characteristic
25	C₂×#63242	C₂²	1	1	yes	characteristic, complemented
26	C₁₇×#123	C₄	1	1	yes	characteristic
27	#10587623	C₄	1	1	yes	characteristic
28	#10587615	C₄	1	1	yes	characteristic
29	#10587582	C₄	1	1	yes	characteristic
30	#10587582	C₄	1	1	yes	characteristic
31	#10587579	C₂²	1	1	yes	characteristic
32	#10587579	C₂²	1	1	yes	characteristic
33	#10587570	C₄	1	1	yes	characteristic
34	#10587570	C₄	1	1	yes	characteristic
35	#10587568	C₄	1	1	yes	characteristic
36	#10587567	C₂²	1	1	yes	characteristic
37	#10587567	C₂²	1	1	yes	characteristic
38	#10587563	C₂²	1	1	yes	characteristic
39	C₂²⋊#3624	C₂²	1	1	yes	characteristic
40	C₂²⋊#3624	C₂²	1	1	yes	characteristic
41	C₂²⋊#3624	C₂²	1	1	yes	characteristic
42	C₂²⋊#3624	C₂²	1	1	yes	characteristic
43	C₂²⋊#3624	C₂²	1	1	yes	characteristic, complemented
44	C₂²⋊#3624	C₂²	1	1	yes	characteristic, complemented
45	#10587544	C₂²	1	1	yes	characteristic, complemented
46	#10587544	C₂²	1	1	yes	characteristic, complemented
47	#10587543	C₂²	1	1	yes	characteristic
48	#10587543	C₂²	1	1	yes	characteristic
49	#10587543	C₂²	1	1	yes	characteristic
50	#10587543	C₂²	1	1	yes	characteristic
51	#10587543	C₂²	1	1	yes	characteristic, complemented
52	#10587543	C₂²	1	1	yes	characteristic, complemented
53	#10587542	C₂²	1	1	yes	characteristic, complemented
54	#10587542	C₂²	1	1	yes	characteristic, complemented
55	C₄⋊#3624	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊#3624	C₂²	1	1	yes	characteristic, complemented
57	C₄×#3624	C₂²	1	1	yes	characteristic, complemented
58	C₄×#3624	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	C₂×C₄	1	1	yes	characteristic
60	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₇	C₂³	1	1	yes	characteristic
67	C₂²×Dic₁₇	C₂³	1	1	yes	characteristic
68	#63254	—	4	1	no	—
69	#63254	—	4	1	no	—
70	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
71	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄×D₁₇	—	4	1	no	—
73	C₂×C₄×D₁₇	—	4	1	no	—
74	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
80	#63247	—	4	1	no	—
81	C₂×#3624	—	4	1	no	—
82	C₂×#3624	C₂³	1	1	yes	characteristic
83	C₂×#3624	C₂³	1	1	yes	characteristic
84	C₂×#3624	C₂³	1	1	yes	characteristic
85	C₂×#3624	C₂³	1	1	yes	characteristic
86	C₂×#3624	C₂³	1	1	yes	characteristic
87	C₂×#3624	C₂³	1	1	yes	characteristic
88	C₂×#3624	C₂³	1	1	yes	characteristic
89	C₂×#3624	C₂³	1	1	yes	characteristic
90	#63243	—	4	1	no	—
91	#63243	—	4	1	no	—
92	#63243	—	4	1	no	—
93	#63242	—	4	1	no	—
94	#63242	—	4	1	no	—
95	#63242	—	4	1	no	—
96	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
98	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
99	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
100	C₂²⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
101	C₂²⋊Dic₁₇	C₂×C₄	1	1	yes	characteristic
102	#63227	C₂×C₄	1	1	yes	characteristic
103	#63227	C₂×C₄	1	1	yes	characteristic
104	#63227	C₂×C₄	1	1	yes	characteristic
105	#63227	C₂×C₄	1	1	yes	characteristic
106	#63227	C₂×C₄	1	1	yes	characteristic
107	#63227	C₂×C₄	1	1	yes	characteristic
108	#63225	C₂×C₄	1	1	yes	characteristic
109	#63225	C₂×C₄	1	1	yes	characteristic
110	#63225	C₂×C₄	1	1	yes	characteristic
111	#63225	C₂×C₄	1	1	yes	characteristic
112	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
113	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
114	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×D₁₇	—	2	2	no	—
117	C₂²×D₁₇	—	4	1	no	—
118	C₂²×D₁₇	—	4	1	no	—
119	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
122	D₄×C₁₇	—	4	1	no	—
123	C₂×C₄×C₁₇	—	4	1	no	—
124	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²⋊D₁₇	—	2	2	no	—
130	C₂²⋊D₁₇	—	4	1	no	—
131	C₂²⋊D₁₇	—	4	1	no	—
132	C₂²⋊D₁₇	—	4	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	2	1	no	—
135	C₂×Dic₁₇	—	2	1	no	—
136	C₂×Dic₁₇	—	2	1	no	—
137	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
139	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×Dic₁₇	C₂⁴	1	1	yes	characteristic
142	D₆₈	—	2	2	no	—
143	D₆₈	—	4	1	no	—
144	C₄×D₁₇	—	2	1	no	—
145	C₄×D₁₇	—	2	1	no	—
146	C₄×D₁₇	—	2	1	no	—
147	C₄×D₁₇	—	2	1	no	—
148	C₄×D₁₇	—	2	1	no	—
149	C₄×D₁₇	—	2	1	no	—
150	C₄×D₁₇	—	4	1	no	—
151	C₄×D₁₇	—	4	1	no	—
152	Dic₃₄	—	4	1	no	—
153	#3624	—	2	1	no	—
154	#3624	—	2	1	no	—
155	#3624	—	2	1	no	—
156	#3624	—	2	1	no	—
157	#3624	—	2	1	no	—
158	#3624	—	2	1	no	—
159	#3624	—	2	1	no	—
160	#3624	—	2	1	no	—
161	#2976	—	17	1	no	maximal, Sylow
162	C₂²×C₁₇	—	2	1	no	—
163	C₂²×C₁₇	—	2	1	no	—
164	C₂²×C₁₇	—	2	1	no	—
165	C₂²×C₁₇	—	2	1	no	—
166	C₂²×C₁₇	—	2	1	no	—
167	C₂²×C₁₇	—	2	1	no	—
168	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
169	C₂×D₁₇	—	2	1	no	—
170	C₂×D₁₇	—	2	1	no	—
171	C₂×D₁₇	—	2	1	no	—
172	C₂×D₁₇	—	2	1	no	—
173	C₂×D₁₇	—	2	1	no	—
174	C₂×D₁₇	—	2	1	no	—
175	C₂×D₁₇	—	2	1	no	—
176	C₂×D₁₇	—	2	1	no	—
177	C₂×D₁₇	—	4	4	no	—
178	C₂×D₁₇	#131	1	2	yes	—
179	C₄×C₁₇	—	2	1	no	—
180	C₄×C₁₇	—	2	1	no	—
181	C₄×C₁₇	—	2	1	no	—
182	C₄×C₁₇	—	2	1	no	—
183	C₄×C₁₇	—	2	1	no	—
184	Dic₁₇	—	2	1	no	—
185	Dic₁₇	—	2	1	no	—
186	Dic₁₇	—	2	1	no	—
187	Dic₁₇	—	2	1	no	—
188	Dic₁₇	H(5,𝔽₂)	1	1	yes	characteristic
189	Dic₁₇	H(5,𝔽₂)	1	1	yes	characteristic
190	#525	—	17	1	no	—
191	#436	—	17	1	no	—
192	#436	—	17	1	no	—
193	#435	—	17	1	no	—
194	#435	—	17	1	no	—
195	#435	—	17	1	no	—
196	#435	—	17	1	no	—
197	C₈×D₄	—	17	1	no	—
198	C₈×D₄	—	17	1	no	—
199	#433	—	17	1	no	—
200	#433	—	17	1	no	—
201	#408	—	17	1	no	—
202	#408	—	17	1	no	—
203	#407	—	17	1	no	—
204	C₂×C₂²⋊C₈	—	17	1	no	—
205	C₂×C₁₇	—	4	1	no	—
206	C₂×C₁₇	—	4	1	no	—
207	C₂×C₁₇	#518	1	1	yes	characteristic
208	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
209	C₂×C₁₇	#568	1	1	yes	characteristic
210	D₁₇	—	2	2	no	—
211	D₁₇	—	4	1	no	—
212	D₁₇	—	4	1	no	—
213	C₂×#41	—	17	1	no	—
214	C₂³×C₄	—	17	1	no	—
215	C₂×M₁₆	—	17	1	no	—
216	C₂×M₁₆	—	17	1	no	—
217	C₂×M₁₆	—	17	1	no	—
218	C₂×M₁₆	—	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	#123	—	17	1	no	—
224	#123	#595	1	1	yes	characteristic
225	C₂²⋊Q₈	—	17	1	no	—
226	C₂²⋊Q₈	—	17	1	no	—
227	#121	—	17	1	no	—
228	#121	—	17	1	no	—
229	#120	—	17	1	no	—
230	#120	—	17	1	no	—
231	C₄×D₄	—	17	1	no	—
232	C₄×D₄	—	17	1	no	—
233	C₄×D₄	—	17	1	no	—
234	C₄×D₄	—	17	1	no	—
235	#117	—	17	1	no	—
236	C₄⋊C₈	—	17	1	no	—
237	C₄⋊C₈	—	17	1	no	—
238	C₄⋊C₈	—	17	1	no	—
239	C₄⋊C₈	—	17	1	no	—
240	C₂²⋊C₈	—	17	1	no	—
241	C₂²⋊C₈	—	17	1	no	—
242	C₂²⋊C₈	—	17	1	no	—
243	C₂²⋊C₈	—	17	1	no	—
244	C₂²⋊C₈	—	17	1	no	—
245	C₂²⋊C₈	—	17	1	no	—
246	C₂²⋊C₈	—	17	1	no	—
247	C₂²⋊C₈	—	17	1	no	—
248	C₂²⋊C₈	—	17	1	no	—
249	C₂²⋊C₈	—	17	1	no	—
250	C₂²⋊C₈	—	17	1	no	—
251	C₂²⋊C₈	—	17	1	no	—
252	#97	—	17	1	no	—
253	#97	—	17	1	no	—
254	C₄×C₈	—	17	1	no	—
255	C₄×C₈	—	17	1	no	—
256	C₁₇	#2976	1	1	yes	characteristic, complemented, Sylow
257	C₂⁴	—	17	1	no	—
258	#41	—	68	1	no	—
259	#41	—	68	1	no	—
260	C₂×Q₈	—	17	1	no	—
261	C₂×D₄	—	17	1	no	—
262	C₂×D₄	—	17	1	no	—
263	C₂×D₄	—	17	1	no	—
264	C₂×D₄	—	17	1	no	—
265	C₂×D₄	—	17	1	no	—
266	C₂×D₄	—	17	1	no	—
267	C₂×D₄	C₂×#595	1	1	yes	characteristic
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	—	17	1	no	—
273	C₂²×C₄	—	17	1	no	—
274	C₂²×C₄	—	17	1	no	—
275	C₂²×C₄	—	68	1	no	—
276	C₂²×C₄	—	68	1	no	—
277	C₂²×C₄	C₂×#595	1	1	yes	characteristic
278	M₁₆	—	68	1	no	—
279	M₁₆	—	68	1	no	—
280	M₁₆	—	68	1	no	—
281	M₁₆	—	68	1	no	—
282	C₂×C₈	—	17	1	no	—
283	C₂×C₈	—	17	1	no	—
284	C₂×C₈	—	17	1	no	—
285	C₂×C₈	—	17	1	no	—
286	C₂×C₈	—	17	1	no	—
287	C₂×C₈	—	17	1	no	—
288	C₂×C₈	—	17	1	no	—
289	C₂×C₈	—	17	1	no	—
290	C₂×C₈	—	68	1	no	—
291	C₂×C₈	—	68	1	no	—
292	C₂×C₈	—	68	1	no	—
293	C₂×C₈	—	68	1	no	—
294	C₄⋊C₄	—	17	1	no	—
295	C₄⋊C₄	—	17	1	no	—
296	C₄⋊C₄	—	17	1	no	—
297	C₄⋊C₄	—	17	1	no	—
298	C₄⋊C₄	C₂×#595	1	1	yes	characteristic
299	C₄⋊C₄	C₂×#595	1	1	yes	characteristic
300	C₂²⋊C₄	—	17	1	no	—
301	C₂²⋊C₄	—	17	1	no	—
302	C₂²⋊C₄	—	17	1	no	—
303	C₂²⋊C₄	—	17	1	no	—
304	C₂²⋊C₄	—	17	1	no	—
305	C₂²⋊C₄	—	17	1	no	—
306	C₂²⋊C₄	—	17	1	no	—
307	C₂²⋊C₄	C₂×#595	1	1	yes	characteristic
308	C₂²⋊C₄	C₂×#595	1	1	yes	characteristic
309	C₂²⋊C₄	C₂×#595	1	1	yes	characteristic
310	C₄²	—	17	1	no	—
311	C₄²	—	17	1	no	—
312	C₂³	—	17	1	no	—
313	C₂³	—	17	1	no	—
314	C₂³	—	17	1	no	—
315	C₂³	—	34	2	no	—
316	C₂³	—	68	1	no	—
317	C₂³	—	68	1	no	—
318	C₂³	C₂²×#595	1	1	yes	characteristic
319	C₂³	C₂²×#595	1	1	yes	characteristic
320	Q₈	—	68	1	no	—
321	D₄	—	4	1	no	—
322	D₄	—	34	2	no	—
323	D₄	—	34	2	no	—
324	D₄	—	68	1	no	—
325	D₄	—	68	1	no	—
326	D₄	—	68	1	no	—
327	D₄	—	68	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	17	1	no	—
330	C₂×C₄	—	17	1	no	—
331	C₂×C₄	—	17	1	no	—
332	C₂×C₄	—	17	1	no	—
333	C₂×C₄	—	17	1	no	—
334	C₂×C₄	—	34	1	no	—
335	C₂×C₄	—	34	1	no	—
336	C₂×C₄	—	34	1	no	—
337	C₂×C₄	—	34	1	no	—
338	C₂×C₄	—	34	1	no	—
339	C₂×C₄	—	34	1	no	—
340	C₂×C₄	—	34	1	no	—
341	C₂×C₄	—	34	1	no	—
342	C₂×C₄	—	34	1	no	—
343	C₂×C₄	—	34	1	no	—
344	C₂×C₄	—	68	1	no	—
345	C₂×C₄	—	68	1	no	—
346	C₂×C₄	C₂²×#595	1	1	yes	characteristic
347	C₂×C₄	C₂²×#595	1	1	yes	characteristic
348	C₂×C₄	C₂²×#595	1	1	yes	characteristic
349	C₂×C₄	C₂²×#595	1	1	yes	characteristic
350	C₂×C₄	C₂²×#595	1	1	yes	characteristic
351	C₈	—	34	1	no	—
352	C₈	—	34	1	no	—
353	C₈	—	34	1	no	—
354	C₈	—	34	1	no	—
355	C₈	—	34	1	no	—
356	C₈	—	34	1	no	—
357	C₈	—	34	1	no	—
358	C₈	—	34	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	17	2	no	—
366	C₂²	—	34	1	no	—
367	C₂²	—	34	1	no	—
368	C₂²	—	34	1	no	—
369	C₂²	—	34	1	no	—
370	C₂²	—	34	1	no	—
371	C₂²	—	34	1	no	—
372	C₂²	—	34	1	no	—
373	C₂²	—	34	1	no	—
374	C₂²	—	68	4	no	—
375	C₂²	C₂³×#595	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	17	1	no	—
382	C₄	—	17	1	no	—
383	C₄	—	34	1	no	—
384	C₄	—	34	1	no	—
385	C₄	—	34	1	no	—
386	C₄	—	34	1	no	—
387	C₂	—	4	1	no	—
388	C₂	—	4	1	no	—
389	C₂	—	34	2	no	—
390	C₂	—	68	1	no	—
391	C₂	—	68	1	no	—
392	C₂	#11763876	1	1	yes	characteristic
393	C₂	D₁₇⋊#131	1	1	yes	characteristic
394	C₂	#11763888	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₁₇×#123	C₄
prime radical	#11763317	C₂
cosocle	C₂×Dic₁₇	C₂⁴
hypercenter	#123	#595
hypocenter	C₁₇	#2976
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. #11763843	C₂	semidirect
2. #11763842	C₂	semidirect
3. #11763841	C₂	semidirect
4. #11763839	C₂	semidirect
5. #11763839	C₂	semidirect
6. #11763838	C₂	semidirect
7. #11763837	C₂	semidirect
8. #11763836	C₂	semidirect
9. #11763833	C₂	semidirect
10. #11763833	C₂	semidirect
11. #11763783	C₂	semidirect
12. #11763782	C₂	semidirect
13. #11763780	C₂	semidirect
14. #11763777	C₂	semidirect
18. C₂×#63247	C₂²	semidirect
19. C₂²×#3624	C₂²	semidirect
22. C₂×#63243	C₂²	semidirect
25. C₂×#63242	C₂²	semidirect
43. C₂²⋊#3624	C₂²	semidirect
44. C₂²⋊#3624	C₂²	semidirect
45. #10587544	C₂²	semidirect
46. #10587544	C₂²	semidirect
51. #10587543	C₂²	semidirect
52. #10587543	C₂²	semidirect
53. #10587542	C₂²	semidirect
54. #10587542	C₂²	semidirect
55. C₄⋊#3624	C₂²	semidirect
56. C₄⋊#3624	C₂²	semidirect
57. C₄×#3624	C₂²	semidirect
58. C₄×#3624	C₂²	semidirect
256. C₁₇	#2976	semidirect

Maximal subgroups

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

subgroup	index
#11763843	2
#11763842	2
#11763841	2
#11763839	2
#11763838	2
#11763837	2
#11763836	2
#11763833	2
#11763783	2
#11763782	2
#11763780	2
#11763777	2
#11763317	2
#2976	17