Subgroups — #17978684

Order 4352.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#63216	C₂	1	1	yes	maximal, characteristic, complemented
3	#14545168	C₂	1	1	yes	maximal, characteristic, complemented
4	#14545164	C₂	1	2	yes	maximal, complemented
5	D₄×#63216	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#63216	C₂	1	2	yes	maximal, complemented
7	#14545149	C₂	1	2	yes	maximal, complemented
8	#14545145	C₂	1	1	yes	maximal, characteristic, complemented
9	#14544993	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic, complemented
11	#14537760	C₂	1	1	yes	maximal, characteristic
12	C₂×#10587538	C₂²	1	2	yes	—
13	C₂²×#63216	C₂²	1	1	yes	characteristic
14	C₂²×#63216	C₂²	1	2	yes	—
15	C₂²×#63216	C₂²	1	1	yes	characteristic
16	C₄⋊#63216	C₂²	1	1	yes	characteristic
17	C₄⋊#63216	C₂²	1	2	yes	—
18	C₄⋊#63216	C₂²	1	2	yes	—
19	C₄⋊#63216	C₂²	1	1	yes	characteristic
20	C₂²⋊#63216	C₂²	1	2	yes	—
21	C₂²⋊#63216	C₂²	1	2	yes	—
22	C₂²⋊#63216	C₂²	1	2	yes	—
23	#11763697	C₂²	1	2	yes	—
24	#11763697	C₂²	1	2	yes	—
25	C₄×#63216	C₂²	1	1	yes	characteristic
26	C₄×#63216	C₂²	1	2	yes	—
27	C₄×#63216	C₂²	1	1	yes	characteristic
28	C₄×#63216	C₂²	1	1	yes	characteristic
29	C₄×#63216	C₂²	1	1	yes	characteristic
30	#11762940	C₄	1	1	yes	characteristic
31	Q₈×#3622	C₂²	1	1	yes	characteristic
32	#11762887	C₄	1	1	yes	characteristic
33	#11762881	C₄	1	2	yes	—
34	D₄×#3622	C₂²	1	2	yes	—
35	D₄×#3622	C₂²	1	1	yes	characteristic
36	#11762870	C₄	1	2	yes	—
37	#11762866	C₄	1	1	yes	characteristic
38	#11762765	C₂²	1	2	yes	—
39	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
40	C₁₇×#529	C₄	1	1	yes	characteristic
41	C₁₇×#126	C₈	1	2	yes	—
42	C₁₇×#125	C₈	1	1	yes	characteristic
43	C₁₇×#124	C₈	1	1	yes	characteristic
44	C₁₇×#123	C₈	1	2	yes	—
45	C₁₇×C₂²⋊Q₈	C₈	1	1	yes	characteristic
46	C₁₇×#121	C₈	1	1	yes	characteristic
47	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₁₇	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₁₇×#117	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
52	C₂×#63223	C₂×C₄	1	2	yes	—
53	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
54	C₂²×#3622	C₂³	1	1	yes	characteristic
55	C₂²×#3622	C₂³	1	2	yes	—
56	#10587538	—	2	4	no	—
57	C₂×#63216	—	2	2	no	—
58	C₂×#63216	—	2	4	no	—
59	C₂×#63216	—	2	2	no	—
60	C₂×#63216	C₂³	1	1	yes	characteristic
61	C₂×#63216	C₂³	1	1	yes	characteristic
62	C₂×#63216	C₂³	1	2	yes	—
63	C₂×#63216	C₂³	1	2	yes	—
64	C₂×#63216	C₂³	1	1	yes	characteristic
65	C₂×#63216	C₂³	1	1	yes	characteristic
66	C₂²⋊#3622	C₂×C₄	1	2	yes	—
67	C₂²⋊#3622	C₂×C₄	1	2	yes	—
68	C₂²⋊#3622	C₂×C₄	1	2	yes	—
69	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
70	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
71	C₄⋊#3622	C₂×C₄	1	2	yes	—
72	C₄⋊#3622	C₂×C₄	1	2	yes	—
73	#10587474	C₂×C₄	1	2	yes	—
74	#10587474	C₂×C₄	1	2	yes	—
75	C₄×#3622	C₂×C₄	1	1	yes	characteristic
76	C₄×#3622	C₂×C₄	1	1	yes	characteristic
77	C₄×#3622	C₂³	1	1	yes	characteristic
78	C₄×#3622	C₂³	1	2	yes	—
79	C₄×#3622	C₂³	1	1	yes	characteristic
80	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
81	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
82	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
87	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
88	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
89	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
90	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
91	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
92	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
93	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
94	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
95	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
96	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
97	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
98	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
100	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
103	#63223	—	2	4	no	—
104	C₂×#3622	—	2	2	no	—
105	C₂×#3622	—	2	1	no	—
106	C₂×#3622	—	2	1	no	—
107	C₂×#3622	—	2	2	no	—
108	C₂×#3622	—	2	2	no	—
109	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
110	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
111	C₂×#3622	C₂²×C₄	1	2	yes	—
112	C₂×#3622	C₂²×C₄	1	2	yes	—
113	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
114	C₂×#3622	C₂⁴	1	1	yes	characteristic
115	#63216	—	2	1	no	—
116	#63216	—	2	1	no	—
117	#63216	—	2	2	no	—
118	#63216	—	2	2	no	—
119	#63216	#41	1	2	yes	—
120	#63216	#41	1	2	yes	—
121	#33480	—	17	1	no	maximal, Sylow
122	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
123	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
124	Q₈×C₁₇	—	2	2	no	—
125	D₄×C₁₇	—	2	4	no	—
126	D₄×C₁₇	—	2	2	no	—
127	C₂×C₄×C₁₇	—	2	2	no	—
128	C₂×C₄×C₁₇	—	2	1	no	—
129	C₂×C₄×C₁₇	—	2	2	no	—
130	C₂×C₄×C₁₇	—	2	1	no	—
131	C₂×C₄×C₁₇	—	2	2	no	—
132	C₂×C₄×C₁₇	—	2	2	no	—
133	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
142	#3622	—	2	1	no	—
143	#3622	—	2	2	no	—
144	#3622	—	2	1	no	—
145	#3622	—	2	2	no	—
146	#3622	#131	1	2	yes	—
147	#3622	C₂×#41	1	1	yes	characteristic
148	#3622	C₂×#41	1	1	yes	characteristic
149	#2966	—	17	1	no	—
150	#2183	—	17	1	no	—
151	Q₈×C₁₆	—	17	1	no	—
152	#2169	—	17	1	no	—
153	#2168	—	17	2	no	—
154	D₄×C₁₆	—	17	1	no	—
155	D₄×C₁₆	—	17	2	no	—
156	#2164	—	17	2	no	—
157	#2163	—	17	1	no	—
158	#2108	—	17	2	no	—
159	C₂×C₄×C₁₆	—	17	1	no	—
160	C₂²×C₁₇	—	2	1	no	—
161	C₂²×C₁₇	—	2	1	no	—
162	C₂²×C₁₇	—	2	2	no	—
163	C₂²×C₁₇	—	2	2	no	—
164	C₂²×C₁₇	—	2	2	no	—
165	C₂²×C₁₇	#504	1	2	yes	—
166	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	2	1	no	—
169	C₄×C₁₇	—	2	1	no	—
170	C₄×C₁₇	—	2	2	no	—
171	C₄×C₁₇	—	2	2	no	—
172	C₄×C₁₇	—	2	1	no	—
173	C₄×C₁₇	—	2	2	no	—
174	C₄×C₁₇	#504	1	2	yes	—
175	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
176	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
177	#529	#595	1	1	yes	characteristic
178	C₂×M₃₂	—	17	2	no	—
179	C₂²×C₁₆	—	17	1	no	—
180	C₂²×C₁₆	—	17	1	no	—
181	C₂²×C₁₆	—	17	2	no	—
182	#446	—	17	1	no	—
183	C₈×Q₈	—	17	1	no	—
184	#436	—	17	1	no	—
185	#435	—	17	2	no	—
186	C₈×D₄	—	17	2	no	—
187	C₈×D₄	—	17	1	no	—
188	#433	—	17	2	no	—
189	#432	—	17	1	no	—
190	#405	—	17	2	no	—
191	C₂×C₄×C₈	—	17	1	no	—
192	C₄⋊C₁₆	—	17	1	no	—
193	C₄⋊C₁₆	—	17	1	no	—
194	C₄⋊C₁₆	—	17	2	no	—
195	C₄⋊C₁₆	—	17	2	no	—
196	C₂²⋊C₁₆	—	17	2	no	—
197	C₂²⋊C₁₆	—	17	2	no	—
198	C₂²⋊C₁₆	—	17	2	no	—
199	#346	—	17	2	no	—
200	#346	—	17	2	no	—
201	C₄×C₁₆	—	17	1	no	—
202	C₄×C₁₆	—	17	1	no	—
203	C₄×C₁₆	—	17	1	no	—
204	C₄×C₁₆	—	17	1	no	—
205	C₄×C₁₆	—	17	2	no	—
206	C₂×C₁₇	—	2	2	no	—
207	C₂×C₁₇	—	4	2	no	—
208	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
209	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
210	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
211	C₂×M₁₆	—	17	2	no	—
212	C₂²×C₈	—	17	1	no	—
213	C₂²×C₈	—	17	2	no	—
214	C₂²×C₈	—	17	1	no	—
215	#126	#3624	1	2	yes	—
216	#125	#3624	1	1	yes	characteristic
217	#124	#3624	1	1	yes	characteristic
218	#123	#3624	1	2	yes	—
219	C₂²⋊Q₈	#3624	1	1	yes	characteristic
220	#121	#3624	1	1	yes	characteristic
221	C₄×Q₈	C₂×#595	1	1	yes	characteristic
222	C₄×D₄	C₂×#595	1	2	yes	—
223	C₄×D₄	C₂×#595	1	1	yes	characteristic
224	#117	C₂×#595	1	2	yes	—
225	C₂×C₄²	C₂×#595	1	1	yes	characteristic
226	M₃₂	—	34	4	no	—
227	C₂×C₁₆	—	17	1	no	—
228	C₂×C₁₆	—	17	1	no	—
229	C₂×C₁₆	—	17	1	no	—
230	C₂×C₁₆	—	17	1	no	—
231	C₂×C₁₆	—	17	2	no	—
232	C₂×C₁₆	—	17	2	no	—
233	C₂×C₁₆	—	34	2	no	—
234	C₂×C₁₆	—	34	4	no	—
235	C₂×C₁₆	—	34	2	no	—
236	C₄⋊C₈	—	17	1	no	—
237	C₄⋊C₈	—	17	1	no	—
238	C₄⋊C₈	—	17	2	no	—
239	C₄⋊C₈	—	17	2	no	—
240	C₂²⋊C₈	—	17	2	no	—
241	C₂²⋊C₈	—	17	2	no	—
242	C₂²⋊C₈	—	17	2	no	—
243	#97	—	17	2	no	—
244	#97	—	17	2	no	—
245	C₄×C₈	—	17	1	no	—
246	C₄×C₈	—	17	1	no	—
247	C₄×C₈	—	17	1	no	—
248	C₄×C₈	—	17	1	no	—
249	C₄×C₈	—	17	2	no	—
250	C₁₇	#33480	1	1	yes	characteristic, complemented, Sylow
251	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
252	C₂×D₄	C₂×#3624	1	2	yes	—
253	C₂×D₄	C₂×#3624	1	1	yes	characteristic
254	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
255	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
256	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
257	C₂²×C₄	C₂²×#595	1	2	yes	—
258	M₁₆	—	34	4	no	—
259	C₂×C₈	—	17	1	no	—
260	C₂×C₈	—	17	1	no	—
261	C₂×C₈	—	17	1	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	17	2	no	—
264	C₂×C₈	—	17	2	no	—
265	C₂×C₈	—	34	1	no	—
266	C₂×C₈	—	34	1	no	—
267	C₂×C₈	—	34	2	no	—
268	C₂×C₈	—	34	2	no	—
269	C₂×C₈	—	34	2	no	—
270	C₄⋊C₄	C₂×#3624	1	2	yes	—
271	C₄⋊C₄	C₂×#3624	1	2	yes	—
272	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
273	C₄⋊C₄	C₂×#3624	1	2	yes	—
274	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
275	C₄⋊C₄	C₂×#3624	1	2	yes	—
276	C₂²⋊C₄	C₂×#3624	1	2	yes	—
277	C₂²⋊C₄	C₂×#3624	1	2	yes	—
278	C₂²⋊C₄	C₂×#3624	1	2	yes	—
279	C₂²⋊C₄	C₂×#3624	1	2	yes	—
280	C₂²⋊C₄	C₂×#3624	1	2	yes	—
281	C₄²	C₂×#3624	1	1	yes	characteristic
282	C₄²	C₂×#3624	1	1	yes	characteristic
283	C₄²	C₂²×#595	1	1	yes	characteristic
284	C₄²	C₂²×#595	1	1	yes	characteristic
285	C₄²	C₂²×#595	1	2	yes	—
286	C₁₆	—	17	2	no	—
287	C₁₆	—	17	2	no	—
288	C₁₆	—	34	1	no	—
289	C₁₆	—	34	1	no	—
290	C₁₆	—	34	2	no	—
291	C₁₆	—	34	2	no	—
292	C₂³	C₂²×#3624	1	2	yes	—
293	C₂³	C₂²×#3624	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
304	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
305	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
306	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
307	C₂×C₄	C₂²×#3624	1	2	yes	—
308	C₂×C₄	C₂²×#3624	1	2	yes	—
309	C₂×C₄	C₂²×#3624	1	2	yes	—
310	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
311	C₂×C₄	C₂³×#595	1	1	yes	characteristic
312	C₈	—	17	1	no	—
313	C₈	—	17	1	no	—
314	C₈	—	17	2	no	—
315	C₈	—	34	1	no	—
316	C₈	—	34	2	no	—
317	C₈	—	34	1	no	—
318	C₈	—	34	2	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#11763867	1	2	yes	—
325	C₂²	C₂³×#3624	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	#11763867	1	2	yes	—
334	C₄	#41×#595	1	1	yes	characteristic
335	C₄	D₁₇⋊#131	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	#41×#3624	1	1	yes	characteristic
339	C₂	C₂×#11763867	1	1	yes	characteristic
340	C₂	C₂×#11763867	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#529	C₄
prime radical	C₁₇×#121	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#529	#595
hypocenter	C₁₇	#33480
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
2. Q₈×#63216	C₂	semidirect
3. #14545168	C₂	semidirect
4. #14545164	C₂	semidirect
5. D₄×#63216	C₂	semidirect
6. D₄×#63216	C₂	semidirect
7. #14545149	C₂	semidirect
8. #14545145	C₂	semidirect
9. #14544993	C₂	semidirect
10. C₂×C₄×#63216	C₂	semidirect
250. C₁₇	#33480	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
Q₈×#63216	2
#14545168	2
#14545164	2
D₄×#63216	2
#14545149	2
#14545145	2
#14544993	2
C₂×C₄×#63216	2
#14537760	2
#33480	17