Subgroups — #17978822

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#14545141	C₂	1	1	yes	maximal, characteristic, complemented
9	#14545119	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	2	yes	—
14	C₂²×#63216	C₂²	1	2	yes	—
15	C₄⋊#63216	C₂²	1	1	yes	characteristic
16	C₄⋊#63216	C₂²	1	2	yes	—
17	C₄⋊#63216	C₂²	1	2	yes	—
18	C₄⋊#63216	C₂²	1	1	yes	characteristic
19	C₄⋊#63216	C₂²	1	2	yes	—
20	C₄⋊#63216	C₂²	1	1	yes	characteristic
21	C₄⋊#63216	C₂²	1	1	yes	characteristic
22	C₂²⋊#63216	C₂²	1	1	yes	characteristic
23	C₂²⋊#63216	C₂²	1	2	yes	—
24	C₂²⋊#63216	C₂²	1	2	yes	—
25	C₂²⋊#63216	C₂²	1	1	yes	characteristic
26	#11763697	C₂²	1	2	yes	—
27	C₄×#63216	C₂²	1	2	yes	—
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	—	4	2	no	—
57	C₂×#63216	—	4	2	no	—
58	C₂×#63216	—	4	2	no	—
59	C₂×#63216	C₂³	1	2	yes	—
60	C₂×#63216	C₂³	1	1	yes	characteristic
61	C₂×#63216	C₂³	1	2	yes	—
62	C₂×#63216	C₂³	1	1	yes	characteristic
63	C₂×#63216	C₂³	1	1	yes	characteristic
64	C₂×#63216	C₂³	1	1	yes	characteristic
65	C₂²⋊#3622	C₂×C₄	1	2	yes	—
66	C₂²⋊#3622	C₂×C₄	1	2	yes	—
67	C₂²⋊#3622	C₂×C₄	1	2	yes	—
68	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
69	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
70	C₄⋊#3622	C₂×C₄	1	2	yes	—
71	C₄⋊#3622	C₂×C₄	1	2	yes	—
72	#10587474	C₂×C₄	1	2	yes	—
73	#10587474	C₂×C₄	1	2	yes	—
74	C₄×#3622	C₂×C₄	1	1	yes	characteristic
75	C₄×#3622	C₂×C₄	1	1	yes	characteristic
76	C₄×#3622	C₂³	1	1	yes	characteristic
77	C₄×#3622	C₂³	1	1	yes	characteristic
78	C₄×#3622	C₂³	1	2	yes	—
79	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
80	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
81	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
86	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
87	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
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	1	yes	characteristic
92	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
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	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
98	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
102	#63223	—	4	2	no	—
103	C₂×#3622	—	2	1	no	—
104	C₂×#3622	—	2	2	no	—
105	C₂×#3622	—	2	1	no	—
106	C₂×#3622	—	2	2	no	—
107	C₂×#3622	—	4	1	no	—
108	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
114	#63216	—	2	1	no	—
115	#63216	—	2	2	no	—
116	#63216	—	2	2	no	—
117	#63216	—	2	1	no	—
118	#63216	—	2	1	no	—
119	#63216	—	2	1	no	—
120	#33508	—	17	1	no	maximal, Sylow
121	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
122	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₁₇	—	2	2	no	—
124	D₄×C₁₇	—	2	4	no	—
125	D₄×C₁₇	—	2	2	no	—
126	C₂×C₄×C₁₇	—	2	2	no	—
127	C₂×C₄×C₁₇	—	2	1	no	—
128	C₂×C₄×C₁₇	—	2	2	no	—
129	C₂×C₄×C₁₇	—	2	1	no	—
130	C₂×C₄×C₁₇	—	2	2	no	—
131	C₂×C₄×C₁₇	—	2	2	no	—
132	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
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	2	yes	—
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	1	yes	characteristic
140	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
141	#3622	—	2	1	no	—
142	#3622	—	2	2	no	—
143	#3622	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3622	—	2	2	no	—
146	#3622	H(5,𝔽₂)	1	1	yes	characteristic
147	#3622	2₋¹⁺⁴	1	1	yes	characteristic
148	#2966	—	17	1	no	—
149	#2183	—	17	1	no	—
150	Q₈×C₁₆	—	17	1	no	—
151	#2169	—	17	1	no	—
152	#2168	—	17	2	no	—
153	D₄×C₁₆	—	17	1	no	—
154	D₄×C₁₆	—	17	2	no	—
155	#2164	—	17	2	no	—
156	#2162	—	17	1	no	—
157	#2151	—	17	2	no	—
158	C₂×C₄⋊C₁₆	—	17	1	no	—
159	C₂²×C₁₇	—	2	1	no	—
160	C₂²×C₁₇	—	2	1	no	—
161	C₂²×C₁₇	—	2	2	no	—
162	C₂²×C₁₇	—	2	2	no	—
163	C₂²×C₁₇	—	2	2	no	—
164	C₂²×C₁₇	#504	1	2	yes	—
165	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	2	1	no	—
168	C₄×C₁₇	—	2	1	no	—
169	C₄×C₁₇	—	2	2	no	—
170	C₄×C₁₇	—	2	2	no	—
171	C₄×C₁₇	—	2	1	no	—
172	C₄×C₁₇	—	2	2	no	—
173	C₄×C₁₇	#504	1	2	yes	—
174	C₄×C₁₇	#520	1	1	yes	characteristic
175	C₄×C₁₇	#568	1	1	yes	characteristic
176	#529	#595	1	1	yes	characteristic
177	C₂×M₃₂	—	17	2	no	—
178	C₂²×C₁₆	—	17	2	no	—
179	C₂²×C₁₆	—	17	2	no	—
180	#446	—	17	1	no	—
181	C₈×Q₈	—	17	1	no	—
182	#436	—	17	1	no	—
183	#435	—	17	2	no	—
184	C₈×D₄	—	17	2	no	—
185	C₈×D₄	—	17	1	no	—
186	#433	—	17	2	no	—
187	#432	—	17	1	no	—
188	#405	—	17	2	no	—
189	C₂×C₄×C₈	—	17	1	no	—
190	C₄⋊C₁₆	—	17	2	no	—
191	C₄⋊C₁₆	—	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	C₄⋊C₁₆	—	17	1	no	—
197	C₂²⋊C₁₆	—	17	1	no	—
198	C₂²⋊C₁₆	—	17	1	no	—
199	C₂²⋊C₁₆	—	17	2	no	—
200	C₂²⋊C₁₆	—	17	2	no	—
201	#346	—	17	2	no	—
202	C₄×C₁₆	—	17	1	no	—
203	C₄×C₁₆	—	17	1	no	—
204	C₄×C₁₆	—	17	2	no	—
205	C₂×C₁₇	—	2	2	no	—
206	C₂×C₁₇	—	4	2	no	—
207	C₂×C₁₇	#2988	1	1	yes	characteristic
208	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
209	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
210	C₂×M₁₆	—	17	2	no	—
211	C₂²×C₈	—	17	1	no	—
212	C₂²×C₈	—	17	2	no	—
213	C₂²×C₈	—	17	1	no	—
214	#126	#3624	1	2	yes	—
215	#125	#3624	1	1	yes	characteristic
216	#124	#3624	1	1	yes	characteristic
217	#123	#3624	1	2	yes	—
218	C₂²⋊Q₈	#3624	1	1	yes	characteristic
219	#121	#3624	1	1	yes	characteristic
220	C₄×Q₈	C₂×#595	1	1	yes	characteristic
221	C₄×D₄	C₂×#595	1	2	yes	—
222	C₄×D₄	C₂×#595	1	1	yes	characteristic
223	#117	C₂×#595	1	2	yes	—
224	C₂×C₄²	C₂×#595	1	1	yes	characteristic
225	M₃₂	—	68	2	no	—
226	C₂×C₁₆	—	17	1	no	—
227	C₂×C₁₆	—	17	1	no	—
228	C₂×C₁₆	—	17	1	no	—
229	C₂×C₁₆	—	17	2	no	—
230	C₂×C₁₆	—	17	1	no	—
231	C₂×C₁₆	—	17	2	no	—
232	C₂×C₁₆	—	68	2	no	—
233	C₂×C₁₆	—	68	2	no	—
234	C₄⋊C₈	—	17	1	no	—
235	C₄⋊C₈	—	17	1	no	—
236	C₄⋊C₈	—	17	2	no	—
237	C₄⋊C₈	—	17	2	no	—
238	C₂²⋊C₈	—	17	2	no	—
239	C₂²⋊C₈	—	17	2	no	—
240	C₂²⋊C₈	—	17	2	no	—
241	#97	—	17	2	no	—
242	#97	—	17	2	no	—
243	C₄×C₈	—	17	1	no	—
244	C₄×C₈	—	17	1	no	—
245	C₄×C₈	—	17	1	no	—
246	C₄×C₈	—	17	2	no	—
247	C₄×C₈	—	17	1	no	—
248	C₁₇	#33508	1	1	yes	characteristic, complemented, Sylow
249	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
250	C₂×D₄	C₂×#3624	1	2	yes	—
251	C₂×D₄	C₂×#3624	1	1	yes	characteristic
252	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
253	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
254	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
255	C₂²×C₄	C₂²×#595	1	2	yes	—
256	M₁₆	—	68	2	no	—
257	C₂×C₈	—	17	1	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	1	no	—
260	C₂×C₈	—	17	1	no	—
261	C₂×C₈	—	17	2	no	—
262	C₂×C₈	—	17	2	no	—
263	C₂×C₈	—	34	1	no	—
264	C₂×C₈	—	34	2	no	—
265	C₂×C₈	—	34	1	no	—
266	C₂×C₈	—	34	2	no	—
267	C₂×C₈	—	68	1	no	—
268	C₄⋊C₄	C₂×#3624	1	2	yes	—
269	C₄⋊C₄	C₂×#3624	1	2	yes	—
270	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
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	C₂²⋊C₄	C₂×#3624	1	2	yes	—
275	C₂²⋊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	C₄²	C₂×#3624	1	1	yes	characteristic
280	C₄²	C₂×#3624	1	1	yes	characteristic
281	C₄²	C₂²×#595	1	1	yes	characteristic
282	C₄²	C₂²×#595	1	1	yes	characteristic
283	C₄²	C₂²×#595	1	2	yes	—
284	C₁₆	—	34	1	no	—
285	C₁₆	—	34	2	no	—
286	C₁₆	—	34	2	no	—
287	C₁₆	—	34	1	no	—
288	C₁₆	—	34	1	no	—
289	C₁₆	—	34	1	no	—
290	C₂³	C₂²×#3624	1	2	yes	—
291	C₂³	C₂²×#3624	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
302	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
303	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
304	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
305	C₂×C₄	C₂²×#3624	1	2	yes	—
306	C₂×C₄	C₂²×#3624	1	2	yes	—
307	C₂×C₄	C₂²×#3624	1	2	yes	—
308	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
309	C₂×C₄	C₂³×#595	1	1	yes	characteristic
310	C₈	—	17	1	no	—
311	C₈	—	17	1	no	—
312	C₈	—	34	1	no	—
313	C₈	—	34	2	no	—
314	C₈	—	34	1	no	—
315	C₈	—	34	1	no	—
316	C₈	—	34	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	#11763867	1	2	yes	—
323	C₂²	C₂³×#3624	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
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	#11763867	1	2	yes	—
332	C₄	#11763878	1	1	yes	characteristic
333	C₄	#11763888	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	4	2	no	—
336	C₂	#14545699	1	1	yes	characteristic
337	C₂	C₂×#11763867	1	1	yes	characteristic
338	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	#33508
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. #14545141	C₂	semidirect
9. #14545119	C₂	semidirect
10. C₂×C₄⋊#63216	C₂	semidirect
248. C₁₇	#33508	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
#14545141	2
#14545119	2
C₂×C₄⋊#63216	2
#14537760	2
#33508	17