Subgroups — #17978678

Order 4352.

558 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	2	yes	maximal, complemented
2	#14545168	C₂	1	2	yes	maximal, complemented
3	#14545164	C₂	1	4	yes	maximal, complemented
4	#14545141	C₂	1	2	yes	maximal, complemented
5	#14545141	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10587538	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×#3622	C₂	1	1	yes	maximal, characteristic
9	C₂×#10587538	C₂²	1	4	yes	—
10	C₂²×#63216	C₂²	1	2	yes	—
11	C₄⋊#63216	C₂²	1	4	yes	—
12	C₄⋊#63216	C₂²	1	2	yes	—
13	C₂²⋊#63216	C₂²	1	2	yes	—
14	C₂²⋊#63216	C₂²	1	4	yes	—
15	#11763697	C₂²	1	4	yes	—
16	C₄×#63216	C₂²	1	2	yes	—
17	C₄×#63216	C₂²	1	1	yes	characteristic
18	C₄×#63216	C₂²	1	1	yes	characteristic
19	C₄×#63216	C₂²	1	2	yes	—
20	Q₈×#3622	C₄	1	1	yes	characteristic
21	Q₈×#3622	C₂²	1	1	yes	characteristic
22	D₄×#3622	C₄	1	1	yes	characteristic
23	D₄×#3622	C₄	1	2	yes	—
24	D₄×#3622	C₂²	1	2	yes	—
25	D₄×#3622	C₂²	1	1	yes	characteristic
26	#11762862	C₄	1	2	yes	—
27	#11762862	C₄	1	1	yes	characteristic
28	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
29	C₂×C₄×#3622	C₂²	1	2	yes	—
30	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
31	C₂×#41×C₁₇	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₁₇	C₈	1	2	yes	—
35	C₄×D₄×C₁₇	C₈	1	1	yes	characteristic
36	C₄×D₄×C₁₇	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₁₇×#117	C₈	1	2	yes	—
39	C₁₇×#117	C₈	1	1	yes	characteristic
40	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
43	C₂²×#3622	C₂×C₄	1	2	yes	—
44	C₂²×#3622	C₂³	1	1	yes	characteristic
45	C₂²×#3622	C₂³	1	2	yes	—
46	#10587538	—	2	8	no	—
47	C₂×#63216	—	2	4	no	—
48	C₂×#63216	C₂³	1	2	yes	—
49	C₂×#63216	C₂³	1	4	yes	—
50	C₂×#63216	C₂³	1	2	yes	—
51	C₂²⋊#3622	C₂×C₄	1	2	yes	—
52	C₂²⋊#3622	C₂×C₄	1	2	yes	—
53	C₂²⋊#3622	C₂×C₄	1	2	yes	—
54	C₄⋊#3622	C₂×C₄	1	2	yes	—
55	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#3622	C₂×C₄	1	2	yes	—
57	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
58	C₄×#3622	C₂×C₄	1	1	yes	characteristic
59	C₄×#3622	C₂×C₄	1	2	yes	—
60	C₄×#3622	C₂×C₄	1	1	yes	characteristic
61	C₄×#3622	C₂×C₄	1	2	yes	—
62	C₄×#3622	C₂³	1	1	yes	characteristic
63	C₄×#3622	C₂³	1	2	yes	—
64	C₄×#3622	C₂³	1	1	yes	characteristic
65	#41×C₁₇	—	2	4	no	—
66	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
76	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
78	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
80	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
81	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
82	C₄²×C₁₇	C₂×C₈	1	2	yes	—
83	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₇	C₂×C₈	1	2	yes	—
85	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
86	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×#3622	—	2	2	no	—
90	C₂×#3622	—	2	4	no	—
91	C₂×#3622	—	2	2	no	—
92	C₂×#3622	—	2	4	no	—
93	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
94	C₂×#3622	C₂²×C₄	1	2	yes	—
95	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
96	C₂×#3622	C₂²×C₄	1	2	yes	—
97	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
98	C₂×#3622	C₂⁴	1	1	yes	characteristic
99	#63216	—	2	2	no	—
100	#63216	—	2	4	no	—
101	#63216	#41	1	4	yes	complemented
102	#33479	—	17	1	no	maximal, Sylow
103	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
104	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
105	Q₈×C₁₇	—	2	2	no	—
106	D₄×C₁₇	—	2	4	no	—
107	D₄×C₁₇	—	2	2	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	2	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	2	no	—
112	C₂×C₄×C₁₇	—	2	4	no	—
113	C₂×C₄×C₁₇	—	2	2	no	—
114	C₂×C₄×C₁₇	—	2	4	no	—
115	C₂×C₄×C₁₇	M₃₂	1	4	yes	complemented
116	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
125	#3622	—	2	1	no	—
126	#3622	—	2	2	no	—
127	#3622	—	2	1	no	—
128	#3622	—	2	2	no	—
129	#3622	#131	1	2	yes	—
130	#3622	C₂×#41	1	2	yes	—
131	C₈×#41	—	17	1	no	—
132	#2183	—	17	2	no	—
133	#2169	—	17	2	no	—
134	#2168	—	17	4	no	—
135	#2162	—	17	2	no	—
136	#2162	—	17	1	no	—
137	C₄×M₃₂	—	17	2	no	—
138	C₂×C₄×C₁₆	—	17	1	no	—
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	—	2	2	no	—
142	C₂²×C₁₇	—	2	2	no	—
143	C₂²×C₁₇	—	2	2	no	—
144	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
145	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
146	C₄×C₁₇	—	2	2	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	2	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	2	no	—
151	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
152	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
153	C₄×C₁₇	#504	1	2	yes	—
154	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
155	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
156	C₄×#41	#595	1	1	yes	characteristic
157	C₂×M₃₂	—	17	4	no	—
158	C₂²×C₁₆	—	17	2	no	—
159	C₈×Q₈	—	17	1	no	—
160	C₈×Q₈	—	17	1	no	—
161	C₈×D₄	—	17	2	no	—
162	C₈×D₄	—	17	1	no	—
163	C₈×D₄	—	17	1	no	—
164	C₈×D₄	—	17	2	no	—
165	#431	—	17	2	no	—
166	#431	—	17	1	no	—
167	C₂×C₄×C₈	—	17	1	no	—
168	C₂×C₄×C₈	—	17	2	no	—
169	C₄⋊C₁₆	—	17	4	no	—
170	C₄⋊C₁₆	—	17	2	no	—
171	C₂²⋊C₁₆	—	17	2	no	—
172	C₂²⋊C₁₆	—	17	4	no	—
173	#346	—	17	4	no	—
174	C₄×C₁₆	—	17	1	no	—
175	C₄×C₁₆	—	17	1	no	—
176	C₄×C₁₆	—	17	2	no	—
177	C₄×C₁₆	—	17	2	no	—
178	C₂×C₁₇	—	2	2	no	—
179	C₂×C₁₇	—	4	2	no	—
180	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
181	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
182	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
183	C₂×#41	#3624	1	1	yes	characteristic
184	C₂²×C₈	—	17	1	no	—
185	C₂²×C₈	—	17	2	no	—
186	C₂²×C₈	—	17	1	no	—
187	C₂²×C₈	—	17	2	no	—
188	C₄×Q₈	#3624	1	1	yes	characteristic
189	C₄×Q₈	C₂×#595	1	1	yes	characteristic
190	C₄×D₄	#3624	1	2	yes	—
191	C₄×D₄	#3624	1	1	yes	characteristic
192	C₄×D₄	C₂×#595	1	2	yes	—
193	C₄×D₄	C₂×#595	1	1	yes	characteristic
194	#117	#3624	1	2	yes	—
195	#117	#3624	1	1	yes	characteristic
196	C₂×C₄²	C₂×#595	1	2	yes	—
197	C₂×C₄²	C₂×#595	1	1	yes	characteristic
198	M₃₂	—	34	8	no	—
199	C₂×C₁₆	—	17	2	no	—
200	C₂×C₁₆	—	17	2	no	—
201	C₂×C₁₆	—	17	4	no	—
202	C₂×C₁₆	—	34	4	no	—
203	C₄⋊C₈	—	17	2	no	—
204	C₄⋊C₈	—	17	1	no	—
205	C₄⋊C₈	—	17	2	no	—
206	C₄⋊C₈	—	17	1	no	—
207	C₂²⋊C₈	—	17	2	no	—
208	C₂²⋊C₈	—	17	2	no	—
209	C₂²⋊C₈	—	17	2	no	—
210	C₄×C₈	—	17	1	no	—
211	C₄×C₈	—	17	2	no	—
212	C₄×C₈	—	17	1	no	—
213	C₄×C₈	—	17	2	no	—
214	C₄×C₈	—	17	1	no	—
215	C₄×C₈	—	17	2	no	—
216	C₄×C₈	—	17	1	no	—
217	C₁₇	#33479	1	1	yes	characteristic, complemented, Sylow
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
220	C₂×D₄	C₂×#3624	1	2	yes	—
221	C₂×D₄	C₂×#3624	1	1	yes	characteristic
222	C₂²×C₄	C₂×#3624	1	2	yes	—
223	C₂²×C₄	C₂×#3624	1	2	yes	—
224	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
225	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
226	C₂²×C₄	C₂²×#595	1	2	yes	—
227	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
228	C₂×C₈	—	17	1	no	—
229	C₂×C₈	—	17	1	no	—
230	C₂×C₈	—	17	1	no	—
231	C₂×C₈	—	17	2	no	—
232	C₂×C₈	—	17	1	no	—
233	C₂×C₈	—	17	2	no	—
234	C₂×C₈	—	34	2	no	—
235	C₂×C₈	—	34	4	no	—
236	C₂×C₈	—	34	2	no	—
237	C₂×C₈	—	34	4	no	—
238	C₄⋊C₄	C₂×#3624	1	2	yes	—
239	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
240	C₄⋊C₄	C₂×#3624	1	2	yes	—
241	C₄⋊C₄	C₂×#3624	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×#3624	1	2	yes	—
243	C₂²⋊C₄	C₂×#3624	1	2	yes	—
244	C₂²⋊C₄	C₂×#3624	1	2	yes	—
245	C₄²	C₂×#3624	1	2	yes	—
246	C₄²	C₂×#3624	1	1	yes	characteristic
247	C₄²	C₂×#3624	1	2	yes	—
248	C₄²	C₂×#3624	1	1	yes	characteristic
249	C₄²	C₂²×#595	1	2	yes	—
250	C₄²	C₂²×#595	1	1	yes	characteristic
251	C₄²	C₂²×#595	1	1	yes	characteristic
252	C₁₆	—	17	4	no	—
253	C₁₆	—	34	2	no	—
254	C₁₆	—	34	4	no	—
255	C₂³	C₂²×#3624	1	2	yes	—
256	C₂³	C₂²×#3624	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	#10587538	1	4	yes	complemented
268	C₂×C₄	C₂²×#3624	1	2	yes	—
269	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
270	C₂×C₄	C₂²×#3624	1	2	yes	—
271	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
272	C₂×C₄	C₂²×#3624	1	2	yes	—
273	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
274	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
275	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
276	C₂×C₄	C₂³×#595	1	1	yes	characteristic
277	C₈	—	17	2	no	—
278	C₈	—	17	2	no	—
279	C₈	—	34	1	no	—
280	C₈	—	34	2	no	—
281	C₈	—	34	1	no	—
282	C₈	—	34	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂×#10587538	1	2	yes	—
289	C₂²	C₂³×#3624	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	C₂×#10587538	1	2	yes	—
296	C₄	C₂×#10587538	1	2	yes	—
297	C₄	#11763867	1	2	yes	—
298	C₄	#41×#595	1	1	yes	characteristic
299	C₄	D₁₇⋊#131	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#41×#3624	1	1	yes	characteristic
303	C₂	C₂²×#10587538	1	1	yes	characteristic
304	C₂	C₂×#11763867	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×#595
Fitting	C₄×#41×C₁₇	C₄
prime radical	C₂×#41×C₁₇	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₄×#41	#595
hypocenter	C₁₇	#33479
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
2. #14545168	C₂	semidirect
3. #14545164	C₂	semidirect
4. #14545141	C₂	semidirect
5. #14545141	C₂	semidirect
6. C₄×#10587538	C₂	semidirect
7. C₂×C₄×#63216	C₂	semidirect
101. #63216	#41	semidirect
115. C₂×C₄×C₁₇	M₃₂	semidirect
217. C₁₇	#33479	semidirect
267. C₂×C₄	#10587538	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
#14545168	2
#14545164	2
#14545141	2
C₄×#10587538	2
C₂×C₄×#63216	2
#41×#3622	2
#33479	17