Subgroups — C₄⋊C₈×#3633 #17981622

Order 4352.

550 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#3633	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3633	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#3633	C₂	1	2	yes	maximal
4	(C₂×#3633)⋊C₈	C₂	1	2	yes	maximal
5	C₄⋊C₈×#595	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3633	C₄	1	2	yes	—
7	C₂×C₄×#3633	C₄	1	2	yes	—
8	C₂×C₄×#3633	C₄	1	2	yes	—
9	C₂×C₄×#3633	C₂²	1	2	yes	—
10	C₂×C₄×#3633	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3633	C₂²	1	1	yes	characteristic
12	(C₂×#3633)⋊C₄	C₄	1	2	yes	—
13	C₈×#3633	—	2	4	no	—
14	#11763810	C₄	1	1	yes	characteristic
15	C₂×C₈×#595	C₂²	1	2	yes	—
16	C₄²×#595	C₂²	1	1	yes	characteristic
17	#11763674	C₄	1	2	yes	—
18	C₄⋊C₈×D₁₇	C₄	1	1	yes	characteristic
19	C₂²×#3633	C₂×C₄	1	4	yes	—
20	C₂²×#3633	C₂×C₄	1	2	yes	—
21	C₂²×#3633	C₂×C₄	1	2	yes	—
22	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#595	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×#595	C₂³	1	1	yes	characteristic
28	C₂×#63242	C₂×C₄	1	2	yes	—
29	C₄⋊#3633	C₈	1	8	yes	complemented
30	C₄×#3633	—	2	4	no	—
31	C₄×#3633	—	2	2	no	—
32	C₄×#3633	—	2	1	no	—
33	C₄×#3633	—	2	1	no	—
34	C₄×#3633	—	2	2	no	—
35	C₄×#3633	C₈	1	8	yes	complemented
36	C₄×#3633	D₄	1	2	yes	—
37	C₄×#3633	Q₈	1	2	yes	—
38	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
39	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
40	#10587544	C₈	1	2	yes	complemented
41	C₄⋊#3624	C₈	1	2	yes	complemented
42	C₈×#595	—	2	2	no	—
43	C₈×#595	—	2	2	no	—
44	C₁₇×C₄⋊C₈	C₈	1	1	yes	characteristic, complemented
45	#10587485	C₈	1	2	yes	complemented
46	C₄⋊#3622	C₈	1	1	yes	characteristic, complemented
47	C₂²×#595	C₄²	1	1	yes	characteristic
48	C₂²×#595	C₄²	1	1	yes	characteristic
49	C₂²×#595	C₂²×C₄	1	1	yes	characteristic
50	C₂²×#595	C₂²×C₄	1	1	yes	characteristic
51	C₂×#3633	—	2	2	no	—
52	C₂×#3633	—	2	4	no	—
53	C₂×#3633	—	2	2	no	—
54	C₂×#3633	—	2	4	no	—
55	C₂×#3633	C₄⋊C₄	1	4	yes	—
56	C₂×#3633	C₂×C₈	1	4	yes	—
57	C₂×#3633	C₂×C₈	1	8	yes	—
58	C₂×#3633	C₂×C₈	1	4	yes	—
59	C₂×#3633	M₁₆	1	4	yes	—
60	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
61	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
62	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×#3624	C₂×C₈	1	4	yes	—
64	C₄×#595	—	2	2	no	—
65	C₄×#595	—	2	2	no	—
66	C₄×#595	—	2	1	no	—
67	C₄×#595	—	2	1	no	—
68	C₄×#595	C₂×C₈	1	4	yes	—
69	C₄×#595	C₂×C₈	1	4	yes	—
70	C₄×#595	C₂×D₄	1	1	yes	characteristic
71	C₄×#595	C₂×Q₈	1	1	yes	characteristic
72	C₄×#595	#41	1	1	yes	characteristic
73	C₄×#595	#41	1	1	yes	characteristic
74	#63242	—	2	2	no	—
75	#63242	—	2	2	no	—
76	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
77	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
78	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
79	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
80	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
81	C₂×#3622	C₂×C₈	1	2	yes	—
82	C₈×D₁₇	—	2	2	no	—
83	C₈×D₁₇	—	2	2	no	—
84	C₈×C₄⋊C₈	—	17	1	no	maximal, Sylow
85	C₂²×D₁₇	C₂×C₄²	1	1	yes	characteristic
86	C₂×#595	—	2	1	no	—
87	C₂×#595	—	2	1	no	—
88	C₂×#595	—	2	1	no	—
89	C₂×#595	—	2	1	no	—
90	C₂×#595	C₄×C₈	1	2	yes	—
91	C₂×#595	C₄×C₈	1	2	yes	—
92	C₂×#595	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×#595	#117	1	1	yes	characteristic
94	C₂×#595	C₄×D₄	1	1	yes	characteristic
95	C₂×#595	C₄×Q₈	1	1	yes	characteristic
96	C₂×#595	C₂²×C₈	1	2	yes	—
97	C₂×#595	C₂²×C₈	1	2	yes	—
98	C₂×#595	C₂×M₁₆	1	2	yes	—
99	C₂×#595	#131	1	2	yes	—
100	#3633	—	2	4	no	—
101	#3633	—	2	8	no	—
102	#3633	C₄⋊C₈	1	8	yes	direct product factor
103	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
107	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
108	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
109	C₄×D₁₇	—	2	1	no	—
110	C₄×D₁₇	—	2	1	no	—
111	C₄×D₁₇	C₄×C₈	1	2	yes	—
112	C₄×D₁₇	C₄×C₈	1	2	yes	—
113	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
114	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
115	C₄×D₁₇	#131	1	1	yes	characteristic
116	C₄×D₁₇	#131	1	1	yes	characteristic
117	#3624	—	2	4	no	—
118	C₈×C₁₇	—	2	2	no	—
119	#3622	—	2	2	no	—
120	C₄×C₄⋊C₈	—	17	1	no	—
121	C₄×C₄⋊C₈	—	17	1	no	—
122	C₄²×C₈	—	17	1	no	—
123	C₂×C₈²	—	17	2	no	—
124	#1273	—	17	2	no	—
125	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
126	C₂×D₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂×D₁₇	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂×D₁₇	C₂×C₄×C₈	1	1	yes	characteristic
129	C₂×D₁₇	C₄×M₁₆	1	1	yes	characteristic
130	C₂×D₁₇	#405	1	1	yes	characteristic
131	C₂×D₁₇	#405	1	1	yes	characteristic
132	#595	—	2	2	no	—
133	#595	—	2	2	no	—
134	#595	C₂×C₄⋊C₈	1	2	yes	—
135	#595	#431	1	2	yes	—
136	#595	C₈×D₄	1	2	yes	—
137	#595	C₈×Q₈	1	2	yes	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	C₈²	1	2	yes	complemented
140	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
141	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
142	Dic₁₇	—	2	1	no	—
143	Dic₁₇	C₈²	1	2	yes	complemented
144	Dic₁₇	C₈×D₄	1	1	yes	characteristic
145	Dic₁₇	C₈×Q₈	1	1	yes	characteristic
146	C₂×C₄⋊C₈	—	17	1	no	—
147	C₂×C₄⋊C₈	—	17	2	no	—
148	C₂×C₄⋊C₈	—	17	1	no	—
149	C₂×C₄×C₈	—	17	1	no	—
150	C₂×C₄×C₈	—	17	2	no	—
151	C₂×C₄×C₈	—	17	2	no	—
152	C₂×C₄×C₈	—	17	2	no	—
153	C₂×C₄×C₈	—	17	2	no	—
154	C₂×C₄×C₈	—	17	1	no	—
155	C₄³	—	17	1	no	—
156	#336	—	17	2	no	—
157	#336	—	17	2	no	—
158	C₈²	—	34	4	no	—
159	C₂×C₁₇	C₂×C₈²	1	1	yes	characteristic
160	C₂×C₁₇	C₈×M₁₆	1	1	yes	characteristic
161	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
162	D₁₇	C₈×M₁₆	1	1	yes	characteristic
163	D₁₇	#1450	1	1	yes	characteristic
164	D₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
165	D₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₈	—	17	2	no	—
167	C₂²×C₈	—	17	2	no	—
168	C₂²×C₈	—	17	4	no	—
169	C₂²×C₈	—	17	2	no	—
170	C₂²×C₈	—	17	2	no	—
171	C₂×C₄²	—	17	1	no	—
172	C₂×C₄²	—	17	1	no	—
173	C₂×C₄²	—	17	1	no	—
174	C₂×C₄²	—	17	1	no	—
175	C₂×C₄²	—	17	1	no	—
176	C₂×C₄²	—	17	1	no	—
177	C₂×C₄²	—	17	1	no	—
178	C₄⋊C₈	—	17	8	no	—
179	C₄⋊C₈	—	17	2	no	—
180	C₄⋊C₈	—	17	2	no	—
181	C₄⋊C₈	—	17	2	no	—
182	C₄⋊C₈	—	17	1	no	—
183	C₄⋊C₈	#3633	1	1	yes	characteristic, direct product factor
184	C₄×C₈	—	17	8	no	—
185	C₄×C₈	—	17	2	no	—
186	C₄×C₈	—	17	2	no	—
187	C₄×C₈	—	34	2	no	—
188	C₄×C₈	—	34	2	no	—
189	C₄×C₈	—	34	1	no	—
190	C₄×C₈	—	34	1	no	—
191	C₄×C₈	—	34	2	no	—
192	C₄×C₈	—	34	2	no	—
193	C₄×C₈	—	34	4	no	—
194	C₁₇	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
195	C₂²×C₄	—	17	1	no	—
196	C₂²×C₄	—	17	1	no	—
197	C₂²×C₄	—	17	1	no	—
198	C₂²×C₄	—	17	1	no	—
199	C₂²×C₄	—	17	1	no	—
200	C₂²×C₄	—	17	1	no	—
201	C₂²×C₄	—	17	1	no	—
202	C₂×C₈	—	17	4	no	—
203	C₂×C₈	—	17	4	no	—
204	C₂×C₈	—	17	4	no	—
205	C₂×C₈	—	17	4	no	—
206	C₂×C₈	—	17	4	no	—
207	C₂×C₈	—	17	8	no	—
208	C₂×C₈	—	17	2	no	—
209	C₂×C₈	—	34	2	no	—
210	C₂×C₈	—	34	2	no	—
211	C₂×C₈	—	34	2	no	—
212	C₂×C₈	—	34	2	no	—
213	C₂×C₈	—	34	4	no	—
214	C₂×C₈	—	34	2	no	—
215	C₂×C₈	—	34	2	no	—
216	C₂×C₈	—	34	4	no	—
217	C₂×C₈	C₂×#3633	1	2	yes	—
218	C₄²	—	17	4	no	—
219	C₄²	—	17	4	no	—
220	C₄²	—	17	1	no	—
221	C₄²	—	17	1	no	—
222	C₄²	—	17	1	no	—
223	C₄²	—	17	1	no	—
224	C₄²	—	17	1	no	—
225	C₄²	—	17	1	no	—
226	C₄²	—	17	1	no	—
227	C₄²	—	34	2	no	—
228	C₄²	—	34	2	no	—
229	C₄²	—	34	1	no	—
230	C₄²	—	34	1	no	—
231	C₄²	C₂×#3633	1	1	yes	characteristic
232	C₂³	—	17	1	no	—
233	C₂×C₄	—	17	1	no	—
234	C₂×C₄	—	17	2	no	—
235	C₂×C₄	—	17	2	no	—
236	C₂×C₄	—	17	1	no	—
237	C₂×C₄	—	17	1	no	—
238	C₂×C₄	—	17	1	no	—
239	C₂×C₄	—	17	2	no	—
240	C₂×C₄	—	17	2	no	—
241	C₂×C₄	—	17	1	no	—
242	C₂×C₄	—	17	1	no	—
243	C₂×C₄	—	17	2	no	—
244	C₂×C₄	—	17	2	no	—
245	C₂×C₄	—	17	1	no	—
246	C₂×C₄	—	17	1	no	—
247	C₂×C₄	—	17	1	no	—
248	C₂×C₄	—	17	1	no	—
249	C₂×C₄	—	17	2	no	—
250	C₂×C₄	—	17	2	no	—
251	C₂×C₄	—	17	1	no	—
252	C₂×C₄	—	34	1	no	—
253	C₂×C₄	—	34	1	no	—
254	C₂×C₄	—	34	1	no	—
255	C₂×C₄	—	34	1	no	—
256	C₂×C₄	—	34	1	no	—
257	C₂×C₄	—	34	1	no	—
258	C₂×C₄	C₄×#3633	1	1	yes	characteristic
259	C₂×C₄	C₄×#3633	1	1	yes	characteristic
260	C₂×C₄	C₂²×#3633	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	17	8	no	—
263	C₈	—	34	4	no	—
264	C₈	—	34	8	no	—
265	C₈	—	34	4	no	—
266	C₈	—	34	2	no	—
267	C₂²	—	17	1	no	—
268	C₂²	—	17	1	no	—
269	C₂²	—	17	1	no	—
270	C₂²	—	17	1	no	—
271	C₂²	—	17	1	no	—
272	C₂²	—	17	1	no	—
273	C₂²	C₂×C₄×#3633	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	17	2	no	—
276	C₄	—	17	2	no	—
277	C₄	—	17	2	no	—
278	C₄	—	17	2	no	—
279	C₄	—	17	1	no	—
280	C₄	—	17	1	no	—
281	C₄	—	17	2	no	—
282	C₄	—	34	2	no	—
283	C₄	—	34	2	no	—
284	C₄	—	34	1	no	—
285	C₄	C₈×#3633	1	2	yes	complemented
286	C₄	D₄×#3633	1	1	yes	characteristic
287	C₄	Q₈×#3633	1	1	yes	characteristic
288	C₂	—	17	1	no	—
289	C₂	—	17	1	no	—
290	C₂	—	17	1	no	—
291	C₂	—	17	1	no	—
292	C₂	C₂×C₈×#3633	1	1	yes	characteristic
293	C₂	M₁₆×#3633	1	1	yes	characteristic
294	C₂	C₄⋊C₄×#3633	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3633
commutator	C₂×C₁₇	C₂×C₈²
Fitting	C₁₇×C₄⋊C₈	C₈
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂²×D₁₇	C₂×C₄²
cosocle	C₂×C₄×#595	C₂³
hypercenter	C₄⋊C₈	#3633
hypocenter	C₁₇	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. C₄⋊#3633	C₈	semidirect
35. C₄×#3633	C₈	semidirect
40. #10587544	C₈	semidirect
41. C₄⋊#3624	C₈	semidirect
44. C₁₇×C₄⋊C₈	C₈	semidirect
45. #10587485	C₈	semidirect
46. C₄⋊#3622	C₈	semidirect
102. #3633	C₄⋊C₈	direct (N × H)
139. C₄×C₁₇	C₈²	semidirect
143. Dic₁₇	C₈²	semidirect
183. C₄⋊C₈	#3633	direct (N × H)
194. C₁₇	C₈×C₄⋊C₈	semidirect
285. C₄	C₈×#3633	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#3633	2
C₄²×#3633	2
C₂×C₈×#3633	2
(C₂×#3633)⋊C₈	2
C₄⋊C₈×#595	2
C₈×C₄⋊C₈	17

Subgroups — C4⋊C8×#3633 #17981622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₈×#3633 #17981622