Subgroups — C₈×#10587478 #16925554

Order 4352.

446 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587478	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762561	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
5	#14526625	C₂	1	1	yes	maximal, characteristic
6	#14526623	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587478	C₄	1	2	yes	—
9	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
10	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	—	2	1	no	—
13	C₄×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
14	#11762571	C₄	1	2	yes	—
15	#11762563	—	2	1	no	—
16	#11762561	C₄	1	4	yes	complemented
17	C₈×#3622	C₄	1	4	yes	complemented
18	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₇	—	2	1	no	—
20	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#336	—	2	1	no	—
23	C₂²×C₈×C₁₇	—	2	1	no	—
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₇	—	2	1	no	—
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₁₇	—	2	1	no	—
31	C₂²×#3622	—	2	1	no	—
32	C₂²×#3622	—	2	1	no	—
33	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
34	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
35	C₁₇×C₄⋊C₈	—	2	1	no	—
36	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₇	—	4	2	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₇	Q₈	1	2	yes	—
40	C₁₇×#95	—	2	1	no	—
41	#10587502	—	2	1	no	—
42	C₈×Dic₁₇	—	4	2	no	—
43	#10587478	C₈	1	8	yes	direct product factor
44	C₄⋊#3622	—	2	1	no	—
45	C₄⋊#3622	C₂×C₄	1	2	yes	—
46	C₄×#3622	—	2	1	no	—
47	C₄×#3622	C₂×C₄	1	2	yes	—
48	C₄×#3622	C₂×C₄	1	2	yes	—
49	C₄×#3622	C₂×C₄	1	2	yes	—
50	C₂²×C₄×C₁₇	—	2	2	no	—
51	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
54	C₂²×Dic₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	—	2	1	no	—
56	C₂×C₈×C₁₇	—	2	1	no	—
57	C₂×C₈×C₁₇	—	2	2	no	—
58	C₂×C₈×C₁₇	—	4	1	no	—
59	C₂×C₈×C₁₇	—	4	1	no	—
60	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₇	D₈	1	2	yes	—
63	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
64	C₄⋊C₄×C₁₇	—	2	2	no	—
65	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
66	C₄²×C₁₇	—	4	1	no	—
67	C₄²×C₁₇	—	4	1	no	—
68	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₇	#41	1	1	yes	characteristic
72	C₄⋊Dic₁₇	—	2	2	no	—
73	C₄⋊Dic₁₇	C₂×C₈	1	4	yes	—
74	C₄×Dic₁₇	—	4	1	no	—
75	C₄×Dic₁₇	—	4	1	no	—
76	C₂×#3622	—	2	4	no	—
77	C₂×#3622	—	2	2	no	—
78	C₂×#3622	—	2	2	no	—
79	C₂×#3622	—	2	2	no	—
80	C₂×#3622	—	4	1	no	—
81	C₂×#3622	—	4	1	no	—
82	C₂×#3622	C₄²	1	2	yes	—
83	C₂×#3622	C₄²	1	2	yes	—
84	C₂×#3622	C₂×C₈	1	4	yes	—
85	C₈×#107	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	—	2	2	no	—
88	C₂×C₄×C₁₇	—	2	4	no	—
89	C₂×C₄×C₁₇	—	4	2	no	—
90	C₂×C₄×C₁₇	—	4	2	no	—
91	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₁₇	#131	1	2	yes	—
99	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
102	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
103	C₂×Dic₁₇	—	2	4	no	—
104	C₂×Dic₁₇	—	4	2	no	—
105	C₂×Dic₁₇	—	4	2	no	—
106	C₈×C₁₇	—	2	2	no	—
107	C₈×C₁₇	—	4	2	no	—
108	C₈×C₁₇	#107	1	4	yes	complemented
109	#3622	—	2	4	no	—
110	#3622	—	2	2	no	—
111	#3622	—	4	2	no	—
112	#3622	C₄×C₈	1	4	yes	complemented
113	C₄×#107	—	17	1	no	—
114	C₈×C₄⋊C₄	—	17	1	no	—
115	C₈×C₄⋊C₄	D₁₇	1	1	yes	characteristic
116	C₂×#335	—	17	1	no	—
117	C₂×C₈²	—	17	1	no	—
118	#1279	—	17	1	no	—
119	#1279	—	17	1	no	—
120	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₁₇	C₈×D₄	1	2	yes	—
122	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
123	C₂²×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
124	C₂²×C₁₇	#443	1	2	yes	—
125	C₄×C₁₇	—	2	1	no	—
126	C₄×C₁₇	—	2	1	no	—
127	C₄×C₁₇	—	4	4	no	—
128	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
129	C₄×C₁₇	#405	1	1	yes	characteristic
130	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
131	C₄×C₁₇	#427	1	1	yes	characteristic
132	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
133	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
134	C₄×C₁₇	C₈×Q₈	1	2	yes	—
135	Dic₁₇	—	4	4	no	—
136	C₂×#107	—	17	2	no	—
137	C₂×C₄⋊C₈	—	17	1	no	—
138	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
139	C₂×C₄×C₈	—	2	1	no	—
140	C₂×C₄×C₈	—	17	1	no	—
141	C₂×C₄×C₈	—	17	1	no	—
142	C₂×C₄×C₈	—	34	1	no	—
143	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
144	C₄×C₄⋊C₄	—	17	1	no	—
145	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
146	#340	—	17	2	no	—
147	#336	—	2	1	no	—
148	#336	—	34	1	no	—
149	#335	—	17	4	no	—
150	C₈²	—	17	4	no	—
151	C₂×C₁₇	C₈×D₈	1	2	yes	—
152	C₂×C₁₇	C₈×Q₁₆	1	2	yes	—
153	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₁₇	#1772	1	1	yes	characteristic
155	C₂×C₁₇	C₄×#107	1	1	yes	characteristic
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	17	1	no	—
158	C₂²×C₈	—	17	1	no	—
159	C₂²×C₈	—	34	1	no	—
160	C₂²×C₈	—	34	1	no	—
161	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
163	C₂×C₄⋊C₄	—	17	2	no	—
164	C₂×C₄⋊C₄	C₄×D₁₇	1	2	yes	—
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	—	34	1	no	—
167	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
168	#107	—	17	8	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	17	2	no	—
171	C₄⋊C₈	—	34	1	no	—
172	C₄⋊C₈	C₄×D₁₇	1	2	yes	—
173	C₄×C₈	—	4	2	no	—
174	C₄×C₈	—	17	2	no	—
175	C₄×C₈	—	17	2	no	—
176	C₄×C₈	—	17	2	no	—
177	C₄×C₈	—	34	1	no	—
178	C₄×C₈	—	68	2	no	—
179	C₄×C₈	Dic₃₄	1	2	yes	—
180	C₄×C₈	C₄×D₁₇	1	2	yes	—
181	#95	—	2	1	no	—
182	#95	—	34	1	no	—
183	C₁₇	C₈×#107	1	1	yes	characteristic, complemented, Sylow
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	34	2	no	—
186	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
187	C₂²×C₄	#63252	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	17	2	no	—
195	C₂×C₈	—	17	4	no	—
196	C₂×C₈	—	17	2	no	—
197	C₂×C₈	—	34	4	no	—
198	C₂×C₈	—	34	2	no	—
199	C₂×C₈	—	34	2	no	—
200	C₂×C₈	—	34	2	no	—
201	C₂×C₈	—	68	1	no	—
202	C₂×C₈	—	68	1	no	—
203	C₂×C₈	#63225	1	2	yes	—
204	C₂×C₈	#63225	1	2	yes	—
205	C₂×C₈	#63228	1	2	yes	—
206	C₂×C₈	#63231	1	2	yes	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	17	4	no	—
209	C₄⋊C₄	—	34	2	no	—
210	C₄⋊C₄	C₈×D₁₇	1	4	yes	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	68	1	no	—
214	C₄²	—	68	1	no	—
215	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
216	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
217	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
218	C₄²	#63252	1	1	yes	characteristic
219	C₂³	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	34	4	no	—
225	C₂×C₄	—	68	2	no	—
226	C₂×C₄	—	68	2	no	—
227	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
228	C₂×C₄	C₄×Dic₃₄	1	1	yes	characteristic
229	C₂×C₄	C₄²×D₁₇	1	1	yes	characteristic
230	C₂×C₄	#10587558	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
232	C₂×C₄	#10587587	1	2	yes	—
233	C₂×C₄	C₂×#63225	1	1	yes	characteristic
234	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
235	C₂×C₄	C₂×#63228	1	1	yes	characteristic
236	C₂×C₄	C₂×#63231	1	1	yes	characteristic
237	C₂×C₄	#10587636	1	1	yes	characteristic
238	C₂×C₄	#10587636	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	4	2	no	—
241	C₈	—	17	4	no	—
242	C₈	—	34	4	no	—
243	C₈	—	34	2	no	—
244	C₈	—	68	2	no	—
245	C₈	#10587478	1	4	yes	direct product factor
246	C₂²	C₄×#63225	1	1	yes	characteristic
247	C₂²	C₈×C₂²⋊D₁₇	1	2	yes	—
248	C₂²	C₄×#63228	1	1	yes	characteristic
249	C₂²	C₄×#63231	1	1	yes	characteristic
250	C₂²	#11762924	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	68	4	no	—
255	C₄	C₄×C₈×D₁₇	1	1	yes	characteristic
256	C₄	#11762763	1	1	yes	characteristic
257	C₄	C₂×#10587478	1	1	yes	characteristic
258	C₄	#11762845	1	1	yes	characteristic
259	C₄	C₄×#63228	1	1	yes	characteristic
260	C₄	C₄×#63231	1	1	yes	characteristic
261	C₄	C₈×Dic₃₄	1	2	yes	—
262	C₂	C₈×#63228	1	2	yes	—
263	C₂	C₈×#63231	1	2	yes	—
264	C₂	C₈×#63225	1	1	yes	characteristic
265	C₂	#14528664	1	1	yes	characteristic
266	C₂	C₄×#10587478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₁₇
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₈×C₄⋊C₄×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₄×D₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₁₇
hypocenter	C₁₇	C₈×#107
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
16. #11762561	C₄	semidirect
17. C₈×#3622	C₄	semidirect
43. #10587478	C₈	direct (N × H)
108. C₈×C₁₇	#107	semidirect
112. #3622	C₄×C₈	semidirect
183. C₁₇	C₈×#107	semidirect
245. C₈	#10587478	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#10587478	2
C₈×C₄⋊Dic₁₇	2
C₂×#11762561	2
C₂×C₈×#3622	2
#14526625	2
#14526623	2
C₈×C₄⋊C₄×C₁₇	2
C₈×#107	17

Subgroups — C8×#10587478 #16925554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#10587478 #16925554