Subgroups — C₂×Q₁₆×#10596842 #34034521

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10596842	C₂	1	2	yes	maximal
2	Q₁₆×#10596842	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989677	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊#10596842	C₂	1	2	yes	maximal
5	C₂×C₈×#10596842	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×#64408	C₂	1	1	yes	maximal, characteristic
7	Q₈×#10596842	—	2	4	no	—
8	Q₈×#10596842	C₂²	1	8	yes	—
9	C₂×C₄⋊#10596842	—	2	2	no	—
10	C₂×C₄⋊#10596842	C₂²	1	2	yes	—
11	C₂×C₄×#10596842	—	2	2	no	—
12	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596842	C₂²	1	1	yes	characteristic
14	#17989677	C₂²	1	4	yes	—
15	Q₈⋊#10596842	C₂²	1	8	yes	—
16	C₈×#10596842	C₂²	1	4	yes	—
17	C₂×Q₈×#64408	C₂²	1	2	yes	—
18	Q₁₆×#64408	C₄	1	4	yes	—
19	Q₁₆×#64408	C₂²	1	4	yes	—
20	C₂×#14556660	C₄	1	1	yes	characteristic
21	C₂×#14556654	C₄	1	2	yes	—
22	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×D₇₃	C₄	1	1	yes	characteristic
24	C₂²×#10596842	—	2	2	no	—
25	C₂²×#10596842	—	2	1	no	—
26	C₂²×#10596842	D₄	1	1	yes	characteristic
27	C₂²×#10596842	D₄	1	1	yes	characteristic
28	C₄⋊#10596842	—	2	8	no	—
29	C₄⋊#10596842	—	2	2	no	—
30	C₄⋊#10596842	C₂³	1	4	yes	—
31	C₄×#10596842	—	2	8	no	—
32	C₄×#10596842	—	2	1	no	—
33	C₄×#10596842	D₄	1	2	yes	—
34	C₄×#10596842	C₂³	1	2	yes	—
35	C₄×#10596842	C₂³	1	2	yes	—
36	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
37	Q₈×#64408	—	2	2	no	—
38	Q₈×#64408	—	2	2	no	—
39	Q₈×#64408	C₂×C₄	1	4	yes	—
40	Q₈×#64408	C₂³	1	4	yes	—
41	C₂×C₄⋊#64408	—	2	2	no	—
42	C₂×C₄⋊#64408	C₂×C₄	1	2	yes	—
43	C₂×C₄×#64408	—	2	2	no	—
44	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
45	#14556660	C₂×C₄	1	4	yes	—
46	#14556654	C₂×C₄	1	8	yes	—
47	C₈×#64408	C₂×C₄	1	2	yes	—
48	C₈×#64408	C₂³	1	2	yes	—
49	C₂×Q₈×D₇₃	C₂×C₄	1	2	yes	—
50	C₂×Dic₂₉₂	C₈	1	1	yes	characteristic, complemented
51	C₂×#11924018	C₈	1	2	yes	complemented
52	Q₁₆×D₇₃	C₈	1	4	yes	complemented
53	Q₁₆×D₇₃	C₂×C₄	1	4	yes	—
54	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×Q₁₆×C₇₃	C₈	1	1	yes	characteristic, complemented
56	C₂×#10596842	—	2	4	no	—
57	C₂×#10596842	—	2	2	no	—
58	C₂×#10596842	—	2	2	no	—
59	C₂×#10596842	—	2	4	no	—
60	C₂×#10596842	—	4	4	no	—
61	C₂×#10596842	Q₁₆	1	4	yes	direct product factor
62	C₂×#10596842	C₂×D₄	1	2	yes	—
63	C₂×#10596842	C₂×D₄	1	2	yes	—
64	C₂²×#64408	—	2	2	no	—
65	C₂²×#64408	C₂×D₄	1	1	yes	characteristic
66	C₂²×#64408	#41	1	1	yes	characteristic
67	#11924036	—	2	1	no	—
68	#11924036	C₂²×C₄	1	2	yes	—
69	C₂×#10596833	—	2	1	no	—
70	C₄⋊#64408	—	2	4	no	—
71	C₄⋊#64408	—	2	4	no	—
72	C₄⋊#64408	—	2	2	no	—
73	C₄⋊#64408	C₂²×C₄	1	4	yes	—
74	C₄×#64408	—	2	4	no	—
75	C₄×#64408	—	2	4	no	—
76	C₄×#64408	C₂²×C₄	1	1	yes	characteristic
77	C₄×#64408	C₂×D₄	1	1	yes	characteristic
78	C₄×#64408	#41	1	1	yes	characteristic
79	C₄×#64408	C₂⁴	1	1	yes	characteristic
80	C₂×Dic₁₄₆	—	2	2	no	—
81	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
82	Q₈×D₇₃	—	2	2	no	—
83	Q₈×D₇₃	—	2	2	no	—
84	Q₈×D₇₃	C₂×C₈	1	4	yes	—
85	Q₈×D₇₃	C₂²×C₄	1	4	yes	—
86	C₂×C₄×D₇₃	—	2	2	no	—
87	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
88	Dic₂₉₂	C₂×C₈	1	4	yes	complemented
89	#11924018	C₂×C₈	1	8	yes	complemented
90	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
91	C₈×D₇₃	C₂×C₈	1	2	yes	—
92	C₈×D₇₃	C₂²×C₄	1	2	yes	—
93	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
94	Q₁₆×C₇₃	C₂×C₈	1	4	yes	complemented
95	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
96	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
97	C₂×#64408	—	2	2	no	—
98	C₂×#64408	—	2	4	no	—
99	C₂×#64408	—	4	4	no	—
100	C₂×#64408	C₄×D₄	1	1	yes	characteristic
101	C₂×#64408	C₄×D₄	1	1	yes	characteristic
102	C₂×#64408	C₂×Q₁₆	1	2	yes	—
103	C₂×#64408	#135	1	2	yes	—
104	C₂×#64408	C₂²×D₄	1	1	yes	characteristic
105	C₂×#64408	C₂×#41	1	1	yes	characteristic
106	#10596842	—	2	4	no	—
107	#10596842	—	2	2	no	—
108	#10596842	—	4	4	no	—
109	#10596842	C₂×Q₁₆	1	4	yes	direct product factor
110	Q₈×C₇₃	—	2	2	no	—
111	Q₈×C₇₃	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×Dic₇₃	—	2	2	no	—
115	C₂×Dic₇₃	#131	1	1	yes	characteristic
116	C₄×D₇₃	—	2	4	no	—
117	C₄×D₇₃	—	2	4	no	—
118	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
119	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
120	C₄×D₇₃	#131	1	1	yes	characteristic
121	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
122	Dic₁₄₆	—	2	2	no	—
123	Dic₁₄₆	—	2	4	no	—
124	Dic₁₄₆	—	2	4	no	—
125	Dic₁₄₆	C₂²×C₈	1	4	yes	—
126	#10596833	—	2	2	no	—
127	C₈×C₇₃	C₂²×C₈	1	2	yes	—
128	#10596831	C₂²×C₈	1	2	yes	—
129	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
130	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
131	C₂×D₇₃	C₄×Q₁₆	1	2	yes	—
132	C₂×D₇₃	#443	1	2	yes	—
133	C₂×D₇₃	C₂×C₄×D₄	1	1	yes	characteristic
134	#64408	—	2	1	no	—
135	#64408	—	2	1	no	—
136	#64408	—	4	4	no	—
137	#64408	C₄×Q₁₆	1	2	yes	—
138	#64408	C₂²×Q₁₆	1	1	yes	characteristic
139	#64408	C₂×#135	1	1	yes	characteristic
140	C₄×C₇₃	—	2	4	no	—
141	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
142	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
143	Dic₇₃	—	2	4	no	—
144	Dic₇₃	C₈×D₄	1	1	yes	characteristic
145	Dic₇₃	C₂×#131	1	1	yes	characteristic
146	C₂×C₈×Q₁₆	—	73	1	no	maximal, Sylow
147	C₂×C₇₃	C₈×Q₁₆	1	2	yes	complemented
148	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
149	D₇₃	C₈×Q₁₆	1	2	yes	complemented
150	D₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
151	D₇₃	C₂×#443	1	1	yes	characteristic
152	C₂×C₈×Q₈	—	73	2	no	—
153	C₂×C₄×Q₁₆	—	73	1	no	—
154	C₈×Q₁₆	—	73	8	no	—
155	C₂×#335	—	73	1	no	—
156	C₂×Q₈⋊C₈	—	73	2	no	—
157	C₂×C₈²	—	73	1	no	—
158	C₇₃	C₂×C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₁₆	—	73	1	no	—
160	C₂×C₄×Q₈	—	73	2	no	—
161	C₈×Q₈	—	73	8	no	—
162	C₈×Q₈	—	146	4	no	—
163	C₄×Q₁₆	—	73	4	no	—
164	C₄×Q₁₆	—	73	4	no	—
165	C₂×#107	—	73	1	no	—
166	C₂×C₄⋊C₈	—	73	2	no	—
167	C₂×C₄⋊C₈	—	146	2	no	—
168	C₂×#103	—	73	2	no	—
169	C₂×C₄×C₈	—	73	1	no	—
170	C₂×C₄×C₈	—	73	1	no	—
171	C₂×C₄×C₈	—	73	1	no	—
172	C₂×C₄×C₈	—	146	2	no	—
173	#335	—	73	4	no	—
174	Q₈⋊C₈	—	73	8	no	—
175	C₈²	—	73	4	no	—
176	C₂²×Q₈	—	73	2	no	—
177	C₂×Q₁₆	—	73	2	no	—
178	C₂×Q₁₆	—	73	4	no	—
179	C₂×Q₁₆	—	73	4	no	—
180	C₂×Q₁₆	—	73	1	no	—
181	C₂×Q₁₆	#10596842	1	1	yes	characteristic, direct product factor
182	C₂²×C₈	—	73	1	no	—
183	C₂²×C₈	—	73	1	no	—
184	C₂²×C₈	—	73	1	no	—
185	C₂²×C₈	—	73	1	no	—
186	C₂²×C₈	—	146	2	no	—
187	C₂²×C₈	—	146	1	no	—
188	C₄×Q₈	—	73	4	no	—
189	C₄×Q₈	—	73	4	no	—
190	C₄×Q₈	—	146	2	no	—
191	C₄×Q₈	—	146	2	no	—
192	C₂×C₄⋊C₄	—	73	2	no	—
193	C₂×C₄⋊C₄	—	146	2	no	—
194	C₂×C₄²	—	73	1	no	—
195	C₂×C₄²	—	146	2	no	—
196	#107	—	73	4	no	—
197	C₄⋊C₈	—	73	4	no	—
198	C₄⋊C₈	—	146	2	no	—
199	C₄⋊C₈	—	146	8	no	—
200	#103	—	73	8	no	—
201	C₄×C₈	—	73	2	no	—
202	C₄×C₈	—	73	2	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₄×C₈	—	73	2	no	—
206	C₄×C₈	—	146	1	no	—
207	C₄×C₈	—	146	8	no	—
208	C₂×Q₈	—	73	4	no	—
209	C₂×Q₈	—	73	4	no	—
210	C₂×Q₈	—	73	2	no	—
211	C₂×Q₈	—	146	2	no	—
212	C₂×Q₈	—	146	2	no	—
213	C₂×Q₈	—	146	2	no	—
214	C₂×Q₈	C₂×#10596842	1	2	yes	—
215	C₂²×C₄	—	73	1	no	—
216	C₂²×C₄	—	73	1	no	—
217	C₂²×C₄	—	73	1	no	—
218	C₂²×C₄	—	146	2	no	—
219	C₂²×C₄	—	146	2	no	—
220	Q₁₆	—	73	8	no	—
221	Q₁₆	—	73	4	no	—
222	Q₁₆	C₂×#10596842	1	4	yes	direct product factor
223	C₂×C₈	—	292	4	no	—
224	C₂×C₈	—	73	4	no	—
225	C₂×C₈	—	73	2	no	—
226	C₂×C₈	—	73	1	no	—
227	C₂×C₈	—	73	2	no	—
228	C₂×C₈	—	73	2	no	—
229	C₂×C₈	—	73	2	no	—
230	C₂×C₈	—	73	2	no	—
231	C₂×C₈	—	146	4	no	—
232	C₂×C₈	—	146	1	no	—
233	C₂×C₈	—	146	1	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	—	146	2	no	—
236	C₂×C₈	—	146	4	no	—
237	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
238	C₄⋊C₄	—	73	4	no	—
239	C₄⋊C₄	—	146	2	no	—
240	C₄⋊C₄	—	146	4	no	—
241	C₄⋊C₄	—	146	4	no	—
242	C₄²	—	73	1	no	—
243	C₄²	—	73	1	no	—
244	C₄²	—	73	1	no	—
245	C₄²	—	73	1	no	—
246	C₄²	—	146	4	no	—
247	C₄²	—	146	4	no	—
248	C₂³	—	73	1	no	—
249	Q₈	—	2	2	no	—
250	Q₈	—	73	4	no	—
251	Q₈	—	146	4	no	—
252	Q₈	—	146	4	no	—
253	Q₈	—	146	2	no	—
254	Q₈	C₂²×#10596842	1	4	yes	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	292	4	no	—
257	C₂×C₄	—	73	1	no	—
258	C₂×C₄	—	73	1	no	—
259	C₂×C₄	—	73	2	no	—
260	C₂×C₄	—	73	2	no	—
261	C₂×C₄	—	73	1	no	—
262	C₂×C₄	—	73	1	no	—
263	C₂×C₄	—	73	1	no	—
264	C₂×C₄	—	73	1	no	—
265	C₂×C₄	—	73	1	no	—
266	C₂×C₄	—	73	1	no	—
267	C₂×C₄	—	73	1	no	—
268	C₂×C₄	—	146	4	no	—
269	C₂×C₄	—	146	4	no	—
270	C₂×C₄	—	146	4	no	—
271	C₂×C₄	—	146	2	no	—
272	C₂×C₄	—	146	2	no	—
273	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
274	C₈	—	292	4	no	—
275	C₈	—	73	4	no	—
276	C₈	—	73	2	no	—
277	C₈	—	146	2	no	—
278	C₈	—	146	2	no	—
279	C₈	—	146	4	no	—
280	C₈	C₂²×#10596842	1	2	yes	—
281	C₂²	—	73	1	no	—
282	C₂²	—	73	1	no	—
283	C₂²	—	73	2	no	—
284	C₂²	—	73	2	no	—
285	C₂²	D₄×#10596842	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	292	4	no	—
288	C₄	—	73	1	no	—
289	C₄	—	73	1	no	—
290	C₄	—	73	2	no	—
291	C₄	—	73	1	no	—
292	C₄	—	73	1	no	—
293	C₄	—	146	1	no	—
294	C₄	—	146	1	no	—
295	C₄	—	146	4	no	—
296	C₄	D₄×#10596842	1	1	yes	characteristic
297	C₄	C₂³×#10596842	1	1	yes	characteristic
298	C₂	—	73	1	no	—
299	C₂	—	73	1	no	—
300	C₂	—	73	2	no	—
301	C₂	Q₁₆×#10596842	1	2	yes	direct product factor
302	C₂	C₂×D₄×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#10596842
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₄	C₂³×#10596842
Fitting	C₂×Q₁₆×C₇₃	C₈
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×#64408	C₂⁴
hypercenter	C₂×Q₁₆	#10596842
hypocenter	C₇₃	C₂×C₈×Q₁₆
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₁₆×#10596842	C₂	direct (N × H)
50. C₂×Dic₂₉₂	C₈	semidirect
51. C₂×#11924018	C₈	semidirect
52. Q₁₆×D₇₃	C₈	semidirect
55. C₂×Q₁₆×C₇₃	C₈	semidirect
61. C₂×#10596842	Q₁₆	direct (N × H)
88. Dic₂₉₂	C₂×C₈	semidirect
89. #11924018	C₂×C₈	semidirect
94. Q₁₆×C₇₃	C₂×C₈	semidirect
109. #10596842	C₂×Q₁₆	direct (N × H)
147. C₂×C₇₃	C₈×Q₁₆	semidirect
149. D₇₃	C₈×Q₁₆	semidirect
158. C₇₃	C₂×C₈×Q₁₆	semidirect
181. C₂×Q₁₆	#10596842	direct (N × H)
222. Q₁₆	C₂×#10596842	direct (N × H)
301. C₂	Q₁₆×#10596842	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10596842	2
Q₁₆×#10596842	2
C₂×#17989677	2
C₂×Q₈⋊#10596842	2
C₂×C₈×#10596842	2
C₂×Q₁₆×#64408	2
C₂×C₈×Q₁₆	73

Subgroups — C2×Q16×#10596842 #34034521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₁₆×#10596842 #34034521