Subgroups — C₂×#41×#10596833 #34031516

Order 18688.

2566 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#10596833	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#10596833	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×#10596833	C₂	1	6	yes	maximal, complemented
4	C₂×#17989603	C₂	1	3	yes	maximal, complemented
5	C₂²×C₄×#10596833	C₂	1	3	yes	maximal, complemented
6	C₂×#41×Dic₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10596833	C₂²	1	6	yes	—
8	Q₈×#10596833	C₂²	1	16	yes	complemented
9	D₄×#10596833	C₂²	1	48	yes	complemented
10	#17989603	C₂²	1	24	yes	complemented
11	C₂×C₄⋊#10596833	C₂²	1	6	yes	—
12	C₂×C₂²⋊#10596833	C₂²	1	6	yes	—
13	C₂×C₄×#10596833	C₂²	1	3	yes	—
14	C₂×C₄×#10596833	C₂²	1	6	yes	—
15	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10596833	C₂²	1	24	yes	complemented
17	#41×Dic₇₃	C₄	1	8	yes	—
18	#41×Dic₇₃	C₂²	1	8	yes	—
19	C₂×Q₈×Dic₇₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
21	C₂×D₄×Dic₇₃	C₄	1	3	yes	—
22	C₂×D₄×Dic₇₃	C₂²	1	3	yes	—
23	C₂×#14556550	C₄	1	3	yes	—
24	C₂²×C₄×Dic₇₃	C₂²	1	3	yes	—
25	C₂²×#41×C₇₃	C₄	1	1	yes	characteristic
26	C₂²×#10596833	—	2	24	no	—
27	C₂²×#10596833	C₂³	1	6	yes	—
28	C₂²×#10596833	C₂³	1	24	yes	—
29	C₂²×#10596833	C₂³	1	2	yes	—
30	C₄⋊#10596833	C₂³	1	24	yes	—
31	C₂²⋊#10596833	C₂³	1	24	yes	—
32	C₄×#10596833	C₂³	1	24	yes	—
33	C₄×#10596833	C₂³	1	4	yes	—
34	C₄×#10596833	C₂³	1	12	yes	—
35	C₂³×Dic₇₃	C₂×C₄	1	3	yes	—
36	C₂³×Dic₇₃	C₂³	1	3	yes	—
37	Q₈×Dic₇₃	C₂×C₄	1	8	yes	—
38	Q₈×Dic₇₃	C₂×C₄	1	4	yes	—
39	Q₈×Dic₇₃	C₂³	1	4	yes	—
40	D₄×Dic₇₃	C₂×C₄	1	24	yes	—
41	D₄×Dic₇₃	C₂×C₄	1	12	yes	—
42	D₄×Dic₇₃	C₂³	1	12	yes	—
43	#14556550	C₂×C₄	1	24	yes	—
44	C₂×C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
45	C₂×C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
46	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	6	yes	—
47	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
48	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
49	C₂×C₄×Dic₇₃	C₂×C₄	1	12	yes	—
50	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
51	C₂×C₄×Dic₇₃	C₂³	1	3	yes	—
52	C₂×C₄×Dic₇₃	C₂³	1	12	yes	—
53	C₂×#41×C₇₃	C₈	1	16	yes	complemented
54	C₂×#41×C₇₃	C₂×C₄	1	8	yes	—
55	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₄×C₇₃	C₂×C₄	1	3	yes	—
57	C₂³×C₄×C₇₃	C₂×C₄	1	3	yes	—
58	C₂×#10596833	—	2	12	no	—
59	C₂×#10596833	—	2	48	no	—
60	C₂×#10596833	#41	1	8	yes	direct product factor
61	C₂×#10596833	C₂⁴	1	12	yes	—
62	C₂×#10596833	C₂⁴	1	4	yes	—
63	C₂²×Dic₇₃	—	2	12	no	—
64	C₂²×Dic₇₃	—	2	12	no	—
65	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
66	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
67	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
68	C₂²×Dic₇₃	C₂²×C₄	1	12	yes	—
69	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₇₃	C₂⁴	1	6	yes	—
71	C₂²×Dic₇₃	C₂⁴	1	1	yes	characteristic
72	C₄⋊Dic₇₃	C₂²×C₄	1	12	yes	—
73	C₄⋊Dic₇₃	C₂²×C₄	1	12	yes	—
74	C₂²⋊Dic₇₃	C₂²×C₄	1	24	yes	—
75	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
76	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
77	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
78	C₄×Dic₇₃	C₂²×C₄	1	6	yes	—
79	C₄×Dic₇₃	C₂⁴	1	2	yes	—
80	C₄×Dic₇₃	C₂⁴	1	6	yes	—
81	C₂⁴×C₇₃	C₂²×C₄	1	3	yes	—
82	#41×C₇₃	C₂×C₈	1	64	yes	complemented
83	C₂×Q₈×C₇₃	C₂×C₈	1	8	yes	complemented
84	C₂×Q₈×C₇₃	C₂²×C₄	1	4	yes	—
85	C₂×D₄×C₇₃	C₂×C₈	1	24	yes	complemented
86	C₂×D₄×C₇₃	C₂²×C₄	1	12	yes	—
87	C₂²×C₄×C₇₃	C₂×C₈	1	24	yes	complemented
88	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
89	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
90	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂³×C₇₃	—	2	6	no	—
92	C₂³×C₇₃	—	2	6	no	—
93	C₂³×C₇₃	C₂²×C₈	1	12	yes	—
94	C₂³×C₇₃	C₂³×C₄	1	6	yes	—
95	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
96	Q₈×C₇₃	C₂²×C₈	1	16	yes	complemented
97	D₄×C₇₃	C₂²×C₈	1	48	yes	complemented
98	C₂×C₄×C₇₃	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₇₃	C₂²×C₈	1	12	yes	—
100	C₂×C₄×C₇₃	C₂²×C₈	1	48	yes	complemented
101	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₇₃	C₂³×C₄	1	6	yes	—
103	C₂×Dic₇₃	—	2	6	no	—
104	C₂×Dic₇₃	—	2	6	no	—
105	C₂×Dic₇₃	—	2	12	no	—
106	C₂×Dic₇₃	—	2	24	no	—
107	C₂×Dic₇₃	—	2	12	no	—
108	C₂×Dic₇₃	#131	1	4	yes	—
109	C₂×Dic₇₃	C₂³×C₄	1	6	yes	—
110	C₂×Dic₇₃	C₂³×C₄	1	6	yes	—
111	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₇₃	C₂³×C₄	1	2	yes	—
113	C₂×Dic₇₃	C₂×#41	1	4	yes	—
114	C₂×Dic₇₃	C₂⁵	1	1	yes	characteristic
115	#10596833	—	2	12	no	—
116	#10596833	C₂×#41	1	8	yes	direct product factor
117	C₂²×C₇₃	—	2	6	no	—
118	C₂²×C₇₃	—	2	24	no	—
119	C₂²×C₇₃	—	2	6	no	—
120	C₂²×C₇₃	C₄×#41	1	2	yes	—
121	C₂²×C₇₃	C₂³×C₈	1	12	yes	—
122	C₂²×C₇₃	C₂³×C₈	1	2	yes	—
123	C₂²×C₇₃	C₂×#131	1	2	yes	—
124	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
125	C₄×C₇₃	C₂³×C₈	1	4	yes	—
126	C₄×C₇₃	C₂³×C₈	1	12	yes	—
127	Dic₇₃	—	2	6	no	—
128	Dic₇₃	—	2	6	no	—
129	Dic₇₃	C₄×#41	1	2	yes	—
130	Dic₇₃	C₂×#131	1	4	yes	—
131	Dic₇₃	C₂²×#41	1	2	yes	—
132	C₂×C₈×#41	—	73	1	no	maximal, Sylow
133	C₂×C₇₃	—	2	12	no	—
134	C₂×C₇₃	C₈×#41	1	4	yes	complemented
135	C₂×C₇₃	C₂×C₄×#41	1	1	yes	characteristic
136	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
137	C₂×C₇₃	C₂²×#131	1	1	yes	characteristic
138	C₂×C₄×#41	—	73	1	no	—
139	C₈×#41	—	73	16	no	—
140	C₂×C₈×Q₈	—	73	2	no	—
141	C₂×C₈×D₄	—	73	6	no	—
142	C₂×#431	—	73	3	no	—
143	C₂²×C₄×C₈	—	73	3	no	—
144	C₇₃	C₂×C₈×#41	1	1	yes	characteristic, complemented, Sylow
145	C₂²×#41	#64408	1	1	yes	characteristic
146	C₂³×C₈	—	73	6	no	—
147	C₄×#41	—	73	8	no	—
148	C₄×#41	—	73	8	no	—
149	C₂×C₄×Q₈	—	73	1	no	—
150	C₂×C₄×Q₈	—	73	1	no	—
151	C₂×C₄×D₄	—	73	3	no	—
152	C₂×C₄×D₄	—	73	3	no	—
153	C₂×#117	—	73	3	no	—
154	C₂²×C₄²	—	73	3	no	—
155	C₈×Q₈	—	73	16	no	—
156	C₈×D₄	—	73	48	no	—
157	#431	—	73	24	no	—
158	C₂×C₄⋊C₈	—	73	6	no	—
159	C₂×C₂²⋊C₈	—	73	6	no	—
160	C₂×C₄×C₈	—	73	1	no	—
161	C₂×C₄×C₈	—	73	3	no	—
162	C₂×C₄×C₈	—	73	6	no	—
163	C₂×C₄×C₈	—	73	24	no	—
164	C₂×#41	#10596833	1	16	yes	direct product factor
165	C₂×#41	C₂×#64408	1	8	yes	—
166	C₂²×Q₈	C₂×#64408	1	1	yes	characteristic
167	C₂²×D₄	C₂×#64408	1	3	yes	—
168	C₂³×C₄	—	73	3	no	—
169	C₂³×C₄	—	73	3	no	—
170	C₂³×C₄	C₂×#64408	1	3	yes	—
171	C₂²×C₈	—	73	2	no	—
172	C₂²×C₈	—	73	6	no	—
173	C₂²×C₈	—	73	24	no	—
174	C₂²×C₈	—	146	24	no	—
175	C₄×Q₈	—	73	4	no	—
176	C₄×Q₈	—	73	8	no	—
177	C₄×Q₈	—	73	4	no	—
178	C₄×D₄	—	73	12	no	—
179	C₄×D₄	—	73	24	no	—
180	C₄×D₄	—	73	12	no	—
181	#117	—	73	24	no	—
182	C₂×C₄⋊C₄	—	73	3	no	—
183	C₂×C₄⋊C₄	—	73	3	no	—
184	C₂×C₂²⋊C₄	—	73	6	no	—
185	C₂×C₄²	—	73	3	no	—
186	C₂×C₄²	—	73	3	no	—
187	C₂×C₄²	—	73	12	no	—
188	C₂×C₄²	—	73	12	no	—
189	C₂×C₄²	—	73	3	no	—
190	C₂×C₄²	—	73	1	no	—
191	C₄⋊C₈	—	73	24	no	—
192	C₂²⋊C₈	—	73	24	no	—
193	C₄×C₈	—	73	4	no	—
194	C₄×C₈	—	73	12	no	—
195	C₄×C₈	—	73	24	no	—
196	C₂⁴	C₂²×#64408	1	3	yes	—
197	#41	C₂×#10596833	1	64	yes	direct product factor
198	C₂×Q₈	C₂×#10596833	1	8	yes	complemented
199	C₂×Q₈	C₂²×#64408	1	4	yes	—
200	C₂×D₄	C₂×#10596833	1	24	yes	complemented
201	C₂×D₄	C₂²×#64408	1	12	yes	—
202	C₂²×C₄	—	73	1	no	—
203	C₂²×C₄	—	73	1	no	—
204	C₂²×C₄	—	73	3	no	—
205	C₂²×C₄	—	73	3	no	—
206	C₂²×C₄	—	73	6	no	—
207	C₂²×C₄	—	73	12	no	—
208	C₂²×C₄	—	73	6	no	—
209	C₂²×C₄	—	146	12	no	—
210	C₂²×C₄	—	146	12	no	—
211	C₂²×C₄	C₂×#10596833	1	24	yes	complemented
212	C₂²×C₄	C₂²×#64408	1	3	yes	—
213	C₂²×C₄	C₂²×#64408	1	12	yes	—
214	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
215	C₂×C₈	—	73	8	no	—
216	C₂×C₈	—	73	4	no	—
217	C₂×C₈	—	73	12	no	—
218	C₂×C₈	—	146	48	no	—
219	C₂×C₈	—	146	12	no	—
220	C₄⋊C₄	—	73	12	no	—
221	C₄⋊C₄	—	73	12	no	—
222	C₂²⋊C₄	—	73	24	no	—
223	C₄²	—	73	2	no	—
224	C₄²	—	73	12	no	—
225	C₄²	—	73	12	no	—
226	C₄²	—	73	6	no	—
227	C₄²	—	73	2	no	—
228	C₄²	—	73	6	no	—
229	C₂³	—	2	6	no	—
230	C₂³	—	2	6	no	—
231	C₂³	C₂²×#10596833	1	12	yes	—
232	C₂³	C₂³×#64408	1	6	yes	—
233	C₂³	C₂³×#64408	1	1	yes	characteristic
234	Q₈	C₂²×#10596833	1	16	yes	complemented
235	D₄	C₂²×#10596833	1	48	yes	complemented
236	C₂×C₄	—	73	4	no	—
237	C₂×C₄	—	73	4	no	—
238	C₂×C₄	—	73	1	no	—
239	C₂×C₄	—	73	1	no	—
240	C₂×C₄	—	73	2	no	—
241	C₂×C₄	—	73	6	no	—
242	C₂×C₄	—	73	6	no	—
243	C₂×C₄	—	146	12	no	—
244	C₂×C₄	—	146	12	no	—
245	C₂×C₄	—	146	24	no	—
246	C₂×C₄	—	146	6	no	—
247	C₂×C₄	—	146	6	no	—
248	C₂×C₄	C₂²×#10596833	1	4	yes	—
249	C₂×C₄	C₂²×#10596833	1	12	yes	—
250	C₂×C₄	C₂²×#10596833	1	48	yes	complemented
251	C₂×C₄	C₂³×#64408	1	2	yes	—
252	C₂×C₄	C₂³×#64408	1	6	yes	—
253	C₈	—	73	8	no	—
254	C₈	—	146	12	no	—
255	C₂²	—	2	6	no	—
256	C₂²	—	2	6	no	—
257	C₂²	—	2	24	no	—
258	C₂²	#41×#64408	1	2	yes	—
259	C₂²	C₂³×#10596833	1	12	yes	—
260	C₂²	C₂³×#10596833	1	2	yes	—
261	C₂²	D₇₃⋊#131	1	2	yes	—
262	C₂²	C₂⁴×#64408	1	1	yes	characteristic
263	C₄	—	73	2	no	—
264	C₄	—	73	2	no	—
265	C₄	—	73	4	no	—
266	C₄	—	146	6	no	—
267	C₄	—	146	6	no	—
268	C₄	C₂³×#10596833	1	4	yes	—
269	C₄	C₂³×#10596833	1	12	yes	—
270	C₂	—	2	12	no	—
271	C₂	#41×#10596833	1	4	yes	direct product factor
272	C₂	C₂×#41×#64408	1	1	yes	characteristic
273	C₂	C₂⁴×#10596833	1	1	yes	characteristic
274	C₂	C₂×D₇₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#64408
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#64408
Fitting, prime radical	C₂²×#41×C₇₃	C₄
socle	C₂³×C₇₃	C₂³×C₄
cosocle	C₂×Dic₇₃	C₂⁵
hypercenter	C₂²×#41	#64408
hypocenter	C₇₃	C₂×C₈×#41
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #41×#10596833	C₂	direct (N × H)
2. C₂×Q₈×#10596833	C₂	semidirect
3. C₂×D₄×#10596833	C₂	semidirect
4. C₂×#17989603	C₂	semidirect
5. C₂²×C₄×#10596833	C₂	semidirect
8. Q₈×#10596833	C₂²	semidirect
9. D₄×#10596833	C₂²	semidirect
10. #17989603	C₂²	semidirect
16. C₂×C₄×#10596833	C₂²	semidirect
53. C₂×#41×C₇₃	C₈	semidirect
60. C₂×#10596833	#41	direct (N × H)
82. #41×C₇₃	C₂×C₈	semidirect
83. C₂×Q₈×C₇₃	C₂×C₈	semidirect
85. C₂×D₄×C₇₃	C₂×C₈	semidirect
87. C₂²×C₄×C₇₃	C₂×C₈	semidirect
96. Q₈×C₇₃	C₂²×C₈	semidirect
97. D₄×C₇₃	C₂²×C₈	semidirect
100. C₂×C₄×C₇₃	C₂²×C₈	semidirect
116. #10596833	C₂×#41	direct (N × H)
134. C₂×C₇₃	C₈×#41	semidirect
144. C₇₃	C₂×C₈×#41	semidirect
164. C₂×#41	#10596833	direct (N × H)
197. #41	C₂×#10596833	direct (N × H)
198. C₂×Q₈	C₂×#10596833	semidirect
200. C₂×D₄	C₂×#10596833	semidirect
211. C₂²×C₄	C₂×#10596833	semidirect
234. Q₈	C₂²×#10596833	semidirect
235. D₄	C₂²×#10596833	semidirect
250. C₂×C₄	C₂²×#10596833	semidirect
271. C₂	#41×#10596833	direct (N × H)

Maximal subgroups

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

subgroup	index
#41×#10596833	2
C₂×Q₈×#10596833	2
C₂×D₄×#10596833	2
C₂×#17989603	2
C₂²×C₄×#10596833	2
C₂×#41×Dic₇₃	2
C₂×C₈×#41	73

Subgroups — C2×#41×#10596833 #34031516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#41×#10596833 #34031516