Subgroups — C₂×#22624975 #34012128

Order 18688.

670 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556665	C₂	1	2	yes	maximal
2	#22624975	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	—	2	4	no	—
5	C₂×#14556665	—	2	2	no	—
6	C₂×#14556665	C₄	1	8	yes	complemented
7	C₂×#14556665	C₂²	1	4	yes	—
8	C₂²×#11924031	—	2	2	no	—
9	C₂³×#10596831	C₄	1	1	yes	characteristic
10	C₂³×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596831	—	2	4	no	—
12	C₂×C₄×#10596831	C₄	1	2	yes	—
13	C₂×C₄×#10596831	C₂²	1	2	yes	—
14	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic
15	#14556665	—	2	8	no	—
16	#14556665	—	2	4	no	—
17	#14556665	C₂×C₄	1	8	yes	complemented
18	C₂×#11924031	—	2	8	no	—
19	C₂×#11924031	—	2	4	no	—
20	C₂²×#10596831	—	2	2	no	—
21	C₂²×#10596831	—	2	2	no	—
22	C₂²×#10596831	C₈	1	2	yes	—
23	C₂²×#10596831	C₈	1	2	yes	—
24	C₂²×#10596831	C₂×C₄	1	4	yes	—
25	C₂²×#10596831	C₂×C₄	1	2	yes	—
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	D₄	1	4	yes	—
28	C₂²×#10596831	D₄	1	1	yes	characteristic
29	C₂²×#10596831	D₄	1	1	yes	characteristic
30	C₂²×#10596831	Q₈	1	1	yes	characteristic
31	C₂²×#10596831	Q₈	1	1	yes	characteristic
32	C₂²×#10596831	C₂³	1	1	yes	characteristic
33	C₄×#10596831	—	2	4	no	—
34	C₄×#10596831	—	2	4	no	—
35	C₂³×C₄×C₇₃	—	2	1	no	—
36	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₃	—	2	2	no	—
38	C₂×C₄²×C₇₃	—	2	1	no	—
39	C₂×C₄²×C₇₃	C₈	1	4	yes	—
40	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
41	#11924031	—	2	8	no	—
42	C₂×#10596831	—	2	4	no	—
43	C₂×#10596831	—	2	4	no	—
44	C₂×#10596831	—	2	2	no	—
45	C₂×#10596831	—	2	2	no	—
46	C₂×#10596831	C₄²	1	2	yes	—
47	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
48	C₂×#10596831	C₂²⋊C₄	1	4	yes	—
49	C₂×#10596831	C₂²⋊C₄	1	4	yes	—
50	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
51	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#10596831	C₄⋊C₄	1	4	yes	—
53	C₂×#10596831	C₄⋊C₄	1	4	yes	—
54	C₂×#10596831	C₂×C₈	1	4	yes	—
55	C₂×#10596831	C₂²×C₄	1	2	yes	—
56	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
57	C₂×#10596831	C₂×D₄	1	2	yes	—
58	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
59	C₂⁴×C₇₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	—	2	8	no	—
61	C₂²×C₄×C₇₃	—	2	4	no	—
62	C₂²×C₄×C₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₇₃	—	2	4	no	—
74	C₄²×C₇₃	—	2	2	no	—
75	C₄²×C₇₃	C₂×C₈	1	4	yes	—
76	C₂³×C₇₃	—	2	4	no	—
77	C₂³×C₇₃	#97	1	2	yes	—
78	C₂³×C₇₃	C₂²⋊C₈	1	2	yes	—
79	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
80	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
81	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₃	—	2	16	no	—
83	C₂×C₄×C₇₃	—	2	8	no	—
84	C₂×C₄×C₇₃	—	2	4	no	—
85	C₂×C₄×C₇₃	#95	1	2	yes	—
86	C₂×C₄×C₇₃	#95	1	2	yes	—
87	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
88	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
89	C₂×C₄×C₇₃	#97	1	2	yes	—
90	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
92	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
103	#10596831	—	2	4	no	—
104	#10596831	#95	1	2	yes	—
105	#10596831	#95	1	2	yes	—
106	#10596831	C₂×C₂²⋊C₄	1	2	yes	—
107	#10596831	C₂×C₄⋊C₄	1	2	yes	—
108	C₂²×C₇₃	—	2	4	no	—
109	C₂²×C₇₃	—	2	4	no	—
110	C₂²×C₇₃	—	2	2	no	—
111	C₂²×C₇₃	—	2	2	no	—
112	C₂²×C₇₃	#336	1	2	yes	—
113	C₂²×C₇₃	#336	1	2	yes	—
114	C₂²×C₇₃	#364	1	4	yes	—
115	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
116	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₇₃	—	2	8	no	—
119	C₄×C₇₃	#336	1	2	yes	—
120	C₄×C₇₃	#336	1	2	yes	—
121	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
122	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
123	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂×#1376	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	—	2	1	no	—
127	C₂×C₇₃	—	2	2	no	—
128	C₂×C₇₃	—	2	1	no	—
129	C₂×C₇₃	#1376	1	4	yes	complemented
130	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
131	C₂×C₇₃	C₂×#364	1	2	yes	—
132	C₂²×M₃₂	—	73	2	no	—
133	C₂²×C₄×C₈	—	73	1	no	—
134	#1376	—	73	4	no	—
135	C₇₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₈	—	73	1	no	—
137	C₂³×C₈	—	73	1	no	—
138	C₂²×C₄²	#64408	1	1	yes	characteristic
139	C₂×M₃₂	—	73	8	no	—
140	C₂×M₃₂	—	73	4	no	—
141	C₂×M₃₂	—	146	4	no	—
142	C₂×M₃₂	—	146	2	no	—
143	C₂²×C₁₆	—	146	2	no	—
144	C₂×C₄×C₈	—	73	2	no	—
145	C₂×C₄×C₈	—	73	2	no	—
146	C₂×C₄×C₈	—	146	4	no	—
147	C₂³×C₄	—	2	1	no	—
148	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
149	C₂²×C₈	—	73	4	no	—
150	C₂²×C₈	—	73	4	no	—
151	C₂²×C₈	—	73	1	no	—
152	C₂²×C₈	—	73	1	no	—
153	C₂²×C₈	—	73	1	no	—
154	C₂²×C₈	—	73	1	no	—
155	C₂²×C₈	—	73	1	no	—
156	C₂²×C₈	—	73	1	no	—
157	C₂²×C₈	—	73	2	no	—
158	C₂²×C₈	—	73	2	no	—
159	C₂²×C₈	—	73	2	no	—
160	C₂²×C₈	—	146	2	no	—
161	C₂²×C₈	—	146	2	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	#10596833	1	4	yes	—
165	C₂×C₄²	C₂×#64408	1	2	yes	—
166	M₃₂	—	73	8	no	—
167	M₃₂	—	146	8	no	—
168	M₃₂	—	146	4	no	—
169	C₂×C₁₆	—	146	8	no	—
170	C₂×C₁₆	—	146	4	no	—
171	C₄×C₈	—	146	4	no	—
172	C₄×C₈	—	146	4	no	—
173	C₂⁴	#11924037	1	1	yes	characteristic
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	C₄×#64408	1	2	yes	—
178	C₂²×C₄	C₂²⋊#64408	1	2	yes	—
179	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
181	C₂²×C₄	C₂×#10596833	1	2	yes	—
182	C₂²×C₄	#11924036	1	1	yes	characteristic
183	C₂²×C₄	#11924036	1	1	yes	characteristic
184	C₂²×C₄	#11924037	1	1	yes	characteristic
185	C₂²×C₄	#11924038	1	2	yes	—
186	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
187	C₂×C₈	—	73	4	no	—
188	C₂×C₈	—	73	4	no	—
189	C₂×C₈	—	73	4	no	—
190	C₂×C₈	—	73	4	no	—
191	C₂×C₈	—	73	2	no	—
192	C₂×C₈	—	73	2	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	73	1	no	—
196	C₂×C₈	—	73	1	no	—
197	C₂×C₈	—	73	1	no	—
198	C₂×C₈	—	73	1	no	—
199	C₂×C₈	—	73	4	no	—
200	C₂×C₈	—	146	2	no	—
201	C₂×C₈	—	146	4	no	—
202	C₂×C₈	—	146	4	no	—
203	C₂×C₈	—	146	2	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	C₂×#10596833	1	4	yes	—
207	C₁₆	—	146	8	no	—
208	C₂³	—	2	4	no	—
209	C₂³	#14556646	1	2	yes	—
210	C₂³	C₂²⋊#10596833	1	2	yes	—
211	C₂³	#14556658	1	1	yes	characteristic
212	C₂³	#14556658	1	1	yes	characteristic
213	C₂³	C₂×#11924037	1	1	yes	characteristic
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	(C₂×#64408)⋊C₄	1	2	yes	—
218	C₂×C₄	(C₂×#64408)⋊C₄	1	2	yes	—
219	C₂×C₄	C₄×#10596833	1	2	yes	—
220	C₂×C₄	C₄×#10596833	1	2	yes	—
221	C₂×C₄	#14556646	1	2	yes	—
222	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
223	C₂×C₄	#14556648	1	2	yes	—
224	C₂×C₄	#14556648	1	2	yes	—
225	C₂×C₄	C₄⋊#10596833	1	2	yes	—
226	C₂×C₄	C₄⋊#10596833	1	2	yes	—
227	C₂×C₄	#14556658	1	1	yes	characteristic
228	C₂×C₄	#14556658	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
230	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
232	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
233	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
234	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
235	C₈	—	73	2	no	—
236	C₈	—	73	2	no	—
237	C₈	—	73	2	no	—
238	C₈	—	73	2	no	—
239	C₈	—	146	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	#17989453	1	2	yes	—
245	C₂²	#17989453	1	2	yes	—
246	C₂²	#17989510	1	4	yes	—
247	C₂²	C₂×#14556646	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊#10596833	1	1	yes	characteristic
249	C₂²	C₂×#14556658	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	#17989453	1	2	yes	—
252	C₄	#17989453	1	2	yes	—
253	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
254	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
255	C₄	C₂×#14556648	1	1	yes	characteristic
256	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	2	1	no	—
260	C₂	#22624975	1	4	yes	direct product factor
261	C₂	C₂×#17989453	1	1	yes	characteristic
262	C₂	C₂×#17989510	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂²×C₄²×C₇₃	C₄
socle	C₂³×C₇₃	C₄⋊C₈
prime radical	C₂⁴×C₇₃	M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂²×C₄²	#64408
hypocenter	C₇₃	C₂×#1376
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #22624975	C₂	direct (N × H)
6. C₂×#14556665	C₄	semidirect
17. #14556665	C₂×C₄	semidirect
129. C₂×C₇₃	#1376	semidirect
135. C₇₃	C₂×#1376	semidirect
260. C₂	#22624975	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556665	2
#22624975	2
C₂²×C₄×#10596831	2
C₂×#1376	73

Subgroups — C2×#22624975 #34012128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624975 #34012128