Subgroups — C₂×#22624979 #34012138

Order 18688.

630 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#22624979	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#14556501	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	—	2	2	no	—
5	C₂×#14556665	—	2	4	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	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×#14556501	—	2	2	no	—
12	C₂×#14556501	—	2	2	no	—
13	C₂×#14556501	C₄	1	2	yes	—
14	C₂×#14556501	C₂²	1	2	yes	—
15	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
16	#14556665	—	2	8	no	—
17	#14556665	—	2	4	no	—
18	#14556665	C₂×C₄	1	8	yes	complemented
19	C₂×#11924031	—	2	8	no	—
20	C₂×#11924031	—	2	4	no	—
21	C₂³×Dic₇₃	C₈	1	2	yes	—
22	C₂²×#10596831	C₂×C₄	1	4	yes	—
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10596831	D₄	1	2	yes	—
25	C₂²×#10596831	D₄	1	2	yes	—
26	C₂²×#10596831	D₄	1	2	yes	—
27	C₂²×#10596831	Q₈	1	2	yes	—
28	C₂²×#10596831	C₂³	1	1	yes	characteristic
29	C₂×C₄×Dic₇₃	—	2	1	no	—
30	C₂×C₄×Dic₇₃	—	2	1	no	—
31	C₂×C₄×Dic₇₃	—	2	1	no	—
32	C₂×C₄×Dic₇₃	—	2	1	no	—
33	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
34	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
35	#14556501	—	2	4	no	—
36	#14556501	—	2	4	no	—
37	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	—	2	1	no	—
39	C₂²×C₈×C₇₃	—	2	1	no	—
40	C₂²×C₈×C₇₃	—	2	2	no	—
41	C₂²×C₈×C₇₃	C₈	1	4	yes	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
43	#11924031	—	2	8	no	—
44	C₂²×Dic₇₃	—	2	2	no	—
45	C₂²×Dic₇₃	—	2	2	no	—
46	C₂²×Dic₇₃	—	4	4	no	—
47	C₂²×Dic₇₃	C₂×C₈	1	4	yes	—
48	C₂×#10596831	C₄²	1	2	yes	—
49	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
50	C₂×#10596831	C₂²⋊C₄	1	4	yes	—
51	C₂×#10596831	C₂²⋊C₄	1	4	yes	—
52	C₂×#10596831	C₄⋊C₄	1	2	yes	—
53	C₂×#10596831	C₄⋊C₄	1	4	yes	—
54	C₂×#10596831	C₄⋊C₄	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	C₄×Dic₇₃	—	2	2	no	—
60	C₄×Dic₇₃	—	2	2	no	—
61	C₄×Dic₇₃	—	2	2	no	—
62	C₄×Dic₇₃	—	2	2	no	—
63	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
71	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	—	2	4	no	—
75	C₂×C₈×C₇₃	—	2	4	no	—
76	C₂×C₈×C₇₃	—	2	2	no	—
77	C₂×C₈×C₇₃	—	2	2	no	—
78	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
79	C₂³×C₇₃	—	2	4	no	—
80	C₂³×C₇₃	C₄×C₈	1	2	yes	—
81	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
82	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
84	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
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	#97	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	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
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	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₇₃	—	2	4	no	—
104	C₂×Dic₇₃	—	4	4	no	—
105	C₂×Dic₇₃	—	4	4	no	—
106	C₂×Dic₇₃	—	4	4	no	—
107	C₈×C₇₃	—	2	4	no	—
108	#10596831	#95	1	4	yes	—
109	#10596831	C₂×C₂²⋊C₄	1	2	yes	—
110	#10596831	C₂×C₄⋊C₄	1	2	yes	—
111	C₂²×C₇₃	—	2	2	no	—
112	C₂²×C₇₃	—	2	2	no	—
113	C₂²×C₇₃	—	2	4	no	—
114	C₂²×C₇₃	—	2	4	no	—
115	C₂²×C₇₃	#336	1	2	yes	—
116	C₂²×C₇₃	#336	1	2	yes	—
117	C₂²×C₇₃	#349	1	4	yes	—
118	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₄×C₇₃	#336	1	2	yes	—
122	C₄×C₇₃	#336	1	2	yes	—
123	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
124	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
125	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
127	Dic₇₃	—	4	4	no	—
128	C₂×#1377	—	73	1	no	maximal, Sylow
129	C₂×C₇₃	—	2	2	no	—
130	C₂×C₇₃	—	2	1	no	—
131	C₂×C₇₃	—	2	1	no	—
132	C₂×C₇₃	#1377	1	4	yes	complemented
133	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
134	C₂×C₇₃	C₂×#349	1	2	yes	—
135	C₂²×M₃₂	—	73	2	no	—
136	C₂²×#97	—	73	1	no	—
137	#1377	—	73	4	no	—
138	C₇₃	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₈	—	73	1	no	—
140	C₂³×C₈	#64408	1	1	yes	characteristic
141	C₂²×C₄²	—	73	1	no	—
142	C₂×M₃₂	—	73	8	no	—
143	C₂×M₃₂	—	73	4	no	—
144	C₂×M₃₂	—	146	2	no	—
145	C₂×M₃₂	—	146	4	no	—
146	C₂²×C₁₆	—	146	2	no	—
147	C₂×#97	—	73	2	no	—
148	C₂×#97	—	73	2	no	—
149	C₂×#97	—	146	2	no	—
150	C₂×#97	—	146	2	no	—
151	C₂³×C₄	—	73	2	no	—
152	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	73	4	no	—
157	C₂²×C₈	—	73	2	no	—
158	C₂²×C₈	—	73	2	no	—
159	C₂²×C₈	—	73	2	no	—
160	C₂²×C₈	—	73	2	no	—
161	C₂²×C₈	—	73	1	no	—
162	C₂²×C₈	—	73	1	no	—
163	C₂²×C₈	#10596833	1	4	yes	—
164	C₂²×C₈	C₂×#64408	1	2	yes	—
165	C₂×C₄²	—	73	2	no	—
166	C₂×C₄²	—	73	2	no	—
167	C₂×C₄²	—	146	1	no	—
168	C₂×C₄²	—	146	1	no	—
169	C₂×C₄²	—	146	1	no	—
170	C₂×C₄²	—	146	1	no	—
171	M₃₂	—	73	8	no	—
172	M₃₂	—	146	8	no	—
173	M₃₂	—	146	4	no	—
174	C₂×C₁₆	—	146	8	no	—
175	C₂×C₁₆	—	146	4	no	—
176	#97	—	146	4	no	—
177	#97	—	146	4	no	—
178	C₂⁴	#11924036	1	1	yes	characteristic
179	C₂²×C₄	—	292	4	no	—
180	C₂²×C₄	—	73	4	no	—
181	C₂²×C₄	—	146	2	no	—
182	C₂²×C₄	—	146	2	no	—
183	C₂²×C₄	C₄×#64408	1	2	yes	—
184	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊#64408	1	2	yes	—
187	C₂²×C₄	C₂×#10596833	1	2	yes	—
188	C₂²×C₄	#11924036	1	1	yes	characteristic
189	C₂²×C₄	#11924037	1	2	yes	—
190	C₂²×C₄	#11924038	1	1	yes	characteristic
191	C₂²×C₄	#11924038	1	1	yes	characteristic
192	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	73	4	no	—
198	C₂×C₈	—	73	4	no	—
199	C₂×C₈	—	73	4	no	—
200	C₂×C₈	—	73	4	no	—
201	C₂×C₈	—	73	2	no	—
202	C₂×C₈	—	73	2	no	—
203	C₂×C₈	—	73	2	no	—
204	C₂×C₈	—	73	2	no	—
205	C₂×C₈	—	73	2	no	—
206	C₂×C₈	—	73	1	no	—
207	C₂×C₈	—	73	1	no	—
208	C₂×C₈	C₂×#10596833	1	4	yes	—
209	C₄²	—	146	2	no	—
210	C₄²	—	146	2	no	—
211	C₄²	—	146	2	no	—
212	C₄²	—	146	2	no	—
213	C₁₆	—	146	8	no	—
214	C₂³	—	2	4	no	—
215	C₂³	C₄×#10596833	1	2	yes	—
216	C₂³	#14556648	1	1	yes	characteristic
217	C₂³	#14556648	1	1	yes	characteristic
218	C₂³	#14556658	1	2	yes	—
219	C₂³	C₂×#11924036	1	1	yes	characteristic
220	C₂×C₄	—	292	4	no	—
221	C₂×C₄	—	292	4	no	—
222	C₂×C₄	—	292	4	no	—
223	C₂×C₄	—	146	4	no	—
224	C₂×C₄	(C₂×#64408)⋊C₄	1	2	yes	—
225	C₂×C₄	(C₂×#64408)⋊C₄	1	2	yes	—
226	C₂×C₄	C₄×#10596833	1	2	yes	—
227	C₂×C₄	#14556646	1	2	yes	—
228	C₂×C₄	#14556646	1	2	yes	—
229	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
230	C₂×C₄	C₂²⋊#10596833	1	2	yes	—
231	C₂×C₄	#14556648	1	1	yes	characteristic
232	C₂×C₄	#14556648	1	1	yes	characteristic
233	C₂×C₄	C₄⋊#10596833	1	2	yes	—
234	C₂×C₄	C₄⋊#10596833	1	2	yes	—
235	C₂×C₄	#14556658	1	2	yes	—
236	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
239	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
240	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
241	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	73	2	no	—
244	C₈	—	73	4	no	—
245	C₈	—	73	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	#17989453	1	2	yes	—
251	C₂²	#17989453	1	2	yes	—
252	C₂²	#17989484	1	4	yes	—
253	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
254	C₂²	C₂×#14556648	1	1	yes	characteristic
255	C₂²	C₂×#14556658	1	1	yes	characteristic
256	C₄	—	292	4	no	—
257	C₄	#17989453	1	2	yes	—
258	C₄	#17989453	1	2	yes	—
259	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
260	C₄	C₂×#14556646	1	1	yes	characteristic
261	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
262	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#22624979	1	4	yes	direct product factor
267	C₂	C₂×#17989453	1	1	yes	characteristic
268	C₂	C₂×#17989484	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂²⋊C₈
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	C₂×#1377
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #22624979	C₂	direct (N × H)
6. C₂×#14556665	C₄	semidirect
18. #14556665	C₂×C₄	semidirect
132. C₂×C₇₃	#1377	semidirect
138. C₇₃	C₂×#1377	semidirect
266. C₂	#22624979	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
#22624979	2
C₂²×#14556501	2
C₂×#1377	73

Subgroups — C2×#22624979 #34012138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624979 #34012138