Subgroups — C₂×#22626304 #34030160

Order 18688.

538 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626304	C₂	1	8	yes	maximal, direct product factor
2	C₂×#17989510	C₂	1	2	yes	maximal, complemented
3	C₂×#17989483	C₂	1	2	yes	maximal
4	C₂×#17989479	C₂	1	2	yes	maximal, complemented
5	C₂×#17988543	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₂²	1	4	yes	—
7	#17989510	C₂²	1	8	yes	complemented
8	#17989483	C₂²	1	8	yes	—
9	#17989479	C₂²	1	8	yes	complemented
10	C₂²×#11924012	C₂²	1	1	yes	characteristic
11	C₂³×#10596831	C₄	1	1	yes	characteristic
12	#17988543	C₄	1	4	yes	—
13	#17988543	C₂²	1	4	yes	—
14	C₂×#14556502	C₄	1	1	yes	characteristic
15	C₂×#14556502	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₂×#117×C₇₃	C₄	1	1	yes	characteristic
19	#14556665	—	2	4	no	—
20	#14556665	C₂³	1	8	yes	—
21	C₂×#11924031	—	2	4	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	C₂×C₄	1	2	yes	—
25	C₂×#11924012	C₂³	1	2	yes	—
26	C₂²×#10596831	—	2	2	no	—
27	C₂²×#10596831	—	2	2	no	—
28	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10596831	C₂×C₄	1	4	yes	—
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10596831	C₂³	1	1	yes	characteristic
33	#14556502	C₂×C₄	1	4	yes	—
34	#14556502	C₂×C₄	1	2	yes	—
35	#14556502	C₂³	1	2	yes	—
36	C₄×#10596831	C₂×C₄	1	4	yes	—
37	C₄×#10596831	C₂×C₄	1	2	yes	—
38	C₄×#10596831	C₂³	1	2	yes	—
39	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#117×C₇₃	—	2	2	no	—
41	#117×C₇₃	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
44	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	#11924031	—	2	8	no	—
47	#11924012	—	2	4	no	—
48	#11924012	—	2	4	no	—
49	C₂×#10596831	—	2	1	no	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	—	2	8	no	—
54	C₂×#10596831	M₁₆	1	2	yes	—
55	C₂×#10596831	M₁₆	1	2	yes	—
56	C₂×#10596831	C₂²×C₄	1	2	yes	—
57	C₂×#10596831	C₂²×C₄	1	2	yes	—
58	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
59	C₂×#10596831	C₂²×C₄	1	2	yes	—
60	C₂×#10596831	#41	1	2	yes	—
61	C₂×#10596831	#41	1	2	yes	—
62	C₂×#10596831	C₂⁴	1	1	yes	characteristic
63	C₂⁴×C₇₃	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	—	2	1	no	—
65	C₂²×C₄×C₇₃	—	2	1	no	—
66	C₂²×C₄×C₇₃	—	2	1	no	—
67	C₂²×C₄×C₇₃	—	2	1	no	—
68	C₂²×C₄×C₇₃	—	2	1	no	—
69	C₂²×C₄×C₇₃	—	2	1	no	—
70	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₄⋊C₄×C₇₃	—	2	4	no	—
75	C₂²⋊C₄×C₇₃	—	2	4	no	—
76	C₄²×C₇₃	—	2	1	no	—
77	C₄²×C₇₃	—	2	1	no	—
78	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
80	C₂³×C₇₃	—	2	1	no	—
81	C₂³×C₇₃	—	2	1	no	—
82	C₂³×C₇₃	—	4	2	no	—
83	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂³×C₇₃	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	2	no	—
90	C₂×C₄×C₇₃	—	2	2	no	—
91	C₂×C₄×C₇₃	—	2	2	no	—
92	C₂×C₄×C₇₃	—	2	2	no	—
93	C₂×C₄×C₇₃	—	4	2	no	—
94	C₂×C₄×C₇₃	—	4	2	no	—
95	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
102	#10596831	—	2	2	no	—
103	#10596831	—	2	2	no	—
104	#10596831	#117	1	2	yes	—
105	#10596831	C₂×M₁₆	1	4	yes	—
106	#10596831	C₂×#41	1	2	yes	—
107	C₂²×C₇₃	—	2	2	no	—
108	C₂²×C₇₃	—	2	2	no	—
109	C₂²×C₇₃	—	4	2	no	—
110	C₂²×C₇₃	—	4	2	no	—
111	C₂²×C₇₃	—	4	2	no	—
112	C₂²×C₇₃	#432	1	1	yes	characteristic
113	C₂²×C₇₃	#432	1	1	yes	characteristic
114	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
115	C₄×C₇₃	—	2	2	no	—
116	C₄×C₇₃	—	4	2	no	—
117	C₄×C₇₃	—	4	2	no	—
118	C₄×C₇₃	#432	1	1	yes	characteristic
119	C₄×C₇₃	#432	1	1	yes	characteristic
120	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
121	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×#2166	—	73	1	no	maximal, Sylow
123	C₂×C₇₃	—	2	1	no	—
124	C₂×C₇₃	—	2	1	no	—
125	C₂×C₇₃	—	4	2	no	—
126	C₂×C₇₃	#2166	1	2	yes	complemented
127	C₂×C₇₃	C₂×#432	1	1	yes	characteristic
128	C₂×#405	—	73	1	no	—
129	#2166	—	73	8	no	—
130	C₂×#364	—	73	2	no	—
131	C₂×#349	—	73	2	no	—
132	C₂×#347	—	73	2	no	—
133	C₇₃	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
134	C₂²×M₁₆	—	73	1	no	—
135	C₂³×C₈	—	73	1	no	—
136	C₂×#117	#64408	1	1	yes	characteristic
137	C₂×M₃₂	—	73	4	no	—
138	#405	—	73	4	no	—
139	#405	—	73	4	no	—
140	C₂×#97	—	73	1	no	—
141	C₂×#97	—	73	1	no	—
142	C₂×C₄×C₈	—	73	1	no	—
143	C₂×C₄×C₈	—	73	1	no	—
144	#364	—	73	8	no	—
145	#349	—	73	8	no	—
146	#347	—	73	8	no	—
147	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
148	C₂×M₁₆	—	73	2	no	—
149	C₂×M₁₆	—	73	2	no	—
150	C₂×M₁₆	—	146	2	no	—
151	C₂×M₁₆	—	146	2	no	—
152	C₂²×C₈	—	73	1	no	—
153	C₂²×C₈	—	73	1	no	—
154	C₂²×C₈	—	73	1	no	—
155	C₂²×C₈	—	73	4	no	—
156	C₂²×C₈	—	73	1	no	—
157	C₂²×C₈	—	146	2	no	—
158	C₂²×C₈	—	146	2	no	—
159	#117	—	2	2	no	—
160	#117	C₂×#64408	1	4	yes	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
164	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
165	M₃₂	—	73	8	no	—
166	M₃₂	—	146	4	no	—
167	C₂×C₁₆	—	146	4	no	—
168	#97	—	73	4	no	—
169	#97	—	73	2	no	—
170	#97	—	73	2	no	—
171	C₄×C₈	—	73	2	no	—
172	C₄×C₈	—	73	4	no	—
173	C₄×C₈	—	73	2	no	—
174	C₂⁴	#11924037	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	#11924037	1	1	yes	characteristic
182	C₂²×C₄	#11924037	1	2	yes	—
183	C₂²×C₄	C₂²×#64408	1	2	yes	—
184	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
185	M₁₆	—	146	4	no	—
186	M₁₆	—	146	4	no	—
187	C₂×C₈	—	73	2	no	—
188	C₂×C₈	—	73	2	no	—
189	C₂×C₈	—	73	2	no	—
190	C₂×C₈	—	73	1	no	—
191	C₂×C₈	—	73	1	no	—
192	C₂×C₈	—	73	2	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	73	2	no	—
196	C₂×C₈	—	146	8	no	—
197	C₂×C₈	—	146	1	no	—
198	C₂×C₈	—	146	1	no	—
199	C₂×C₈	—	146	1	no	—
200	C₂×C₈	—	146	1	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	C₂²×#64408	1	2	yes	—
206	C₄²	C₂²×#64408	1	2	yes	—
207	C₁₆	—	146	8	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	4	2	no	—
211	C₂³	C₂×#11924037	1	1	yes	characteristic
212	C₂³	C₂×#11924037	1	2	yes	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	#14556670	1	1	yes	characteristic
224	C₂×C₄	#14556670	1	1	yes	characteristic
225	C₂×C₄	C₂×#11924037	1	2	yes	—
226	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
227	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
228	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
229	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
230	C₈	—	73	2	no	—
231	C₈	—	73	2	no	—
232	C₈	—	73	4	no	—
233	C₈	—	146	2	no	—
234	C₈	—	146	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	#17989607	1	1	yes	characteristic
241	C₂²	#17989607	1	1	yes	characteristic
242	C₂²	C₂²×#11924037	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	#17989607	1	1	yes	characteristic
247	C₄	#17989607	1	1	yes	characteristic
248	C₄	C₂×#14556670	1	1	yes	characteristic
249	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	4	2	no	—
253	C₂	#22626304	1	2	yes	direct product factor
254	C₂	C₂×#17989607	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556670
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×#117×C₇₃	C₄
socle	C₂²×C₇₃	#432
prime radical	C₂⁴×C₇₃	M₁₆
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×#117	#64408
hypocenter	C₇₃	C₂×#2166
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22626304	C₂	direct (N × H)
2. C₂×#17989510	C₂	semidirect
4. C₂×#17989479	C₂	semidirect
7. #17989510	C₂²	semidirect
9. #17989479	C₂²	semidirect
126. C₂×C₇₃	#2166	semidirect
133. C₇₃	C₂×#2166	semidirect
253. C₂	#22626304	direct (N × H)

Maximal subgroups

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

subgroup	index
#22626304	2
C₂×#17989510	2
C₂×#17989483	2
C₂×#17989479	2
C₂×#17988543	2
C₂×#2166	73

Subgroups — C2×#22626304 #34030160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626304 #34030160